John R. Ellis

Glenn Research Center, Cleveland, Ohio

Ali Abul-Aziz

Cleveland State University, Cleveland, Ohio

Specimen Designs for Testing Advanced

Aeropropulsion Materials Under

In-Plane Biaxial Loading

NASA/TM—2003-212090

January 2003
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John R. Ellis

Glenn Research Center, Cleveland, Ohio

Ali Abul-Aziz

Cleveland State University, Cleveland, Ohio

Specimen Designs for Testing Advanced

Aeropropulsion Materials Under

In-Plane Biaxial Loading

NASA/TM—2003-212090

January 2003

National Aeronautics and

Space Administration

Glenn Research Center
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Available electronically at http://gltrs.grc.nasa.gov
NASA/TM—2003-212090 1 

Specimen Designs for Testing Advanced Aeropropulsion Materials 

Under In-Plane Biaxial Loading 

John R. Ellis 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Ali Abul-Aziz 

Cleveland State University 

Cleveland, Ohio 44115 

1. Summary 

 A design study was undertaken to develop specimen designs for testing advanced aeropropulsion 

materials under in-plane biaxial loading. Materials of interest in this work include titanium-based alloys 

in both monolithic and composite forms and also nickel- and cobalt- based superalloys in plate form. The 

focus of initial work was on developing a specimen design suitable for deformation and strength tests to 

be conducted under monotonic loading. The type of loading initially assumed in this study was the 

special case of equibiaxial, tensile loading. A specimen design was successfully developed after a 

lengthy design and optimization process with overall dimensions of 12 by12 by 0.625 in., and a gage area 

of 3.875 by 3.875 by 0.080 in. Perhaps the most important feature of the design was the four sets of 

flexures which were incorporated to partially decouple the applied biaxial loading. This essential design 

feature resulted in a complex specimen configuration, presenting a challenge to manufacture and leading 

to high manufacturing costs. Subsequently, the scope of the work was extended to include the 

development of a second design tailored for tests involving cyclic loading. One obvious consideration 

here is buckling stability as these experiments involve both tensile and compressive loading. A specimen 

design suitably tailored to meet these requirements was successfully developed with overall dimensions 

of 12 by 12 by 0.500 in. and a gage area of 2.375 by 2.375 by 0.050 in. Compared to the earlier design, 

details of the design were simpler in the gripped region and also in the flexure region, but, overall, the 

geometry of the specimen remained complex. Finally, an investigation was made to determine whether 

the specimen designs developed in this study could be used without modification to investigate general 

forms of biaxial loading. It was concluded that the two specimen designs can be used to investigate 

deformation behavior under general forms of biaxial loading, provided measurement and observation is 

limited to the 1.0-in.-diameter circular region at the specimen center. However, the situation is more 

complicated in the case of experiments investigating strength and fracture behavior. For best results, it 

was concluded that specimen designs need to be optimized and tailored to meet the loading requirements 

of individual research programs. 

2. Introduction 

 One technique for investigating material behavior under complex stress states is to use in-plane 

biaxial loading. Using this approach, cruciform specimens fabricated from plate material are gripped at 

four locations and loaded along two orthogonal axes as illustrated in figure 1. Servohydraulic loading 

systems are used in this application which are similar to those used for uniaxial testing. Thus, the 

technique has the advantage that the loading arrangement is relatively straightforward and uses 
NASA/TM—2003-212090 2 

equipment which has seen extensive development over the past thirty years. Also, the test method allows 

a wide range of biaxial stress states to be investigated with minimum complication from the load 

application viewpoint. For these reasons, the test method has been used to generate a sizable body of 

biaxial test data for both monolithic and composite materials (refs. 1 to 29). 

 One difficulty facing these investigations has been the selection and/or development of the most 

suitable specimen design for the particular program. It should be noted that consensus standards do not 

exist for this method of testing and so the experimentalist is faced with a wide range of possibilities. A 

major complication here is that use of the cruciform specimen configuration and associated gripping 

fixtures results in “coupling” between the two loading directions. In the present research, specimens are 

positioned in the load frame using four hydraulic grips which rigidly constrain the specimen over the 

gripped regions. It follows that loading applied in one direction is partially reacted by the specimen and 

partially by the grips associated with loading in the second direction. One method of minimizing this 

effect is to use specimen designs which incorporate fairly complicated arrangements of flexures as 

illustrated in figure 2. It has been demonstrated that flexures with low bending stiffness in the plane of 

loading can be used to minimize the constraint imposed by specimen gripping. Also, it has been shown 

that the geometry of these flexures can be optimized to give near-uniform stress/strain conditions in the 

gage area for specific biaxial loading conditions. 

 One obvious disadvantage of using flexures is that regions of high stress concentration can be 

introduced into specimens in close proximity to the gage area. Of particular concern are stress 

concentrations at the ends and intersection points of the flexures. This raises the possibility that failure 

can be initiated outside of the gage area in regions where stress/strain conditions are ill-defined. 

Traditionally, this problem has been addressed by incorporating a gage area within which specimen 

thickness is reduced significantly from the value in the gripped regions. In the case of conventional 

structural alloys, experience has shown that thickness reduction factors as high as ten are needed to 

achieve acceptable performance. That is failure initiating within the gage area where stress/strain 

conditions are both relatively uniform and relatively well defined. 

 The primary goal of the present work was to develop specimen designs for testing advanced 

aeropropulsion materials under in-plane biaxial loading. Materials of interest in this research included 

nickel- and cobalt-based superalloys and also titanium-based alloys in both monolithic and composite 

forms. The plan was to first conduct a series of tests under monotonic loading to investigate deformation 

and strength behavior in titanium alloys in unreinforced plate form. Subsequently, the intent was to 

extend the scope of the study to include cyclic loading with the aim of investigating fatigue and fracture 

in the same materials and product forms. The most immediate goal of this work was to develop specimen 

designs which would be fully compatible with the in-plane biaxial testing systems recently installed at 

the NASA Glenn Research Center (GRC) (ref. 27). 

3. Specimen Designs for Testing Under Monotonic Loading 

 The focus of initial work was on developing a specimen design suitable for deformation and strength 

testing under monotonic loading. One requirement here was for a specimen with a relatively large gage 

area to meet instrumentation needs. A further requirement was to keep the overall specimen thickness to 

a minimum to keep manufacturing costs within acceptable limits. The approach adopted in developing an 

initial specimen design was to use the design proposed by Brown & Miller (ref. 18) as a starting point. 
NASA/TM—2003-212090 3 

Note that it was not possible to use the exact same design in this work because of a number of differences 

in testing and material requirements. These requirements are outlined in the following along with the 

results of design and analysis work. 

3.1 Initial Design 

 The first step in the design process was to select overall specimen dimensions. Consideration was 

given here to a number of test system features and characteristics including: 

 1. Load capacity of specimen grips, hydraulic actuators, load cells, and load frame. 

 2. Size envelope available within the load frame for specimen installation and gripping. 

 3. Gripping arrangement and size restrictions imposed by the hydraulic specimen grips. 

 4. The size and type of the heating system available for planned elevated temperature testing. 

 5. Specimen gage area needed to meet strain measurement and other instrumentation needs. 

 Preliminary design and analysis showed that the overall specimen dimensions best meeting testing 

requirements were 12 by 12 by 0.500 in. (fig. 3). Note that a relatively large gage area, 5 by 5 in., was 

selected for the monotonic specimen design as it would allow specimens to be highly instrumented with 

minimum installation difficulty. 

 As noted earlier, in-plane biaxial testing requires that the plate specimens be gripped at four 

locations. The gripping area provided for each specimen grip was 6 by 1.500 in. which meets the size 

requirements and load transfer requirements of the hydraulic grips available for this work. Inspection of 

figure 3 will show that the depth of sections available for gripping is 1.800 in. Additional material was 

provided to avoid gripping the specimen in close proximity to the flexures to avoid adverse interactions 

between stress concentrations resulting from gripping with those at the ends of the flexures resulting 

from biaxial loading. 

 The initial flexure configuration shown in figure 3 resulted from close inspection of the Brown and 

Miller design. Also, the design was based in part on an early series of stress analyses aimed at checking 

the load capacity of the design. Specifically, the width and length selected for the flexures was 0.188 and 

1.387 in., and the spacing was 0.375 in. in 12 locations and 0.438 in. in the remaining 24 locations. 

Regarding the thickness of the gage area, consideration was given to the intended application for the 

advanced metallics of interest in this study. It was anticipated that in aeropropulsion service, the 

thickness of the plate material would most likely fall in the range 0.050 to 0.100 in. With this in mind, a 

gage area thickness of 0.100 in. was selected as being at the upper end of the intended use range. For 

simplicity, it was decided to use a straight taper in transitioning from the 0.500 in. overall thickness to 

the 0.100 in. gage area thickness (fig. 4). Later in the optimization process, the straight taper was 

replaced by a radius to minimize stress concentrations in the blend region. The remaining dimensions 

were selected using “engineering judgment.” 

3.2 Stress Analysis Details 

 The three dimensional model shown in figures 5 and 6 was generated using the MSC/PATRAN 

computer code (ref. 30). The model consists of 6300 eight-node isoparametric solid brick elements and 

9000 nodes. As indicated in figure 5, a relatively fine mesh was introduced through the thickness to 

capture stress distributions with a reasonable degree of accuracy. Note that only 1/8 of the total specimen 
NASA/TM—2003-212090 4 

was modeled to keep the size of the finite element model within reasonable limits. This was possible 

because of the symmetry of the specimen and because of the straightforward nature of the loading 

conditions applied. Linear elastic stress analyses were conducted using the MARC finite element code, 

version K5.2 (ref. 31). Analytical runs were conducted on the Cray YMP main frame computer at GRC. 

Individual runs required about 500 computer processing units (CPU) to complete. Note that the material 

properties used in this work were handbook values for Inconel 718, and that the results of stress analysis 

are expressed in the form of von Mises equivalent stress unless otherwise stated. 

 As noted earlier, the external loadings assumed in the analyses were applied over eight 6 by 1.5 in. 

areas, including both the top and bottom specimen surfaces. The approach followed was to simulate 

infinitely stiff grips by constraining all surface nodes within the gripped regions to move 0.005 in. in the 

loading direction so as to produce a state of equibiaxial tension in the gage area. This value was selected 

to give von Mises equivalent stress levels of about 50,000 psi at the center of the gage area. Also, the 

surface nodes within the gripped area were constrained to move 0.0002 in. in the thickness direction to 

simulate clamping by the hydraulic grips. 

