Abstract:
A load can be applied to a beam and a property of the load can be calculated. In one example, a first shear gauge can be configured for positioning on a neutral axis of a beam on one side of a force that the beam is subjected to. Similarly, a second shear gauge can be configured for positioning on the neutral axis of the beam on an opposite side of the force to the first shear gauge. A calculator can be configured to identify a characteristic of the force through use of an output of the first shear gauge and through use of an output of the second shear gauge.

Description:
CROSS-REFERENCE 
     This application is a continuation application of, and claims priority to, U.S. patent application Ser. No. 13/545,565 that was filed on Jul. 10, 2012, the entirety of which is hereby incorporated by reference. 
    
    
     GOVERNMENT INTEREST 
     The invention described herein may be manufactured, used, imported, sold, and licensed by or for the Government of the United States of America without the payment to me of any royalty thereon. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates generally to deflection plates to protect load cells from overloading and, more particularly, to a deflection plate for the M16 and M18 Mobile Dynamometers used by the U.S. Army. The present invention also relates to a method for employing strain gages to identify the magnitudes and locations of loads on slender beams using strain gage based methods with application to portable Army bridges. 
     BACKGROUND OF THE INVENTION 
     A mobile dynamometer is a vehicle that can measure the force needed to tow large vehicles such as tanks. A load cell with a capacity of about, for example, 100,000 pounds is connected to a sway bar underneath the dynamometer. However, a method and structure is needed to prevent the load cell from being overloaded should the force suddenly exceed the 100,000 pound capacity. Overloading the load cell can lead to damage. 
     SUMMARY OF THE INVENTION 
     In one embodiment, a system comprises a processor and a non-transitory computer-readable medium. The non-transitory computer-readable medium is configured to store processor-executable instructions that when executed by a processor cause the processor to perform a method. The method comprises using a curve of a beam caused by the beam being subjected to a load to make an identification of a property of the load. The method also comprises causing the property to be outputted after identification is made. 
     In one embodiment, a system comprises a first shear gauge, a second shear gauge, and a calculator. The first shear gauge is configured to be positioned on a neutral axis of a beam on one side of a force that the beam is subjected to. The second shear gauge is configured to be positioned on the neutral axis of the beam on an opposite side of the force to the first shear gauge. The calculator is configured to identify a characteristic of the force through use of an output of the first shear gauge and through use of an output of the second shear gauge, where the characteristic is outputted. 
     In one embodiment, a system comprises a determiner, a recognizer, and a determiner. The determiner is configured to determine a first bending moment curve for a load on a beam and a second bending moment curve for a load on a beam if a second bending moment curve exists. The recognizer is configured to make a recognition if the first bending moment curve is too physically close to the second bending moment curve, where the recognition is based, at least in part, on a possible solution number for a characteristic set of the load on the beam. The decider is configured to make a decision on a characteristic determination method for determination of the characteristic set of the load on the beam, where the decision is based, at least in part, on the recognition, where the determiner, the recognizer, the decider, or a combination thereof are, at least in part, implemented through hardware. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The following detailed description is made with reference to the accompanying drawings, in which: 
         FIG. 1  is a perspective view of a deflection plate in accordance with one preferred embodiment of the present invention; 
         FIG. 2  is a front view of the deflection plate of  FIG. 1 ; 
         FIG. 2A  is a front view with detailed dimensions of the deflection plate of  FIG. 1 ; 
         FIG. 3  is a side view of the deflection plate of  FIG. 1 ; 
         FIG. 3A  is a side view with detailed dimensions of the deflection plate of  FIG. 1 ; 
         FIG. 3B  is an top end view with detailed dimensions of the deflection plate of 
         FIG. 4  is a perspective view of a deflection plate in accordance with an alternative preferred embodiment of the present invention; 
         FIG. 5  is a front view of the deflection plate of  FIG. 4 ; 
         FIG. 5A  is a front view with detailed dimensions of the deflection plate of  FIG. 4 ; 
         FIG. 6  is a side view of the deflection plate of  FIG. 4 ; 
         FIG. 6A  is a side view with detailed dimensions of the deflection plate of  FIG. 4 ; 
         FIG. 6B  is an top end view with detailed dimensions of the deflection plate of  FIG. 4 ; 
         FIG. 7  is a perspective assembly view including the deflection plate of  FIG. 1  coupled to a load cell in accordance with one preferred embodiment of the present invention; 
         FIG. 8  is an assembly side view of the deflection plate of  FIG. 7 ; 
         FIG. 9  is an assembly front view of the deflection plate of  FIG. 7 ; 
         FIG. 10  is a perspective overall underneath view including the deflection plate of  FIG. 7  coupled to a load cell; 
         FIG. 11  is a perspective overall underneath view including the deflection plate of  FIG. 7  coupled to a load cell which is coupled to a stopper; 
         FIG. 12  is a perspective top view of a U.S. Army M16 Heavy Dynamometer coupled to a Stryker vehicle via a deflection plate in accordance with one preferred embodiment of the present invention; 
         FIG. 13  is an enlarged close up perspective top view of the coupling of the U.S. Army M16 Heavy Dynamometer to the Stryker vehicle of  FIG. 12 ; 
         FIG. 14  is side view of the coupling of the U.S. Army M16 Heavy Dynamometer to the Stryker vehicle of  FIG. 12 ; 
         FIG. 15  is an enlarged close up side view of the coupling of the U.S. Army M16 Heavy Dynamometer to the Stryker vehicle of  FIG. 12 ; 
         FIG. 16  is a graph illustrating the stress and the safety factor on a 100 kip (where one kip is equal to 1,000 pounds) deflection plate along its vertical length in accordance with one preferred embodiment of the present invention; 
         FIG. 17  is a graph illustrating the stress and the safety factor on a 75 kip deflection plate along its vertical length in accordance with an alternative preferred embodiment of the present invention; 
         FIG. 18  is a picture of a portable army bridge on the BCS at the US Army Aberdeen Test Center; 
         FIG. 19  is a picture of an army vehicle crossing a portable army bridge; 
         FIG. 20  illustrates the components of a strain gage based transducer, the illustration courtesy of Vishay Measurements; 
         FIG. 21  shows a uniaxial strain gage consisting of one continuous wire, the illustration courtesy of Vishay Measurements; 
         FIG. 22  is a picture of an experimental setup of a cantilever beam strain gage based force transducer in Ref. [20]; 
         FIG. 23  is a side view of the beam in  FIG. 22  that can determine the magnitude of the force F using two strain gages at point&#39;s a and b; 
         FIG. 24  is a picture of the strain gages for the beam in  FIG. 22 ; 
         FIG. 25  is a free body diagram of a beam with pinned boundary conditions, and two strain gages on each side of the force F2; 
         FIG. 26  is a shear force diagram for the beam in  FIG. 25 ; 
         FIG. 27  shows calculated (solid lines) and actual (dashed lines) bending moment curves for the beam in  FIG. 25 , where the force F2 is located inside the weight area; 
         FIG. 28  is a free body diagram for a case where the force F4 is located outside the force transducer; 
         FIG. 29  shows calculated (solid lines) and actual (dashed lines) bending moment curves for the case in  FIG. 28 , where the force F4 is located outside the force transducer; 
         FIG. 30  is a free body diagram for a case where the force F3 is outside the weight area, but inside the force transducer; 
         FIG. 31  shows calculated (solid line) and actual (dashed lines) bending moment curves for the case in  FIG. 30 , where the force F3 is outside the weight area, but inside the force transducer; 
         FIG. 32  illustrates a Wheatstone bridge from the strain gages in  FIG. 25 ; 
         FIG. 33  is a free body diagram for a beam with a uniformly distributed force within the weight area and pinned boundary conditions; 
         FIG. 34  is a shear force diagram for the case in  FIG. 33 ; 
         FIG. 35  shows calculated (solid lines) and actual (dashed lines) bending moment curves, and the bending moment curve over the length of the distributed force (bold line), for the case in  FIG. 33 , where the distributed force is within the weight area; 
         FIG. 36  is a free body diagram for pinned, firm rest, soft rest, pinned-fixed, and fixed boundary conditions; 
         FIG. 37  is a free body diagram for a beam with soft rest boundary conditions; 
         FIG. 38  is a bending moment diagram for pinned, firm rest, soft rest, pinned-fixed, and fixed boundary conditions; 
         FIG. 39  shows a known nonzero force Fn placed immediately after strain gage epsilon sub n with all the other forces on the beam being zero; 
         FIG. 40  is a picture of an experimental setup of a steel beam with rest boundary conditions; 
         FIG. 41  is a picture of a close-up guide block at end abutment for the beam in  FIG. 40 ; 
         FIG. 42  is a picture of a load applied via thin string on beam in  FIG. 40 ; 
         FIG. 43  shows a cross section of the steel beam in  FIG. 40 ; 
         FIG. 44  is a side view of the steel beam in  FIG. 40  with four numbered strain gage locations; 
         FIG. 45  is a comparison of the theoretical and calculated bending moment diagrams for the beam in  FIG. 40 ; 
         FIG. 46  is a side view of the steel beam in  FIG. 40  with 12 numbered strain gage locations; 
         FIG. 47  is a comparison of the theoretical and calculated bending moment diagrams for the beam in  FIG. 46 ; 
         FIG. 48  is a picture of an experimental setup of an aluminum beam with a distributed load and rest boundary conditions; 
         FIG. 49  is a cross section of the beam in  FIG. 48 ; 
         FIG. 50  is a side view of the beam in  FIG. 48  with numbered strain gage locations; 
         FIG. 51  is a comparison of the theoretical and calculated bending moment diagrams for the beam in  FIG. 48  with firm rest boundary conditions; 
         FIG. 52  is a picture of an experimental setup of an aluminum beam with fixed boundary conditions; 
         FIG. 53  is a top view of bolted clamp at end abutment for the beam in  FIG. 52 ; 
         FIG. 54  is an end view showing cross section of the beam in  FIG. 52  along with bolted clamp assembly; 
         FIG. 55  is a side view of the aluminum beam in  FIG. 52  with numbered strain gage locations; 
         FIG. 56  is a picture of an experimental setup of the aluminum beam with firm rest boundary conditions; 
         FIG. 57  is a picture of an experimental setup of the aluminum beam with soft rest boundary conditions; 
         FIG. 58  is a comparison of the theoretical and calculated bending moment diagrams for the beam in  FIG. 56  with firm rest boundary conditions; 
         FIG. 59  is a comparison of the theoretical and calculated bending moment diagrams for the beam in  FIG. 57  with soft rest boundary conditions; 
         FIG. 60  is a picture of an experimental setup of the aluminum beam with firm rest-fixed boundary conditions; 
         FIG. 61  is a comparison of the theoretical and calculated bending moment diagrams for the beam in  FIG. 60  with firm rest-fixed boundary conditions; 
         FIG. 62  is a comparison of the theoretical and calculated bending moment diagrams for the beam in  FIG. 60  with fixed boundary conditions; 
         FIG. 63  is a picture of an experimental setup of a continuously tapered aluminum beam with rest boundary conditions; 
         FIG. 64  is a side view of the continuously tapered aluminum beam in  FIG. 63 ; 
         FIG. 65  is an end view of the continuously tapered aluminum beam in  FIG. 63 ; 
         FIG. 66  is a side view of the continuously tapered aluminum beam in  FIG. 63  with numbered strain gage locations; 
         FIG. 67  shows measured strains, the theoretical bending moment diagram, and the calculated bending moment diagram for a 1,201 N force located at position 4.5; 
         FIG. 68  is a free body diagram of a beam with multiple forces on the beam; 
         FIG. 69  is a shear force diagram for the beam in  FIG. 68 ; 
         FIG. 70  is a bending moment diagram for the beam in  FIG. 68 ; 
         FIG. 71  shows two possible bending moment diagrams for two forces separated by one uniaxial strain gage location, as shown with solid and dashed lines; 
         FIG. 72  illustrates a stress element at a 0 degree angle from the x axis; 
         FIG. 73  illustrates a stress element at a 45 degree angle from the x axis; 
         FIG. 74  is a picture of a uniaxial strain gage mounted on the bottom surface of a beam; 
         FIG. 75  is a picture of an experimental setup of an aluminum beam with rest boundary conditions; 
         FIG. 76  is a picture of a Vishay P3 strain indicator (left) and a Vishay SB-10 switch unit (right); 
         FIG. 77  is a cross section of the aluminum beam in  FIG. 75 ; 
         FIG. 78  is a side view of the aluminum beam in  FIG. 75  with numbered strain gage locations; 
         FIG. 79  is a comparison of the theoretical and calculated bending moment diagrams for the beam in  FIG. 75 ; 
         FIG. 80  is a picture of an experimental setup of nine forces on the aluminum beam; 
         FIG. 81  is a comparison of the theoretical and calculated bending moment diagrams for the beam in  FIG. 80 ; 
         FIG. 82  is a picture of an experimental setup of two forces on the aluminum beam in  FIG. 75  separated by one strain gage location; 
         FIG. 83  is a picture of a shear gage mourned on the neutral axis of the aluminum beam in  FIG. 82 ; 
         FIG. 84  is a comparison of the theoretical and calculated bending moment diagrams for the case in Sec. 3.3.2; 
         FIG. 85  is a comparison of the theoretical and calculated shear force diagrams for the case in Sec. 3.3.2; 
         FIG. 86  shows a free body diagram of a non-continuous beam with pinned boundary conditions and two strain gages on each side of the force F2; 
         FIG. 87  is a shear force diagram for the beam in  FIG. 86 ; 
         FIG. 88  shows calculated (solid lines) and actual (dashed lines) bending moment curves for the beam in  FIG. 86 , where the force F2 is located inside the weight area; 
         FIG. 89  shows a known nonzero force Fn placed immediately after strain gage epsilon sub n with all the other forces on the beam being zero; 
         FIG. 90  is a picture of an experimental setup of a continuously tapered aluminum beam with a series of welded joints and rest boundary conditions; 
         FIG. 91  is a picture of a close-up view of some welded joints in the beam in  FIG. 90 ; 
         FIG. 92  is a side view of the beam in  FIG. 90 ; 
         FIG. 93  is an end view of the beam in  FIG. 90  (dimensions in mm); 
         FIG. 94  is a side view of the beam in  FIG. 90  with numbered strain gage locations (dimensions in mm); 
         FIG. 95  is a color presentation of a predicted stress pattern from Finite Element Analysis (FEA) on the beam in  FIG. 90  with an 801 N force at position 4.5; 
         FIG. 96  is a color presentation of a bonded interface for welded joints in the FEA model; 
         FIG. 97  is a color presentation of a contact interface between the top flange and web in the FEA model; 
         FIG. 98  shows comparisons of the theoretical strains for firm rest and soft rest boundary conditions, the measured strains, and the predicted strains from FEA for firm rest and soft rest boundary conditions; 
         FIG. 99  shows comparisons of the theoretical bending moment and the calculated bending moments, both before and after calibration, for an 801 N force at position 4.5; 
         FIG. 100  is a picture of an experimental setup of an aluminum 6061-T6 beam with a bolted joint and rest boundary conditions; 
         FIG. 101  is a picture of a close-up view of the bolted joint in the beam in  FIG. 100 ; 
         FIG. 102  is a close-up view of the bolted joint in the beam in  FIG. 100  (dimensions in mm); 
         FIG. 103  is an end view of the beam in  FIG. 100  (dimensions in mm); 
         FIG. 104  is a side view of the beam in  FIG. 100  with numbered strain gage locations (dimensions in mm); 
         FIG. 105  is a color presentation of a predicted stress pattern from FEA on the beam in  FIG. 100  with pinned boundary conditions, zero preloads in the bolts, and a 445 N force at position 4.5; 
         FIG. 106  is a color presentation of a predicted stress pattern from FEA on a prismatic beam similar to the beam in  FIG. 100 , under a 445 N force at position 4.5; 
         FIG. 107  is a color presentation of predicted deflections from FEA on the beam in  FIG. 105  for a 445 N force at position 6.5; 
         FIG. 108  is a color presentation of predicted deflections from FEA on the beam in  FIG. 106  for a 445 N force at position 6.5; 
         FIG. 109  shows strain response from FEA on the beam in  FIG. 100  with increasing magnitudes of a force at position 6.5; 
         FIG. 110  shows comparisons of raw strains, preload strains, and zeroed strains for a bolt tension of 1.7 kN; locations of strain gages 1 through 6 are indicated; 
         FIG. 111  shows comparisons of raw strains, preload strains, and zeroed strains for a bolt tension of 8.5 kN; locations of strain gages 1 through 6 are indicated; 
         FIG. 112  shows comparisons of raw strains, preload strains, and zeroed strains for a bolt tension of 17.0 kN; locations of strain gages 1 through 6 are indicated; 
         FIG. 113  shows strain response for the beam in  FIG. 100  with different bolt torque values; 
         FIG. 114  is a comparison of the theoretical and calculated bending moment diagrams for the beam in  FIG. 100  with different bolt torque values; 
         FIG. 115  shows comparisons of the theoretical strains for firm rest boundary conditions, the measured strains, and the zeroed strains from FEA for soft rest boundary conditions with the updated Young&#39;s modulus; 
         FIG. 116  is a comparison of the theoretical bending moment diagram and the calculated bending moment diagram from a global scaling factor, for a 445 N force at position 9.5 on the beam in  FIG. 100 ; 
         FIG. 117  is a comparison of the theoretical and calculated bending moment diagrams for the beam in  FIG. 100 ; 
         FIG. 118  is a picture of an experimental setup of a half aluminum and half steel beam with a bolted joint and rest boundary conditions; 
         FIG. 119  illustrates a cross section of the half aluminum beam in  FIG. 118 ; 
         FIG. 120  illustrates a cross section of the half steel beam in  FIG. 118 ; 
         FIG. 121  is a picture of a close-up view of the bolted joint in the beam in  FIG. 118 ; 
         FIG. 122  is a side view of the bolted joint in the beam in  FIG. 118  (dimensions in mm); 
         FIG. 123  is a close-up view of the bolted joint in the beam in  FIG. 118  (dimensions in mm); 
         FIG. 124  is a side view of the beam in  FIG. 118  with numbered strain gage locations (dimensions in mm); 
         FIG. 125  is a color presentation of a predicted stress pattern from FEA on the beam in  FIG. 118  with pinned boundary conditions, zero preloads in the bolts, and a 445 N force at position 2.5; 
         FIG. 126  is a comparison of the theoretical and calculated bending moments from a global scaling factor for a 445 N force at position 9.5 for the beam in  FIG. 118 ; 
         FIG. 127  is a comparison of the theoretical and calculated bending moment diagrams for the beam in  FIG. 118 ; 
         FIG. 128  is a picture of an experimental setup of a portable army bridge on the BCS at the US Army Aberdeen Test Center; 
         FIG. 129  is a schematic of the portable army bridge with ten Whiffles on the BCS; 
         FIG. 130  is a side view of the portable army bridge on the BCS, showing the Whiffle locations (dimensions in meters); 
         FIG. 131  is an end view of the BCS from abutment A, showing the bridge, a Whiffle, pads, and hydraulic cylinders; 
         FIG. 132  is a side view of a Whiffle (dimensions in meters); 
         FIG. 133  is a cross section of the portable army bridge showing strain gage locations; 
         FIG. 134  is a comparison of the theoretical and calculated bending moment diagrams for the roadside of the portable army bridge on the BCS; 
         FIG. 135  is a comparison of the theoretical and calculated bending moment diagrams for the curbside of the portable army bridge on the BCS; and 
         FIG. 136  shows a tank crossing a truss bridge. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     In the description herein with respect to the drawings, any reference to direction is purely informative and is employed for ease of reference. It is not intended as limiting the scope of the claims herein. 
     The deflection plate or anchor arm plate of the present invention is for use in the M16 Heavy Mobile Dynamometer Vehicle (manufactured by Barnes and Reinecke in the late 1970&#39;s) and the M18 Medium Mobile Dynamometer Vehicle (manufactured also by Barnes and Reinecke, in the late 1980&#39;s). These vehicles are designed for testing power drawbar effect and resistance to towing characteristics of tracked and wheeled vehicles. 
     Referring to  FIG. 1 , there is shown one preferred embodiment of a high strength deflection plate  10  according to the present invention. The deflection plate  10 , as shown, generally in  FIGS. 1, 2 and 3 , is a 100,000 pound (referred to here as 100 kip) embodiment which has a generally rectangular configuration having a flat back surface  12  and a front surface  14  as shown in  FIG. 3  which is not flat along its entire length from top to bottom. The top surface  16  and the bottom surface  18  of the deflection place  10  are also preferably flat. Generally the front surface  14  is tapered such that the front surface  14  adjacent the top surface  16  is thicker than the front surface  14  adjacent the bottom surface  18 . 
     The deflection plate  10  as shown in  FIG. 3  and also in  FIG. 16  is formed of preferably five segments of varying thickness and shape. As shown in  FIG. 16 , these are segments  20 ,  22 ,  24 ,  26  and  28 . Segments  20  and  28  have flat front surfaces  30  and  38 , respectively. Segments  22 ,  24  and  26  have tapered front surfaces  32 ,  34  and  36 , respectively. Generally each of these tapered segments has decreasing cross-sectional thicknesses toward the bottom segment  28  which has the smallest cross-sectional thickness. Hence segment  20  has the largest thickness and segment  28  has the smallest thickness. The deflection plate  10  has a right side wall  40  and a left side wall  42  which are preferably flat. Segment  28  has bottom inwardly tapered flat wall portions  44  and  46  as shown in  FIGS. 1, 2 and 3 . 
     Segment  20  has four similar bores  48  as shown in  FIGS. 1, 2 and 3  to receive suitably dimensioned and configured similar bolts  50  as shown, for example, in  FIGS. 7 through 10 . Segment  28  has a single bore  52  to receive a single suitably dimensioned and configured bolt  54  that is preferably larger in cross-section than bolts  50 . 
     Referring now to  FIGS. 4, 5 and 6 , there is shown an alternative preferred embodiment of a deflection plate  100  also according to the present invention. The deflection plate  100  is a 75 kip (75,000 pound) embodiment which has generally like configuration as does deflection plate  10 . However the 75 kip embodiment  100  has different tapered segments and different sizes and dimensions than those of the 100 kip embodiment  10 . 
     A comparison of these tapered segments and dimensions for the deletion plates  10  and  100  are shown in  FIGS. 16 and 17 . For example, in deflection plate  10  segment  20  has a thickness of about 3.85 inches. Segment  120  of deflection plate  100  has a thickness of about 3.34 inches. In general, deflection plate  100  has smaller dimensions than deflection plate  10 . But both have the same height of about 17.5 inches and a width of about 6.00 inches so that they can each be bolted in place to the same surrounding structural components of the undercarriage of the Mobile Dynamometer Vehicle as shown in  FIGS. 7 through 10 . More detailed dimensions of a preferred embodiment for deflection plate  10  are shown in  FIGS. 2A, 3A and 3B  and for deflection plate  100  are shown in  FIGS. 2A, 3A and 3B . 
     The deflection plate  100  also has a generally rectangular configuration having a flat back surface  112  and a front surface  114  as shown in  FIG. 4  which is not flat along its entire length from top to bottom. The top surface  116  and the bottom surface  118  of the deflection place  10  are preferably flat. As was the case with deflection plate  10 , deflection plate  100  similarly has a front surface  114  that is tapered such that the front surface  114  adjacent the top surface  116  is thicker than the front surface  114  adjacent the bottom surface  118 . 
     The deflection plate  100  as shown in  FIG. 6  and also in  FIG. 17  is formed of preferably five segments of varying thickness and shape. As shown in  FIG. 17 , these are segments  120 ,  122 ,  124 ,  126  and  128 . Segments  120  and  128  have flat front surfaces  130  and  138 , respectively. Segments  122 ,  124  and  126  have tapered front surfaces  132 ,  134  and  136 , respectively. Generally each of these tapered segments has decreasing cross-sections thicknesses toward the bottom segment  128  which has the smallest cross-sectional thickness. Hence segment  120  has the largest thickness and segment  128  has the smallest thickness. The deflection plate  100  has a right side wall  140  and a left side wall  142  which are preferably flat. Segment  128  has bottom inwardly tapered flat wall portions  144  and  146  as shown in  FIGS. 4, 5 and 6 . 
     Segment  120  has four similar bores  148  as shown in  FIGS. 4, 5 and 6  to receive suitably dimensioned and configured similar bolts (not shown) which are comparable to bolts  50  shown, for example, in  FIGS. 7 through 10 . Segment  128  has a single bore  152  to receive a single suitably dimensioned and configured bolt (not shown) that is preferably larger in cross-section than the bolts for bores  148  and is comparable to bolt  54  also shown in  FIGS. 7 through 10 . 
     Referring now to the embodiment shown in  FIGS. 7 through 9 , the deflection plate  10  is mounted to a lower support plate  200  that includes an L shaped member having a horizontal portion  202  and a vertical portion  204 . As shown in  FIGS. 7 and 8 , the deflection plate  10  is positioned between a front bushing or restraining plate  206  and a rear bushing or restraining plate  208  which is sandwiched between back surface  12  and the front surface of the vertical portion  204  of the L shaped member of lower support plate  200 . An upper support plate  210  is positioned against the back surface of the vertical portion  204 . These components are fastened together with the bolts  50  that extend through comparably sized and aligned holes (not shown) in each of these components and also bores  48  of deflection plate  10 . The ends  212  of the bolts  50  extend past the back surface of the upper support plate  210 . The holes of at least upper support plate  210  can have grooves to receive the grooves on the bolts  50  so as to fasten these components together with the deflection plate  10 . 
     The upper support plate  210  can preferably be attached to a horizontal plate  214  and a transverse plate  216  as shown in  FIGS. 7 and 8 . These plates collectively form a nut fixture to received and secure the bolts  50 . A semi-ring  218  can be affixed to the upper surface of horizontal plate  214 . 
     As shown in  FIG. 7 , a load cell  220  is mounted to the deflection plate  10  through a front clevis  222  into which is positioned a front eye bolt  224  extending from the front of the load cell  220 . A front mourning bolt  226  couples the front eye bolt  224  to the front clevis  222 . The front mounting bolt  226  is kept in place by two cross pins  228  and corresponding catch rings  230  on opposite ends of the front mounting bolt  226  as shown in  FIG. 9 . The bolt  54  is secured to the clevis  222  which has a bore (not shown) to receive the bolt  54 . A bushing  55  is positioned between bolt  54  and the front surface  38  of deflection plate  10 . 
     The load cell  220  also has a rear eye bolt or ball joint rod end  232  extending from the front of the load cell  220  as shown in  FIG. 8 . Threaded adapters  233  positioned on each side of the load cell  220  are used in coupling the load cell  220  to the clevis  222  and the rear eye bolt  232 . The rear eye bolt  232  is configured to receive a rear mounting bolt  234 . In this manner, the load cell  220  can be coupled to a sway bar or load beam  236  as shown in  FIGS. 10 and 11 . The sway bar or load beam  236  has a clevis  238  which allows for coupling the rear eye bolt  232  by the rear mounting bolt  234 . The cross pins  228  and corresponding catch rings  230  help to maintain the rear mounting bolt in place. A rocker arm  240  is coupled to the clevis  238  with a retaining bolt  242 . The load beam  236  is coupled at it rearward portion to a towbar cross-member  244  which is part of the tow mechanism  246  shown in  FIGS. 12 through 15 . These latter drawings demonstrate a US Army mobile Dynamometer  248  hooked up to and towing a Stryker vehicle  250 . 
     Referring now to  FIG. 11 , the sway bar or load beam  236  includes a stopper plate  252  which is positioned between a front stop member  254  and a rear stop member  256 . Both the front stop member  254  and rear stop member  256  are coupled or mounted on support members  258  and  260 , respectively as shown in  FIG. 11 , and which axe mounted to the undercarriage  262  of mobile Dynamometer  248 . As shown in  FIG. 11 , the stopper plate  252  during the towing and testing operation can be spaced from the front stop member  254  and the rear stop member  256 . 
     The high strength deflection plates  10  and  100  are designed such that they will not yield under the design force but can deflect, preferably 0.25 inches at the 100,000 pound capacity of the load cell  220 . At this point the sway bar or load beam  236  can hit a mechanical stopper  256  directly attached to the frame or undercarriage  262  of the mobile dynamometer  248 . In this manner, the load cell  220  is allowed to measure the towing force up to but not exceeding about 100,000 pounds. As noted above, the deflection plates  10  and  100  are tapered such that the stress remains approximately constant throughout most of the entire plate. This allows the maximum deflection possible since there is no unnecessary material. In one preferred embodiment, the material for high strength and low modulus of elasticity can be Aluminum 7075-T651. 
     As shown in  FIGS. 16 and 17 , the deflection plates are designed to generally have a safety factor (S.F.) between 1.1 and 1.2 in at least the upper four segments  20 ,  22 ,  24  and  26  of deflection plate  10  and the upper four segments  120 ,  122 ,  124  and  126  of deflection plate  100 . The safety factor is the yield strength (shown as sigma in psi in  FIGS. 16 and 17 ) divided by the design stress. In  FIGS. 16 and 17 , the topmost horizontal broken line shows a S.F. of 1.0. The next lower horizontal broken line represents a S.F. of 1.1 while the next lower horizontal broken line represents a S.F. of 1.2 as also indicated in the legend boxes in  FIGS. 16 and 17 . The larger the safety factor, the less likely the part will fail. The yield strength of aluminum 7075 is about 73,000 psi. For example a safety factor of 2 would be a design stress of 73,000/2=36,500 psi. In this case, the S.F.=1.1 corresponds to a design stress of 66,364 psi and the S.F.=1.2 corresponds to a design stress of 60,833 psi. 
     The design criteria which determine the thickness of the material and points of slope change of preferred embodiments of deflection plates  10  and  100  was for the stress generally throughout the part to be between a S.F. of 1.2 and 1.1 and all times. If the safety factor dropped below 1.1 the part would be in danger of failing and if the safety factor was greater than 1.2, there is more material than needed. 
     In  FIGS. 16 and 17 , the thicknesses of deflection plates  10  and  100  are shown in relative relation to the safety factors determined along the various segments of the deflection plates  10  and  100 . However, the full vertical heights or lengths of the deflection plates are not shown. Basically, as can be seen from  FIGS. 2A and 3A , the x axis in  FIGS. 16 and 17  starts at the single main bolt hole  52  and  152  at the left. For deflection plate  10 , from x=0 to x=1.25″, the thickness is 1.4″. At x=1.25″, the slope changes and at x=2″, the thickness is 1.78″. At x=4.1″, the thickness is 2.55″. At x=9.31″, the thickness is 3.85″. Finally at 9.81″ the thickness remains 3.85″ and the graphs in  FIGS. 16 and 17  end. That is the point where the front bushing or restraining plate  206  begins as shown in  FIG. 7 . 
     Although a load cell can be employed as described above in a Mobile Dynamometer, load cells are not always available and they are not cost effective to directly measure the vehicle weights on a bridge by using load cells. However, strain gages can be employed to identify the magnitudes and locations of loads on slender beams using strain gage based methods with application to portable Army bridges. 
     According to this aspect of the present invention, unique strain gage based methods were developed to identify magnitudes and locations of loads on a non-continuous, non-homogenous, slender beam with variable cross sections, welded and bolted joints, and pinned, firm rest, soft rest, pinned-fixed, and fixed boundary conditions. Four uniaxial strain gages mounted to the bottom surface of the beam created a force transducer capable of identifying the magnitude and location of a load inside the weight area. By combining individually scaled strain gage outputs, the bending moment diagram was constructed. For the case of multiple loads separated by two or more strain gage locations, uniaxial strain gages forming multiple force transducers can still identify the magnitudes and locations of all the loads. A calibration method was developed to account for the discrepancies between the theoretical and actual scaling factors arising from stress concentrations and unpredictable stress patterns in the beams due to the presence of the joints. The strain gage based force transducer methodology was experimentally validated on prismatic beams with firm rest, soft rest, firm rest-fixed, and fixed boundary conditions; an aluminum beam with a bolted joint; and a half aluminum and half steel beam with two different cross sections and a bolted joint. It was also experimentally validated on a continuous aluminum beam with a linearly varying cross section and rest boundary conditions, a tapered aluminum beam with a series of welded joints, and a full scale portable army bridge at the US Army Aberdeen Test Center. The force transducer methodology is independent of the boundary conditions of the beam and the error from strain gage drift due to uniform thermal expansion on a prismatic beam can cancel out. 
     When there are multiple loads separated by only one strain gage location, the problem is ill posed for the force transducer methodology. Another method has been developed using two shear gages mounted on the neutral axis of the beam, one on each side of a load, to identify the magnitude of the load in this case. A combination of two uniaxial strain gages and two shear gages, with one uniaxial strain gage and one shear gage at the same location on each side of a load, can be used to identify the location of the load. The strain gage based methods were experimentally validated on a prismatic beam with rest boundary conditions. 
     In the discussion which follows, the list of variables includes:
     [ε]: u×1 matrix that represents measured strains at candidate locations   u: number of rows in matrix [ε]   [P]: v×1 matrix that represents independent input forces   v: number of rows in matrix [P]   [A]: u×v coefficient matrix that depends on the geometric and material properties   F: force on a beam   x: horizontal position measured from left boundary   a and b: points where strain is measured on a cantilever beam   d a : location of strain gage a measured from left boundary   d ab : distance between strain gages a and b   d F : location of force F measured from left boundary   M a  and M b : bending moments at points a and b   ε a  and ε b : measured strains at points a and b   V: shear force   M: bending moment   σ: stress   γ: distance from the neutral axis of the beam to the point where stress is calculated   l: area moment of inertia of a cross section   E: Young&#39;s modulus   ε: strain   A and B: left and right end abutments, respectively, of a beam supported at two ends   R A  and R B : reaction forces at left and right boundaries, respectively   n, h, i, j, and: strain gage number   ε n : measured strain at strain gage n   F n : for located to the right of strain gage n   p n : position of force F n  measured from left boundary   d n : position of strain gage n measured from left boundary   f n : position of force F n  measured from strain gage n   d nj : distance between strain gages n and j   a nj : relative position of force F n  between strain gages n and j   a n : notation for relative position of force F n  between strain gages n and j when the two strain gages are adjacent   M n : bending moment at strain gage n   V n : shear force at strain gage n   E n : Young&#39;s modulus at strain gage n   l n : area moment of inertia of a cross section at strain gage n   γ n : distance from the neutral axis of the beam to the point where stress is calculated at stain gage n   γ n : scaling factor at strain gage n   γ: global scaling factor   β n : calibration factor at strain gage n   β: global calibration factor   FT hijk : force transducer formed from strain gages h, i, j, and k   FT i : notation for force transducer FT hijk  when the four strain gages are adjacent   d: distance between two adjacent strain gages when all are equally spaced   w: uniformly distributed force   d w : length of uniformly distributed force w   F w : resultant force of uniformly distributed force w   d L  and d R : distances between the force F and the left and right boundaries, respectively   d A  and d B : distances between the left and right reaction forces R A  and R B , and the edges of the soft abutments, respectively   D A  and D B : lengths of the beam sitting on the left and right soft abutments, respectively   M A  and M B : reaction moments at left and right boundaries, respectively   L: span of beam   N: number of forces on a beam   [F]: q×1 matrix that represents all the known forces   q: number of rows in matrix [F]   [γ]: m×1 matrix that represents the scaling factors   m: number of rows in matrix [γ]   [ε]: q×m matrix that represents the measured strains from strain gages forming the force transducers   λ F : maximum force error   λ 0 : maximum zero error   λ T : total error   C: user defined positive parameter to calculate total error λ T      Δp n : difference between a known force position and the calculated one   τ: shear stress   σ 1  and σ 2 : principal stresses   ε 1  and ε 2 : principal strains   v: Poisson&#39;s ratio   Q: first moment of area   t: thickness of cross section   D: shear gage output   k: constant that changes the bending moment and strain relationship   T: bolt torque   P: tensile load in a bolt   K: bolt torque constant   Φ: major bolt diameter
 
