NASA Technical Memorandum 101565 

Characterization of the Bending Stiffness of 

Large Space Structure Joints 

K. Chauncey Wu 

May 1989 

& W 

1 

(NASA-TH-101565) CHARACTERIZATION OF THE N89-24676 

BENDING STIFFNESS OP LARGE SPACE STRUCTUBE 

JOINTS [NASA. Langley Research Center) 

CSCL 20R Unclas 18 P 

G3/39 0216747 

NASA 

National Aeronautics and 

Space Administration 

Langley Research Center 

Hampton, Virginia 23665 
CHARACTERIZATION OF THE 

BENDING STIFFNESS OF LARGE SPACE STRUCTURE JOINTS 

K. Chauncey Wu 

NASA Langley Research Center 

Hampton, VA 23665-5225 

INTRODUCTION 

Extensive studies have been conducted to determine the axial stiffness 

characteristics of large space structure truss joints (references 1 and 2) because 

axial loading of the truss members dominates the structural response. However, 

the bending behavior of such truss joints is not understood as thoroughly. It is 

important to understand and quantify the bending stiffness of joints used in this 

class of structure, since the joint bending stiffness will have a strong influence on 

both the buckling load and natural frequency of individual truss members, as 

shown in reference 3. A consewative joint bending stiffness of zero may be 

assumed to estimate these values but the results may be misleading, since a 

relatively small joint bending stiffness can have a large effect on the behavior of 

the truss member. Past experience has shown that dynamic characterization of 

space trusses is extremely difficult if the member frequencies and truss 

frequencies are not widely separated (references 4 and 5). Based on this 

experience, it is desirable to have individual member frequencies which are 

higher than the fundamental frequencies of the complete truss structure. By 

quantifying the joint bending behavior, the truss member properties may be 

accurately predicted. 

. 

It is desirable for the truss joints in large space structures to have a linear axial 

load-deflection response. In order to accomplish this goal, the joint bending 

stiffnesses (El) tend to be non-linear due to complicated internal mechanisms and 

load paths in the joint. It is difficult to use analytical methods exclusively to 

predict the joint bending stiffness because of this non-linear behavior. This paper 

presents a straightfotward method of estimating the bending stiffness of truss joint 

concepts by using a combination of classical beam theory and experimental 

methods. This technique is demonstrated for a mechanically preloaded joint 

used in an experiment to evaluate the potential for automated assembly of large 
To estimate the joint bending stiff ness from experimentally measured deflections, 

an analysis of a beam with a variable cross-section loaded in three-point bending 

was performed (see figure 3). The derivation of a closed-form expression for the 

.lateral deflection of the beam is shown in the appendix. The deflection at the 

beam mid-span (x = L/2) is: 

space structures. Since this joint is to be used in a structurally redundant truss, 

the operational bending loads are expected to be small when compared to the 

axial loads. Consequently, characterizing the joint bending stiff ness for applied 

bending moments of between 30 and 120 in-lbs is expected to cover the range of 

moments encountered in service. 

JOINT D ESC RI PTl ON 

The hardware used in this study is a mechanically preloaded, side-latching joint 

concept designed for automated structural assembly. The joint, shown in figures 

1 and 2, is designed to provide a linear axial load-deflection response and 

eliminate any free play after assembly. The joint is assembled by inserting the 

connector into the receptacle (see figure 1) and given a compressive preload by 

applying a fixed torque to the locking nut, driving a wedge which pulls the 

connector and receptacle together and compresses a series of Belleville washers 

inside the joint (figure 2). The upper limit of the linear stiffness range under 

tensile loading is equal to the maximum compressive load in the Belleville 

washers. By loading the joint in tension, the compressive preload across the 

jointkeceptacle interface is linearly reduced. After the preload is exceeded, the 

tensile load is taken up by the connector and transferred through the joint into the 

strut. When a compressive load is applied to the joint, the stresses at the 

interface contact surfaces are increased. Since these contact surfaces do not 

extend completely around the connector, the bending stiffness of this joint is likely 

to depend on the orientation of the joint symmetry plane with respect to the 

applied loading. A joint was bonded to each end of a 68.7 inch-long 

graphite/epoxy strut to form a complete truss member. 

ANALYSIS 

2 
3 

In equation 1, the lateral deflection of the beam is y, the axial position along the 

beam is x and the applied mid-span load is P. L is the distance between the 

supports (81.4 in) and a is the distance from the support to the jointlstrut interface 

(35.7 in). The bending stiffness of the graphitelepoxy strut is Els and EIj is the 

effective bending stiffness of the two joints and the node. All of the terms in 

equation 1 are known or experimentally determined constants with the exception 

of the joint bending stiffness. Thus, equation 1 may be rearranged to yield: 

It should be noted that this analysis is based on linear beam bending theory. 

Consequently, the term Y/p in equation 2 can be determined from the slope of the 

experimental mid-span load-deflection curve. In addition to the boundary 

conditions shown in figure 3, other assumptions inherent in this analysis are that 

the beam is symmetric with respect to its mid-span and that the applied loading 

passes through the shear center of the joint cross-section, preventing axial twist 

in the beam. 