3.3 Stress Analysis Results 

 Results of finite element analyses performed using the above approach are given in figures 7 to 9. 

Considering overall conditions in the gage area, it can be seen that: 

 1. The stress distribution is relatively uniform, falling for the most part within the range 43,600 to 

48,800 psi. 

 2. “Through thickness” variations in stress distribution are relatively minor (see figs. 7 and 8). 

 3. Stress levels are significantly below average in the corner region and are significantly above 

average at the ends of the center-most flexures. 

 A more detailed analysis of the results showed that the maximum stress in the gage area occurred at 

location N (fig. 7) and was about 53,000 psi. These analyses also showed that the minimum stress in the 

gage area occurred at location P and was about 39,400 psi. Thus, the deviation of stress over the entire 

gage area was within ±15 percent of the mean at the start of the optimization process. 

 Considering the balance of the specimen (fig. 9), the maximum stress was found to occur at location 

D and was 69,700 psi. As expected, a large stress concentration also was found to exist at the corners 

joining the two sets of flexures, location F in figure 9. Here, the maximum stress was determined to be 

64,000 psi. Note that the “hidden regions” in this figure were examined in a similar manner using 

isometric projections to allow stress concentrations to be quantified throughout the part. These 

examinations showed that the maximum stress in these regions was 55,000 psi and occurred at location G 

(fig. 7). Overall, the results summarized above were viewed as being very encouraging and appeared to 

confirm the effectiveness of the Brown and Miller approach to specimen design. However, it was 

apparent that additional work was needed to improve the uniformity of stress in the gage area and also to 

reduce stress concentrations at the ends of the flexures. 
NASA/TM—2003-212090 5 

3.4 Optimization Method 

 The optimization method used in achieving these goals focused on two straightforward design 

requirements. 

 1. The maximum stress in the specimen should occur within the gage area. 

 2. The distribution of stress within the gage area should be relatively uniform, for instance, within 

±5 percent of the mean. 

 The optimization process simply involved making systematic changes to the specimen design and 

repeating stress analysis as necessary until the two design requirements had been met. This process was 

performed manually since the finite element code available for this work did not include a formal 

optimization package. 

 More specifically, the approach followed in developing a fully optimized design involved conducting 

the work in three stages. Consideration was given first to the thickness transition details. Of primary 

interest here was whether acceptable specimen performance in terms of stress uniformity in the gage area 

could be obtained using a single step reduction in specimen thickness. Clearly, this is the preferred 

approach as it results in a simpler design with low associated machining costs. The design variables of 

interest during this stage of the process were the gage area thickness and the thickness transition radius. 

Second, consideration was given to the detailed design of the flexures. Inspection of figure 3 will show 

that there are a large number of possibilities available with the subject design which can be used to 

influence the stress distribution in the gage area. These include the number, the size, the spacing and the 

configuration of the flexures. 

 Up to this stage of the process, no attempt was made to incorporate blend radii at the ends of the 

flexures. The goal here was to keep the size of the finite element model within reasonable limits during 

the early stages of the work. The final stage of the optimization process was to include such radii so as to 

get a more realistic picture of the stress concentrations at the ends of the flexures. It was anticipated that 

inclusion of these radii would produce only local effects on the stress distribution and would not have a 

marked effect on the gage area stress distribution. Clearly, the possibility existed that the above 

procedure would have to be repeated if less than ideal choices of dimensions and configurations were 

made at any stage of the optimization process. 

3.5 Final Design 

 As expected, development of a final design using the above process proved to be lengthy and 

involved over 80 design interactions. Early in the process, it became apparent that the two design goals 

could not be achieved using the 12 by 12 by 0.500 in. configuration selected at the outset of this work. To 

correct this situation, it was decided to increase the overall thickness of the specimen to 0.625 in., primarily 

with the aim of reducing stress levels at the ends of the flexures. Also, it proved necessary to adopt a twostep reduction in gage area thickness. Again, this approach was adopted because it proved impossible to 

meet the design requirements using the single-step approach. Apart from these differences, the plan was 

to make minor changes to the flexure design to further improve stress uniformity in the gage area. 

 The finite element model used in achieving a fully optimized design is shown in figures 10 and 11. 

Here it can be seen that the configuration of the final design, designated no. 228.1, is similar in overall 

configuration to the initial design. In addition to the changes noted above, it can be seen that the width of 
NASA/TM—2003-212090 6 

the center-most flexures was reduced and steps were added to the outermost flexures. The effect of these 

changes in terms of gage section stress distribution can be seen in figure 12. In this case, stress levels 

over the major part of the gage area fell within the range 54,000 to 59,200 psi. This increase over values 

reported earlier resulted from the increase in overall specimen thickness and the “fixed displacement” 

mode of specimen loading used to simulate biaxial testing. More detailed analysis of the results showed 

that the maximum stress, 58,300 psi, occurred at the corner and that the minimum stress, 53,000 psi, 

occurred at the center line. The corresponding deviation is about ±5 percent which means design 

no. 228.1 meets the design goal regarding stress uniformity in the gage area. 

 The stress distribution in the flexures is shown in figure 13. Detailed analysis showed that the 

maximum stress at locations D, E, and F fell in the range of 57,700 to 58,600 psi. The maximum stress in 

the specimen was found to be 59,500 psi and occurred at location C. This value exceeds the maximum 

gage section stress by about 2 percent which is a significant improvement over earlier designs. Given this 

encouraging result, the design study progressed to its final stage and blend radii were incorporated at the 

ends of the flexures. As noted earlier, the intent here was to refine the finite element model so as to 

obtain a more meaningful measure of local stresses at these locations. 

 The approach adopted in modeling these radii is shown in figure 14 for the case of the blend radius at 

the corner fillet region. About 80 additional elements were used to model the 0.063 in. radius shown. The 

effect of these radii on the local stress distribution is shown in figure 15. As expected, the local stress 

distribution in the corner fillet region is modified significantly by the addition of the blend radius. 

Detailed analysis showed that the maximum stress in this region was 55,800 psi which represents a 

4 percent reduction from the earlier value. Also, it can be seen that the maximum stress location was 

shifted from the corner along the length of the flexure. Further analysis showed that use of larger radii 

produced a negative result in that stress levels in the flexures were increased over the above value. For 

this reason, the 0.063 in. radius was selected for use with the subject design. 

 The above stress values are summarized along with earlier data in table I. These data clearly show 

the performance gains at the various stages of design and optimization process. It can also be seen that 

specimen design no. 228.2 meets the design requirements regarding uniformity of gage area stress and 

location of maximum stress in the part. However, it should be noted that this level of performance is met 

with little if any margin of conservatism. 

 Details of specimen design no. 228.2 in final form are given in figures 16 to 18. It can be seen that 

the optimization process resulted in a number of changes to overall specimen dimensions. Specifically, 

the effective gage area dimension was reduced to 3.780 by 3.780 in. and the inner corner fillet radii was 

reduced to 0.575 in. Corresponding changes in the thickness transition details can be seen in figure 17. 

The final flexure configuration is shown in figure 18 and can be seen to be complex. Inspection of these 

figures will show that blend radii of 0.063 in. were used at the critical corner locations and also in 

locations where the width of flexures was “stepped.” Also, it can be seen that a blend radius of 0.125 in. 

was used at the ends of the flexures in a total of 152 locations. The relative merits of this and subsequent 

specimen designs are discussed later in the paper. 

4. Specimen Designs for Testing Under Cyclic Loading 

 The second design exercise was aimed at developing a specimen suitable for investigating material 

behavior under cyclic loading. As noted earlier, the types of experiment of interest in this research 

included low-cyclic fatigue tests and fatigue crack growth tests. One important consideration here is 
NASA/TM—2003-212090 7 

buckling stability since these experiments involve both tensile and compressive loadings. As a result of 

preliminary analysis, the gage area was reduced to 2.375 by 2.375 in. to better meet cyclic testing 

requirements. One advantage of reducing the size of the gage area was that it was possible to incorporate 

a number of significant design changes at the outset based on experience gained thus far into the design 

study. These changes are described in the following along with the results of design and analysis work. 

4.1 Initial Design 

 Details of the initial specimen configuration, designated no. 232.1, are given in figures 19 and 20. 

The overall dimensions chosen as best meeting cyclic testing requirements were 12 by 12 by 0.500 in. 

Based on experience gained with the earlier designs, a “two-step” reduction in thickness was used to 

obtain the 2.375 by 2.375 in. gage area as illustrated in figure 20. Here it can be seen that the thickness of 

the gage area and the intermediate step were 0.080 and 0.150 in., and a single thickness transition radius, 

0.500 in., was used throughout. The remaining gage area dimensions for design no. 232.1 were again 

selected using “engineering judgment.” 

 With regard to specimen gripping, inspection of figure 19 shows that the area available for gripping, 

4.125 by 1.500 in., is also reduced from the earlier value. Checkout experiments conducted on the earlier 

design showed that a gripping area of about 6.0 in2

 met the loading requirements of the planned test 

programs. This result allowed the width of the material available for gripping to be reduced to 4.125 in. 

and the specimen configuration to be simplified in the gripped region. A further modification to the 

gripped regions is the incorporation of sixteen 0.063-in.-wide slots. The goal here was to provide “relief” 

which would allow localized stresses due to gripping to be distributed more uniformly. Also, the 

incorporation of slots was expected to be advantageous in minimizing thermal stresses in the gripped 

region during elevated temperature tests. 

 One important advantage associated with reducing the gage area is that it allowed the length of the 

flexures to be increased to 2.000 in. This represents a 40 percent increase over the value used in the 

earlier design. Inspection of figure 19 will show that the number of flexures used in each loading arm 

was ten and that the width of individual flexures was 0.188 in. It can also be seen in this figure that blend 

radii were not included at the ends and intersection points of the flexures. As before, the plan was to 

incorporate these radii once the optimum flexure configuration had seen established. 