1.1 Motivation
   

     Load identification on large scale bridges has many applications in both the public and military sectors. The Bridge Crossing Simulator (BCS), which is operated by the US Army Aberdeen Test Center at Aberdeen Proving Ground in Maryland, USA, tests portable army bridges for fatigue failure by simulating thousands of vehicle crossings. The current methodology involves setting up a bridge on dirt abutments and measuring strains from the bridge that are collected from vehicles with various crossing speeds. Next, the bridge is set up on the BCS with wood abutments and straps, and hydraulic cylinders are used to apply loads to the bridge through multiple load application devices, referred to as “Whiffles”. The strain time history is used in a control system feedback loop for each corresponding cylinder in order to simulate various crossing speeds of vehicles on the bridge. This method can be subject to uncertainties since the actual vehicle loads are not known from the field data. A new load identification methodology can be used to identify cylinder loads that need to be applied, from measured strains from live crossings of vehicles. 
     1.2 Previous Work 
     A load identification methodology can also be used to determine the life span of a commercial bridge given the strain gage data of vehicles that cross it [1]. The frequency and the weights of vehicles that cross a bridge are often an estimated quantity by structural designers. Load cells are not always available and it is not cost effective to directly measure the vehicle weights using load cells [2]. By developing a measurement technique using strain gages, one can identify specific axle loads of a vehicle without using a scale [3]. Previous methods transform a bridge into a dynamic scale when multiple vehicles were crossing it [1-9]. This is primarily an inverse problem since the loads are unknown, but the response is known [10]. 
     By measuring strain at one location, the system “AXWAY” in Ref. [4] develops an influence line by summing the individual strain curves from known vehicle axle weights and performing a best fit analysis. The influence line is dependent on vehicle speed, axle spacing, axle weights, and gross vehicle weight [4]. A limitation of this method is that a bridge can only have one vehicle on it at a time. Deflection measurements can also be used, where the dynamic beam response is calculated [11]. According to the previous studies, the load identification method using deflection data is more prone to error than that using bending moment data [5]. Another approach involves modeling a vehicle/bridge system and using acceleration measurements to calculate the dynamic response of the system [6]. However, when comparing the strain model to the acceleration model, lower error was obtained from the strain model [12]. A limitation of deflection based methods and acceleration based methods is that the boundary conditions must be accurately known. 
     A Reaction Force Detection (RFD) system to detect lameness in cows was developed by measuring four reaction forces on a plate using load cells [13]. Calibration is performed along with statistical methods to determine when forces are reduced in a limb of a lame cow. When a single limb is on a plate, the RFD calculates the magnitude and position of the force. A limitation of the method is that when multiple limbs are on the plate, the RFD calculates the magnitude and position of the resultant force, and cannot decipher individual forces. 
     A transducer converts one form of energy to another [14]. The interest here is to convert mechanical energy to an electrical signal that represents the load applied. Components of a force transducer include an applied load, the spring response, strain gages, and a Wheatstone bridge circuit, as shown in  FIG. 20  (illustration courtesy of Vishay Measurements) [15, 17]. Strain gages are useful when stress measurements are desired, since stress cannot be directly measured, but strain can be measured [16]. Strain is defined as the change in length over original length, and does not have units [16]. A strain gage consists of a continuous wire that is bonded to a material for measurement through a rigorous application procedure, with a uniaxial strain gage that measures strain in one direction shown in  FIG. 21  (illustration courtesy of Vishay Measurements) [17]. When the material deforms, each leg of the wire also deforms. The wire changes its cross section and, more importantly, electrical resistance. When the strain gage is connected to a Wheatstone bridge, the voltage amplifies the resistance change, which can be converted to a strain measurement. Typical strain measurements are very small and reported in microstrain (με). One με is defined as one unit of deformation per 1,000,000 units of original length [14]. Even though strain is unitless, the precision of the measurement is high, since for a strain gage with an active strain gage length of 6.35 millimeters, one με is a deformation of 6.35 nanometers. Strain gages are a useful measurement technique for most metal structures such as aluminum and steel. Because of the porous nature of concrete, it is difficult to correctly bond a strain gage. However, it is possible to measure strains on large scale concrete bridges by mounting aluminum members to concrete and applying strain gages to the aluminum members [10]. 
     A general challenge of an inverse problem is that it is often ill-posed [8]. Given measurement quantities can often lead to multiple solutions for input loads [18]. Hence a final design of a force transducer should be robust enough to pinpoint a unique solution. It has been shown multiple times that a structure itself can be turned into a force transducer using strain measurements [4,5,10]. A general formulation is given by Masroor and Zachary [19]:
 
[ε]=[ A ][ P ]  (1.1)
 
where [ε] is a u×1 matrix that represents measured strains at candidate locations, [P] is a v×1 matrix that represents independent input forces, and [A] is a u×v coefficient matrix that depends on the geometric and material properties. The coefficient matrix is determined experimentally through known inputs P and outputs ε. The inverse problem associated with Eq. (1.1) can be solved:
 
[ P ]=[ A ] −1 [ε]  (1.2)
 
where [A] −1  is the generalized inverse of [A]. The challenge of properly setting up this method is to first determine appropriate strain and load locations so that Eq. (1.2) is well-posed, since not all locations can provide independent information.
 