 The details of the finite element analyses conducted on design no. 232.1 were identical to those 

described earlier. The three dimensional model shown in figure 21 was generated using PATRAN, and 

linear elastic stress analyses were conducted using the MARC finite element code. As before, all surface 

nodes in the gripped regions were constrained to move 0.005 in. in the X and Y directions to produce a 

state of equibiaxial tension in the gage area. Also, surface nodes in the gripped region were constrained 

to move 0.0002 in. in the thickness direction to simulate clamping in the hydraulic grips. As in the case 

of the earlier work, elastic constants used in these analyses were handbook values for Inconel 718. 

 The results of the finite element analyses are given in figures 22 to 25. As in the case of the earlier 

designs, the stress distribution over the major part of the gage area is relatively uniform, falling within 

the range 48,800 psi to 54,000 psi (fig. 23). More detailed analyses showed that the maximum stress, 

55,000 psi, occurred at location P and that the minimum stress, 44,500 psi, occurred at location M. Thus, 

the deviation of stress in the gage area prior to optimization was within ±11 percent of the mean. 

Considering the balance of the specimen, figures 24 and 25, the maximum stress was found to occur at 

location D and was about 55,000 psi. The next highest stresses occurred at locations B and C and were 
NASA/TM—2003-212090 8 

54,800 psi and 54,000 psi. Interestingly, the stress levels at the intersection point of the flexures, location 

F in figure 24, were relatively low in this particular design and were not viewed as being problematical. 

Based on these results, the primary focus of the optimization process was on improving the uniformity of 

stress within the gage area. 

4.2 Final Design 

 The optimization process leading to the final design was relatively straightforward and involved 37 

design iterations. In this case, the optimization process consisted of making systematic changes to the 

design of the flexures and determining the effect of these changes on the stress distribution in the gage 

area. As noted earlier, the goal was to get the distribution of stress in the gage area within ±5 percent of 

the mean. The procedure followed was simply to remove material from the flexures until the above target 

had been achieved. The flexures were treated in sequence starting at the centerline and working out 

toward the corner between loading arms. In the event, it proved necessary to modify eight of the flexures 

in each of the four loading arms. 

 One of the finite element models used in developing a final design is shown in figure 26. This 

particular design, designated no. 269.1, does not include blend radii at the ends of the flexures. The 

results of stress analyses performed on this particular design are shown in figures 27 to 30. The stress 

distribution over the entire gage area falls within the range, 43,600 psi to 48,800 psi. A more detailed 

analysis showed that the maximum stress, 46,600 psi, occurred at location P and that the minimum stress, 

45,000 psi, occurred at location M. In this case, the deviation about the mean was ±2 percent which 

meets the original design requirement. The stress distribution in the balance of the specimen is shown in 

figures 29 and 30. The maximum stress in the flexures was found to occur at location D and was 

46,975 psi. The next highest stresses were found to occur at locations A, B, and C and fall within the 

range, 45,375 to 46,500 psi. These results were viewed as being encouraging and the subject flexure 

configuration was adopted for the final design. 

 As before, the final stage of the design process was to incorporate blend radii. This procedure was 

followed and a series of finite element analyses were performed to establish optimum radius values. In 

summary, best results in terms of uniformity of stress in the gage area and local stress values were 

obtained using a radius of 0.125 in. at the four corner locations and a 0.063 in. radius in all other 

locations. One of the finite element models used in establishing this result is shown in figure 31. This 

particular model contains 9600 eight-node, isoparametric, solid brick elements and 13,300 nodes with 

three degrees of freedom. Initial results showed that the maximum stresses in the flexures were about the 

same as those in the gage area. To add a margin of conservatism, the decision was made to reduce the 

thickness of the gage area from 0.080 to 0.050 in. This necessitated some minor changes to the 

configuration of the gage area as will be described later in the paper. 

 Stress analysis results for the modified design, designated 269.2, are shown in figures 32 and 33. The 

stress distribution in the gage area is relatively uniform, falling within the range 54,000 psi to 59,200 psi. 

Detailed analysis showed that the maximum stress, 58,800 psi, occurred at location P and that the 

minimum stress, 54,000 psi, occurred at location M. In this case, the deviation of stress about the mean is 

about ±5 percent which meets the original design goal. The stress distribution in the flexure region is 

shown in figures 34 and 35. The maximum stress in this region occurred at locations A and B and was 

55,000 psi. The next highest stresses occurred at locations C and D and were 54,500 psi and 53,200 psi. 

The above stresses are summarized in table II. Here, it can be seen that the ratio of the maximum stress in 

the flexures over the maximum stress in the gage area is 0.94 for specimen design no. 269.2. Thus, this 
NASA/TM—2003-212090 9 

particular design met both design requirements regarding stress uniformity in the gage area and location 

of maximum stress in the specimen. 

 Details of specimen design no. 269.2 in final form are given in figures 36 and 37. Final dimensions 

for the gage area were 2.300 by 2.300 in. and the thickness was 0.050 in. The thickness transition details 

can be seen to be similar to those used earlier with the exception of some minor changes needed to 

accommodate the 0.050 in. thickness. The final flexure configuration was relatively straight forward and 

unlike earlier designs, did not involve the use of “steps.” The eight outer-most flexures were 0.188 in. 

wide and the balance were 0.125 in. wide. The four central slots were 0.250 in. wide and the remaining 

slots were 0.313 in. wide. As noted earlier, the radius used in the corner locations was 0.125 in. and a 

radius of 0.063 in. was used in all remaining locations. Overall, the final specimen design was thought to 

be relatively simple and a significant improvement over earlier designs. 

5.0 Performance Under General Forms of Biaxial Loading 

 One limiting feature of the design/optimization work described this far is that it focused entirely on a 

single form of biaxial loading. This was the special case of equibiaxial tensile loading. One advantage of 

using this type of loading was that it caused symmetrical stress states to be introduced into the 

specimens. This simplified the optimization process by effectively reducing the number of variables 

involved. Regardless of such efforts, the process of developing fully optimized designs remained 

complex and time consuming. Given this result, it was apparent that major savings in time and effort 

would be realized if it could be demonstrated that the equibiaxial specimen design could be used without 

modification to investigate more general forms of biaxial loading. Possible issues to be addressed here 

included the location of the maximum stress and also the uniformity of stress in the specimen gage area. 

5.1 Stress Analysis Details 

 The specimen configuration selected for further study was specimen no. 269.2, details of which are 

given in figures 36 and 37. One goal in conducting this work was to maintain a reference state of stress at 

the center of the specimen gage area so as to allow meaningful comparison of performance under the 

various loading conditions investigated. The approach adopted in achieving this goal is shown 

schematically in figure 38. Here it can be seen that a single value of von Mises equivalent stress, 

50,000 psi, was used throughout. Further, six stress ratios (θ) in the range ±45° were selected to 

investigate specimen performance over a single quadrant of biaxial stress space. In the case of isotropic 

materials, stress states in the remaining quadrants can be inferred from the symmetry of the von Mises 

ellipse. The components of stress, σx and σy, corresponding to the six reference conditions were 

calculated in a straightforward manner using the relationships shown in figure 38 and the results of these 

calculations are summarized in table III. It remained to establish the grip displacements needed to achieve 

the reference stress states in planned finite element analyses. 

 As noted earlier, loading in the subject in-plane biaxial tests is introduced into specimens by means 

of four hydraulic grips. These grips rigidly constrain the gripped region of the specimen over 1.5 by 

4.125 in2

 areas (fig. 19). This was simulated in earlier finite element analyses by constraining all surface 

nodes in the gripped regions to displace predetermined amounts in the two loading directions. The plan 

was to use the same general approach in the present investigation. To determine the required grip 

displacements, it was assumed that stress components at the center of the gage area, σx and σy, are related 

to corresponding grip displacements, ∆x and ∆y, by the expressions: 
NASA/TM—2003-212090 10 

∆x = K1 σx + K2 σy (1) 

∆y = K1 σy + K2 σx (2) 

Where K1 and K2 are constants. 

 Since K1 and K2 were unknown, it was necessary to conduct a preliminary finite element analysis to 

effectively calibrate these expressions. The approach was to apply known grip displacements to the 

model and to calculate the corresponding values of σx and σy. It was then possible to solve equations (1) 

and (2) for K1 and K2. 

 Regarding the boundary conditions and loading used in the preliminary analysis, clamping in the 

rigid grips was simulated as before by constraining surface nodes in the gripped regions to displace the 

same amount in a given loading direction. Loading was introduced into the finite element model by 

applying a simulated grip displacement of 0.005 in. in the Y direction. The condition in the transverse 

direction was “gripped” but free-to-displace in this direction. The value of ∆x calculated as a result of this 

loading was –0.0000567 in. and the calculated values of σy and σx were 52,523 and 2078 psi. These 

values along with the known value of ∆y were substituted in the equations (1) and (2) and solved for K1 

and K2 with the following results: 

 K1 = 9.5371×10–8 (in.)(psi)–1 

 K2 = –4.85267×10–9 (in.)(psi)–1

At this stage, it was possible to solve equations (1) and (2) to determine the required grip displacements. 

These values are summarized in table III along with the target stress values. 

5.2 Stress Analysis Results 

 The results obtained in subsequent finite element analyses are shown in figures 39 to 44. The first 

application of these data was to assess the accuracy of the stress analysis approach described above. This 

simply involved determining the stress components, σx and σy, at the center of the gage area and 

comparing them to the target values. The stresses calculated for the six loading cases are summarized in 

table III along with the earlier data. Overall, the two data sets were found to be in good agreement. This 

result provided some degree of confidence in the analysis approach adopted. 

 In the case of experiments investigating strength and fracture behavior, the focus is on the magnitude 

and the location of the maximum stress in the specimen. The results shown in figures 39 to 44 were 

analyzed further from this particular viewpoint. In this case, stress states were analyzed along individual 

axes to identify the magnitude and location of the maximum stress. The three axes considered in these 

analyses were the x, y, and 45° axes. The results of these analyses are summarized in tables IV to IX. 

Here, the six components of stress are given for particular locations in the gage area along with the 

corresponding value of von Mises equivalent stress. This level of detail was provided to facilitate 

comparison with experimental data to be determined in planned specimen evaluation experiments. 

Further analysis of these data was performed to establish the magnitude and the location of the maximum 

value of von Mises equivalent stress in the specimen. The results of these analyses are summarized in 

table X. 
NASA/TM—2003-212090 11 

 Finally, the same data is shown in graphical form in figures 45 to 50. One goal of presenting the data 

in this form was to allow more quantitative analysis of the data. These plots were used for example to 

establish regions in the gage area where stress values fell within ±5 percent of the mean. Determination 

of such regions of relatively uniform stress was of interest in establishing the optimum location for 

instrumentation to be used in planned deformation experiments. These results along with data 

summarized in tables IV to X are discussed in detail in the following section of the paper. 

6.0 Discussion 

 As noted in the introduction, the aim of this research was to develop specimen designs suitable for 

testing advanced metallic materials under in-plane biaxial loading. A range of experiments was planned 

in this research including equibiaxial tension tests and equibiaxial fatigue tests. In the case of tests 

involving monotonic loading, the requirement was for a relatively large gage area which would allow 

extensive instrumentation with minimum installation difficulty. In addition, it was required that the stress 

distribution in the gage area should be reasonably uniform, for instance, within ±5 percent of the mean. 

This was to ensure that in-situ measurements were made in relatively uniform stress/strain fields. Also, it 

was required that the maximum stress in the specimen should occur within this gage area, preferably with 

some reasonable margin of conservatism. The aim here was to ensure that failure would be initiated 

within the gage area where stress/strain conditions are relatively well defined. 

 The results summarized in table I show that these requirements were met in part during the first 

optimization exercise. At the start of this process, the deviation of stress within the gage area of design 

no. 146.1 was ±15 percent. Also, the ratio of the maximum stress in the flexures over the maximum stress 

in the gage area, termed R ratio in table I, was 1.32. A lengthy optimization process involving over 80 

design iterations resulted in the development of a number of improved designs. In the case of specimen 

design no. 228.2, the deviation in gage area stress was ±5 percent and the R ratio was 1.00. Perhaps the 

most significant design change made during this process was that a two-step reduction in thickness was 

used to achieve the 3.875 by 3.875 in. gage area. Also, the thickness of the gage area was reduced to 

0.080 in. and the overall thickness of the specimen was increased to 0.625 in. These changes were made 

reluctantly since the two-step feature adds complexity to the design and leads to increased manufacturing 

costs. 

 The most time consuming aspect of the optimization process for design no. 228.2 was in developing 

an improved flexure configuration. One obvious difficulty here is that there are a large number of 

variables involved including the number, the size, the spacing, and the shape of the flexures. Another 

difficulty is that the stress distribution in the gage area is highly sensitive to small changes in the flexure 

configuration. Details of specimen design no. 228.2, including the final flexure configuration, are given 

in figures 16 to 18. One feature of this design is that the two outer-most flexures in each loading arm 

incorporate “steps,” resulting in a relatively complex design. 

 Turning to the development of the specimen design for cyclic loading, the process in this case was 

less protracted, primarily because the specimen design selected at the outset featured a number of 

improvements. These improvements were based on experience gained during the first design exercise and 

also on the results of evaluation experiments conducted on the earlier specimen design. Regarding the 

more important design changes, a “two-step” reduction in specimen thickness was used from the outset to 

obtain the 2.375 by 2.375 in. gage area. One advantage of this approach was that it allowed the length of 

the flexures to be increased by about 40 percent. Also, the reduced gage area offered improved buckling 

stability in experiments involving both tensile and compressive loadings. In addition to meeting design 
NASA/TM—2003-212090 12 

goals regarding uniformity of gage area stress and location of the maximum stress, the aim was to 

develop a simpler design with improved margins of conservatism. 

 The results shown in table II indicate that these requirements were met in total during the second 

design exercise. In the case of specimen design no. 232.1, the deviation of stress within the gage area is 

±11 percent and the R ratio is 1.00. Another encouraging result was that the stresses at the four corner 

locations were low relative to the average stress in the gage area. Given this starting point, the primary 

focus of the optimization effort was on refining the flexure design to improve the uniformity of stress in 

the gage area. The final specimen design, designated no. 269.2, was developed after 37 design iterations 

and met all of the performance requirements. In the case of the design with no blend radii, no. 269.1, the 

deviation of gage area stress was ±2 percent and the R ratio was 1.00. Clearly this represents a major 

improvement over earlier designs in regard to the uniformity of the gage area stress. 

 The final stage of the optimization process was to perform a series of analyses to establish optimum 

blend radius values. Best results were obtained using a 0.125 in. radius at the four corner locations and a 

0.063 in. radius in all other locations. However, analysis showed that the R ratio for this particular 

configuration was still about 1.00. It was decided, therefore, to reduce the thickness of the gage area to 

0.050 in. to achieve some margin of conservatism. Details of the fully optimized design, designated 

no. 269.2, are given in figures 36 and 37. Stress analysis results for this design are summarized in 

table II. Here it can be seen that the deviation of gage area stress was ±5 percent and that the R ratio was 

0.94. Thus, design no. 269.2 meets the design goals regarding uniformity of gage area stress and location 

of maximum stress. Further, inspection of figure 36 will show that the flexure configuration did not 

include “steps,” resulting in a less complicated design. Based on these results, design no. 269.2 appeared 

to both meet and exceed all requirements for planned experiments involving cyclic loading. 

 The design work this far was focused on the development of specimens optimized for testing under 

equibiaxial loading. Given the length and complexity of the optimization process, one obvious question is 

whether these designs can be used for testing under more general forms of loading without compromising 

the validity of the results. Further analyses addressing this specific question were performed on specimen 

design no. 269.2. As described earlier, the goal was to load the specimen so as to produce given reference 

states of stress in the central elements. Loading was introduced into the specimen using the displacement 

controlled mode described earlier. Equations relating grip displacement, specimen stiffness, and local 

stress in the central reference elements were calibrated using the results of finite element analyses. Also 

as described earlier, these relationships were used to calculate the grip displacements needed to achieve 

the required stress states. These displacements are summarized in table III along with back-calculated 

values of stress obtained from finite element analyses conducted to check the accuracy of the procedure. 

Comparison of the two data sets showed that the finite element results were within ±2 percent of the 

target values in the case of σx and were within ±4 percent in the case of σy. Similarly, the values of von 

Mises equivalent stress were found to be within ±1 percent of the target value. These results were judged 

to be acceptable for the present comparative study and detailed analyses were conducted investigating the 

six stress ratios selected for the study. 

 The results of these analyses are shown in figures 39 to 44 in the form of color plots. The same data 

for the X, Y and 45° axes are shown in graphical form in figures 45 to 50 and in tabular form in tables IV 

to X. These data were used first to assess the uniformity of stress over the entire specimen gage area with 

the results shown in table X. As described earlier, the optimized form of specimen no. 269.2 was 

developed using equibiaxial tensile loading which corresponds to stress ratio (θ) = +45.0°. Not 

surprisingly, the stress distribution for this loading case shown in figure 39 is fairly uniform with stress 

levels falling within ±3 percent of the mean over the entire gage area. Analysis of the data shown in 
NASA/TM—2003-212090 13 

figures 40 to 44 showed that the deviations of stress are significantly higher for the other stress ratios. 

More specifically, it can be seen in table X that the deviations for the other five stress ratios fall in the 

range ±11 to ±19 percent. Clearly, these results do not meet the original design requirement that stresses 

within the gage area should be uniform within ±5 percent of the mean. 

 As would be expected, a much improved situation holds for stress distributions within a more limited 

area at the center of the specimen’s gage area. The data shown in figures 45 to 50 were analyzed further 

to determine the area within which the stress distributions met the original design requirement. This 

analysis showed that the ±5 percent target was met within a 1.0-in.-diameter circular region at the center 

of the specimen gage area. This result suggests that the equibiaxial design can be used to investigate 

deformation behavior under general forms of biaxial loading, provided measurement and observation is 

limited to this central 1.0-in.-diameter region. 

 In the case of experiments investigating strength and fracture behavior, the focus is on the magnitude 

and the location of the maximum stress in the specimen. The results shown in figures 39 to 44 were 

analyzed further with this particular viewpoint in mind. In this case, stress states were analyzed along 

individual axes to identify the magnitude and the location of the maximum stress. The results of these 

analyses for the X, Y and 45° axes are summarized in table X. As expected, the location of the maximum 

stress is highly dependent on the loading direction or stress ratio. In the case of equibiaxial loading, 

θ = 45.0°, the maximum stress occurred on the 45.0° axis at the outer perimeter of the gage area. For 

stress ratios of 0°, 15.0° and 30.0°, the maximum stress location fell on the x-axis, again at the gage 

section outer perimeter. Interestingly, this pattern of behavior was not repeated for negative stress ratios. 

For θ = –22.5° and –45.0°, the maximum stress was located at the specimen center. Clearly, the behavior 

described above complicates interpretation of any strength or fracture tests involving general forms of 

biaxial loading. 

 In summary, this study showed that the equibiaxial specimen designs can be used without 

modification to investigate deformation behavior under general forms of biaxial loading. This is 

assuming that all measurements and observations are limited to the 1.0-in.-diameter circular region at the 

specimen center. It was also shown that the equibiaxial design can be used within limits for other forms 

of testing for stress ratios of +45.0°, 0°, –22.5°, and –45.0°. This recognized that data for θ = 0° can be 

generated under uniaxial loading using a simpler test setup. In the case of the negative stress ratios, the 

location of the maximum stress occurs at the specimen center within a region of uniform specimen 

thickness. This facilitates measurement of both strain and temperature at the critical location as 

stress/strain conditions in this region were shown to be relatively uniform. For the remaining stress 

ratios, θ = +15.0° and +30.0°, some modifications to the specimen design are necessary. For best results, 

it appears that specimen designs need to be optimized and tailored to meet the loading requirements of 

individual research programs. 

7.0 Conclusions 

 The following conclusions were drawn from this design study aimed at developing specimen designs 

for testing advanced aeropropulsion materials under in-plane biaxial loading. 

 1. A specimen design was developed for deformation and strength tests with overall dimensions, 12 

by 12 by 0.625 in. and gage section dimensions, 3.875 by 3.875 by 0.080 in. The configuration of 

the specimen was complex, presenting a challenge to manufacture and leading to high 

manufacturing costs. The design criteria regarding uniformity of stress in the gage area and the 
NASA/TM—2003-212090 14 

location of the maximum stress in the specimen were both met with this design, but with little, if 

any, margin of conservatism. 