1.3 New Methods
 
     Novel strain gage based methods are developed to identify the magnitudes and locations of loads on beams. Chapter 2 introduces a new force transducer methodology that will calculate the magnitude and location of a single load on a beam using four uniaxial strain gages mounted on the bottom surface of the beam. Section 2.1 discusses a cantilever beam force transducer developed by Vishay using two strain gages [20]. Section 2.2.1 introduces a new force transducer methodology to determine the magnitude of a load on a beam. The load can be a concentrated load, distributed load, or any combination of concentrated and distributed loads. The concept of firm and soft rest boundary conditions is discussed along with pinned, pinned-fixed and fixed boundary conditions, where the methodology is independent of boundary conditions. Section 2.2.2 utilizes the force transducer methodology to determine the location of a load. Section 2.2.3 discusses causes of error in strain gage based transducers and how the methodology developed here mitigates the errors. A novel calibration procedure is developed to systematically determine the scaling factors of strain gages on a beam. Section 2.3 applies the force transducer methodology to four different experiments. Section 2.3.1 shows the force transducer methodology applied to a prismatic steel beam with rest boundary conditions to accurately determine the magnitude and location of a load. Section 2.3.2 shows the results for various boundary conditions for a prismatic aluminum beam. Section 2.3.3 shows a distributed load on a prismatic aluminum beam with rest boundary conditions. Section 2.3.4 shows the force transducer methodology applied to a continuously tapered aluminum beam with rest boundary conditions. 
     Chapter 3 expands the force transducer methodology to multiple loads separated by two strain gage locations (Sec. 3.2.1), and introduces a new strain gage based method using shear gages mounted on the neutral axis of a beam and the uniaxial strain gages (Sec. 3.2.2). The corresponding experimental results for a prismatic aluminum box beam with rest boundary conditions are shown for the force transducer methodology (Sec. 3.3.1) and strain gage methodology using shear gages and uniaxial strain gages (Sec. 3.3.2). 
     Chapter 4 applies the force transducer methodology to beams with welded and bolted joints with application to a portable army bridge. Section 4.2.1 discusses the effect of welded or bolted joints on strain gages in close proximity to joints. The previously developed calibration method accounts for stress concentrations due to joints (Sec. 4.2.2). Section 4.3.1 shows the experimental results for a continuously tapered aluminum beam with a series of welded joints. Section 4.3.2 shows the experimental results for an aluminum beam with a bolted joint in the middle. Section 4.3.3 shows the experimental results for a beam that is half aluminum and half steel. Section 4.3.4 shows the application of the force transducer methodology to a portable army bridge on the BCS. 
     Chapter 5 reviews the force transducer methodology and the strain gage based methodology using shear gages and uniaxial strain gages developed and provides future research directions. Section 5.1 summarizes the experimental results for Chapters 2 through 4. Section 5.2 describes the contributions, advantages, and limitations of the methods. Section 5.3 discusses future work for the methods developed here. 
     Chapter 2: Identifying the Magnitude and Location of a Load on a Slender Beam Using a Strain Gage Based Force Transducer 
     Abstract 
     A unique strain gage based method is developed to identify the magnitude and location of a load on a slender beam with non-homogeneous material, variable cross sections, and pinned, firm rest, soft rest, pinned-fixed, and fixed boundary conditions. Four uniaxial strain gages are mounted to the bottom surface of the beam, and the bending moment diagram of the beam can be constructed using measured strains on the beam. By combining individually scaled strain gage outputs, the magnitude and location of the load can be accurately identified. The strain gage based force transducer methodology is experimentally validated on prismatic beams with firm rest, soft rest, firm rest-fixed, and fixed boundary conditions, and a continuously tapered beam with rest boundary conditions. The force transducer methodology is independent of the boundary conditions of the beam and the error from strain gage drift due to uniform thermal expansion on a prismatic beam can cancel out. 
     2.1 Introduction 
     A strain gage based force transducer has been used to identify the magnitude of a force F on a slender cantilever beam [20], as shown in  FIGS. 22 to 24 . Two uniaxial strain gages, which measure strains along the x direction, are located between the load and the fixed boundary, as shown in  FIGS. 23 and 24 . For an Euler-Bernoulli beam, the relationship between the shear force V and the bending moment M is [21] 
                       dM   ⁡     (   x   )       dx     =     V   ⁡     (   x   )               (   2.1   )               
The bending stress in the beam can be calculated from
 
                   σ   =     My   I             (   2.2   )               
where γ is the distance from the neutral axis of the beam to the point where stress is calculated, and l is the area moment of inertia of the cross section [22]. The stress σ is proportional to the strain ε through the Young&#39;s modulus ε [23]:
 
σ= Eε   (2.3)
 
Hence the bending moment M is proportional to the strain ε:
 
                   M   =       EI   y     ⁢   ɛ             (   2.4   )               
Since V(x)=F and dM(x)/dx can be calculated using the finite difference method, which is an exact method here since the corresponding bending moment curve is a line, by Eq. (2.1), one has
 
                         M   b     -     M   a         d   ab       =   F           (   2.5   )               
where M a  and M b  are the bending moments at points a and b, respectively, and d ab  is the distance between strain gages a and b ( FIG. 23 ). Applying Eq. (2.4) at points a and b and using Eq. (2.5) yield
 
                   F   =       EI     y   ⁢           ⁢     d   ab         ⁢     (       ɛ   b     -     ɛ   a       )               (   2.6   )               
where ε a  and ε b  are the measured strains at points a and b, respectively. Equation (2.6) indicates that the force F can be calculated using the measured strains at points a and b. Note that it is not necessary to measure the strains of the cantilever beam from the applied load to the free end, since there is no strain in that section. Hence only two strain gages are needed to identify the magnitude of a load on the cantilever beam. The concept of the cantilever beam strain gage based force transducer is extended in this work to slender beams with non-homogeneous materials, variable cross sections, and pinned, firm rest, soft rest, pinned-fixed, and fixed boundary conditions. In addition, methods to identify the location of a load on a slender beam are developed.
 
2.2 Theory
 
2.2.1 Identification of the Magnitude of a Load
 
     Consider a beam with non-homogeneous material, variable cross sections, and pinned boundary conditions. While a strain gage based force transducer for the cantilever beam has only two strain gages, that for a beam with reaction forces at two boundaries has four strain gages, with two strain gages on each side of a load to calculate the slopes of the bending moment curves on the two sides of the load. The free body diagram shown in  FIG. 25  has a force F 2  located inside a designated weight area between strain gages 2 and 3, where R A  and R B  are the reaction forces at the two boundaries; in this work, the index n for a force F n  denotes the force is located to the right of strain gage ε n , where n=1,2,3,4 is the strain gage number. The standard sign conventions for the shear force and bending moment diagrams are used [21]. The left shear force curve in  FIG. 26  has a constant value R A , and the right shear force curve has a constant value −R B . Since a bending moment curve is the integral of a shear force curve, the slope of the left bending moment curve is R A , and that of the right bending moment curve is −R B , as shown in  FIG. 27 . Subtracting the latter from the former, and using the force equilibrium equation and Eq. (2.1), yield
 
Slope Left −Slope Right   =R   A   +R   B   =F   2   (2.7)
 
                           M   2     -     M   1         d   12       -         M   4     -     M   3         d   34         =     F   2             (   2.8   )               
where M 1  through M 4  are the bending moments at strain gages 1 through 4, d 12  is the distance between strain gages 1 and 2, and d 34  is the distance between strain gages 3 and 4. Note that Eqs. (2.2) through (2.4) apply to both prismatic and non-prismatic beams. Using Eq. (2.4) in Eq. (2.8) yields the general force transducer equation:
 
                     FT   1234     =       FT   2     =       F   2     =       -       ɛ   1     ⁡     (         E   1     ⁢     I   1           y   1     ⁢     d   12         )         +       ɛ   2     ⁡     (         E   2     ⁢     I   2           y   2     ⁢     d   12         )       +       ɛ   3     ⁡     (         E   3     ⁢     I   3           y   3     ⁢     d   34         )       -       ɛ   4     ⁡     (         E   4     ⁢     I   4           y   4     ⁢     d   34         )                     (   2.9   )               
where FT 1234  denotes the output of the force transducer formed by strain gages 1 through 4, and it can also be denoted by FT 2 , in which the subscript is the strain gage number immediately to the left of the force F 2 , when the four strain gages are adjacent to each other; this notation is used consistently elsewhere. One can define an individual scaling factor γ n  for strain gage ε n , and apply it to Eq. (2.9):
 
     
       
         
           
             
               
                 
                   
                     γ 
                     n 
                   
                   = 
                   
                     
                       
                         E 
                         n 
                       
                       ⁢ 
                       
                         I 
                         n 
                       
                     
                     
                       y 
                       n 
                     
                   
                 
               
               
                 
                   ( 
                   2.10 
                   ) 
                 
               
             
           
         
       
     
                     FT   2     =       F   2     =       -       ɛ   1     ⁡     (       γ   1       d   12       )         +       ɛ   2     ⁡     (       γ   2       d   12       )       +       ɛ   3     ⁡     (       γ   3       d   34       )       -       ɛ   4     ⁡     (       γ   4       d   34       )                   (   2.11   )               
One can also define a calibration factor β n  corresponding to the scaling factor γ n :
 
                       β   n     =           γ   n       d   nj       ⁢           ⁢   for   ⁢           ⁢   n     &lt;   j       ,       β   n     =           γ   n       d   jn       ⁢           ⁢   for   ⁢           ⁢   n     &gt;   j               (   2.12   )               
where j is also the strain gage number, and d nj  or d jn  is the distance between adjacent strain gages that are not at the end points of the weight area. Note that the distance between strain gages 2 and 3, denoted by d 23 , does not appear in Eqs. (2.9) and (2.11); hence it does not affect determination of the magnitude of the force. Furthermore, as long as the force F 2  is located inside the weight area, Eqs. (2.9) and (2.11) hold true. When the force F 4  is outside the force transducer, as shown in  FIG. 28 , the force transducer will read zero since the slopes measured from the two pairs of strain gages are the same and will cancel out when subtracted ( FIG. 29 ).
 
     The last possibility is for a force located outside the weight area, but inside the force transducer, i.e., the force is between strain gages 1 and 2 or between strain gages 3 and 4;  FIG. 30  shows the case when the force is between strain gages 3 and 4. The strain measurements lead to a partially correct measured bending moment diagram, as shown in  FIG. 31 . Let 
                     α   nj     =       f   n       d   nj               (   2.13   )               
be the relative position of the force, where f n  is the position of the force measured from the left adjacent strain gage, and d nj  is the distance between strain gages n and j. When j=n+1, a nj  can be denoted as a n . For the case in  FIG. 30 , when a 3 =0, the output of FT 2  is F 3 , and when a 3 =1, the output of FT 2  is 0. The calculated magnitude of the force varies linearly with the position of the force between strain gages 3 and 4. A similar method applies if the force is between strain gages 1 and 2. Since a n  is defined from left to right and not symmetrically outward from the weight area, one has
 
 FT   2 =(1− a   3 ) F   3   (2.14)
 
 FT   2   =a   1   F   1   (2.15)
 
If the position of the force is known, the magnitude of the force can be calculated from Eq. (2.14) or (2.15), and vice versa.
 
     For the case of a homogenous, prismatic beam with unequal strain gage spacing, γ 1 =γ 2 =γ 3 =γ 4 =γ, where γ is the global scaling factor for all the strain gages, and Eq. (2.11) becomes 
                     FT   2     =       F   2     =     γ   ⁡     (           -     ɛ   1       +     ɛ   2         d   12       +         ɛ   3     -     ɛ   4         d   34         )                 (   2.16   )               
When d 12 =d 34 =d, β 1 =β 2 =β 3 =β 4 =β, where β is the global calibration factor, and Eq. (2.16) becomes
 
 FT   2   =F   2 =β(−ε 1 +ε 2 +ε 3 −ε 4 )  (2.17)
 
The output measurements of the strain gages on the beam in  FIG. 25  can be combined within one circuit via a Wheatstone bridge, as shown in  FIG. 32 . Opposite legs of the bridge add together and adjacent legs subtract [15]. While in theory a single output of the circuit is possible, it will not be used for this methodology. If one strain gage is not functioning properly or one wire is damaged, the entire circuit is destroyed. This makes troubleshooting difficult [3]. It is better practice to combine the strain gage outputs algebraically.
 
     It should be noted that a beam can have many strain gages, and the four strain gages of a force transducer do not have to be adjacent to each other for Eq. (2.11). Any combination of strain gages ε h , ε i , ε j , and ε k , can form a force transducer: 
                     FT   hijk     =       F   i     =       -       ɛ   h     ⁡     (       γ   h       d   hi       )         +       ɛ   i     ⁡     (       γ   i       d   hi       )       +       ɛ   j     ⁡     (       γ   j       d   jk       )       -       ɛ   k     ⁡     (       γ   k       d   jk       )                   (   2.18   )               
where the subscript of FT hijk  is the combination of the strain gage numbers since the strain gages forming the force transducer are not adjacent to each other. The first two strain gages of a force transducer, measuring the left half slope of a bending moment diagram, and the next two strain gages, measuring the right half slope of the bending moment diagram, should be adjacent to each other, since a load outside the weight area, but inside the force transducer, can only yield a partially correct calculated force.
 
     The methodology developed in this work is also applicable to a distributed load within the weight area, as shown in  FIG. 33 , where a uniformly distributed force w is applied over a length d w . The shear force diagram in  FIG. 34  shows that the slope of the shear force curve over the length d w  is equal to the distributed force w. The drop in the shear force curves across the distributed force is equal to the resultant force F w =wd w , which is the same as that for the case of a concentrated force F 2 , as shown in  FIG. 26 . The comparison of the bending moment diagrams for the concentrated force and the distributed force, whose centroid is located at the same position as the concentrated force, is shown in  FIG. 35 . It can be seen from  FIG. 27  that the intersection of the left and right bending moment curves is the position of the concentrated force F 2 , and it is also the position of the centroid of the distributed force w, as shown in  FIG. 35 . This concept can be extended to a distributed load of any shape. A distributed load outside the force transducer can be treated as the case in  FIG. 28 , and a distributed load between strain gages 1 and 2 or between strain gages 3 and 4 can be treated as the case in  FIG. 30 . A distributed load that crosses a strain gage boundary is in effect a multiple load problem, which is beyond the scope of the current chapter and will be discussed in Chapter 3. Finally, a combination of concentrated and distributed loads follows the same methodology as that for a single load case as long as they do not cross a strain gage boundary. 
     Let d L  and d R  be the distances between the force F and the left and right boundaries, respectively, as shown in  FIG. 36 . There are two types of a rest boundary condition: a firm rest boundary condition, where the reaction force acts at the edge of a firm abutment, and a soft rest boundary condition ( FIG. 37 ) [24], where the reaction force is distributed over a soft abutment, which can be represented by a concentrated reaction force at the edge of the abutment and a reaction moment ( FIG. 36 ). The distances between the left and right reaction forces R A  and R B , and the edges of the soft abutments, are d A  and d B , respectively, and the lengths of the beam sitting on the left and right soft abutments are D A  and D B , respectively ( FIG. 37 ). In this study, a firm rest boundary is similar to a pinned boundary, and a soft rest boundary is similar to a fixed boundary, except that the direction of the reaction moment for the soft rest boundary is opposite to that of the fixed boundary, and the slope of a fixed boundary is zero while that of a soft rest boundary may not be zero. With a positive external force in the upward direction and a positive external moment in the counterclockwise direction, the reaction forces and moments for pinned, firm rest, soft rest, pinned-fixed, and fixed boundary conditions are shown in Table 2.1, where a statically indeterminate analysis for the pinned-fixed and fixed boundary conditions is used [25]. 
     
       
         
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 2.1 
               
             
             
               
                   
               
               
                 Reaction forces and moments for pinned, firm rest, soft rest, pinned- 
               
               
                 fixed, and fixed boundary conditions 
               
             
          
           
               
                 Boundary  
                   
                   
                   
                   
                   
               
               
                 Conditions 
                 R A   
                 R B   
                 M A   
                 M B   
                 R A  + R B   
               
               
                   
               
               
                 Pinned or Firm Rest 
                 
                   
                     
                       
                         
                           Fd 
                           R 
                         
                         L 
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           Fd 
                           L 
                         
                         L 
                       
                     
                   
                 
                 0 
                 0 
                 F 
               
               
                   
               
               
                 Soft Rest 
                 
                   
                     
                       
                         
                           F 
                           ⁡ 
                           
                             ( 
                             
                               
                                 d 
                                 R 
                               
                               + 
                               
                                 d 
                                 B 
                               
                             
                             ) 
                           
                         
                         
                           L 
                           + 
                           
                             d 
                             A 
                           
                           + 
                           
                             d 
                             B 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           F 
                           ⁡ 
                           
                             ( 
                             
                               
                                 d 
                                 L 
                               
                               + 
                               
                                 d 
                                 A 
                               
                             
                             ) 
                           
                         
                         
                           L 
                           + 
                           
                             d 
                             A 
                           
                           + 
                           
                             d 
                             B 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         - 
                         
                           
                             
                               Fd 
                               A 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   d 
                                   R 
                                 
                                 + 
                                 
                                   d 
                                   B 
                                 
                               
                               ) 
                             
                           
                           
                             L 
                             + 
                             
                               d 
                               A 
                             
                             + 
                             
                               d 
                               B 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             Fd 
                             B 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 d 
                                 L 
                               
                               + 
                               
                                 d 
                                 A 
                               
                             
                             ) 
                           
                         
                         
                           L 
                           + 
                           
                             d 
                             A 
                           
                           + 
                           
                             d 
                             B 
                           
                         
                       
                     
                   
                 
                 F 
               
               
                   
               
               
                 Pinned- Fixed 
                 
                   
                     
                       
                         
                           
                             Fd 
                             R 
                             2 
                           
                           
                             2 
                             ⁢ 
                             
                               L 
                               3 
                             
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               d 
                               L 
                             
                             + 
                             
                               2 
                               ⁢ 
                               L 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             Fd 
                             L 
                           
                           
                             2 
                             ⁢ 
                             
                               L 
                               3 
                             
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               3 
                               ⁢ 
                               
                                 L 
                                 2 
                               
                             
                             - 
                             
                               d 
                               L 
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 0 
                 
                   
                     
                       
                         
                           - 
                           
                             
                               
                                 Fd 
                                 L 
                               
                               ⁢ 
                               
                                 d 
                                 R 
                               
                             
                             
                               2 
                               ⁢ 
                               
                                 L 
                                 2 
                               
                             
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               d 
                               L 
                             
                             + 
                             L 
                           
                           ) 
                         
                       
                     
                   
                 
                 F 
               
               
                   
               
               
                 Fixed 
                 
                   
                     
                       
                         
                           
                             Fd 
                             R 
                             2 
                           
                           
                             L 
                             3 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               3 
                               ⁢ 
                               
                                 d 
                                 L 
                               
                             
                             + 
                             
                               d 
                               R 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             Fd 
                             L 
                             2 
                           
                           
                             L 
                             3 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               d 
                               L 
                             
                             + 
                             
                               3 
                               ⁢ 
                               
                                 d 
                                 R 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             Fd 
                             L 
                           
                           ⁢ 
                           
                             d 
                             R 
                             2 
                           
                         
                         
                           L 
                           2 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         - 
                         
                           
                             
                               Fd 
                               L 
                               2 
                             
                             ⁢ 
                             
                               d 
                               R 
                             
                           
                           
                             L 
                             2 
                           
                         
                       
                     
                   
                 
                 F 
               
               
                   
               
             
          
         
       
     
     The shear force diagrams for the firm rest, soft rest, pinned-fixed and fixed boundary conditions are the same as that for the pinned boundary conditions in  FIG. 26 . Similarly, the slopes of the left and right bending moment curves are still R A  and −R B , respectively, as shown in  FIG. 38 . Positive and negative strains are expected at the bottom surfaces of the beams near the soft rest and fixed boundaries due to the positive and negative ( FIG. 38 ) bending moments there, respectively. The sum of the reaction forces R A  and R B  will always equal the applied force, as shown in Table 2.1; hence Eqs. (2.7) through (2.18) still apply regardless of the boundary conditions. While inaccurate boundary conditions can create a discrepancy in the strains, the magnitude of the load can still be accurately calculated. 
     2.2.2 Identification of the Location of a Load 
     When a force is located inside a weight area, the position of the force can be identified using the strain gage readings. Since the position of the intersection point of the left and right bending moment curves in  FIG. 27  corresponds to the position of the force, the relative position a 2  of the force F 2  can be calculated by simultaneously solving the equations for the left and right bending moment curves: 
                     α   2     =         f   2       d   23       =           d   12     ⁢       d   23     ⁡     (         γ   3     ⁢     ɛ   3       -       γ   4     ⁢     ɛ   4         )         +       d   12     ⁢       d   34     ⁡     (         γ   3     ⁢     ɛ   3       -       γ   2     ⁢     ɛ   2         )                 d   23     ⁢       d   34     ⁡     (         γ   2     ⁢     ɛ   2       -       γ   1     ⁢     ɛ   1         )         +       d   12     ⁢       d   23     ⁡     (         γ   3     ⁢     ɛ   3       -       γ   4     ⁢     ɛ   4         )                       (   2.19   )               
The position of the force is not affected by the boundary conditions. Since the shear force diagram in  FIG. 26  is the same for pinned, firm rest, soft rest, pinned-fixed, and fixed boundary conditions, the discontinuity in the shear force diagram due to the presence of a concentrated force always occurs at the intersection point of the two bending moment curves. The position of the centroid of a distributed force can also be calculated from Eq. (2.19), a potential application of which is to identify the longitudinal center of gravity of a vehicle on a bridge. For the case of a homogenous, prismatic beam with unequal strain gage spacing, Eq. (2.19) becomes
 
                     α   2     =           d   12     ⁢       d   23     ⁡     (       ɛ   3     -     ɛ   4       )         +       d   12     ⁢       d   34     ⁡     (       ɛ   3     -     ɛ   2       )                 d   23     ⁢       d   34     ⁡     (       ɛ   2     -     ɛ   1       )         +       d   12     ⁢       d   23     ⁡     (       ɛ   3     -     ɛ   4       )                     (   2.20   )               
When d 12 =d 23 =d 34 =d, Eq. (2.20) becomes
 
                     α   2     =         -     ɛ   2       +     2   ⁢     ɛ   3       -     ɛ   4           -     ɛ   1       +     ɛ   2     +     ɛ   3     -     ɛ   4                 (   2.21   )               
The absolute position of the force p n  can be obtained from Eq. (2.13) ( FIG. 2.4 ):
 
 p   n   =d   n   +a   nj   d   nj   (2.22)
 
where d n  is the absolute position of the strain gage immediately to the left of the force, measured from the left boundary.
 