 2. A second specimen design was developed for tests involving cyclic loading with overall 

dimensions, 12 by 12 by 0.500 in., and with gage area dimensions, 2.375 by 2.375 by 0.050 in. 

Details of this design were simpler in the gripped region and also in the flexure region compared 

to the earlier design but, overall, the configuration of the specimen remained complex. This design 

met the various design criteria adopted for the study, but did so with relatively small margins of 

conservatism. 

 3. The two specimen designs were optimized for the special case of equibiaxial, tensile loading. 

Further study showed that the designs can be used without modification to investigate deformation 

behavior under general forms of biaxial loading, provided measurement and observation is limited 

to the 1.0-in.-diameter circular region at the specimen center. In the case of experiments 

investigating strength and fracture, for best results, specimen designs need to be optimized and 

tailored to meet the loading requirements of individual research programs. 

 4. The optimization process used in this study involved making systematic changes to the specimen 

design and repeating stress analysis as necessary until the various design requirements had been 

met. This process was performed manually since the finite element analysis code available for this 

work did not include a formal optimization package. This procedure proved to be both time 

consuming and inefficient as a large number of design iterations were needed to achieve a fully 

optimized design. 

Future Work 

 The scope of this study will be extended to investigate the feasibility of using simple specimen 

designs and reusable fixturing for in-plane biaxial tests planned for advanced composite materials. Fiber 

reinforced composite systems of interest include: polymeric matrix composites; metal and intermetallic 

matrix composites; and ceramic matrix composites. 

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NASA/TM—2003-212090 15 

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TABLE I.—STRESS DISTRIBUTION IN SPECIMEN DESIGNS FOR TESTING UNDER MONOTONIC LOADING 

Gage Area Stress Max. Stress in Flexures at Location Indicated, 

Specimen psi 

No. & Type Max., 

psi 

Min., 

psi Deviation C* D E F 

Ratio of 

Maximum 

Stresses 

(R)+

146.1 

(no blend 

radii) 

53,000 

(N*) 

39,400 

(P) ±15 -------- 69,700 -------- 64,000 1.32 

228.1 

(no blend 

radii) 

58,300 

(P) 

53.000 

(M) ±5 59,500 58,600 57,700 58,100 1.02 

228.2 

(with blend 

radii) 

57,750 

(P) 

52,000 

(M) ±5 52,500 49,500 57,750 55,800 1.00 

Notes: *Reference locations are given in figure 7. +

R = ratio of maximum stress in flexures over maximum stress in gage area. 
NASA/TM—2003-212090 17 

TABLE II.—STRESS DISTRIBUTION IN SPECIMEN DESIGNS FOR TESTING UNDER CYCLIC LOADING 

Gage Area Stress 

Max. Stress in Flexures at Location 

Indicated, 

psi Specimen 

No. & Type Max., 

psi 

Min., 

psi Deviation A* B C D 

Ratio of 

Maximum 

Stresses 

(R)+

232.1 

(no blend 

radii) 

55,000 

(P*) 

44,500 

(M) ±11 53,000 54,800 54,000 55,000 1.00 

269.1 

(no blend 

radii) 

46,600 

(P) 

45,000 

(M) ±2 46,500 45,375 45,375 46,975 1.00 

269.2 

(with blend 

radii) 

58,800 

(P) 

54,000 

(M) ±5 55,000 55,000 54,500 53,200 0.94 

Notes: * Reference locations are given in figure 7. +

R = ratio of maximum stress in flexures over maximum stress in gage area. 

TABLE III.—GLOBAL DISPLACEMENTS (∆X AND ∆Y) GIVING REQUIRED STRESS RATIO (θ) AND 

VON MISES EQUIVALENT STRESS (σ) IN CENTRAL REFERENCE ELEMENTS 

Stress Calculated Values Finite Element Results 

Ratio, 

θ

σx, 

psi 

σy, 

psi 

∆x, 

in. 

∆y, 

in. 

σx, 

psi 

σy, 

psi 

σ, 

psi 

45.0 50,000 50,000 0.004526 0.004526 49,929 50,089 50,011 

30.0 57,506 33,201 0.005324 0.002887 57,501 33,085 49,978 

15.0 55,767 14,943 0.005246 0.001155 56,259 14,644 50,542 

0.0 50,000 0 0.004768 -0.000243 49,790 -601 50,084 

-22.5 39,705 -16,446 0.003867 -0.001761 39,474 -16,929 50,136 

-45.0 28,868 -28,868 0.002893 -0.002893 28,392 -29,883 50,473 
NASA/TM—2003-212090 18 

Location Element Coordinates Stress Component, psi Von Mises

No. x y z σxx σyy σzz σxy σyz σxz Equiv. Stress,

psi

Stress distribution along center line in Y-direction

9321 0.058 0.139 0.016 49929 50089 -1 -83 1 -3 50011

Uniform 9320 0.058 0.300 0.016 49559 50453 19 -179 -1 -2 49994

Gage 9319 0.058 0.462 0.016 48921 51081 22 -279 2 -4 50017

Area 9317 0.061 0.784 0.016 46808 53166 29 -504 26 -4 50270

9316 0.063 0.946 0.016 45287 54641 -66 -651 -94 1 50698

9315 0.066 1.107 0.016 43576 56477 325 -828 321 -7 50972

9136 0.068 1.211 0.017 41219 52197 1243 -868 3047 -48 46802

Thickness 9135 0.068 1.259 0.022 38142 41158 1699 -737 5289 -93 39155

Transition 9134 0.068 1.306 0.030 35248 30520 1325 -620 5469 -84 33221

9133 0.068 1.352 0.041 32940 22557 -334 -540 3814 -59 30259

Stress distribution along center line in X-direction

9077 0.139 0.058 0.016 50091 49927 -1 -83 -3 1 50011

Uniform 9070 0.300 0.058 0.016 50456 49557 19 -179 -2 -1 49994

Gage 9063 0.462 0.058 0.016 51083 48919 22 -279 -4 2 50017

Area 9056 0.623 0.059 0.016 51982 48002 20 -385 -4 -7 50096

9049 0.784 0.061 0.016 53167 46807 29 -504 -4 26 50271

9039 1.107 0.066 0.016 56479 43575 325 -827 -7 321 50974

8866 1.211 0.068 0.017 52199 41218 1243 -868 -48 3047 46803

Thickness 8862 1.259 0.068 0.022 41159 38140 1699 -737 -93 5289 39156

Transition 8858 1.306 0.068 0.030 30521 35247 1325 -619 -84 5470 33221

8854 1.352 0.068 0.041 22557 32939 -335 -540 -59 3814 30259

Stress distribution along 45° diagonal

9373 0.174 0.247 0.022 49862 50164 -25 -445 3 -2 50045

9380 0.290 0.355 0.022 49813 50228 -32 -1062 5 -6 50088

9387 0.406 0.464 0.022 49775 50279 -40 -1946 7 -6 50184

Uniform 9394 0.522 0.576 0.022 49741 50325 -50 -3101 9 -10 50375

Gage 9399 0.638 0.688 0.022 49701 50359 -42 -4511 11 -7 50686

Area 9404 0.754 0.802 0.022 49623 50375 -109 -6108 10 -15 51221

9656 0.844 0.888 0.022 49739 50440 -90 -7362 -2 -2 51779

9655 0.906 0.945 0.022 49826 50420 -26 -8255 25 6 52151

9654 0.968 1.003 0.022 49814 50307 -70 -9126 -90 -86 52566

9653 1.029 1.063 0.022 50067 50420 134 -9719 334 274 52872

9584 1.075 1.112 0.024 48244 48003 809 -11200 2483 2021 51462

Thickness 9583 1.107 1.148 0.031 43197 41802 1247 -13692 4197 3419 48522

Transition 9582 1.138 1.183 0.042 38168 36046 2094 15306 4577 3649 45125

9581 1.168 1.218 0.057 32091 29639 480 -16383 2780 2159 42154

TABLE IV.—STRESS DISTRIBUTION IN SPECIMEN NO. 269.2: STRESS RATIO θ = 45.0° 
NASA/TM—2003-212090 19 

TABLE V.—STRESS DISTRIBUTION IN SPECIMEN NO. 269.2: STRESS RATIO θ = 30.0°

Location Element Coordinates Stress Component, psi Von Mises

No. x y z σxx σyy σzz σxy σyz σxz Equiv. Stress,

psi

Stress distribution along center line in Y-direction

9321 0.058 0.139 0.016 57479 33084 9 -80 1 -3 49960

Uniform 9320 0.058 0.300 0.016 56951 33427 31 -163 -0.73 -3 49542

Gage 9319 0.058 0.462 0.016 56050 34001 35 -254 1 -5 48878

Area 9317 0.061 0.784 0.016 53070 35914 38.22 -468 18 -5 46880

9316 0.063 0.946 0.016 50932 37287 -42 -614 -66 -0.49 45723

9315 0.066 1.107 0.016 48426 39013 250 -790 225 -7.18 44258

9136 0.068 1.211 0.017 45846 36356 892 -829 2120 -46 41250

Thickness 9135 0.068 1.259 0.022 43276 28795 1186 -695 3707 -89 37614

Transition 9134 0.068 1.309 0.030 40894 21446 930 -571 3844 -87 35264

9133 0.068 1.352 0.041 38964 15926 -244 -483 2688 -65 34467

Stress distribution along center line in X-direction

9077 0.139 0.058 0.016 57649 32961 -11 -70 -2 1 50107

Uniform 9070 0.300 0.058 0.016 57966 32819 4 -161 -0.87 -0.87 50344

Gage 9063 0.462 0.058 0.016 58529 32564 5 -251 -2 2 50792

Area 9056 0.623 0.059 0.016 59336 32194 3 -343 -2 -7 51449

9049 0.784 0.061 0.016 60394 31717 14 -444 -3 28 52318

9044 0.946 0.063 0.016 61693 31099 -77 -565 2 -104 53506

9039 1.107 0.066 0.016 63292 30505 338 -709 -6 356 54557

8866 1.211 0.068 0.017 58195 28816 1360 -743 -40 3400 49633

Thickness 8862 1.259 0.068 0.022 45760 25811 1891 -640 -79 5874 39404

Transition 8858 1.306 0.068 0.030 33836 22953 1471 -552 -64 6064 30414

8854 1.352 0.068 0.041 24934 20702 -362 -496 -41 4221 24634

Stress distribution along 45° diagonal

9373 0.174 0.247 0.022 57307 33230 -17 -410 3 -1 49861

9380 0.29 0.355 0.022 57098 33426 -22 -969 5 -5 49737

9387 0.406 0.464 0.022 56841 33673 -29 -1771 6 -5 49630

Uniform 9394 0.522 0.576 0.022 56535 33972 -38 -2821 8 -9 49572

Gage 9399 0.638 0.688 0.022 56163 34322 -25 -4098 10 -5 49575

Area 9404 0.754 0.802 0.022 55634 34646 -130 -5499 9 -15 49707

9656 0.844 0.888 0.022 55555 34978 -98 -6663 1 0.22 50089

9655 0.906 0.945 0.022 55481 35156 -26 -7480 18 8 50338

9654 0.968 1.003 0.022 55328 35214 -72 -8271 -58 -96 50644

9653 1.029 1.063 0.022 55565 35372 116 -8837 215 323 50968

9584 1.075 1.112 0.024 53207 34053 706 -10242 1637 2361 49590

Thickness 9583 1.107 1.148 0.031 47070 30351 1095 -12570 2734 4047 46594

Transition 9582 1.138 1.183 0.042 41196 26715 1849 -14051 2858 4508 43210

9581 1.168 1.218 0.057 34087 22378 360 -14911 1632 2805 39805
NASA/TM—2003-212090 20 

 TABLE VI.—STRESS DISTRIBUTION IN SPECIMEN NO. 269.2: STRESS RATIO θ = 15.0° 

Location Element Coordinates Stress Component, psi Von Mises

No. x y z σxx σyy σzz σxy σyz σxz Equiv. Stress,

psi

Stress distribution along center line in Y-direction

9321 0.058 0.139 0.016 56259 14645 16 -66 0.9 -3 50542

Uniform 9320 0.058 0.300 0.016 55672 14923 36 -127 -0.51 -4 49888

Gage 9319 0.058 0.462 0.016 54677 15374 41 -199 1 -5 48812

Area 9317 0.061 0.784 0.016 51386 16882 41 -375 10 -5 45341

9316 0.063 0.946 0.016 49027 17986 -18 -501 -35 -1 42983

9315 0.066 1.107 0.016 46201 19390 156 -653 117 -6 40077

9136 0.068 1.211 0.017 43780 18429 483 -685 1071 -39 37752

Thickness 9135 0.068 1.259 0.022 41971 14747 605 -567 1906 -74 36584

Transition 9134 0.068 1.306 0.030 40328 11098 479 -454 1989 -79 35919

9133 0.068 1.352 0.041 38968 8326 -137 -371 1400 -62 35727

Stress distribution along center line in X-direction

9077 0.139 0.058 0.016 56412 14570 -18 -50 -0.84 1 50734

Uniform 9070 0.300 0.058 0.016 56646 14637 -9 -124 0.5 -0.74 50937

Gage 9063 0.462 0.058 0.016 57077 14735 -10 -193 -0.13 2 51329

Area 9056 0.623 0.059 0.016 57695 14869 -12 -261 0.07 -7 51893

9049 0.784 0.061 0.016 58499 15053 0.13 -333 -0.8 27 52616

9044 0.946 0.063 0.016 59470 15270 -76 -414 2 -98 53553

9039 1.107 0.066 0.016 60646 15687 304 -509 -4 338 54322

8866 1.211 0.068 0.017 55527 14778 1278 -533 -28 3246 49280

Thickness 8862 1.259 0.068 0.022 43562 12228 1802 -467 -56 5586 38887

Transition 8858 1.306 0.068 0.030 32135 9771 1398 -415 -38 5759 29286

8854 1.352 0.068 0.041 23623 7858 -337 -387 -21 4002 22250

Stress distribution along 45° diagonal

9373 0.174 0.247 0.022 56023 14830 -9 -324 2 -0.3 50286

9380 0.29 0.355 0.022 55711 15108 -10 -759 3 -4 49929

9387 0.406 0.464 0.022 55310 15482 -17 -1383 5 -4 49495

Uniform 9394 0.522 0.576 0.022 54822 15948 -24 -2202 6 -4 49010

Gage 9399 0.638 0.688 0.022 54230 16509 -8 -3195 8 -3 48473

Area 9404 0.754 0.802 0.022 53407 17041 -129 -4245 7 -13 47919

9656 0.844 0.888 0.022 53184 17546 -91 -5179 4 2 47857

9655 0.906 0.945 0.022 52991 17861 -23 -5824 10 9 47792

9654 0.968 1.003 0.022 52750 18049 -65 -6444 -23 -92 47807

9653 1.029 1.063 0.022 52951 18220 86 -6914 88 323 48051

9584 1.075 1.112 0.024 50455 17983 529 -8075 720 2342 46274

Thickness 9583 1.107 1.148 0.031 44202 16844 827 -9972 1160 4051 42373

Transition 9582 1.138 1.183 0.042 38384 15447 1406 -11156 1054 4646 38571

9581 1.168 1.218 0.057 31333 13400 213 -11729 463 2983 34310
NASA/TM—2003-212090 21 

 TABLE VII.—STRESS DISTRIBUTION IN SPECIMEN NO. 269.2: STRESS RATIO θ = 0.0°

Location Element Coordinates Stress Component, psi Von Mises

No. x y z σxx σyy σzz σxy σyz σxz Equiv. Stress,

psi

Stress distribution along center line in Y-direction

9321 0.058 0.139 0.016 49790 -601 20 -51 0.63 -3 50084

Uniform 9320 0.058 0.300 0.016 49211 -395 37 -90 -0.3 -4 49392

Gage 9319 0.058 0.462 0.016 48231 -74 42 -141 0.3 -5 48249

Area 9317 0.061 0.784 0.016 44999 1002 39 -273 3 -5 44490

9316 0.063 0.946 0.016 42687 1805 3 -371 -8 -2 41819

9315 0.066 1.107 0.016 39878 2840 70 -491 24 -5 38511

9136 0.068 1.211 0.017 37800 3244 130 -515 184 -30 36231

Thickness 9135 0.068 1.259 0.022 36733 2820 112 -419 376 -56 35362

Transition 9134 0.068 1.306 0.030 35798 2289 96 -325 410 -65 34672

9133 0.068 1.352 0.041 34992 1842 -44 -252 302 -53 34141

Stress distribution along center line in X-direction

9077 0.139 0.058 0.016 50068 -211 -17 -77 1.2 -0.57 50197

Uniform 9070 0.300 0.058 0.016 50070 -418 -18 -86 1.4 -0.6 50290

Gage 9063 0.462 0.058 0.016 50368 -68 -20 -134 1 1 50412

Area 9056 0.623 0.059 0.016 50794 427 -22 -179 1 -6 50594

9049 0.784 0.061 0.016 51346 1084 -10 -224 0.62 23 50820

9044 0.946 0.063 0.016 51997 1897 -68 -270 2.7 -84 51114

9039 1.107 0.066 0.016 52770 3020 250 -324 -2 292 51199

8866 1.211 0.068 0.017 48153 2778 1094 -340 -17 2816 46527

Thickness 8862 1.259 0.068 0.022 37707 815 1561 -307 -35 4832 37483

Transition 8858 1.306 0.068 0.030 27763 -1099 1209 -286 -17 4976 29095

8854 1.352 0.068 0.041 20369 -2566 -286 -280 -5 3454 22721

Stress distribution along 45° diagonal

9373 0.174 0.247 0.022 49528 -403 -1 -234 1 0.4 49733

9380 0.29 0.355 0.022 49165 -88 -1 -539 2 -3 49220