     Similarly, the location of a load on a cantilever beam in  FIG. 23  can be identified. Let d a  be the distance between the strain gage at point a to the fixed boundary. The absolute position of the force F, measured from the fixed end, is 
                     d   F     =       d   α     +         ɛ   a     ⁢     d   ab           ɛ   a     -       (         E   b     ⁢     I   b     ⁢     y   a           E   a     ⁢     I   a     ⁢     y   B         )     ⁢     ɛ   b                     (   2.23   )               
For the case of a homogenous, prismatic beam, Eq. (2.23) becomes
 
                     d   F     =       d   a     +         ɛ   a     ⁢     d   ab           ɛ   a     -     ɛ   b                   (   2.24   )               
2.2.3 Error Mitigation and Calibration
 
     It is known that strain gages will drift when temperature changes within the material [14]. In fact, this is typically the main cause for error when dealing with strain gages [26]. For an outdoor setup, the ambient temperature can change over 10° C. throughout the course of the day. Depending on the material, this can cause a change of 1° C. or more for the actual material temperature through convection. Aluminum 6061-T6 has a coefficient of thermal expansion of 23.6 με/° C. Hence a 1° C. shift in temperature of the material can introduce a 23.6 με drift in all the strain gages simultaneously. If this happens, the force transducer will still respond correctly. When combining four strain gages in a typical strain gage based transducer in a Wheatstone bridge or mathematically, the temperature effects can cancel out when uniform strain gage drift occurs [27]. For the force transducer in Eq. (2.11), when γ 1 =γ 2  and γ 3 =γ 4 , the same shift in all four strain gages cancels out. Non-uniform thermal expansion often occurs when the sun causes a part of a structure to expand more than some other part due to radiation. For non-uniform thermal expansion or when γ 1 =γ 2  and γ 3 =γ 4  are not satisfied, such as the case of a non-homogeneous and/or non-prismatic beam, error can be eliminated by zeroing strain gages before readings are taken. 
     One of the inherent features of the force transducer methodology is that it possesses an absolute error instead of a relative error for the strain gages, which is due to the accuracy of the equipment and the sensitivity of measurements. For example, each strain gage is accurate to within 2 to 3 με regardless of whether it is reading 5 or 500 με. It is recommended to use optimal strain readings between 1,000 and 1,500 με when designing a force transducer to increase the signal to noise ratio [15]. A method to increase the output signal is to introduce local notching near a strain gage [28]. 
     As with any transducer design, calibration is expected and plays a vital role in developing a methodology [29]. The geometric and material properties can deviate from theoretical values. In the case of extreme temperatures, the Young&#39;s Modulus can change by 1 to 3% [14]. An accepted calibration method for an over-determined problem is to minimize the least squares error between the experimental and calculated magnitudes of loads [8]. Another accepted method is to reduce the maximum error. 
     The calibration procedure is set up so that a single known nonzero force F n , where n=2, 3, . . . , N+1, is placed between strain gages ε n  and ε n+1 , which is inside the weight area of the force transducer FT n  that consists of strain gages ε n−1  through ε n+2 , and all the other forces on the beam are zero; all the strains are recorded ( FIG. 39 ). There are N+3 strain gages on the beam that create N force transducers, each of which consists of four adjacent strain gages, in order to identify the magnitudes of the N nonzero forces. Equations (2.11) and (2.12) are modified to reflect the N force transducers created: 
     
       
         
           
             
               
                 
                   
                     FT 
                     n 
                   
                   = 
                   
                     
                       F 
                       n 
                     
                     = 
                     
                       
                         - 
                         
                           
                             ɛ 
                             
                               n 
                               - 
                               1 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 γ 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                               
                                 d 
                                 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   , 
                                   n 
                                 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           ɛ 
                           n 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               γ 
                               n 
                             
                             
                               d 
                               
                                 
                                   n 
                                   - 
                                   1 
                                 
                                 , 
                                 n 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           ɛ 
                           
                             n 
                             + 
                             1 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               γ 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                             
                               d 
                               
                                 
                                   n 
                                   + 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   + 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
                           ɛ 
                           
                             n 
                             + 
                             2 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               γ 
                               
                                 n 
                                 + 
                                 2 
                               
                             
                             
                               d 
                               
                                 
                                   n 
                                   + 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   + 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2.25 
                   ) 
                 
               
             
           
         
       
     
     For each applied force F n , the N force transducers FT n  (n=2, 3, . . . , N+1), each involving four scaling factors γ n−1  through γ n+2 , are used. It should be noted that the goal of the force transducer here is not only to identify a load inside the weight area, but also to identify a zero load outside the force transducer. Hence both the loaded and unloaded zones are used in calibration. All the force transducers used, except FT n , are expected to read zero. The force transducer FT n−1  and FT n+1  are not used because the force F n  is outside their weight areas, but inside the force transducers, which can only provide a partial response. The calibration equation is
 
[ F ]=[ε][γ]  (2.26)
 
where [F] is a q×1 matrix, in which q=N 2 −2N+2, that represents all the known forces, [ε] is a q×m matrix, in which m=N+3, that represents the measured strains from strain gages forming the force transducers, and [γ] is an m×1 matrix that represents the scaling factors. An example of Eq. (2.26) for N=5 is
 
                     [           F   2                 F   4     =   0                 F   5     =   0                 F   6     =   0               F   3                 F   5     =   0                 F   6     =   0                 F   2     =   0               F   4                 F   6     =   0                 F   2     =   0                 F   3     =   0               F   5                 F   2     =   0                 F   3     =   0                 F   4     =   0               F   6           ]     =             [             -     ɛ   1       /     d   12               ɛ   2     /     d   12               ɛ   3     /     d   34               -     ɛ   4       /     d   34           0       0       0       0           0       0           -     ɛ   3       /     d   34               ɛ   4     /     d   34               ɛ   5     /     d   56               -     ɛ   6       /     d   56           0       0           0       0       0           -     ɛ   4       /     d   45               ɛ   5     /     d   45               ɛ   6     /     d   67               -     ɛ   7       /     d   67           0           0       0       0       0           -     ɛ   5       /     d   56               ɛ   6     /     d   56               ɛ   7     /     d   78               -     ɛ   8       /     d   78               0           -     ɛ   2       /     d   23               ɛ   3     /     d   23               ɛ   4     /     d   45               -     ɛ   5       /     d   45           0       0       0           0       0       0           -     ɛ   4       /     d   45               ɛ   5     /     d   45               ɛ   6     /     d   67               -     ɛ   7       /     d   67           0           0       0       0       0           -     ɛ   5       /     d   56               ɛ   6     /     d   56               ɛ   7     /     d   78               -     ɛ   8       /     d   78                   -     ɛ   1       /     d   12               ɛ   2     /     d   12               ɛ   3     /     d   34               -     ɛ   4       /     d   34           0       0       0       0           0       0           -     ɛ   3       /     d   34               ɛ   4     /     d   34               ɛ   5     /     d   56               -     ɛ   6       /     d   56           0       0           0       0       0       0           -     ɛ   5       /     d   56               ɛ   6     /     d   56               ɛ   7     /     d   78               -     ɛ   8       /     d   78                   -     ɛ   1       /     d   12               ɛ   2     /     d   12               ɛ   3     /     d   34               -     ɛ   4       /     d   34           0       0       0       0           0           -     ɛ   2       /     d   23               ɛ   3     /     d   23               ɛ   4     /     d   45               -     ɛ   5       /     d   45           0       0       0           0       0       0           -     ɛ   4       /     d   45               ɛ   5     /     d   45               ɛ   6     /     d   67               -     ɛ   7       /     d   67           0               -     ɛ   1       /     d   12               ɛ   2     /     d   12               ɛ   3     /     d   34               -     ɛ   4       /     d   34           0       0       0       0           0           -     ɛ   2       /     d   23               ɛ   3     /     d   23               ɛ   4     /     d   45               -     ɛ   5       /     d   45           0       0       0           0       0           -     ɛ   3       /     d   34               ɛ   4     /     d   34               ɛ   5     /     d   56               -     ɛ   6       /     d   56           0       0           0       0       0       0           -     ɛ   5       /     d   56               ɛ   6     /     d   56               ɛ   7     /     d   78               -     ɛ   8       /     d   78             ]     ⁡     [           γ   1               γ   2               γ   3               γ   4               γ   5               γ   6               γ   7               γ   8           ]                 (   2.27   )               
where the partitions show five loading scenarios, and the measured strains within each partition are given different signs and divided by the distances between the strain gages according to Eq. (2.25). In Eq. (2.27), the nonzero force F 2  is used first to calibrate force transducer FT 2 ; zero forces are simultaneously used in other weight areas to calibrate force transducers FT 4 , FT 5 , and FT 6 . The nonzero force F 3  is used next to calibrate force transducer FT 3 , with zero forces in other weight areas used to calibrate force transducers FT 5  and FT 6 . This procedure is repeated through the nonzero force F 6 . Equation (2.26) is under-determined when N≦4, and over-determined when N≧4. The calibration procedure should be set up so that Eq. (2.26) is over-determined. The absolute value of the difference between the known force and the output of the force transducer with the force inside its weight area, divided by the known force, is referred to as the force error. The absolute values of the outputs of the other force transducers with the force outside the force transducers, which are expected to read zero, divided by the known force, are referred to as the zero errors. The maximum force error is denoted by λ F , and the maximum zero error is denoted by λ 0 . The total error λ T  is defined by
 
λ T   =Cλ   F +λ 0   (2.28)
 
where C is a user defined positive parameter. The theoretical γ n  (n=1, 2, . . . , N+3) are used as the starting point for calibration, and the Microsoft Excel Solver is used to adjust all γ n  to minimize λ T  with the constraints γ n &gt;0. The program can be run several times by adjusting C to achieve desired force and zero errors. Acceptable error limits must be defined given economic considerations, sensitivity of measurement equipment, and specific application [30]. The goal for the experimental data in Sec. 2.3 was for the force errors to be within 5%, and the zero errors within 10%. The position error is defined by |Δp n |/L, where Δp n  is the difference between a known force position and the calculated one. By applying the calibration procedure for the magnitudes of the forces, the position errors usually do not need to be calibrated and are usually within 5%. The position of a force can also be calibrated for a non-homogeneous and/or non-prismatic beam using Eq. (2.19). However, the position cannot be calibrated for a homogeneous, prismatic beam, since no geometric and material properties (i.e., γ n ) appear in Eqs. (2.20) and (2.21). Once the seating factors γ n  for all strain gages are determined using known loads, they can be used to calculate the magnitudes and locations of unknown loads.
 
     For the case of a single force transducer on a homogenous, prismatic beam, the magnitude of one known force inside the weight area is needed to determine the global scaling factor γ in Eq. (2.16), as shown in Sec. 2.3.1. For the case of multiple force transducers on the beam, the above calibration procedure can also be used to determine γ, with γ n  in Eq. (2.26) replaced by γ. Alternatively, one force transducer on the beam can be used to determine γ with a known force, and the same scaling factor can be used in another force transducer to calculate an unknown force, as shown in Sec. 2.3.2. When the strain gages are equally spaced, γ n  in Eq. (2.26) can be replaced by β n . 
     2.3 Experimental Results 
     2.3.1 Identifying the Magnitude and Location of a Load on a Steel Beam with Rest Boundary Conditions 
     A quantity of four uniaxial Vishay strain gages (part number CEA-06-250UN-350) were bonded, with equal spacing, along the bottom surface of a steel A36 beam with rest boundary conditions, where the supports for the beam were made of aluminum, as shown in  FIG. 40 . It should be noted that the blocks on the end abutments are only a guide for the beam and does not create fixed boundary conditions, as shown in  FIG. 41 . The weight was applied via dumbbells, supported by a thin string to accurately apply a force at a given position, as shown in  FIG. 42 . The nominal weight of the dumbbells was used, whose weights were within 1.2% of their nominal values. The weight was moved to different positions inside the weight area. A Vishay P3 strain indicator and a Vishay SB-10 switch unit were used to display the strain gage outputs. Since strain is directly proportional to the Young&#39;s modulus, and steel is three times suffer than aluminum, the strain readings will be one-third the values from a comparable aluminum beam. The theoretical Young&#39;s modulus for aluminum is 69 GPa and that for steel is 200 GPa. Unlike the design range of an aluminum beam force transducer, which is between 1,000 to 1,500 με, that of a steel beam force transducer is between 300 to 500 με, which can reduce the signal to noise ratio. The cross section of the steel beam is 19.1 mm by 25.4 mm ( FIG. 43 ) and its side view with strain gage locations is shown in  FIG. 44 . 
     The errors for the uncalibrated magnitude of the force were within 2.2%. Since there is one force transducer on a homogeneous, prismatic beam, there is one global scaling factor γ. A known force F 2 =445 N at a 23 =0.5 was used to determine γ in Eq. (2.16). The errors for the calibrated magnitudes of all the unknown forces were within 0.6%. The position errors for all the forces were within 0.1%, as shown in Table 2.2. The adjusted global scaling factor was 102% of the theoretical global scaling factor. The bending moment diagram was also calculated using measured strains and compared with the theoretical bending moment diagram for the firm rest boundary conditions, as shown in  FIG. 45 . 
     
       
         
               
             
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 2.2 
               
             
             
               
                   
               
               
                 Calculated magnitudes and positions of a force on a steel beam 
               
             
          
           
               
                 Experimental 
                   
                 Uncalibrated 
                 Calibrated 
                 Calibrated 
                 Calibrated 
                   
                   
               
               
                 Force 
                 Experimental 
                 FT 2   
                 FT 2   
                 FT 2   
                 FT 2   
                 Calculated 
                 Position 
               
               
                 (N) 
                 α 23   
                 (N) 
                 (N) 
                 (N) 
                 (Error) 
                 α 23   
                 Error 
               
               
                   
               
               
                 445 
                 0.10 
                 438 
                 1.6% 
                 445 
                 0.0% 
                 0.10 
                 0.0% 
               
               
                 445 
                 0.20 
                 440 
                 1.1% 
                 447 
                 0.6% 
                 0.22 
                 0.1% 
               
               
                 445 
                 0.30 
                 435 
                 2.2% 
                 442 
                 0.6% 
                 0.31 
                 0.0% 
               
               
                 445 
                 0.40 
                 435 
                 2.2% 
                 442 
                 0.6% 
                 0.41 
                 0.1% 
               
               
                 445 
                 0.50 
                 438 
                 1.6% 
                 445 
                 0.0% 
                 0.50 
                 0.0% 
               
               
                 445 
                 0.60 
                 435 
                 2.2% 
                 442 
                 0.6% 
                 0.60 
                 0.0% 
               
               
                 445 
                 0.70 
                 435 
                 2.2% 
                 442 
                 0.6% 
                 0.70 
                 0.0% 
               
               
                 445 
                 0.80 
                 438 
                 1.6% 
                 445 
                 0.0% 
                 0.80 
                 0.0% 
               
               
                 445 
                 0.90 
                 440 
                 1.1% 
                 447 
                 0.6% 
                 0.90 
                 0.0% 
               
               
                   
               
             
          
         
       
     
     While the four strain gages in  FIG. 44  demonstrated the application of a single force transducer, the beam in  FIG. 40  was also used for multiple force transducers, where 12 strain gages were bonded along the bottom surface of the beam as shown in  FIG. 46 . The adjusted global scaling factor from the single force transducer in  FIG. 44  was used, with calculated magnitudes and positions of the forces shown in Table 2.3. The position of a force is the strain gage number n directly to the left of the force added to the corresponding a nj . The maximum force error was 3.7%, the maximum zero error was 5.1%, and the maximum position error was 0.6%. The bending moment diagram for a 445 N force at position 8.5 was calculated using measured strains and compared with the theoretical bending moment diagram for the firm rest boundary conditions, as shown in  FIG. 47 . 
                                                                                         TABLE 2.3                   Calculated magnitudes and positions of a force on a steel beam in FIG. 46            Experimental       Experimental   Calculated   Calculated       Maximum           Force   Experimental   Position   Force   Position   Force   Zero   Position       (N)   Position   (m)   (N)   (m)   Error   Error   Error                    445   2.50   0.313   448   0.306   0.7%   1.1%   0.4%       445   3.50   0.438   456   0.440   2.4%   2.8%   0.2%       445   4.50   0.575   455   0.575   2.4%   3.4%   0.0%       445   5.50   0.700   439   0.691   1.4%   5.1%   0.6%       445   6.50   0.813   461   0.820   3.7%   4.5%   0.5%       445   7.50   0.938   455   0.931   2.2%   1.8%   0.4%       445   8.50   1.063   445   1.063   0.0%   2.1%   0.0%       445   10.50   1.313   432   1.319   2.8%   3.4%   0.4%                    
2.3.2 Identifying the Magnitude and Location of a Distributed Load on an Aluminum Beam with Rest Boundary Conditions
 
     A quantity of twelve uniaxial Vishay strain gages (part number CEA-13-250UN-350) were bonded, with equal spacing, along the bottom surface of an aluminum 6061-T6 beam, as shown in  FIG. 48 , where multiple dumbbells created a distributed load on the beam. The cross section of the beam is 12.7 mm by 305 mm ( FIG. 49 ) and its side view with strain gage locations is shown in  FIG. 50 . A known force F 6 =445 N at a 67 =0.5 was used to determine the global calibration factor γ, which was 97.5% of its theoretical value. A distributed load was placed within the weight area of FT 12BC , with the results shown in Table 2.4. The magnitude and position of the resultant force were calculated from the magnitudes and positions of forces F 5 , F 6 , and F 7  ( FIG. 50 ). The force errors were within 0.8% and the position errors were within 2.3%. The bending moment diagram was calculated using measured strains and compared with the theoretical bending moment diagram for firm rest boundary conditions, for a 1,557 N distributed force at the center of the beam, as shown in  FIG. 51 . 
                                                                             TABLE 2.4                   Calculated magnitudes and positions of forces for       a distributed force within the weight area of FT 12BC              Experi-                                mental   Experi-   Experi-   Experi-       Force           Resultant   mental   mental    mental   FT 12BC      Error    Position       Force (N)   F 5 (N)   F 6 (N)   F 7 (N)   (N)    FT 12BC      Error               1,246   400   445   400   1,248   0.2%   2.3%       1,401   400   601   400   1,396   0.4%   1.0%       1,557   400   756   400   1,544   0.8%   0.6%                    
2.3.3 Identifying the Magnitude and Location of a Load on an Aluminum Beam with Firm Rest, Soft Rest, Firm Rest-Fixed, and Fixed Boundary Conditions
 
     A quantity of six uniaxial Vishay strain gages (part number CEA-13-250UN-350) were bonded, with equal spacing, along the bottom surface of an aluminum 6061-T6 beam, as shown in  FIG. 52 , where the fixed boundary conditions were approximated by bolted clamps ( FIGS. 53 and 54 ). A dumbbell was directly placed on the beam. While the deformation of the beam can cause the load to be distributed over a small area, it has been shown in Sec. 2.2.1 that the magnitude of a distributed load within a weight area can be calculated using the same methodology as that for a concentrated load. The cross section of the aluminum beam is 3.2 mm by 40 mm ( FIG. 54 ) and its side view with strain gage locations is Shown in  FIG. 55 . The force transducer FT 3  was not used because the load was outside the weight area, but inside the force transducer. 
     The beam with firm rest and soft rest boundary conditions is shown in  FIGS. 56 and 57 , respectively, where sponges were used to simulate the soft rest boundaries. The bending moment diagram was calculated using measured strains and compared with the theoretical bending moment diagram for the firm rest boundary conditions, as shown in  FIG. 58 . The accuracy of the experimental firm rest boundary conditions can be quantitatively determined from the equations of the calculated left and right bending moment curves. The equations for the curves are determined by adding trend lines to the calculated bending moments using Microsoft Excel, with a reported coefficient of determination R 2 , where R 2 =1 indicates a perfect correlation to a line. The equation of the left bending moment curve is M=R A x+M A , where M A =0 for a firm rest boundary, and that of the right bending moment curve is M=−R B x+b B , where b B =R B L for a firm rest boundary. The equations of the two bending moment curves can be used to calculate the reaction moments M A  and M B  in  FIG. 58 , which can show whether a rest boundary was a firm rest boundary or a soft rest boundary. The left reaction moment is determined by setting x=0 for the left trend line equation, and the right reaction moment is determined by setting x=L for the right trend line equation. The calculated left and right reaction moments are −0.009 Nm and −0.006 Nm, respectively, which are within 0.4% of the largest bending moment value. The calculated d A  from  FIG. 2.37  is −0.6 mm; consequently d A /D A =−0.8%. The calculated d B  is −0.8 mm; consequently d B /D B =1.0%. The discrepancies between the theoretical and calculated bending moments near a boundary indicate that either the boundary was not an ideal boundary or there was a measurement error. Given that the calculated d A  and d B  were negative, the latter was most likely the case. Since the absolute values of d A /D A  and d B /D B  were within 1%, the rest boundaries were indeed firm rest boundaries, which can also be seen from  FIG. 56 , where the beam contacted the aluminum supports at their edges. Similarly, the theoretical and calculated bending moment diagrams for the soft rest boundary conditions are shown in  FIG. 59 , where evenly distributed reaction forces were assumed for the theoretical bending moment diagram, with d A /D A =d B /D b =50%. The calculated left reaction moment is 0.537 Nm compared with the theoretical value of 0.569 Nm from Table 2.1. The calculated right reaction moment is 0.278 Nm compared with the theoretical value of 0.319 Nm from Table 2.1. The calculated d A /D A , and d B /D B  are 46% and 44%, respectively, which are close to the theoretical values of 50%. A triangular distributed reaction force with a maximum value at the edge of an abutment would give d A /D A =d B /D B =33.3%; the experimental boundary conditions were closer to evenly distributed reaction forces. 
     The experimental setup for the firm rest-fixed boundary conditions are shown in  FIG. 60 . Comparisons of the theoretical and calculated bending moment diagrams for the firm rest-fixed, and fixed boundary conditions are shown in  FIGS. 61 and 62 , respectively. The bolted clamps for the fixed boundaries were designed to be much larger than the test beam ( FIG. 52 ); negative strains were observed near the fixed boundaries, as expected. The calculated left reaction moment for the firm rest-fixed beam in  FIG. 2.40  is −0.04 Nm, which is 0.9% of the largest bending moment value, and the calculated right reaction moment is −1.89 Nm, which is 9.1% lower in magnitude than the reaction moment of the fixed boundary (−2.08 Nm from Table 2.1). The calculated left reaction moment for the fixed beam in  FIG. 62  is −2.70 Nm compared to −2.97 Nm for the left fixed boundary from Table 2.1, and the calculated right reaction moment is −1.43 Nm compared to −1.49 Nm for the right fixed boundary from Table 2.1. The discrepancies between the theoretical and calculated reaction moments in  FIGS. 61 and 62  indicate that the bolted clamps did not completely apply the theoretical reaction moments. While the errors between the theoretical and calculated bending moments are not directly related to those between the theoretical and calculated magnitudes and positions of the forces, they would affect the accuracy in determining the reaction moments at the boundaries. 
     The scaling factors for firm rest, soft rest, firm rest-fixed, and fixed boundary conditions were determined by applying a known force F 4  at position 4.5, and used to calculate the unknown force F 2  at position 2.5. While the scaling factors for the four types of boundary conditions were determined separately for each type of boundary conditions, they were within a tight range of 88.4 to 90.7% of the theoretical γ, showing that the geometric and material properties are independent of the boundary conditions. The calculated magnitudes and positions of the unknown forces are shown in Table 2.5. The errors for the magnitudes of the forces were within 2.3% when the force was inside the weight area of the force transducer FT 2 . When the force was outside the force transducer FT 4 , the force transducer output should read zero, and the zero errors were within 2.5%. The position errors were within 2.3%. 
                                                                                                 TABLE 2.5                   Calculated magnitudes and positions of forces on an       aluminum beam with four types of boundary conditions            Boun-    Experi-    Experi-                    Calcu-           dary   mental   mental                   lated   Posi-       Con-    Force    Posi-    FT 2      FT 2      FT 4      FT 4      Posi-   tion       dition    (N)   tion   (N)    Error    (N)    Error    tion    Error                    Firm   8.9   2.50    8.7   2.3%   0.1   0.6%   2.55   0.8%       Rest   13.3   2.50   13.2   1.3%   −0.1   0.8%   2.53   0.4%           17.8   2.50   17.5   1.4%   0.0   0.0%   2.49   0.2%           22.2   2.50   22.4   0.6%   0.0   0.0%   2.52   0.3%       Soft   8.9   2.50    8.7   1.7%   0.0   0.0%   2.46   0.6%       Rest   13.3   2.50   13.2   1.2%   0.1   0.4%   2.44   1.0%           17.8   2.50   17.6   1.4%   0.1   0.3%   2.50   0.0%           22.2   2.50   22.4   0.6%   0.1   0.5%   2.45   0.9%       Firm   13.3   2.50   13.2   0.8%   −0.1   0.4%   2.64   2.3%       Rest-   17.8   2.50   17.8   0.0%   −0.2   1.2%   2.53   0.5%       Fixed   22.2   2.50   22.7   2.2%   −0.2   0.7%   2.44   1.0%           31.1   2.50   31.9   2.3%   −0.8   2.5%   2.53   0.5%       Fixed   31.1   2.50   31.2   0.1%   0.1   0.2%   2.51   0.1%           35.6   2.50   35.9   0.8%   −0.1   0.3%   2.47   0.4%           40.0   2.50   40.0   0.0%   0.1   0.3%   2.49   0.1%           44.5   2.50   43.9   1.2%   0.0   0.0%   2.53   0.5%                    
2.3.4 Identifying the Magnitude and Location of a Load on a Continuously Tapered Aluminum Beam with Rest Boundary Conditions
 