9387 0.406 0.464 0.022 48690 349 -6 -977 3 -3 48551

Uniform 9394 0.522 0.576 0.022 48106 902 -11 -1553 4 -6 47746

Gage 9399 0.638 0.688 0.022 47395 1574 6 -2247 5 -1 46791

Area 9404 0.754 0.802 0.022 46415 2219 -118 -2940 5 -10 45699

9656 0.844 0.888 0.022 46094 2813 -78 -3615 5 3 45233

9655 0.906 0.945 0.022 45821 3206 -18 -4072 3 9 44874

9654 0.968 1.003 0.022 45525 3479 -53 -4505 5 -79 44606

9653 1.029 1.063 0.022 45664 3651 54 -4856 -15 289 44725

9584 1.075 1.112 0.024 43312 4250 343 -5715 -36 2087 42499

Thickness 9583 1.107 1.148 0.031 37607 5134 543 -7091 -129 3637 37625

Transition 9582 1.138 1.183 0.042 32420 5544 933 -7924 -395 4270 33341

9581 1.168 1.218 0.057 26128 5405 83 -8216 -453 2809 28266
NASA/TM—2003-212090 22 

TABLE VIII.— STRESS DISTRIBUTION IN SPECIMEN NO. 269.2: STRESS RATIO θ = -22.5° 

Location Element Coordinates Stress Component, psi Von Mises

No. x y z σxx σyy σzz σxy σyz σxz Equiv. Stress,

psi

Stress distribution along center line in Y-direction

9321 0.050 0.139 0.016 39334 -17311 23 -29 0.3 -2 50272

Uniform 9320 0.058 0.300 0.016 38806 -17200 35 -42 -0.04 -3 49684

Gage 9319 0.058 0.462 0.016 37918 -17048 40 -66 -0.26 -4 48725

Area 9317 0.061 0.784 0.016 34989 -16532 34 -139 -6 -4 45559

9316 0.063 0.946 0.016 32898 -16120 25 -201 22 -2 43271

9315 0.066 1.107 0.016 30312 -15567 -29 -276 -80 -3 40423

9136 0.068 1.211 0.017 28761 -13692 -266 -290 -805 -18 37620

Thickness 9135 0.068 1.259 0.022 28584 -10502 -439 -226 -1335 -32 35234

Transition 9134 0.068 1.306 0.030 28495 -7565 -332 -159 -1356 -45 33131

9133 0.068 1.352 0.041 28357 -5423 62 -104 -927 -40 31445

Stress distribution along center line in X-direction

9077 0.139 0.058 0.016 39423 -17293 -23 -7 1 0.27 50354

Uniform 9070 0.300 0.058 0.016 39474 -16929 -27 -37 2 -0.35 50136

Gage 9063 0.462 0.058 0.016 39599 -16323 -30 -57 3 1 49816

Area 9056 0.623 0.059 0.016 39780 -15463 -31 -73 3 -4 49371

9049 0.784 0.061 0.016 40006 -14329 -21 -84 2 17 48783

9044 0.946 0.063 0.016 40246 -12910 -54 -87 3 -63 48040

9039 1.107 0.066 0.016 40505 -11081 173 -89 -0.09 221 46987

8866 1.211 0.068 0.017 36742 -10571 815 -94 -3 2151 42952

Thickness 8862 1.259 0.068 0.022 28677 -11753 1189 -100 -9 3670 36328

Transition 8858 1.306 0.068 0.030 21043 -12942 917 -115 8 3771 30315

8854 1.352 0.068 0.041 15384 -13823 -210 -135 14 2612 25732

Stress distribution along 45° diagonal

9373 0.174 0.247 0.022 39061 -17113 7 -114 0.02 1 49871

9380 0.29 0.355 0.022 38671 -16781 9 -252 0.75 -1 49255

9387 0.406 0.464 0.022 38149 -16307 6 -451 1 -0.72 48416

Uniform 9394 0.522 0.576 0.022 37505 -15701 4 -714 2 -3 47369

Gage 9399 0.638 0.688 0.022 36720 -14957 21 -1026 2 1 46092

Area 9404 0.754 0.802 0.022 35645 -14239 -96 -1273 2 -6 44590

9656 0.844 0.888 0.022 35246 -13593 -58 -1615 6 4 43766

9655 0.906 0.945 0.022 34911 -13146 -12 -1832 -5 8 43138

9654 0.968 1.003 0.022 34582 -12800 -36 -2029 36 -61 42610

9653 1.029 1.063 0.022 34654 -12645 16 -2226 -128 233 42590

9584 1.075 1.112 0.024 32618 -11170 116 -2700 -865 1672 39803

Thickness 9583 1.107 1.148 0.031 27887 -8117 195 -3417 -1539 2949 33683

Transition 9582 1.138 1.183 0.042 23733 -5745 352 -3816 -1964 3583 28644

9581 1.168 1.218 0.057 18685 -3773 -66 -3786 -1433 2438 22476
NASA/TM—2003-212090 23 

TABLE IX.—STRESS DISTRIBUTION IN SPECIMEN NO. 269.2: STRESS RATIO θ = -45.0°

Location Element Coordinates Stress Component, psi Von Mises

No. x y z σxx σyy σzz σxy σyz σxz Equiv. Stress,

psi

Stress distribution along center line in Y-direction

9321 0.050 0.139 0.016 28340 -29941 24 -10 -0.011 -1.51 50479

Uniform 9320 0.058 0.300 0.016 27887 -29915 31 0.5 0.17 -2.95 50069

Gage 9319 0.058 0.462 0.016 27128 -29911 36 -0.2 -0.7 -3.5 49419

Area 9317 0.061 0.784 0.016 24629 -29886 28 -21 -12 -3 47287

9316 0.063 0.946 0.016 22847 -29819 42 -49 45 -2.7 45750

9315 0.066 1.107 0.016 20603 -29696 -109 -84 -159 -1.5 43791

9136 0.068 1.211 0.017 19579 -26731 -575 -88 -1567 -6.6 40321

Thickness 9135 0.068 1.259 0.022 20143 -20774 -863 -54 -2654 -10.7 35742

Transition 9134 0.068 1.306 0.030 20753 -15176 -663 -13 -2720 -26 31663

9133 0.068 1.352 0.041 21171 -11043 145 24 -1878 -27 28524

Stress distribution along center line in X-direction

9077 0.139 0.058 0.016 28392 -29883 -24 13 1.59 -0.03 50473

Uniform 9063 0.462 0.058 0.016 28333 -28641 -36 8 3.58 0.65 49342

Gage 9070 0.300 0.058 0.016 28356 -29419 -32 5 3 -0.14 50038

Area 9056 0.623 0.059 0.016 28301 -27535 -37 18 3.7 -2.86 48358

9049 0.784 0.061 0.016 28246 -26079 -29 36 3.28 11.35 47062

9044 0.946 0.063 0.016 28136 -24253 -40 68 2.72 -42.45 45416

9039 1.107 0.066 0.016 27957 -21956 99 108 1.6 149.7 43329

8866 1.211 0.068 0.017 25125 -20861 537 114 8 1473 39950

Thickness 8862 1.259 0.068 0.022 19509 -21331 811 76 13 2491.5 35677

Transition 8858 1.306 0.068 0.030 14238 -21852 622 32 29 2551.9 31879

8854 1.352 0.068 0.041 10350 -22198 -134 -8 29 1760.9 28946

Stress distribution along 45° diagonal

9373 0.174 0.247 0.022 28076 -29756 13 -10 -0.81 1.72 50092

9380 0.29 0.355 0.022 27686 -29429 17 -2 -0.51 -0.16 49473

9387 0.406 0.464 0.022 27159 -28954 16 7 -0.59 0.8 48607

Uniform 9394 0.522 0.576 0.022 26505 -28342 16 17 -0.5 -0.58 47512

Gage 9399 0.638 0.688 0.022 25707 -27586 2 39 -0.18 2.6 46169

Area 9404 0.754 0.802 0.022 24621 -26852 -72 171 0.02 -3 44595

9656 0.844 0.888 0.022 24183 -26203 -37 123 7 5 43649

9655 0.906 0.945 0.022 23817 -25738 -6 116 -11 7 42928

9654 0.968 1.003 0.022 23481 -25356 -20 125 59 -41 42306

9653 1.029 1.063 0.022 23494 -25224 -16 66 -213 173 42206

9584 1.075 1.112 0.024 21843 -23121 -77 -61 -1492 1230 39104

Thickness 9583 1.107 1.148 0.031 18203 -18479 -101 -195 -2600 2203 32314

Transition 9582 1.138 1.183 0.042 15150 -14645 -145 -214 -3129 2799 26818

9581 1.168 1.218 0.057 11433 -11063 -184 78 -2147 1983 20210
TABLE X.

—SUMMARY OF STRESS STATES IN GAGE AREA FOR GENERAL FORMS OF BIAXIAL LOADIN

Note: OP indicates outer perimeter of gage area 

Overall Variation of 

Stress in Gage Area Stress States Along Individual Axes 

X-Axis Y-Axis 45°-Axis Stress ratio, 

θ Average, 

ksi 

Deviation, 

percent Maximum, ksi Average, ksi Maximum, ksi Average, ksi Maximum, ksi Average, ksi 

M

Stre

45.0 51.2 ±3 51.0 50.2 51.0 50.3 52.9 51.2 5230.0 49.8 ±11 54.6 51.9 50.0 47.6 51.0 50.0 54.15.0 49.1 ±18 54.3 52.2 50.5 46.3 50.3 48.7 54.

0 47.6 ±19 51.2 50.7 50.1 45.4 49.7 46.7 51.

-22.5 47.1 ±14 50.4 49.1 50.3 46.3 49.9 45.8 50.-45.0 47.1 ±12 50.5 47.7 50.5 47.8 50.1 45.7 50.

NASA/TM—2003-212090 24 
NASA/TM—2003-212090 25

Load 

frame

Loading

direction

Cruciform

specimen

Loading

direction

Vacuum

chamber

Figure 1.—In plane biaxial test system.
NASA/TM—2003-212090 26

Typical regions

of high stress

concentration

Gage area

Plan view

(Partial section

on centerline)

Overall

thickness

Thickness

transition detail

Reduced

thickness

Flexures

Figure 2.—In-plane biaxial test specimen (ref. 18).

Straight

taper

Gripped area

typ. 4 places
NASA/TM—2003-212090 27

Griped area

typ. 4 places

Note: all dimensions in inches

Figure 3.—Specimen design 146.1: plan view.

0.375

typ. 4 places

12.000

1.500

1.800

6.000

1.387

5.000

0.063

0.438

typ. 24 places

0.375

typ. 8 places

1.000 0.750

0.188

typ. 40 places

y

z

x
NASA/TM—2003-212090 28

Figure 4.—Specimen design 

 146.1: Thickness transition.

Partial section on centerline

0.500

0.250

0.100

Straight

taper

Note: all dimensions in inches
NASA/TM—2003-212090 29

Thickness transition detail

Figure 5.—Finite element model of specimen design 146.1: isometric view.

y

z

x

Figure 6.—Finite element model of specimen design 146.1: plan view.

y

x

z
NASA/TM—2003-212090 30

Figure 7.—Stress distribution on the top and bottom surfaces: specimen design 146.1.

y

x

z

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

Location A

M

N

P

R

B C D EF

G

von Mises

equivalent stress,

psi
NASA/TM—2003-212090 31

Figure 8.—Stress distribution at midsection: specimen design 146.1.

y

x

z

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

von Mises

equivalent stress,

psi
NASA/TM—2003-212090 32

Figure 9.—Stress distribution in flexures: specimen design 146.1.

y

x

z

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

von Mises

equivalent stress,

psi

F

E

D

C

B

Figure 10.—Finite element model of specimen design 228.1: isometric view.

y

z

x

Thickness transition detail
NASA/TM—2003-212090 33

Figure 11.—Finite element model of specimen design 228.1: plan view.

y

x

z

Figure 12.—Stress distribution on the top and bottom surfaces: specimen design 228.1.

y

x

z

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

Location A

M

P

BCDEF

von Mises

equivalent stress,

psi
NASA/TM—2003-212090 34

Figure 13.—Stress distribution flexures: specimen design 228.1.

z

y

x

60000

56125

52250

48375

44500

40625

36750

23875

29000

25125

21250

17375

13500

9625

5750

1875

von Mises

equivalent stress,

psi

F

E

D

C

B

Figure 14.—Finite element model of corner fillet region: specimen design 228.2.

y

z

x
NASA/TM—2003-212090 35

Figure 15.—Stress distribution in corner fillet region: specimen design 228.2.

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

von Mises

equivalent stress,

psi
NASA/TM—2003-212090 36

0.063 rad

typ. 8 places

0.125 rad

typ. 8 places

Figure 16.—Specimen design 228.2: plan view.

0.125 rad

typ. 4 places

all slots

12.000

1.800

6.000

1.387

3.780

0.063

1.000

0.750

0.566

0.063 rad

typ. 4 places

0.250

0.575

Note: all dimensions in inches
NASA/TM—2003-212090 37

Figure 17.—Specimen design 228.2: thickness transition.