     A continuously tapered aluminum 6061-T6 beam with rest boundary conditions, is shown in  FIG. 63 . The beam has a taper angle of 5°, and a width of 25.4 mm as shown in  FIGS. 64 and 65 . A quantity of 12 uniaxial Vishay strain gages (part number CEA-13-250UN-350) were bonded, with equal spacing, along the bottom surface of the beam, and the strain gage locations are shown in  FIG. 66 . The hexadecimal numbering system is used, where the force transducer formed by strain gages 9 through 12 is FT 9ABC  or FT A . Equation (2.11) was used to calculate the theoretical scaling factor for each strain gage at different positions, which have different cross sections. 
     The calibration procedure in Sec. 2.2.3 was used, where a known force of 845 N was applied inside the weight areas of all nine possible force transducers. An unknown force of 1,201 N was then applied inside all the weight areas and calculated using the adjusted individual scaling factors determined from the calibration procedure. The theoretical and adjusted scaling factors for all the loading scenarios are shown in Table 2.6; the average force error was reduced from 5.7% to 2.2% after calibration, and the average zero error was reduced from 6.6% to 4.1% after calibration. The adjusted scaling factors ranged from 67 to 100% of their theoretical values. 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 2.6 
               
             
             
               
                   
               
               
                 Theoretical and adjusted scaling factors for the continuously 
               
               
                 tapered aluminum beam with average force and zero errors 
               
             
          
           
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 Average 
                 Average 
               
               
                 Strain 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 Force  
                 Zero 
               
               
                 Gage  
                 1  
                 2  
                 3  
                 4  
                 5  
                 6  
                 7 
                 8 
                 9  
                 10  
                 11  
                 12  
                 Error 
                 Error 
               
               
                   
               
               
                 Theoretical  
                 0.31 
                 0.55 
                 0.86  
                 1.25  
                 1.70 
                 2.22 
                 2.22  
                 1.70  
                 1.25 
                 0.86 
                 0.55 
                 0.31  
                 5.7% 
                 6.6% 
               
               
                 Adjusted  
                 0.25  
                 0.52 
                 0.81 
                 1.21  
                 1.60  
                 2.21 
                 2.22  
                 1.69  
                 1.23  
                 0.80 
                 0.47  
                 0.27  
                 2.2% 
                 4.1% 
               
               
                   
               
             
          
         
       
     
     While an experimental strain curve for a 1,201 N force at position 4.5 in  FIG. 67  is not a line, after applying Eq. (2.4), the calculated bending moment diagram is similar to the theoretical one for the firm rest boundary conditions. The calculated left and right reaction moments are 13 Nm and 8 Nm, respectively, which are within 2.8% of the largest bending moment value. The calculated d A =16.6 mm with d A /D A =16.3%, and the calculated d B =19.2 mm with d B /D B =18.8%, indicate that the boundary conditions were between firm and soft rest boundary conditions. Any discrepancies between the theoretical and calculated bending moments are not necessarily reflective of the errors of the calculated magnitude and position of a force. 
     A possible source of error was hysteresis and/or temperature drift in the beam. After unloading the beam, it sometimes did not return to zero strain. Another possible source of error is the low signal to noise ratio, since the maximum strain gage reading of 626 με was below the recommended range of 1,000 to 1,500 με. A third source of error was that the vertical cross sections used for the theoretical seating factors are not perpendicular to the neutral axes, which is in violation of die Euler-Bernoulli beam theory [21]. The reason that this shape was chosen was to more accurately represent the shapes of army bridges. A fourth source of error was due to horizontal reaction forces at the boundaries that were not considered in the analysis. While the above sources of error arise before calibration, a fifth possible source of error for a continuously tapered beam was that there were many scaling factors; the prismatic beams in Secs. 2.3.1 through 2.3.3 had only one scaling factor. It is observed from Table 2.6 that the highest scaling factor is almost an order of magnitude greater than the lowest scaling factor, which demonstrates the range of applicability of Eq. (2.11) for a continuously tapered beam. 
     The calibrated magnitudes and positions of the forces from several force transducers are shown in Table 2.7, where the force errors were within 4.0%, the zero errors were within 5.7%, and the position errors were within 2.0%. The force error in each row in Table 2.7 corresponds to the error for the force transducer with the force inside its weight area, and the zero error is the maximum error for the remaining force transducers that are expected to read zero. The force transducers with the forces outside the weight areas, but inside the force transducers, were not used and indicated as “N/A” in Table 2.7. Note that the middle two strain gages of a force transducer can be spaced any distance apart and still accurately calculate the magnitude and position of a force, as shown with FT 459A . 
                                                                                                                 TABLE 2.7                   Calculated magnitudes and positions of       forces for a continuously tapered aluminum beam            Experi-    Experi-                                           mental    mental    FT 2      FT 3     FT 4      FT 459A      FT 9      FT A      Force    Zero   Position       Force (N)    Position    (N)    (N)    (N)    (N)    (N)    (N)    Error    Error    Error                    845   2.50   850   N/A   48   42   48   −48   0.6%   5.7%   0.3%       1,201   2.50   1,199   N/A   45   35   59   −60   0.1%   5.0%   0.1%       845   3.50   N/A   826   N/A   34   13   −5   2.3%   4.1%   0.2%       1,201   3.50   N/A   1,211   N/A   39   41   −43   0.9%   3.6%   1.3%       845   4.50   36   N/A   812   N/A   15   −11   3.9%   4.3%   0.4%       1,201   4.50   19   N/A   1,190   N/A   29   −35   0.9%   2.9%   0.4%       845   5.50   42   −6   N/A   879   12   −3   4.0%   5.0%   1.0%       1,201   5.50   31   13   N/A   1,243   33   −26   3.5%   2.8%   2.0%       845   6.50   −1   48   −6   812   −8   16   4.0%   5.6%   0.6%       1,201   6.50   −16   65   −9   1,155   −22   25   3.9%   5.4%   1.1%       845   7.50   12   21   −14   848   −29   17   0.4%   3.4%   1.5%       1,201   7.50   −3   33   −9   1,203   −30   21   0.1%   2.7%   1.6%       845   8.50   3   23   5   813   N/A   48   3.8%   5.7%   0.9%       1,201   8.50   0   33   −16   1,198   N/A   37   0.3%   3.1%   0.3%       845   9.50   1   22   5   N/A   813   N/A   3.8%   2.6%   0.9%       1,201   9.50   12   26   −18   N/A   1,171   N/A   2.5%   2.2%   0.8%       845   10.50   16   −5   6   33   N/A   878   3.9%   3.9%   1.1%       1,201   10.50   10   11   −1   59   N/A   1,221   1.7%   4.9%   0.7%                    
2.4 Conclusion
 
     A unique method has been developed to identify the magnitude and location of a load on a slender beam that is supported at both ends, using four uniaxial strain gages mounted to the bottom surface of the beam. When the load is located inside the weight area, the magnitude and location of the load can be accurately identified. The force transducer can also identify a zero load outside the force transducer. When the load is outside the weight area, but inside the force transducer, the magnitude of the load can be calculated if its location is known and vice versa. The force transducer methodology can be applied to a distributed load and a combination of concentrated and distributed loads that do not cross a strain gage boundary. While the boundary conditions at the two ends can affect the strain measurements, they do not affect the calculated magnitude and location of a load. The experimented results can be used to determine whether a fixed boundary is an ideal fixed boundary, and whether a rest boundary is a firm rest boundary or a soft rest boundary. A calibration procedure has been developed to globally and individually adjust the scaling factors of the strain gages in calculating the magnitude of a load for prismatic and non-prismatic beams, respectively. Calibration can be performed for the location of a load on a non-homogenous and/or non-prismatic beam, but not on a homogeneous, prismatic beam. 
     Experiments on a prismatic steel beam with rest boundary conditions, a prismatic aluminum beam with firm rest, soft rest, firm rest-fixed, and fixed boundary conditions, and a continuously tapered aluminum beam with rest boundary conditions validated the force transducer methodology. The force errors were within 3.7% after calibration for the prismatic beams and within 4.0% for the continuously tapered beam. The zero errors were within 5.1% for the prismatic beams and within 5.7% for the continuously tapered beam. The location of the load was accurately calculated with an error within 0.1% for the steel beam with a precise string loading. The locations of the loads for the other cases were within an error of 2.3%. 
     Chapter 3: Identifying Magnitudes and Locations of Multiple Loads on a Slender Beam Using Strain Gage Based Methods 
     Abstract 
     Unique strain gage based methods are developed to identify magnitudes and locations of multiple loads on a slender beam. Four uniaxial strain gages mounted to the bottom surface of the beam create a force transducer capable of identifying the magnitude and location of a load inside the weight area. For the case of multiple loads separated by two or more strain gage locations, uniaxial strain gages forming multiple force transducers can still identify the magnitudes and locations of all the loads. However, this creates an ill-posed problem for loads separated by only one strain gage location. A new method has been developed using two shear gages mounted on the neutral axis of the beam, one on each side of a load, to identify the magnitude of the load in this case. A combination of two uniaxial strain gages and two shear gages, with one uniaxial strain gage and one shear gage at the same location on each side of a load, can be used to identify the location of the load. The strain gage based methods are experimentally validated on a prismatic beam with rest boundary conditions. 
     3.1 Introduction 
     The load identification problem is very important to bridge designers since the weights of vehicles that cross a bridge are not always known [2]. Using a structure itself to measure magnitudes and locations of loads is an inverse problem, where the response is measured and the magnitudes and locations of the loads are identified [10]. Many different approaches, such as measurements from accelerations, deflections, and strains, have been used [1-12,18,19,27-31]. Bending moment data from strain gages are less prone to error than deflection data according to a previous study [5]. Likewise, the strain model has been proven better than the acceleration model [12]. For the case of a vehicle crossing a bridge, strain readings from the bridge provide the information to identify the magnitude of the load [3,4,9]. 
     It has been previously shown in Chapter 2 that four uniaxial strain gages mounted to the bottom surface of a slender beam can form a force transducer, which can accurately identify the magnitude and location of a single load on the beam [ 31 ]. An inverse problem is often ill-posed, where the measurements do not lead to a unique solution [7,18]. The force transducer methodology is robust enough to pinpoint a desired solution when the load is located inside the weight area, which is defined as a location between the two middle strain gage locations [31]. When the load is a distributed load or a combination of distributed and concentrated loads, the force transducer can identify the magnitude and location of the resultant load. The methodology has been demonstrated for firm rest, soft rest, firm rest-fixed, and fixed boundary conditions as well as on a continuously tapered beam. 
     The force transducer methodology causes many types of error to cancel out. In the design of a strain gage based transducer, temperature drift usually causes the largest error [26]. Since four uniaxial strain gages are combined to form a force transducer, the temperature drift is negated for the case of a prismatic beam [31]. Another source of error in load identification is inaccurate modeling of the boundary conditions, which is also negated with the methodology [31]. 
     This work extends the single load problem in Ref. [31] to a multiple load problem, since a vehicle always consists of more than one load on the bridge due to multiple axles of the vehicle. Section 3.2.1 examines the theory for the case where there are two or more strain gage locations between two adjacent loads, with the corresponding experimental data shown in Sec. 3.3.1. Section 3.2.2 examines the theory for the case where there is only one strain gage location between two adjacent loads, with the corresponding experimental data shown in Sec. 3.3.2. When there is no strain gage location between two adjacent loads, the magnitude and location of the resultant load can be identified, as indicated in Ref. [31]. 
     3.2 Theory 
     3.2.1 Identification of Magnitudes and Locations of Loads Separated by Two or More Strain Gage Locations 
     Consider a slender beam with pinned boundary conditions, whose reaction forces at the two boundaries are denoted by R A  and R B , with non-homogeneous material, variable cross sections, and N farces F n  on the beam, where n=1, 2, . . . , N, as shown in  FIG. 3.1 . There are N+1 uniaxial strain gages ε n  mounted to the bottom surface of the beam, where n=1, 2, . . . , N+1, with N strain gages immediately to the left of each force and one to the right of the last force. For an Euler-Bernoulli beam, the shear force V ( FIG. 69 ) and the bending moment M ( FIG. 70 ) are related through [21] 
                       dM   ⁡     (   x   )       dx     =     V   ⁡     (   x   )               (   3.1   )               
where x is measured from the left reaction force. Let V n  and M n  be the shear forces and bending moments at the strain gage locations, respectively ( FIGS. 69 and 70 ). Furthermore, the bending moment M is proportional to the strain ε [22]:
 
                   M   =       EI   y     ⁢   ɛ             (   3.2   )               
where γ is the distance from the neutral axis of the beam to the point where the strain is measured, l is the area moment of inertia of the cross section, and E is the Young&#39;s modulus.
 
     The force transducer FT n  formed by adjacent strain gages ε n−1  through ε n+2  in  FIG. 68  contains the weight area located between strain gages ε n  and ε n+1 . One can define a scaling factor γ n , where n=1, 2, . . . , N+1 is the strain gage number, and the corresponding calibration factor β n  for each strain gage [31]: 
     
       
         
           
             
               
                 
                   
                     γ 
                     n 
                   
                   = 
                   
                     
                       
                         E 
                         n 
                       
                       ⁢ 
                       
                         I 
                         n 
                       
                     
                     
                       y 
                       n 
                     
                   
                 
               
               
                 
                   ( 
                   3.3 
                   ) 
                 
               
             
           
         
       
     
                       β   n     =           γ   n       d   nj       ⁢           ⁢   for   ⁢           ⁢   n     &lt;   j       ,       β   n     =           γ   n       d   jn       ⁢           ⁢   for   ⁢           ⁢   n     &gt;   j               (   3.4   )               
where j=1, 2, . . . , N+1 is also the strain gage number, and d nj  or d jn  is the distance between adjacent strain gages that are not at the end points of the weight area. By subtracting the slopes of adjacent bending moment curves, the formulation of the force transducer is obtained [31]:
 
 FT   n =−β n−1 ε n−1 +B n ε n +β n+1 ε n−1 −β n+2 ε n+2   (3.5)
 
The relative position a nj  of each force measured from the left strain gage in  FIG. 3.1  is [31]
 
                     α   nj     =       f   n       d   nj               (   3.6   )               
When j=n+1, i.e., the strain gages are adjacent to each other, a nj  can be denoted by a n . From Eqs. (3.1) and (3.5), the force F n  inside the weight area can be calculated from the force transducer. For the case where the force is outside the weight area, but inside the force transducer, the calculated force varies linearly with the distance between the force and an end point of the weight area [31]. By combining the three forces inside the force transducer, one has [31]
 
 FT   n   =a   n−1   F   n−1   +F   n +(1− a   n+1 ) F   n−1   (3.7)
 
When F n&#39;1 =F n+1 =0, by Eqs. (3.5) and (3.7), one has
 
 FT   n   =F   n =−β n−1 ε n−1 +β n ε n +β n+1 ε n+1 −β n+2 ε n+2   (3.8)
 
It can be seen from Eq. (3.8) that other forces on the beam do not affect the output of FT n . Hence by extending Eq. (3.8) to multiple force transducers, multiple forces can be calculated from the corresponding equations as long as the forces are separated by two or more uniaxial strain gage locations.
 
     It was shown in Ref. [31] that the position of the force F n  can be calculated when F n  is inside the weight area and the other forces are outside the force transducer FT n , by finding the position of the intersection point of two adjacent bending moment curves: 
                     α   n     =           d     n   ,     n   +   1         ⁡     (         β     n   +   1       ⁢     ɛ     n   +   1         -       β     n   +   2       ⁢     ɛ     n   +   2           )       +       γ     n   +   1       ⁢     ɛ     n   +   1         -       γ   n     ⁢     ɛ   n             d     n   ,     n   +   1         ⁡     (     FT   n     )                 (   3.9   )               
The absolute position of the force p n  can be obtained from the position of the left adjacent strain gage d n  in  FIG. 3.1  and Eq. (3.6):
 
 p   n   =d   n   +a   n   d   n,n+1   (3.10)
 
The positions of multiple forces on the beam can be calculated using Eq. (3.10) as long as two uniaxial strain gage locations are on both sides of a force and there are no other forces inside the corresponding force transducer.
 
     It should be noted that the four uniaxial strain gages of a force transducer do not have to be adjacent to each other for Eq. (3.8). Any combination of strain gages ε h , ε i , ε j , and ε k , can form a force transducer: 
                     FT   hijk     =       -       ɛ   h     ⁡     (       γ   h       d   hi       )         +       ɛ   i     ⁡     (       γ   i       d   hi       )       +       ɛ   j     ⁡     (       γ   j       d   jk       )       -       ɛ   k     ⁡     (       γ   k       d   jk       )                 (   3.11   )               
where the subscript of FT hijk  is the combination of the strain gage numbers since the strain gages forming the force transducer are not adjacent to each other. For a homogeneous, prismatic beam, γ h =γ i =γ j =γ k =γ, where γ is the global scaling factor for all the strain gages, and Eq. (3.11) becomes
 
                     FT   hijk     =     γ   ⁡     (           -     ɛ   h       +     ɛ   i         d   hi       +         ɛ   j     -     ɛ   k         d   jk         )               (   3.12   )               
For equal strain gage spacing, d hi =d jk =d, and Eq. (3.12) becomes
 
 FT   hijk =β(−ε k +ε i +ε j −ε k )  (3.13)
 
where β=γ/d is the global calibration factor.
 
3.2.2 Identification of Magnitudes and Locations of Loads Separated by One Strain Gage Location
 
     When two loads are separated by only one strain gage location, the methodology in Sec. 3.2.1 is ill-posed. That exists an infinite number of loading scenarios in reconstructing the bending moment diagram; two possible bending moment diagrams are shown in  FIG. 71 . A force transducer calculates the slopes of two adjacent bending moment curves using two uniaxial strain gages for each curve [31]. The slope of the left bending moment curve in  FIG. 71  is calculated with M 1  and M 2 , and the slope of the right beading moment curve is calculated with M 4  and M 5 . However, the slope of the middle bending moment curve in  FIG. 71  cannot be calculated, because only one point on the curve M 3  is known, and the problem is ill-posed. 
     A uniaxial strain gage mounted to the bottom surface of the beam enables one to calculate the point on the bending moment diagram M n  in  FIG. 3.3  from Eq. (3.2). For most beams, the shear stress has the maximum value at the neutral axis, but can deviate from that if the thickness is significantly smaller at other locations [21]. For a stress element that is aligned with the x axis ( FIG. 72 ), when the shear stress has a maximum value, the bending stress vanishes [14]. The shear strain cannot be directly measured, since the stress element under a pure shear stress τ does not stretch ( FIG. 72 ) [20]. A shear gage consists of two strain gages mounted at ±45° from the x axis [32]. The two principal stresses, σ 1  and σ 2 , can be calculated from the measured principal strains, ε 1  and ε 2 , respectively, at this configuration via Mohr&#39;s circle, as shown in  FIG. 73  [23]: 
                       ɛ   1     =         σ   1     -     v   ⁢           ⁢     σ   2         ɛ       ,       ɛ   2     =         σ   2     -     v   ⁢           ⁢     σ   1         ɛ               (   3.14   )               
where v is the Poisson&#39;s ratio.
 
     The shear stress τ is proportional to the shear force V [21]: 
                   τ   =     VQ   It             (   3.16   )               
where Q is the first moment of area, and t is the thickness of the cross section at the point where the shear stress is calculated. The shear gage is mounted on the neutral axis of the beam in order to get a high signal to noise ratio [15]. The shear gage output D is the difference of the two measured strains:
 
 D=ε   1 −ε 2   (3.17)
 
By Eqs. (3.14)-(3.17), the shear force V is related to the shear gage output D:
 
                   V   =       [     EIt     2   ⁢     Q   ⁡     (     1   +   v     )           ]     ⁢   D             (   3.18   )               
A shear gage mounted on the neutral axis of the beam enables one to calculate the point on the shear force diagram V n  in  FIG. 69  from Eq. (3.18). The magnitude of the force F n  can be obtained from the calculated shear forces to the left and right of the force, V n  and V n+1 , respectively:
 
 F   n   =V   n   −V   n+1   (3.19)
 
     While the magnitude of the force for this loading scenario can be calculated with two adjacent shear gages, the position of the force can be calculated by also using the two corresponding uniaxial strain gages.  FIG. 70  illustrates how the intersection of the bending moment curves to the left and right of the force F n  gives the position p n . The bending moments M n  and M n+1  are calculated using the measurements from the corresponding uniaxial strain gages and Eq. (3.2). The shear forces V n  and V n+1 , which give the slopes of the left and right bending moment curves, are calculated using the corresponding shear gage measurements and Eq. (3.18). By simultaneously solving the equations for two adjacent bending moment curves, the position p n  of the force F n  can be calculated: 
     
       
         
           
             
               
                 
                   
                     p 
                     n 
                   
                   = 
                   
                     
                       
                         
                           V 
                           n 
                         
                         ⁢ 
                         
                           d 
                           n 
                         
                       
                       - 
                       
                         
                           V 
                           
                             n 
                             + 
                             1 
                           
                         
                         ⁢ 
                         
                           d 
                           
                             n 
                             + 
                             1 
                           
                         
                       
                       - 
                       
                         M 
                         n 
                       
                       + 
                       
                         M 
                         
                           n 
                           + 
                           1 
                         
                       
                     
                     
                       
                         V 
                         n 
                       
                       - 
                       
                         V 
                         
                           n 
                           + 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3.20 
                   ) 
                 
               
             
           
         
       
     
     The strain gage based methods are independent of the boundary conditions of the beam, as demonstrated for the case of a single load on a beam with pinned, firm rest, soft rest, pinned-fixed, and fixed boundary conditions [31]. This is the case because the difference between the slopes of the bending moment curves to the right and left of the position p n  of the force F n  in  FIG. 70  is not affected by other forces. Changes in the boundary conditions do not affect load identification just as the other loads on the beam do not either. 
     3.3 Experimental Results 
     3.3.1 Identification of Magnitudes and Locations of Loads Separated by Two Strain Gage Locations on an Aluminum Beam 
     A quantity of 12 uniaxial Vishay strain gages (part number CEA-13-250UN-350,  FIG. 74 ) was bonded, with equal spacing, along the bottom surface of an aluminum 6061-T6 beam with rest boundary conditions, as shown in  FIG. 75 . A Vishay P3 strain indicator and a Vishay SB-10 switch unit were used to display the strain gage outputs ( FIG. 76 ). The loads were applied via dumbbells, whose nominal weights were within 1.2% of their actual values. The cross section of the beam is a 50-8 mm square box with a 3.2 mm wall thickness ( FIG. 77 ), and its side view with strain gage locations is shown in  FIG. 78 . 
     The position of a force F n  is the strain gage number n directly to the left of the force added to the corresponding a n . The hexadecimal numbering system is used, where the force transducer formed by uniaxial strain gages 9 through 12 is FT 9ABC  or FT A . The calibration procedure in Chapter 2 was used, where a known force was placed in each weight area and all the strains were recorded. The errors between the experimental and calculated magnitudes of the forces inside the weight areas of the force transducers are referred to as the force errors, and the errors for the force transducers with the forces outside the force transducers, are referred to as the zero errors. The Microsoft Excel Solver was used to reduce the maximum force and zero errors by changing β with the constraint β&gt;0. The beam was then loaded with two unknown forces, each weighing 222 N, placed inside various weight areas, separated by two or more strain gage locations, and the force transducer results from the adjusted β are shown in Table 3.1. The adjusted global calibration factor was 93% of the theoretical global calibration factor. The force transducers with the forces outside the weight areas, but inside the force transducers, were not used and indicated as “N/A” in Table 3.1. The force errors were within 12.3% before calibration and reduced to within 4.3% after calibration. The zero errors were within 2.6% after calibration. The position errors defined by |Δp n |/L, where L is the span of the beam, as shown in  FIG. 78 , were within 1.1% after calibration. The main source of error is the low signal to noise ratio, since the maximum strain gage reading of 269 με was below the recommended range of 1,000 to 1,500 με for a strain gage reading [15]. Another possible cause of error was that the multiple load problem is more sensitive to response error than the single load problem [5]. 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 3.1 
               
             
             
               
                   
               
               
                 Calculated magnitudes and positions of 
               
               
                 two 222N forces on the aluminum beam 
               
             
          
           
               
                 Experi- 
                   
                   
                   
                   
                   
                   
                   
                 Maximum 
                 Maximum 
                 Maximum 
               
               
                 mental  
                 FT 3   
                 FT 5    
                 FT 6    
                 FT 7    
                 FT 8    
                 FT 9    
                 FT A    
                 Force  
                 Zero 
                 Position 
               
               
                 Position  
                 (N)  
                 (N)  
                 (N)  
                 (N)  
                 (N)  
                 (N)  
                 (N)  
                 Error  
                 Error  
                 Error 
               
               
                   
               
             
          
           
               
                 3.5 &amp; 5.5 
                 213 
                 232 
                 N/A 
                 −4 
                 8 
                 0 
                 0 
                 4.3% 
                 1.7% 
                 0.8% 
               
               
                 3.5 &amp; 6.5 
                 217 
                 N/A 
                 213 
                 N/A 
                 4 
                 4 
                 −4 
                 4.3% 
                 0.9% 
                 0.3% 
               
               
                 3.5 &amp; 7.5 
                 224 
                 0 
                 N/A 
                 224 
                 N/A 
                 −8 
                 0 
                 0.9% 
                 1.7% 
                 1.1% 
               
               
                 3.5 &amp; 8.5 
                 217 
                 4 
                 −11 
                 N/A 
                 232 
                 N/A 
                 −4 
                 4.3% 
                 0.9% 
                 0.5% 
               
               
                 3.5 &amp; 9.5 
                 221 
                 8 
                 −11 
                 −4 
                 N/A 
                 213 
                 N/A 
                 0.9% 
                 2.6% 
                 0.6% 
               
               
                 3.5 &amp; 10.5 
                 221 
                 0 
                 −11 
                 4 
                 4 
                 N/A 
                 221 
                 0.9% 
                 2.6% 
                 0.6% 
               
               
                   
               
             
          
         
       
     
     A comparison of the theoretical and calculated bonding moment diagrams for two 222 N forces at positions 3.5 and 7.5, are shown in  FIG. 79 , where a rest boundary is assumed to be a firm rest boundary for the theoretical bending moment. The discrepancies between the theoretical and calculated bending moment diagrams can indicate that either a rest boundary was not an ideal firm rest boundary or there was a small measurement error for the span of the beam. The errors between the theoretical and calculated bending moments are not directly related to those between the theoretical and calculated magnitudes and positions of the forces. 
     The beam was also loaded with nine weights totaling 1,779 N, as shown in  FIG. 80 . One can calculate the magnitude and position of the resultant force on the beam using the force transducer FT 12BC . The calculated magnitude of the resultant force was 1,768 N, which had a 0.6% error. The calculated position of the resultant force had a 3.7% error. A potential application of this methodology is to identify the longitudinal center of gravity of a vehicle on a bridge. Another potential application is to identify the weight and longitudinal center of gravity of a payload on a bed of a truck or trailer by measuring strains on the frame of the truck or trailer. The theoretical and calculated bending moment diagrams are shown in  FIG. 81 . The theoretical bending moment curve approaches a parabola as one would expect for a distributed load on a beam. The calculated bending moments are close to the theoretical ones. 
     3.3.2 Identification of Magnitudes and Locations of Loads Separated by One Strain Gage Location on the Aluminum Beam 
     A quantity of 12 Vishay shear gages (part number CEA-13-187UV-350) was bonded along the neutral axis of the beam in  FIG. 75  ( FIG. 82 ), at the same x locations as the uniaxial strain gages as shown in  FIG. 78 ; a close-up view of a shear gage is shown in  FIG. 83 . Calibration was performed similar to the case in Sec. 3.3.1, where a single known force was placed inside various weight areas and the maximum force and zero errors were minimized. In this case, the global calibration factors for both the uniaxial and shear gages were adjusted. The adjusted global calibration factors were 93% and 101% of the theoretical global calibration factors for the uniaxial strain gages and shear gages, respectively. The beam was then loaded by two unknown forces at various positions ( FIG. 82 ), with one uniaxial strain gage and one shear gage at the same location between adjacent forces including the reaction forces. Equation (3.19) was used to calculate the magnitudes of the forces using the shear gages, as shown in Table 3.2. The calibrated force errors were within 6.3% and the calibrated zero errors were within 5.0%. By using both the uniaxial strain gages and the shear gages, the positions of the forces were calculated from Eq. (3.20), with the calibrated position errors within 1.0%, as shown in Table 3.3. Comparisons of the theoretical and calculated bending moment and shear force diagrams are shown in  FIGS. 84 and 85 , respectively. Any discrepancies between the theoretical and calculated bending moments do not affect the load identification methodology. 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 3.2 
               
             
             
               
                   
               
               
                 Calculated magnitudes of two forces on the 
               
               
                 aluminum beam using shear gages 
               
             
          
           
               
                 F = 400  
                 F = 445  
                   
                   
                   
                   
                   
                   
                 Maximum 
                 Maximum 
               
               
                 N at 
                 N at 
                 FT 1   
                 FT 2    
                 FT 3    
                 FT 4    
                 FT 5    
                 FT 6    
                 Zero 
                 Position 
               
               
                 Position #1 
                 Position #2 
                 (N)  
                 (N)  
                 (N)  
                 (N)  
                 (N)  
                 (N)  
                 Error  
                 Error 
               
               
                   
               
             
          
           
               
                 1.25 
                 2.50 
                 395 
                 466 
                 −14 
                 14 
                 −14 
                 −7 
                 4.8% 
                 1.7% 
               
               
                 1.50 
                 2.50 
                 395 
                 466 
                 −14 
                 28 
                 −21 
                 −14 
                 4.8% 
                 3.3% 
               
               
                 1.75 
                 2.50 
                 388 
                 466 
                 −7 
                 28 
                 −21 
                 −14 
                 4.8% 
                 3.3% 
               
               
                 3.25 
                 4.50 
                 −21 
                 21 
                 388 
                 473 
                 −21 
                 −14 
                 6.3% 
                 3.3% 
               
               
                 3.50 
                 4.50 
                 −7 
                 14 
                 395 
                 473 
                 −21 
                 −14 
                 6.3% 
                 2.5% 
               
               
                 3.75 
                 4.50 
                 −14 
                 14 
                 388 
                 466 
                 −14 
                 −14 
                 4.8% 
                 2.5% 
               
               
                 5.25 
                 6.50 
                 0 
                 7 
                 −14 
                 28 
                 388 
                 445 
                 3.0% 
                 4.2% 
               
               
                 5.50 
                 6.50 
                 0 
                 7 
                 −14 
                 35 
                 381 
                 452 
                 4.8% 
                 4.2% 
               
               
                 5.75 
                 6.50 
                 0 
                 0 
                 −7 
                 35 
                 395 
                 438 
                 1.6% 
                 5.0% 
               
               
                   
               
             
          
         
       
     
                                           TABLE 3.3                   Calculated positions of two forces on the aluminum beam        using unixial strain gages and shear gages            Experi-   Experi-           Max-       mental   mental   Calculated   Calculated   imum       Position   Position   Position   Position   Position       #1   #2   #1   #2   Error               1.25   2.50   1.27   2.48   0.1%       1.50   2.50   1.27   2.48   0.1%       1.75   2.50   1.68   2.45   0.6%       3.25   4.50   3.13   4.45   1.0%       3.50   4.50   3.41   4.45   0.7%       3.75   4.50   3.65   4.43   0.8%       5.25   6.50   5.14   6.41   0.9%       5.50   6.50   5.40   6.42   0.8%       5.75   6.50   5.68   6.43   0.6%                    
3.4 Conclusion
 
     The strain gage based force transducer methodology in Chapter 2 has been extended to the case of multiple loads on a slender beam. When the loads are separated by two or more strain gage locations, multiple force transducers, each consisting of four uniaxial strain gages mounted to the bottom surface of the beam, can identify the magnitudes and locations of the loads. Each force transducer can also identify a zero load when the loads on the beam are located outside the force transducer. When the loads are separated by one strain gage location, the problem is ill-posed since many solutions exist for the calculated bending moment diagram. However, with two shear gages, one on each side of a load, the magnitude of the load can be identified. When the uniaxial strain gages at the corresponding locations are also used, the position of the load can be identified. When multiple loads are located inside the weight area of a force transducer, the magnitude and location of the resultant load can be identified. The strain gage based methods are independent of the boundary conditions of the beam. 
     Experiments were performed on a prismatic aluminum beam with rest boundary conditions. For the case of two loads separated by two or more strain gage locations, the force errors were within 4.3% after calibration, the zero errors were within 6.3%, and the errors for the relative locations of the loads were within 1.1%. For the case of two loads separated by one strain gage location, the force errors were within 6.3% after calibration, the zero errors were within 5.0%, and the errors for the relative locations of the loads were within 1.0%. 
     Chapter 4: Identifying Magnitudes and Locations of Loads on Slender Beams with Welded and Bolted Joints Using a Strain Gage Based Force Transducer with Application to a Portable Army Bridge 
     Abstract 
     A strain gage based force transducer has been developed to identify magnitudes and locations of loads on non-continuous slender beams with welded and bolted joints. The slopes of the bending moment curves on the two sides of a load are calculated from measured strains on a beam. Four uniaxial strain gages are mourned to the bottom surface of the beam, with two strain gages on each side of the load. A calibration method developed earlier can be used to account for the discrepancies between the theoretical and actual scaling factors arising from stress concentrations and unpredictable stress patterns in the beams due to the presence of the joints. The force transducer methodology is experimentally validated on a continuously tapered aluminum beam with a series of welded joints, an aluminum beam with a constant cross section and a bolted joint, a half aluminum and half steel beam with two different cross sections and a bolted joint, and a full scale portable army bridge at the US Army Aberdeen Test Center. 
     4.1 Introductions 
     Knowing the gross weight of a vehicle that cross a bridge is important in calculating its fatigue life when designing commercial or military bridges [3]. It is often impractical to obtain this information from directly instrumented vehicles [2]. Hence it is better practice to instrument a bridge via strain gages and calculate the input loads from the inverse problem [1-4,7-9]. The fatigue design load, which is the vehicle weight multiplied by an impact factor, is an important value when designing portable army bridges [24]. A load identification method should not only determine the static weight of a vehicle, but also the dynamic interaction between the vehicle and the bridge. 
     A methodology to determine the magnitudes and locations of single and multiple loads on slender beams has been developed by the authors using strain gage based methods [31,33]. It has been experimentally demonstrated on continuous beams with constant cross sections and a continuously tapered beam. It has also been demonstrated on beams with different boundary conditions [31]. While the theory has been proven robust for single and multiple loads for various continuous beams with different boundary conditions, a commercial or military bridge is rarely made from a constant extrusion, and it would have many welded and bolted joints [34]. 
     This paper examines the applications of the strain gage based force transducer methodology in Ref. [31] to various laboratory beams with welded and bolted joints as well as a full scale portable army bridge. Section 4.2.1 describes how the strain gage based force transducer methodology is applied to a non-continuous slender beam, and Sec. 4.2.2 describes the calibration procedure. The experimental results for a continuously tapered aluminum beam with a series of welded joints are shown in Sec. 4.3.1. The experimental results for an aluminum beam, and a half aluminum and half steel beam, with bolted joints at the centers of the beams, are shown in Secs. 4.3.2 and 4.3.3, respectively. Section 4.3.4 shows the application of the strain gage based force transducer methodology to a full scale portable army bridge at the US Army Aberdeen Test Center. 
     4.2 Theory 
     4.2.1 Identification of the Magnitude and Location of a Load on a Non-continuous Slender Beam 
     Consider a non-continuous slender beam with pinned boundaries, where the reaction forces are denoted by R A  and R B , non-homogeneous material, variable cross sections, and a force F 2  on the beam, as shown in  FIG. 86 . There can be one or more welded and/or bolted joints in the beam, and it is assumed that the beam behaves linearly in its response. A strain gage based force transducer, consisting of four uniaxial strain gages, with two strain gages on each side of the force F 2 , is used to accurately identify the magnitude and position of the force when the force is located inside the weight area of the force transducer, which is a location between the two middle strain gages [31]. The force F 2  can be a concentrated force, a distributed force of any shape, or a combination of concentrated and distributed forces with the resultant force being F 2 . For an Euler-Bernoulli beam, the shear force V and the bending moment M are related through [22] 
                       dM   ⁡     (   x   )       dx     =     V   ⁡     (   x   )               (   4.1   )               
where x is measured from the left reaction force. Furthermore, the bending moment M is proportional to the strain ε [23,25,35,36]:
 
                   M   =     k   ⁢           ⁢     EI   y     ⁢   ɛ             (   4.2   )               
where γ is the distance from the neutral axis of the beam to the point where the strain is measured, l is the area moment of inertia of the cross section, ε is the Young&#39;s modulus, and k an unknown constant that changes the bending moment and strain relationship from that of a continuous beam due to the presence of the joints. The constant k depends on the geometric and material properties of the beam and the joints, and the locations of the joints relative to where the strain is measured. It can also depend on the bolt torque T, which is proportional to the tensile load in a bolt P [37]:
 
 T=KΦ   (4.3)
 
where K ranges from 0.1 to 0.2, depending on bolt lubrication and the pitch, and Φ is the major bolt diameter.
 
     The four strain gage measurements ε 1  through ε 4  are needed to calculate the slopes of the bending moment curves on the two sides of the force; the shear force and bending moment diagrams for the beam in  FIG. 86  are shown in  FIGS. 87 and 88 , respectively. By subtracting the slope of the right bending moment curve in  FIG. 88  from that of the left one, the formulation of the force transducer is obtained from Eqs. (4.1) and (4.2) [31]: 
                     FT   1234     =       FT   2     =       F   2     =       -       ɛ   1     ⁡     (         k   1     ⁢     E   1     ⁢     I   1           y   1     ⁢     d   12         )         +       ɛ   2     ⁡     (         k   2     ⁢     E   2     ⁢     I   2           y   2     ⁢     d   12         )       +       ɛ   3     ⁡     (         k   3     ⁢     E   3     ⁢     I   3           y   3     ⁢     d   34         )       -       ɛ   4     ⁡     (         k   4     ⁢     E   4     ⁢     I   4           y   4     ⁢     d   34         )                     (   4.4   )               
where FT 2 , in which the subscript is the strain gage number for the strain gage immediately to the left of the force, is a shorthand notation for FT 1234  when the four strain gages are adjacent to each other, d 22  is the distance between strain gages ε 1  and ε 2 , and d 34  is the distance between strain gages ε 3  and ε 4 . One can define a scaling factor γ n , where n=1,2,3,4 is the strain gage number, and the corresponding calibration factor β n  for each strain gage [31]:
 
     
       
         
           
             
               
                 
                   
                     γ 
                     n 
                   
                   = 
                   
                     
                       
                         k 
                         n 
                       
                       ⁢ 
                       
                         E 
                         n 
                       
                       ⁢ 
                       
                         I 
                         n 
                       
                     
                     
                       γ 
                       n 
                     
                   
                 
               
               
                 
                   ( 
                   4.5 
                   ) 
                 
               
             
           
         
       
     
                       β   n     =           γ   n       d   nj       ⁢           ⁢   for   ⁢           ⁢   n     &lt;   j       ,       β   n     =           γ   n       d   jn       ⁢           ⁢   for   ⁢           ⁢   n     &gt;   j               (   4.6   )               
where j is also the strain gage number, and d nj  or d jn  is the distance between adjacent strain gages that are not at the end points of the weight area. By applying Eq. (4.5) to Eq. (4.4), one has an individual scaling factor for each strain gage in the force transducer FT 2 :
 
                     FT   2     =       F   2     =       -       ɛ   1     ⁡     (       γ   1       d   12       )         +       ɛ   2     ⁡     (       γ   2       d   12       )       +       ɛ   3     ⁡     (       γ   3       d   34       )       -       ɛ   4     ⁡     (       γ   4       d   34       )                   (   4.7   )               
By applying Eq. (4.6) to Eq. (4.7), one has an individual calibration factor for each strain gage in the force transducer FT 2 :
 
 FT   2   =F   2 =−β 1 ε 1 +β 2 ε 2 +β 3 ε 3 −β 4 ε 4   (4.8)
 
     When the force is outside the force transducer, the output of FT 2  will be zero since the slopes of the calculated bending moment curves will cancel out when subtracted. Let 
                     α   nj     =       f   n       d   nj               (   4.9   )               
be the relative position of a force F n , where f n  is the position of the force measured from the left adjacent strain gage. For the case where the force is outside the weight area, but inside the force transducer, the calculated force varies linearly with the distance between the force and an end point of the weight area [31].
 