0.625

0.425

0.250

0.184

0.080

0.150

0.500 rad

typ. 2 places

0.500 rad

typ. 2 places

Note: all dimensions in inches
NASA/TM—2003-212090 38

Figure 18.—Specimen design 228.2: flexure and slot details.

0.694

typ. 16

places

0.250 

typ. 16 places

0.500

0.438

0.375 0.188

0.125

0.437 

typ. 8 places

Flexure

width

Slot

width

Outer flexure details

Note: all dimensions in inches
NASA/TM—2003-212090 39

Gripped area

typ. 4 places

Figure 19.—Specimen design 232.1: plan view.

12.000

1.500

1.938

4.125

2.000

2.375

0.063

0.719 0.719

0.813 0.813

0.813

0.563

0.375

0.250

0.370

0.750

0.063

typ. 16 places

0.188

typ. 40 places

0.250

typ. 36 places

Note: all dimensions in inches
NASA/TM—2003-212090 40

Figure 20.—Specimen design 232.1: thickness transition.

0.500

0.380

0.250

0.184

0.080

0.150

0.500 rad

typ. 2 places

0.500 rad

typ. 2 places

Note: all dimensions in inches
NASA/TM—2003-212090 41

Figure 21.—Finite element model of specimen design 232.1: plan view.

y

x

z

Figure 22.—Stress distribution on the top and bottom surfaces: specimen design 232.1.

y

x

z

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

von Mises

equivalent stress,

psi
NASA/TM—2003-212090 42

Figure 23.—Details of gage area stress distribution: specimen design 232.1.

y

x

z

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

P

von Mises

equivalent stress,

psi

Location A

M

P

BCDEF
NASA/TM—2003-212090 43

Figure 24.—Stress distribution in flexures: specimen design 232.1 viewed on corner.

y

x

z

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

P

von Mises

equivalent stress,

psi

F
NASA/TM—2003-212090 44

Figure 25.—Stress distribution in flexures: specimen design 232.1 viewed on centerline.

y

x

z

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

P

von Mises

equivalent stress,

psi

F

E

D

C

B

A
NASA/TM—2003-212090 45

Figure 26.—Finite element model of specimen design 269.1: plan view.

y

x

z

Figure 27.—Stress distribution on the top and bottom surfaces: specimen design 269.1.

y

x

z

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

von Mises

equivalent stress,

psi
NASA/TM—2003-212090 46

Figure 28.—Details of gage area stress distribution: specimen design 269.1.

y

x

z

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

P

von Mises

equivalent stress,

psi

Location A

M

BCDEF
NASA/TM—2003-212090 47

Figure 29.—Stress distribution in flexures: specimen design 269.1 viewed on corner.

y

x

z

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

von Mises

equivalent stress,

psi
NASA/TM—2003-212090 48

Figure 30.—Stress distribution in flexures: specimen design 269.1 viewed on centerline.

y

x

z

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

von Mises

equivalent stress,

psi

F

E

D

C

B

A
NASA/TM—2003-212090 49

Figure 31.—Finite element model of specimen design 269.2: plan view.

y

x

z

Figure 32.—Stress distribution on the top and bottom surfaces: specimen design 269.2.

y

x

z

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

von Mises

equivalent stress,

psi
NASA/TM—2003-212090 50

Figure 33.—Details of gage area stress distribution: specimen design 269.2.

y

x

z

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

P

von Mises

equivalent stress,

psi

Location A

M

B CD E F
NASA/TM—2003-212090 51

Figure 34.—Stress distribution in flexures: specimen design 269.2 viewed on corner.

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

von Mises

equivalent stress,

psi
NASA/TM—2003-212090 52

Figure 35.—Stress distribution in flexures: specimen design 269.2 viewed on centerline.

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

von Mises

equivalent stress,

psi

E

D

C

B

A
NASA/TM—2003-212090 53

Figure 36.—Specimen design 269.2: plan view.

12.000

1.938

4.125

2.000

2.300 0.063

0.250

0.750

typ. 8 places

0.750

typ. 4 places

0.813

typ. 8 places

0.563

0.345

0.370

0.750

0.063

typ. 16 places

0.188

typ. 8 places

0.063 rad

typ. 4 places

all slots

0.125 rad

typ. 4 places

0.125

typ. 32 places

0.250

typ. 4

places

0.313

typ. 32 places

Note: all dimensions in inches
NASA/TM—2003-212090 54

Figure 37.—Specimen design 269.2: thickness transition.

0.500

0.380

0.250

0.218

0.050

0.150

0.500 rad

typ. 2 places

0.500 rad

typ. 2 places

Note: all dimensions in inches
NASA/TM—2003-212090 55

Figure 38.—Stress ratios () used to investigate general forms of biaxial loading.

Where von Mises equivalent stress ()

= (x – xy + y ) = 50,000 psi

and stress ratio ()

= Tan–1

 (y/x)

Major axis

Minor axis

–45°

45° 30°

15°

–22.5°

6

5

1

2

3

4

y

x



2 2 1/2

–

–
NASA/TM—2003-212090 56

Figure 39.—Details of gage area stress distribution in specimen design 269.2: stress ratio () = 45.0°.

y

x

z

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

von Mises

equivalent stress,

psi
NASA/TM—2003-212090 57

Figure 40.—Details of gage area stress distribution in specimen design 269.2: stress ratio () = 30.0°.

y

x

z

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

von Mises

equivalent stress,

psi
NASA/TM—2003-212090 58

Figure 41.—Details of gage area stress distribution in specimen design 269.2: stress ratio () = 15.0°.

y

x

z

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

von Mises

equivalent stress,

psi
NASA/TM—2003-212090 59

Figure 42.—Details of gage area stress distribution in specimen design 269.2: stress ratio () = 0.0°.

y

x

z

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

von Mises

equivalent stress,

psi
NASA/TM—2003-212090 60

Figure 43.—Details of gage area stress distribution in specimen design 269.2: stress ratio () = –22.5°.

y

x

z

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

von Mises

equivalent stress,

psi
NASA/TM—2003-212090 61

Figure 44.—Details of gage area stress distribution in specimen design 269.2: stress ratio () = –45.0°.

y

x

z

80000

74800

69600

64400

59200

54000

48800

43600

38400

33200

28000

22800

17600

12700

7200

2000

von Mises

equivalent stress,

psi
NASA/TM—2003-212090 62

Figure 45.—Variation of gage area stress in specimen design 269.2 in the x, y, and 45° directions:

 stress ratios () = 45.0°.

Note: The location of thickness transition occurs at the same distance in the x and y direction.

 The distance is reduced in the 45° direction because of the corner fillet radius.

60

50

40

30

20

0.0 0.5 1.0 1.5

Distance from center, in.

von Mises stress, Ksi

X-direction

Y-direction

45°-direction

±5%

Start thickness 

transition

Figure 46.—Variation of gage area stress in specimen design 269.2 in the x, y, and 45° directions:

 stress ratio () = 30.0°.

60

50

40

30

20

0.0 0.5 1.0 1.5

Distance from center, in.

von Mises stress, Ksi

X-direction

Y-direction

45°-direction

±5%

Start thickness 

transition

0.7
NASA/TM—2003-212090 63

Figure 47.—Variation of gage area stress in specimen design 269.2 in the x, y, and 45° directions:

 stress ratio () = 15.0°.

60

50

40

30

20

0.0 0.5 1.0 1.5

Distance from center, in.

von Mises stress, Ksi

X-direction

Y-direction

45°-direction

±5%

Start thickness 

transition

0.6

Figure 48.—Variation of gage area stress in specimen design 269.2 in the x, y, and 45° directions:

 stress ratio () = 0.0°.

60

50

40

30

20

0.0 0.5 1.0 1.5

Distance from center, in.

von Mises stress, Ksi

X-direction

Y-direction

45°-direction

±5%

Start thickness 

transition

0.52
NASA/TM—2003-212090 64

Figure 49.—Variation of gage area stress in specimen design 269.2 in the x, y, and 45° directions:

 stress ratio () = –22.5°.

60

50

40

30

20

0.0 0.5 1.0 1.5

Distance from center, in.

von Mises stress, Ksi

X-direction

Y-direction

45°-direction

±5%

Start thickness 

transition

0.56

Figure 50.—Variation of gage area stress in specimen design 269.2 in the x, y, and 45° directions:

 stress ratio () = –45.0°.

60

50

40

30

20

0.0 0.5 1.0 1.5

Distance from center, in.

von Mises stress, Ksi

X-direction

Y-direction

45°-direction

±5%

Start thickness 

transition

0.57
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NASA TM—2003-212090

E–13362

WBS–22–708–31–19

70

Specimen Designs for Testing Advanced Aeropropulsion Materials Under

In-Plane Biaxial Loading

John R. Ellis and Ali Abul-Aziz

In-Plane biaxial testing; Advanced aeropropulsion materials; Cruciform specimen designs

Unclassified -Unlimited

Subject Category: 26 Distribution: Nonstandard

John R. Ellis, NASA Glenn Research Center, and Ali Abul-Aziz, Cleveland State University, Cleveland, Ohio 44115.

Responsible person, John R. Ellis, organization code 5900, 216–433–3340.

A design study was undertaken to develop specimen designs for testing advanced aeropropulsion materials under in-plane

biaxial loading. The focus of initial work was on developing a specimen design suitable for deformation and strength tests

to be conducted under monotonic loading. The type of loading initially assumed in this study was the special case of

equibiaxial, tensile loading. A specimen design was successfully developed after a lengthy design and optimization

process with overall dimensions of 12 by12 by 0.625 in., and a gage area of 3.875 by 3.875 by 0.080 in. Subsequently, the

scope of the work was extended to include the development of a second design tailored for tests involving cyclic loading.

A specimen design suitably tailored to meet these requirements was successfully developed with overall dimensions of 12

by 12 by 0.500 in. and a gage area of 2.375 by 2.375 by 0.050 in. Finally, an investigation was made to determine whether

the specimen designs developed in this study for equibiaxial, tensile loading could be used without modification to

investigate general forms of biaxial loading. For best results, it was concluded that specimen designs need to be optimized

and tailored to meet the specific loading requirements of individual research programs.