     The position of the force F 2  can be calculated when F 2  is inside the weight area, by finding the position of the intersection point of the two bending moment curves shown in  FIG. 88  [31]: 
                     α   23     =           d   23     ⁡     (         β   3     ⁢     ɛ   3       -       β   4     ⁢     ɛ   4         )       +       γ   3     ⁢     ɛ   3       +       γ   2     ⁢     ɛ   2             d   23     ⁡     (     FT   1234     )                 (   4.10   )               
The absolute position of the force p 2  can be obtained from the position of the left adjacent strain gage d 2  in  FIG. 86  and Eq. (4.9):
 
 p   2   =d   2   +a   23   d   23   (4.11)
 
     The force transducer methodology does not depend on the boundary conditions of the beam, since the sum of the reaction forces always equals the applied force [31]. Furthermore, any reaction moment at a fixed or soft rest boundary of the beam does not change the slope of the bending moment curve, but merely shifts it along the ordinate [25,31]. The methodology can also identify the magnitudes and locations of multiple loads on a slender beam, provided that any two loads are separated by two strain gage locations [33]. 
     4.2.2 Calibration of a Force Transducer for a Non-continuous Slender Beam 
     The theoretical scaling factors γ n  in Eq. (4.7) can be calculated from the geometric and material properties of a continuous beam. Modeling of a joint is often a difficult task involving extensive finite element analysis [37-39]. Instead of relying on the theoretical sealing factors, calibration can be performed using known loads; it is often the preferred method over analytical predictions, given the complexity of bolted joints [29,37]. 
     The calibration procedure is set up so that a single known nonzero force F n , where n=2, 3, . . . , N+1, is placed inside the weight area of the force transducer FT n  that consists of strain gages ε n−1  through ε n+2 , and all the other forces on the beam are zero [31] ( FIG. 89 ); all the strains are measured. There are N+3 strain gages on the beam that create N force transducers, each of which consists of four adjacent strain gages, in order to identify the magnitudes of the N nonzero forces. Equation (4.7) is modified to reflect the N force transducers created: 
     
       
         
           
             
               
                 
                   
                     FT 
                     n 
                   
                   = 
                   
                     
                       F 
                       n 
                     
                     = 
                     
                       
                         - 
                         
                           
                             ɛ 
                             
                               n 
                               - 
                               1 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 γ 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                               
                                 d 
                                 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   , 
                                   n 
                                 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           ɛ 
                           n 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               γ 
                               n 
                             
                             
                               d 
                               
                                 
                                   n 
                                   - 
                                   1 
                                 
                                 , 
                                 n 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           ɛ 
                           
                             n 
                             + 
                             1 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               γ 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                             
                               d 
                               
                                 
                                   n 
                                   + 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   + 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
                           ɛ 
                           
                             n 
                             + 
                             2 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               γ 
                               
                                 n 
                                 + 
                                 2 
                               
                             
                             
                               d 
                               
                                 
                                   n 
                                   + 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   + 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4.12 
                   ) 
                 
               
             
           
         
       
     
     The calibration equation is
 
 [F]=[ε][γ]   (4.13)
 
where [F] is a q×1 matrix, in which q=N 2 −2N+2, that represents all the known forces, including zero forces; [ε] is a q×m matrix, in which m=N+3, that represents the measured strains from strain gages forming the force transducers, and [γ] is an m×1 matrix that represents the scaling factors, which were assumed to be independent of loading. The force errors are defined as the relative errors between the known force magnitudes and the calculated ones from the force transducers, and the zero errors are the relative errors for the outputs of the force transducers that are expected to read zero. The maximum force error is denoted by λ P , and the maximum zero error is denoted by λ 0 . The total error λ T  is defined by
 
λ T   =Cλ   F +λ 0   (4.14)
 
where C is a user defined positive parameter. The theoretical γ n  (n=1, 2, . . . , N+3) are used as the starting point for calibration, and the Microsoft Excel Solver is used to adjust all γ n  to minimize λ T  with the constraints γ n &gt;0; the joint effects can be accounted for by the calibration procedure. The position error is defined by |Δp n |/L, where Δp n  is the difference between a known force position and the calculated one. By applying the calibration procedure for the magnitudes of the forces, the position errors usually do not need to be calibrated [31,33]. When the scaling factors are adjusted, calibration for temperature drift, which can cause the largest error for strain gages [14,26,30], is also performed by zeroing strain gages before readings are taken.
 
4.3 Experimental Results
 
4.3.1 Identifying the Magnitude and Location of a Load on a Continuously Tapered Aluminum Beam with a Series of Welded Joints and Rest Boundary Conditions
 
     A quantity of 12 uniaxial Vishay strain gages (part number CEA-13-250UN-350) were bonded, with equal spacing, along the bottom surface of a continuously tapered, aluminum 6061-T6 beam, with rest boundary conditions, as shown in  FIG. 90 . The theoretical Young&#39;s modulus for aluminum is 69 GPa. The beam was fabricated from a continuously tapered T-shaped beam, with a bent plate acting as the top flange, which is connected to the web with a series of welded joints, to complete an I-shaped beam, as shown in  FIG. 91 . A Vishay P3 strain indicator and a Vishay SB-10 switch unit were used to display the strain gage outputs. The loads were applied via dumbbells, whose nominal weights were within 1.2% of their actual values. The beam has a 4.4° taper ( FIG. 92 ), with its end view shown in  FIG. 93  and its side view with strain gage locations shown in  FIG. 94 . 
     The position of the force is the strain gage number n directly to the left of the force added to the corresponding relative position a nj  of the force. The hexadecimal numbering system is used, where the force transducer formed by uniaxial strain gages 9 through 12 is FT 9ABC  or FT A . The welded joints have an irregular pattern; the position of a strain gage can correspond to a cross section of the beam that may or may not contain a welded joint. In either case, the theoretical scaling factors were for the I-shaped beam cross sections. Finite element analysis (FEA) was performed on the beam in  FIG. 90  with pinned-roller boundary conditions, where a roller boundary condition allows the beam to move freely in the horizontal direction, and an 801 N force at position 4.5, using Pro-Engineer Mechanica to demonstrate the stress concentrations due to the presence of the welded joints ( FIG. 95 ). The welded joints were modeled as a bonded interface ( FIG. 96 ), and the top flange was modeled as a contact interface to the web ( FIG. 97 ). Comparisons of the theoretical strains for firm rest and soft rest boundary conditions, the measured strains, and the predicted strains from FEA for firm rest and soft rest boundary conditions are shown in  FIG. 98 , where the resultant reaction forces were assumed to be 85 and 80 mm away from the edges of the left and right abutments, respectively, for the soft rest boundary conditions [31]. While the theoretical strains for soft rest boundary conditions were within 20.3% of the measured strains, the predicted strains from FEA with soft rest boundary conditions were within 3.8% of the measured strains. 
     Calibration was performed using the method in Sec. 2.2 with C=0.7 in Eq. (4.14); comparisons of the theoretical, FEA, and adjusted γ n , along with the average force and zero errors, are shown in Table 4.1, where soft rest boundary conditions were used in FEA; the theoretical γ n  were for the firm rest boundary conditions and used as the starting point for calibration. The adjusted scaling factors ranged from 34 to 83% of their theoretical values. Known forces of 578 N were used for calibration and unknown forces of 801 N were calculated using the same adjusted scaling factors. The average force error before calibration was 18.9%, and it was reduced to 2.8% after calibration. The average zero error before calibration was 25.6%, and it was reduced to 7.1% after calibration. The theoretical bending moment diagram for firm rest boundary conditions and the calculated bending moment diagrams, both before and after calibration, for an 801 N force at position 4.5, are shown in  FIG. 99 . Errors in the bending moment calculations are not reflected in those in load identification [31]. The errors for the calculated magnitudes and positions of the forces for various leading scenarios are shown in Table 4.2. The force errors were within 4.5% after calibration, the zero errors were within 9.9%, and the position errors were within 2.3%. A source of error was the low signal to noise ratio, since the maximum strain gage reading of 268 με was below the recommended range of 1,000 to 1,500 με for a strain gage reading [15]. While the two neutral axes for the continuously tapered beam are not collinear, which is needed for the Euler-Bernoulli beam theory [21], and the vertical cross sections used for the theoretical scaling factors are not perpendicular to the neural axes, they can be accounted for to some extent through calibration. 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 4.1 
               
             
             
               
                   
               
               
                 Theoretical, FEA, and adjusted scaling factors for the beam in FIG. 90, with the 
               
               
                 corresponding average force and zero errors 
               
             
          
           
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 Average 
                 Average 
               
               
                 Strain 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 Force  
                 Zero 
               
               
                 Gage  
                 1  
                 2  
                 3  
                 4  
                 5  
                 6  
                 7 
                 8 
                 9  
                 10  
                 11  
                 12  
                 Error 
                 Error 
               
               
                   
               
             
          
           
               
                 Theoretical  
                 0.63 
                 0.95 
                 1.30 
                 1.65 
                 2.01 
                 2.39 
                 2.39 
                 2.01 
                 1.65 
                 1.30 
                 0.95 
                 0.63 
                 18.9% 
                 25.6% 
               
               
                 Scaling Factor 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 FEA 
                 0.49 
                 0.82 
                 1.35 
                 1.65 
                 1.90 
                 2.5 
                 2.40 
                 2.04 
                 1.68 
                 1.33 
                 1.00 
                 0.48 
                 18.6% 
                 21.9% 
               
               
                 Scaling Factor 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 Adjusted  
                 0.21 
                 0.52 
                 0.97 
                 1.24 
                 1.56 
                 1.97 
                 1.97 
                 1.60 
                 1.22 
                 0.92 
                 0.57 
                 0.28 
                  2.8% 
                  7.1% 
               
               
                 Scaling Factor 
               
               
                   
               
             
          
         
       
     
                                               TABLE 4.2                   Errors for calculated magnitudes and positions of forces on the        beam in FIG. 90; known forces of 578N were used to        calculate unknown forces of 801N            Experi-                           mental   Experi-           Maximum           Force   mental   Calculated   Force   Zero   Position       (N)   Position   Force (N)   Error   Error   Error               578   2.5   555   4.1%   7.4%   2.3%       801   2.5   764   4.5%   3.8%   2.1%       578   3.5   591   2.3%   5.0%   0.3%       801   3.5   795   0.7%   3.7%   0.5%       578   4.5   601   3.9%   8.7%   2.0%       801   4.5   827   3.3%   6.3%   1.9%       578   5.5   556   3.8%   8.7%   0.7%       801   5.5   799   0.3%   8.8%   0.8%       578   6.5   561   2.9%   8.6%   0.6%       801   6.5   816   1.9%   9.9%   0.4%                    
4.3.2 Identifying the Magnitude and Location of a Load on an Aluminum Beam with a Bolted Joint and Rest Boundary Conditions
 
     A quantity of 12 uniaxial strain gages were bonded along the bottom surface of an aluminum 6061-T6 beam with a bolted joint at the center of the beam and rest boundary conditions, as shown in  FIGS. 100-102 . The cross section of the beam is a 44.5 mm square box with a 3.2 mm wall thickness ( FIG. 103 ), and its side view with strain gage locations is shown in  FIG. 104 . 
     FEA was performed on the beam in  FIG. 100  with pinned-roller boundary conditions, zero preloads in the bolts, and a 445 N force at position 4.5, using Abaqus to demonstrate the stress flow in the beam and the stress concentrations due to the presence of a bolted joint ( FIG. 105 ). The finite element model also had an updated Young&#39;s modulus, which is 94% of the theoretical value, a coefficient of friction of 1.05 between an aluminum joint plate and the aluminum beam, a coefficient of friction of 0.60 between a steel bolt or not and an aluminum plate or beam, and soft rest boundary conditions, where the resultant reaction forces were assumed to be 30 mm away from the edges of the abutments [31]. The stress flow in a similar prismatic beam, under a 445 N force at position 4.5, is shown in  FIG. 106 . It can be seen that the normal flow of stress in  FIG. 105  is interrupted by the bolted joint, which affects the strains in the neighborhood of the bolted joint. A comparison of the deflections for the beam with the bolted joint and a similar prismatic beam are shown in  FIGS. 107 and 108 , respectively. The predicted deflection for the beam with the bolted joint is 17.6 mm and that for the similar prismatic beam is 3.9 mm The higher deflection for the beam with the bolted joint is due to the lower stiffness of the joint plates that have a lower moment of inertia then the beam. The linearity of the beam response can be verified from  FIG. 109 . The Microsoft Excel trend lines display the equations of the lines for the force-strain relationships in  FIG. 109  as well as the coefficients of determination R 2 , where R 2 =1 indicates a perfect correlation to a line. FEA was also performed on the beam with a force of 445 N at position 4.5 and bolt tension values of 1.7, 8.5, and 17.0 kN, which would correspond to bolt torque values of 1.36, 6.78, and 13.56 Nm, respectively, according to Eq. (4.3), where a lubricated bolt was assumed to have K=0.1 and Φ=8 mm. Comparisons of raw strains due to the preloads in the bolts and the applied load, strains due to the preloads in the bolts, referred to as preload strains, and strains due to the applied load, referred to as zeroed strains, which are obtained by subtracting the preload strains from the raw strains, for different bolt tension values are shown in  FIGS. 110-112 . White the bolt torque creates higher preload strains near the bolted joint, the preload strains elsewhere on the beam are almost zero. Hence k n ≅1 in Eq. (4.5) away from the bolted joint, and the corresponding γ n  are close to their theoretical values for a continuous beam. While a higher torque value produces a higher preload strain at strain gage 6, which is closest to the bolted joint, the zeroed strains are almost the same. 
     The bolt torque was varied experimentally to validate the linear response of the beam in  FIG. 100 . The responses of strain gages 2, 4, and 6 are shown in  FIG. 113  for five different magnitudes of the force at position 6.5 and three different bolt torque values (1.36, 6.78, and 13.56 Nm); the strain gages were zeroed after a preload was applied and before readings were taken. All the lines in  FIG. 133  had R 2 &gt;0.9992, which validated that the beam response is linear. For strain gages 2, 4, and 6, the differences among the slopes of the lines for the force-strain relationships were within 3.9%, 1.9%, and 1.6%, respectively, for the three bolt torque values. It was observed that when the bolt torque was changed from 1.36 to 6.78 Nm, strain gages 6 and 7 increased by about 190 με, while the other strain gages that were away from the bolted joint stayed within 2 με, which is consistent with the FEA result. Similarly, when the bolt torque was changed from 6.78 to 13.56 Nm, strain gages 6 and 7 increased by about 150 με, while the other strain gages stayed within 2 με. Comparisons of the theoretical and calculated bending moment diagrams for the three torque values, for a 445 N force at position 4.5, are shown in  FIG. 114 . Comparisons of the theoretical strains for firm rest boundary conditions, the measured strains for the bolt torque of 13.56 Nm, and the corresponding zeroed strains from FEA are shown in  FIG. 115 . There were drops in the measured strains at strain gages 6 and 7 and the corresponding zeroed strains from FEA because the bolted joint stiffens the beam in its neighborhood. 
     The theoretical bending moments for firm rest boundary conditions and the calculated bending moments from a global scaling factor, which is 93% of its theoretical value, for a 445 N force at position 9.5, have noticeable discrepancies at strain gages 6 and 7 due to the stiffening effect of the bolted joint ( FIG. 116 ). Calibration was performed using the method in Sec. 4.2.2 with C=1 in Eq. (4.14), the comparison of the theoretical and adjusted γ n , along with the average force and zero errors, is shown in Table 4.3. The known force of 445 N was placed in five weight areas and the unknown force of 445 N was calculated for the remaining four weight areas using the same scaling factors. The adjusted scaling factors ranged from 93 to 106% of their theoretical values. The average force error before calibration was 14.6%, and it was reduced to 2.8% after calibration. The average zero error before calibration was 31.1%, and it was reduced to 6.0% after calibration. The errors for the calculated magnitudes and positions of the forces for various loading scenarios are shown in Table 4.4. The force errors were within 5.3% after calibration, the zero errors were within 9.6%, and the position errors were within 1.7%. The theoretical and calculated bending moment diagrams for a 445 N force at position 9.5 are shown in  FIG. 117 . 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 4.3 
               
             
             
               
                   
               
               
                 Scaling factors for the beam in FIG. 100 before and after calibration, with the 
               
               
                 corresponding average force and zero errors 
               
             
          
           
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 Average 
                 Average 
               
               
                 Strain 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 Force  
                 Zero 
               
               
                 Gage  
                 1  
                 2  
                 3  
                 4  
                 5  
                 6  
                 7 
                 8 
                 9  
                 10  
                 11  
                 12  
                 Error 
                 Error 
               
               
                   
               
             
          
           
               
                 Theoretical  
                 0.46 
                 0.46 
                 0.46 
                 0.46 
                 0.46 
                 0.46 
                 0.46 
                 0.46 
                 0.46 
                 0.46 
                 0.46 
                 0.46 
                 14.6% 
                 31.1% 
               
               
                 Scaling Factor 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 Adjusted  
                 0.47 
                 0.45 
                 0.46 
                 0.45 
                 0.45 
                 0.47 
                 0.49 
                 0.44 
                 0.45 
                 0.44 
                 0.46 
                 0.43 
                  2.8% 
                  6.0% 
               
               
                 Scaling Factor 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 4.4 
               
             
             
               
                   
               
               
                 Calculated magnitude and position errors of the beam in FIG. 100 
               
             
          
           
               
                 Experi- 
                   
                   
                   
                   
                   
               
               
                 mental 
                 Experi- 
                   
                   
                 Maximum 
                   
               
               
                 Force 
                 mental 
                 Calculated 
                 Force 
                 Zero 
                 Position 
               
               
                 (N) 
                 Position 
                 Force (N) 
                 Error 
                 Error 
                 Error 
               
               
                   
               
               
                 445 
                 2.5 
                 464 
                 4.3% 
                 5.8% 
                 0.7% 
               
               
                 445 
                 3.5 
                 464 
                 4.4% 
                 6.0% 
                 0.7% 
               
               
                 445 
                 4.5 
                 445 
                 0.0% 
                 6.4% 
                 0.6% 
               
               
                 445 
                 5.5 
                 436 
                 1.9% 
                 4.1% 
                 0.7% 
               
               
                 445 
                 6.5 
                 426 
                 4.3% 
                 4.4% 
                 1.3% 
               
               
                 445 
                 7.5 
                 445 
                 0.1% 
                 5.8% 
                 0.5% 
               
               
                 445 
                 8.5 
                 424 
                 4.6% 
                 9.6% 
                 1.7% 
               
               
                 445 
                 9.5 
                 421 
                 5.3% 
                 6.6% 
                 1.6% 
               
               
                 445 
                 10.5  
                 448 
                 0.6% 
                 5.4% 
                 1.3% 
               
               
                   
               
             
          
         
       
     
     A quantity of six uniaxial Vishay strain gages (part number CEA-06-250UN-350) were bonded along the bottom surface of a beam, the left half of which is the same as that in  FIG. 100  and the right half of which is a steel A36 beam, with a bolted joint at the center of the beam ( FIG. 118 ). The theoretical Young&#39;s modulus for steel is 200 GPa. The cross section of the steel part of the beam is a 38.1 mm square box with a 3.2 mm wall thickness ( FIG. 119 ; the joint plates are the same as those in  FIG. 100 , and there is a plate between the top joint plate and the steel part of the beam because of different dimensions of the cross sections of the aluminum and steel parts of the beam ( FIGS. 119-123 ). The side view of the beam with strain gage locations is shown in  FIG. 124 . The experiment here demonstrated the applicability of the methodology to a beam with non-homogenous material, since the Young&#39;s modulus of the aluminum part of the beam differs significantly from that of the steel part of the beam. FEA performed here with Mechanica shows the discontinuity in the stress flow at the joint, similar to that in Sec. 4.3.2, for the beam in  FIG. 4.33  with pinned boundary conditions, zero preloads in the bolts, and a 445 N force at position 2.5 ( FIG. 125 ). It should be noted that there is no stress in the plate between the top joint plate and the top of the steel part of the beam. 
     The theoretical bending moments for firm rest boundary conditions and the calculated bending moments from a global scaling factor, which is 88% of its theoretical values, for a 445 N force at position 9.5, have more discrepancies at strain gages 6 and 7 due to the stiffening effect of the bolted joint ( FIG. 126 ). Calibration was performed using the method in Sec. 4.2.2 with C=0.5 in Eq. (4.14); the comparison of the theoretical and adjusted γ i , along with the average force and zero errors, is shown in Table 4.5. The known force of 445 N was placed in five weight areas and the unknown force of 445 N was calculated for the remaining four weight areas using the same scaling factors. The adjusted scaling factors ranged from 94 to 104% of their theoretical values. The average force error before calibration was 11.6%, and it was reduced to 3.0% after calibration. The average zero error before calibration was 21.5%, and it was reduced to 4.1% after calibration. The errors for the calculated magnitudes and positions of the forces for various loading scenarios are shown in Table 4.6. The force errors were within 4.7% after calibration, the zero errors were within 4.9%, and the position errors were within 1.0%. The theoretical and calculated beading moment diagrams for a 445 N force at position 9.5 are shown in  FIG. 127 . 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 4.5 
               
             
             
               
                   
               
               
                 Scaling factors for the beam in FIG. 118 before and after calibration, with the 
               
               
                 corresponding average force and zero errors 
               
             
          
           
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 Average 
                 Average 
               
               
                 Strain 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 Force  
                 Zero 
               
               
                 Gage  
                 1  
                 2  
                 3  
                 4  
                 5  
                 6  
                 7 
                 8 
                 9  
                 10  
                 11  
                 12  
                 Error 
                 Error 
               
               
                   
               
               
                 Theoretical  
                 0.46 
                 0.46 
                 0.46 
                 0.46 
                 0.46 
                 0.46 
                 0.95 
                 0.95 
                 0.95 
                 0.95 
                 0.94 
                 0.95 
                 11.6% 
                 21.5% 
               
               
                 Adjusted  
                 0.46 
                 0.45 
                 0.44 
                 0.45 
                 0.45 
                 0.48 
                 0.96 
                 0.93 
                 0.90 
                 0.92 
                 0.92 
                 0.99 
                  3.0% 
                  4.1% 
               
               
                   
               
             
          
         
       
     
                                               TABLE 4.6                   Calculated magnitude and position errors of the beam FIG. 118            Experi-                           mental   Experi-           Maximum           Force   mental   Calculated   Force   Zero   Position       (N)   Position   Force (N)   Error   Error   Error               445   2.5   424   4.7%   4.8%   0.6%       445   3.5   426   4.3%   3.0%   1.0%       445   4.5   453   1.8%   4.9%   0.4%       445   5.5   441   0.8%   4.8%   0.4%       445   6.5   431   3.0%   4.9%   0.7%       445   7.5   466   4.7%   4.0%   0.1%       445   8.5   425   4.5%   2.7%   0.5%       445   9.5   448   0.7%   2.6%   0.6%       445   10.5    433   2.8%   2.9%   0.7%                    
4.3.4 Identifying the Magnitude and Location of a Load on a Portable Army Bridge
 
     The force transducer methodology was used to measure the magnitude and location of a load on a portable army bridge on the Bridge Crossing Simulator (BCS), which is operated by the US Army Aberdeen Test Center at Aberdeen Proving Ground in Maryland, USA ( FIG. 128 ). The BCS tests portable army bridges for fatigue failure by simulating thousands of vehicle crossings. The BCS consists of ten pairs of hydraulic cylinders; the two cylinders in each pair are connected by a Whiffle, which applies a load to the bridge via eight pads, as shown in  FIGS. 128-132 . First, a bridge is set up on dirt abutments and measured strains from the bridge are collected from vehicles with various crossing speeds. Next, the bridge is set up on the BCS with wood abutments and straps, which create boundary conditions that are between rest and fixed boundary conditions, and loads from multiple Whiffles are applied to the bridge to simulate various crossing speeds of vehicles. A load identification methodology can be used to identify cylinder loads that need to be applied, from measured strains from live crossings of vehicles. 
     The portable army bridge shown in  FIGS. 128-130  has a span of 46 meters, a height of 1.2 meters, a width of 4.3 meters, and a cross section that consists of four I-shaped beams and a deck, as shown in  FIG. 133 . Twenty uniaxial strain gages were bonded along the bottom flange of the bridge ( FIG. 133 ) at the hydraulic cylinder locations. The bridge consists of eight modules, as shown in  FIGS. 129 and 130 , which are connected by bolted joints, and each module has many welded and bolted joints. The BCS was used to apply two known 89 kN forces at the roadside and curbside matching cylinders in  FIGS. 130 and 131 , at various locations along the bridge. The resultant force from each cylinder consists of four distributed forces at the pads, as shown in  FIG. 132 . Since the force applied on each side is at a strain gage location, a force transducer used to identify the force was formed by two strain gages to the left of the force and two to the right. The zero errors were calculated by force transducers that do not contain the strain gage at the force position. For instance, to identify the magnitude and position of a force at position 3.0, the force transducer FT 1245  was used; the zero errors were calculated by force transducers FT 4578 , FT 5689 , and FT 679A . Calibration was performed independently for each side of the bridge using the method in Sec. 4.2.2; the theoretical scaling factors that were calculated from the cross section in  FIG. 133  were used as the starting point, and C=1 in Eq. (4.14). The adjusted scaling factors are shown in Table 4.7. Since the bridge has many welded and bolted joints and the end modules have variable cross sections, the scaling factors of the strain gages at various locations along the bridge differed significantly, with the maximum to minimum scaling factor ratio being 147%. However, since the bridge is almost symmetric from the roadside to the curbside, the roadside to curbside scaling factor ratio is between 90 to 107%. The adjusted scaling factors ranged from 75 to 110% of their theoretical values. Two unknown forces of 67 kN were applied at the roadside and curbside matching cylinders in four weight areas of the roadside and curbside force transducers, with the results shown in Tables 4.8 and 4.9. The roadside force errors were within 5.3% after calibration, the zero errors were within 13.7%, and the position errors were within 5.3%. The curbside force errors were within 3.8% after calibration, the zero errors were within 13.2%, and the position errors were within 5.1%. The theoretical bending moment diagram for firm rest boundary conditions and the calculated bending moment diagram for two 67 kN forces at position 5.0 are shown in  FIGS. 134 and 135 . The reaction moments at the boundaries were calculated to be negative due to the straps at the abutments. 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 4.7 
               
             
             
               
                   
               
               
                 Scaling factors for the BCS strain gages on the roadside and  
               
               
                 curbside of the portable army bridge and their ratios 
               
             
          
           
               
                 Strain 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 Gage  
                 1  
                 2  
                 3  
                 4  
                 5  
                 6  
                 7 
                 8 
                 9  
                 10 
               
               
                   
               
               
                 Scaling Factors 
                 2,554 
                 1,155 
                 2,428 
                 2,192 
                 1,905 
                 1,944 
                 2,323 
                 2,772 
                 2,493 
                 2,480 
               
               
                 Roadside 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 Scaling Factors 
                 2,631 
                 2,147 
                 2,690 
                 2,176 
                 1,876 
                 2,054 
                 2,174 
                 2,626 
                 2,529 
                 2,496 
               
               
                 Curbside 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 Roadside/Curbside 
                 97% 
                 100% 
                 90% 
                 101% 
                 102% 
                 95% 
                 107% 
                 106% 
                 99% 
                 99% 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 4.8 
               
             
             
               
                   
               
               
                 Calculated magnitude and position errors for the  
               
               
                 roadside of the portable army bridge 
               
             
          
           
               
                 Experi- 
                   
                   
                   
                   
                   
               
               
                 mental 
                 Experi- 
                   
                   
                 Maximum 
                   
               
               
                 Force 
                 mental 
                 Calculated 
                 Force 
                 Zero 
                 Position 
               
               
                 (N) 
                 Position 
                 Force (N) 
                 Error 
                 Error 
                 Error 
               
               
                   
               
               
                 66,723 
                 3.0 
                 65,402 
                 2.0% 
                 13.7% 
                 1.0% 
               
               
                 88,964 
                 3.0 
                 87,153 
                 2.0% 
                 13.2% 
                 1.2% 
               
               
                 66,723 
                 4.0 
                 63,182 
                 5.3% 
                 11.2% 
                 5.1% 
               
               
                 88,964 
                 4.0 
                 86,488 
                 2.8% 
                  9.8% 
                 5.3% 
               
               
                 66,723 
                 5.0 
                 66,178 
                 0.8% 
                  6.7% 
                 2.2% 
               
               
                 88,964 
                 5.0 
                 86,488 
                 2.8% 
                  7.0% 
                 2.3% 
               
               
                 66,723 
                 6.0 
                 63,986 
                 4.1% 
                  0.6% 
                 0.7% 
               
               
                 88,964 
                 6.0 
                 87,480 
                 1.7% 
                  6.8% 
                 0.8% 
               
               
                   
               
             
          
         
       
     
                                               TABLE 4.9                   Calculated magnitude and position errors for the        curbside of the portable army bridge            Experi-                           mental   Experi-           Maximum           Force   mental   Calculated   Force   Zero   Position       (N)   Position   Force (N)   Error   Error   Error               66,723   3.0   65,365   2.0%   12.2%   3.9%       88,964   3.0   85,895   3.5%   12.4%   3.7%       66,723   4.0   67,588   1.3%   11.2%   4.8%       88,964   4.0   91,103   2.4%   10.4%   5.1%       66,723   5.0   66,498   0.3%   13.2%   4.0%       88,964   5.0   85,895   3.5%   10.9%   4.1%       66,723   6.0   69,257   3.8%    7.9%   1.0%       88,964   6.0   90,773   2.0%    7.8%   1.2%                    
4.4 Conclusion
 
     Using four uniaxial strain gages mounted to the bottom surface of a slender beam with welded and bolted joints, the magnitude and location of a load on the beam can be accurately identified. Each force transducer can also identify a zero load when the load on the beam is located outside the force transducer. The force transducer methodology will work for any slender non-continuous, non-homogenous beam with variable cross sections and different types of boundary conditions, as long as the response of the beam is linear. The theoretical scaling factors are calculated from geometric and material properties of the beam, and adjusted during a calibration procedure to account for the joint and other effects. 
     Experiments were performed on slender beams with welded and bolted joints and test boundary conditions. For the continuously tapered aluminum beam with a series of welded joints in Sec. 4.3.1, the force errors were within 4.5% after calibration, the zero errors were within 9.9%, and the position errors were within 2.3%. The adjusted scaling factors differed significantly from the theoretical scaling factors and the calibration procedure reduced the average force error from 18.9% to 2.8%. For the aluminum beam with a bolted joint in Sec. 4.3.2, the force errors were within 5.3% after calibration, the zero errors were within 9.6%, and the position errors were within 1.7%. The adjusted scaling factors were within 7% of the theoretical scaling factors. FEA for zero preloads in the bolts and experiments for different bolt torque values confirmed that the response of the beam was linear. For a half aluminum and half steel beam with a bolted joint in Sec. 4.3.3, the force errors were within 4.7% after calibration, the zero errors were within 4.6%, and the position errors were within 1.0%. The adjusted scaling factors were within 6% of the theoretical scaling factors. The force transducer methodology was validated on a non-continuous and non-homogenous beam with two different cross sections in this case. For the portable army bridge on the BCS in Sec. 4.3.4, the force errors were within 5.3% after calibration, the zero errors were within 13.7%, and the position errors were within 5.3%. While the scaling factors differed significantly along the bridge, the roadside and curbside scaling factors were within 7% of each other. The theoretical scaling factors were a good starting point for calibration in all cases. 
     Chapter 5: Conclusion 
     5.1 Summary 
     A unique strain gage based method has been developed that can identify the magnitude and location of a load within a weight area of a force transducer. When the load is outside the force transducer, a zero load is accurately identified without influence from other loads on the beam. The load can be a concentrated load, distributed load or a combination of each. The goal of the force transducer methodology was to have force errors within 5%, zero errors within 10%, and position errors within 5%. For the experimental results of Chapter 2, the force errors were within 3.7% after calibration for the prismatic beams and within 4.0% for the continuously tapered beam. The zero errors were within 5.1% for the prismatic beams and within 5.7% for the continuously tapered beam. The location of the load was accurately calculated with an error within 0.1% for the steel beam with a precise string loading. The locations of the loads for the other cases were within an error of 2.3%. 
     The force transducer methodology was also applied to multiple loads on a beam that are separated by two strain gage locations. For the experimental results of Chapter 3, the force errors were within 4.3%, the zero errors were within 6.3%, and the position errors were within 1.1%. A new strain gage based method was developed using shear gages mounted on the neutral axis of a beam along with the uniaxial strain gages mounted on the bottom surface of the beam. For the case of two loads separated by one strain gage location, the force errors were within 6.3% after calibration, the zero errors were within 5.0%, and the position errors were within 1.0%. 
     The force transducer methodology was applied to beams with welded and bolted joints in Chapter 4, with stress concentration factors due to the joints being accounted for through the calibration procedure developed here. Experiments were performed on three laboratory beams as well as a portable army bridge on the BCS. For a continuously tapered aluminum beam with a series of welded joints, the force errors were within 4.5%, the zero errors were within 9.9%, and the position errors were within 2.3%. For an aluminum beam with a bolted joint, the force errors were within 5.3%, the zero errors were within 9.6%, and the position errors were within 1.7%. For a half aluminum and half steel beam with a bolted joint, the force errors were within 4.7%, the zero errors were within 4.6%, and the position errors were within 1.0%. For the portable army bridge on the BCS, the force errors were within 5.3%, the zero errors were within 13.7%, and the position errors were within 5.3%. 
     5.2 Contributions, Advantages, and Limitations 
     The force transducer methodology developed here successfully identifies the magnitudes and locations of loads on non-continuous, non-homogenous, slender beams with variable cross sections, welded and bolted joints, and pinned, firm rest, soft rest, pinned-fixed, and fixed boundary conditions, by using four uniaxial strain gages mounted to the bottom surface of the beam, when adjacent loads are separated by two strain gage locations. It is also used to determine the boundary conditions by calculating the bending moment diagram. When there are multiple loads separated by only one strain gage location, another method developed here uses two shear gages mounted on the neutral axis of the beam, one on each side of a load, to identify the magnitude of the load in this case. A combination of two uniaxial strain gages and two shear gages, with one uniaxial strain gage and one shear gage at the same location on each side of a load, is used to identify the location of the load. These methods have not been developed before and are innovative to the field of load identification. The calibration method and corresponding calibration matrix developed here is innovative and useful for strain gage based force transducers. By systematically applying known nonzero loads along the beam in various weight areas, both the nonzero load and nonzero loads are accounted for through calibration. The Microsoft Excel Solver is used to optimize the individual scaling factors of strain gages, while reducing the maximum force and zero errors. 
     The assumptions of the strain gags based methods are:
         1. The beam has a linear response at all strain gage locations.   2. The beam is slender, and the Euler-Bernoulli beam theory is used to developed the strain gage based methods.   3. There exist individual scaling factors that are independent of loading, which can be applied to each strain gage and account for joint effects.
 
The strain gage based methods developed here are robust since:
   1. The methods are independent of boundary conditions.   2. The scaling factor of each strain gage is determined through a calibration procedure regardless of the starting point.   3. Temperature effects can be cancelled out for homogeneous, prismatic beams without having to zero the strain gages before readings are taken. Temperature effects can be cancelled out for non-homogeneous and/or non-prismatic beams by zeroing the strain gages before readings are taken.   4. The beam can be non-continuous and non-homogenous, and have variable cross sections, welded and bolted joints, and different types of boundary conditions.       

     A limitation of the theory at this point is that the beam should be a slender beam governed by the Euler-Bernoulli beam theory. Another limitation of the force transducer methodology is when two forces are separated by one uniaxial strain gage, the problem is ill-posed. However, the additional strain gage based method developed here using shear gages creates a unique solution for that case. 
     5.3 Future Work 
     Future research will be conducting static experiments on the portable army bridge on the BCS using shear gages to determine loads separated by one strain gage location. Additional research will be the application of the strain gage based methods to determine the magnitude and location of dynamic loads on a bridge. As long as the sampling rate of the strain data is sufficiently high, the methods would work for static and dynamic loads. Specifically, the dynamic loading can be used to identify the cylinder forces for the BCS at Aberdeen Test Center. A computer algorithm can be developed to identify the magnitudes and locations of dynamic loads on a bridge by filtering zero loads below a threshold (i.e. 10% of the load), and calculating the vehicle speed. Another type of bridge that can be tested on the BCS is a truss bridge based off of the Bailey Bridge design as shown in  FIG. 136 . The load identification methods developed here can be extended for a truss bridge with periodically changing cross sections. The methods developed here can also be extended to a bridge with multiple supports. 
     The existing methodology can be tested on commercial bridges as well as army bridges. There are additional challenges of commercial bridges that do not exist for the portable army bridge used here: there can be additional supports under the bridge, there can be a nonlinear response such as for a concrete bridge, there can be small strain readings which can yield a low signal to noise ratio, and vehicles can cross at high speeds giving a large dynamic effect. It is also possible that a vehicle that is accelerating can produce large longitudinal loads that were not accounted for in this methodology. 
     Other future research can be developing a more robust method to determine the accuracy of experimental boundary conditions, by recreating the bending moment curve and calculating the reaction moments at the end abutments. The distinction between firm rest and soft rest can be determined using strain data, and can be applied also to military bridges resting on dirt end abutments. The accuracy of a simulated fixed boundary created from a clamp can be determined by calculating the reaction moments. 
     Lastly, future research can focus on non-slender beams that are governed by the Timoshenko beam theory instead of the Euler-Bernoulli beam theory. It is possible that the strain gage based methods developed here can be extended to such non-slender beams if the beam response is linear at the strain gage locations. 
     REFERENCES 
     
         
         [1] Zhu, X. Q. and Law, S. S., “Practical Aspects in Moving Load Identification”, Journal of Sound and Vibration, 2002, V258, pgs. 123-146. 
         [2] Asnachinda, P., Pinkaew, T. and Laman, J. A., “Multiple Vehicle Axle Load Identification from Continuous Bridge Bending Moment Response”, Science Direct, May 2008, pgs. 1-18. 
         [3] Moses, Fred M., “Weigh-In-Motion System Using Instrumented Bridges”, Transportation Engineering Journal, May 1979, pgs. 233-249. 
         [4] Peters, R. J., “AXWAY—A System to Obtain Vehicle Axle Weights”, The 12 th  ARRB Conference, Hobart, Tasmania, August 1984, pgs. 10-18. 
         [5] Yuan, X. R., Cheng, E. L. and Chan, T. H., “Identification of Moving Loads from the Response of Simply Supported Beam”, Proceedings of the International Conference on Structural Dynamics, Vibration, Noise and Control, Hong Kong, December 1995, pgs. 924-929. 
         [6] Bu, J. Q., Law, S. S. and Zhu, X. Q., “Innovative Bridge Condition Assessment from Dynamic Response of a Passing Vehicle”, Journal of Engineering Mechanics, December 2006, pgs. 1372-1379. 
         [7] Law, S. S., Bu, J. Q., Zhu, X. Q. and Chan, S. L., “Moving Load Identification on a Simply Supported Orthotropic Plate”, International Journal of Mechanical Sciences, 2007, V49, pgs. 1262-1275. 
         [8] Rowley, C. W., O&#39;Brien, E. J., Gonzalez, A. and Znidaric, A., “Experimental Testing of a Moving Force Identification Bridge Weigh-In-Motion Algorithm”, Experimental Mechanics, November 2008, pgs. 743-746. 
         [9] Skelton, S. B. and Richardson, J. A., “A Transducer for Measuring Tensile Strains in Concrete Bridge Girders”, Experimental Mechanics, 2006, V46, pgs 325-332. 
         [10] Adams, R. and Doyle, J. F., “Multiple Force Identification for Complex Structures”, Experimental Mechanics, March 2002, pgs. 25-36. 
         [11] Jiu, J., Ma, C., Kung, I. and Lin, D., “Input Force Estimation of a Cantilever Plate by Using a System Identification Technique”, Computer Methods in Applied Mechanics and Engineering, 2000, V190, pgs. 1309-1322. 
         [12] Hillary, B. and Ewins, D. J., “The Use of Strain Gauges in Force Determination and Frequency Response Function Measurements”, Proceedings of the 2 nd  International Modal Analysis Conference and Exhibit, February 1984, pgs. 627-634. 
         [13] Rajkondawar, P., Tasch, U., Lefcourt, A. M., Erez, B., Dyer, R. M., and Varner, M. A., “A System for Identifying Lameness in Dairy Cattle”, American Society of Agricultural and Biological Engineers, 2002, V18, pgs 1-28. 
         [14] The Technical Staff of Measurements Group, Inc., “Strain Gage Based Transducers: Their Design and Construction”, 1988, pgs. 1-79. 
         [15] Vishay Measurements Group, Inc., “Measurement of Force, Torque, and Other Mechanical Variables with Strain Gages”, Technical Publication 2011, pgs. 1-26. 
         [16] Young, W., “Roark&#39;s Formulas for Stress &amp; Strain, 6 th  Edition”, McGraw-Hill, 1989, pg. 52. 
         [17] Vishay Measurements Group, Inc., “Strain Gage Selection: Criteria, Procedures, Recommendations”, Tech Note TN-505-4, November 2010 pgs. 49-64. 
         [18] Moller, P. W., “Load Identification Through Structural Modification”, Journal of Applied Mechanics, Vol. 66, Issue 1, March 1999, pgs. 236-241. 
         [19] Masroor, S. A. and Zachary, L. W., “Designing an All-Purpose Force Transducer, Experimental Mechanics”, Vol. 31, Issue 1, 1991, pgs, 33-35. 
         [20] Vishay Measurements Group, Inc., SC-300 Strain Measuring Systems Workshop, 2003. 
         [21] Craig, Roy R. Jr., “Mechanics of Materials”, John Wiley and Sons, 1996. 
         [22] Nash, W. A., “Schaum&#39;s Outlines: Statics and Mechanics of Materials”, McGraw-Hill Companies, 1992 
         [23] Muvdi, B. B. and McNabb, J. W., “Engineering Mechanics of Materials”, Macmillan Publishing Co., 1980. 
         [24] Kluck, J., Connor, R., and Hombeck, S., “Trilateral Design and Test Code for Military Bridging and Gap-Crossing Equipment”, Military Document Approved for Public Release, 2005, pgs. 1-117. 
         [25] American Forest &amp; Paper Association “Beam Design Formulas with Shear and Moment Diagrams”, November 2007, pgs. 1-20. 
         [26] Vishay Measurements Group, Inc., “Strain Gage Thermal Output and Gage Factor Variation with Temperature”, Tech Note TN-504-1, September 2010, pgs 35-47. 
         [27] Andrae, J. and Sawla, A., “Time Synchronized Measurement of Multi-Bridge Force Transducers”, Measurement, 2001, V29, pgs. 105-111. 
         [28] Blakeborough, A., Clement, D., Williams, M. S. and Woodward, N., “Novel Load Cell for Measuring Axial Force, Shear Force and Bending Moment in Large-Scale Structural Experiments”, Experimental Mechanics, March 2002, pgs. 115-122. 
         [29] Frdderiksen, P. S. and Petersen, T, “On Calibration of Adjustable Strain Transducers”, Experimental Mechanics, September 1996, pgs. 218-223. 
         [30] Dorsey, J., “Homegrown Strain-Gage Transducers”, Experimental Mechanics, July 1977, pgs. 255-260. 
         [31] Bednarz III, E. T., Zhu, W. D. and Smith, S. A. “Identifying the Magnitude and Location of a Load on a Slender Beam Using a Strain Gage Based Force Transducer”, submitted to the Journal of Strain Analysis. 
         [32] Vishay Measurements Group, Inc., “Plane-Shear Measurement with Strain Gages”, Tech Note TN-512-1, September 2010 pgs. 113-118. 
         [33] Bednarz III, E. T., Zhu, W. D. and Smith, S. A., “Identifying Magnitudes and Locations of Multiple Loads on a Slender Beam Using Strain Gage Based Methods”, submitted to the Journal of Strain Analysis. 
         [34] Mahoud, H., “Applying a Fuzzy Logic Expert System in the Selection of Bridge Deck Joints”, Dissertation, University of Central Florida, 1998, pgs 1-474. 
         [35] Groper, M., “Microslip and Macroslip in Bolted Joints”, Experimental Mechanics, 1985, pgs. 171-174. 
         [36] Gaul, L., and Bohlen, S., “Identification of Nonlinear Structural Joint Models and Implementation in Discretized Structure Models”, Conference on Mechanical Vibration and Noise, 1987, pgs. 213-219. 
         [37] Blendulf, B., “Fastening Technology &amp; Bolted/Screwed Joint Design”, seminar, February 2010, pgs. 1-284. 
         [38] He, K. and Zhu, W. D., “Finite Element Modeling of Structures With L-Shaped Beams and Bolted Joints”, Journal of Vibration and Acoustics, February 2011, V133, pgs. 1-13. 
         [39] Song, Y., Hartwigsen, C. J., McFarland, D. M., Vakakis, A. F., and Bergman, L. A., “Simulation of Dynamics of Beam Structures with Bolted Joints Using Adjusted Iwan Beam Elements”, Journal of Sound and Vibration, 2004, V273, pgs. 249-276. 
       
    
     It is to be further understood that other features and modifications to the foregoing detailed description are within the contemplation of the present invention, which is not limited by this detailed description. Those skilled in the art will readily appreciate that any number of configurations of the present invention and numerous modifications and combinations of materials, components, arrangements, and dimensions can achieve the results described herein, without departing from the spirit and scope of this invention. Accordingly, the present invention should not be limited by the foregoing description. Rather the present invention is defined by the appended claims.