JOINT BENDING TEST 

A three-point bending test, shown in figure 4, was conducted to determine the 

experimental load-deflection response at the beam mid-span. The' beam was 

comprised of two complete truss members connected to a node cluster and 

supported at each end with a fixture (see inset, figure 4) which permitted free 

rotation of the beam end. The mechanical joints were attached to receptacles 

which were bolted to the node at a given orientation. The preload in both joints 

was set to the same nominal value by applying a 20 in-lb torque to the locking 

nut. Incremental static loads of 1.56, 3.56 and 5.56 Ibs (corresponding to midspan moments of 31.7, 72.4 and 11 3.1 in-lbs, or Pb4) were applied at the node 

cluster. The resulting deflections of the node were measured with respect to its 

initial position with a mechanical dial gage. The bending stiffness of the 
4 

graphite/epoxy struts (Els, experimentally determined from similar three-point 

bending tests) was found to be 4.862~105 Ib-in? 

The independent parameter in this experiment was chosen to be the orientation 

of the joint symmetry plane with respect to the applied loading axis, shown in 

figure 5. To take advantage of the plane of symmetry in the joint, only orientations 

between 0 and 180 degrees (inclusive) were studied. Seven joint orientations of 

0, 35, 55, 90, 125, 145 and 180 degrees were selected. The joint orientations at 

35, 55, 125 and 145 degrees were convenient because a pin on the receptacle 

could be aligned with the arms of a cruciform slot cut into the node faces. The test 

procedure described above was repeated at each of the seven different joint 

orientation angles. 

The support fixtures used in this experiment differ from ideal pin-roller boundary 

conditions in several ways. The center of rotation of the fixture was not coincident 

with the beam's neutral axis. Consequently, rotation of the strut causes small 

axial and lateral deflections at the supports. Also, axial motion of the struts at the 

supports was unrestrained except by small axial forces generated between the 

strut and the support. The reaction force at the supports was distributed over a 3 

inch section of the strut, rather than at a knife-edge, to prevent local damage to 

the graphite/epoxy material and deflections due to distortion of the strut crosssection. Use of these support fixtures should not adversely affect the results, 

since the forces at the ends of the beam contribute only a small amount to the 

total bending strain energy of the system. 

In addition to the gravity load on the beam, there is a bending moment at each 

support caused by the weight of the portion of the truss member on the opposite 

side of the support from the node. The analytical procedure was used to 

determine the deformed shape of the beam with respect to an arbitrary reference, 

in this case, the beam displacement field due to the constant loads described 

above. Thus, the measured deflections resulted only from the applied loading. 

Also, since the bending stiffnesses of the two joints were assumed to be equal in 

the analysis, application of the experimental procedure would only determine the 

average stiffness of the two joints tested. 
5 

RESULTS AND DISCUSSION 

Load-deflection data was taken at the beam mid-span for three applied moments 

at each of the seven joint Orientation angles studied. A representative plot of the 

beam mid-span deflection versus the applied load is shown in figure 6. The ratio 

of the mid-span deflection and the corresponding applied load (the secant slope) 

from the loading cuwe in figure 6 was used with equation 2 to compute the 

bending stiffness of the joint. The term Y/p in equation 2 is equal to the inverse of 

the secant slope of a load-deflection cuwe for a linear beam. This secant 

formulation is exact for small bending moments, where the secant and tangent 

slopes are equal (in the range where the joint stiffness is linear). However, for 

large moments, the tangent slope is a better value for computing the joint bending 

stiffness than the secant slope. In the non-linear range, the secant slope tends to 

be higher than the tangent, resulting in a higher (and less conservative) 

computed joint bending stiffness. Computation of the secant joint bending 

stiffness in this test is justified based on the small number of experimental data 

points taken and the the uncertainty involved in determining the tangent slope of 

the experimental load-deflection data. If the joint bending stiffness is linear, then 

the secant and tangent slopes are equal and the assumptions made in deriving 

equations 1 and 2 are valid. The test data obtained was repeatable when the 

beam was tested immediately after the joints were locked. As the load cycle was 

repeated, the bending stiffness decreased. This stiffness reduction occured 

because the joint preload was being relieved by the cyclic loading. 

The computed bending stiffness of the joint (normalized by the strut bending 

stiffness) was plotted against the corresponding mid-span deflection for the joint 

orientations and applied loads studied in figure 7. The joint bending stiffness was 

found to vary significantly at a given joint orientation and to decrease as the 

applied moment was increased. The maximum bending stiff ness was 0.844~105 

Ib-in2 (17.4 percent of the strut stiffness) for an applied moment of 31.7 in-lbs and 

a joint orientation angle of 90 degrees. The minimum bending stiffness of 

. 0.301~105 Ib-in2 (6.2 percent of the strut stiffness) occurred for an applied 

moment of 113.1 in-lb and a joint orientation angle of 180 degrees. The joint 

bending stiffness was consistently higher at three of the joint orientation angles 

(35, 55 and 90 degrees). 
6 

During application of the load to the beam, a gap was observed to form at the 

contact surface between the joint and receptacle. This gap first appeared when 

the tensile bending stress at the edge of the jointlreceptacle interface exceeded 

the compressive stress caused by the joint preload. As the applied bending 

moment was increased, the compressive contact area of the interface was 

reduced, reducing the apparent stiffness of the joint. Because the contact area 

was shrinking, the moment arm from the connector shaft to the centroid of the 

compressive contact area was increasing. The higher joint bending stiff nesses 

observed at 35, 55 and 90 degrees occurred at these orientations because the 

centroidal moment arms were longer than the corresponding moment arms at the 

other orientations. The maximum preload generated in the Belleville washers 

also affected the apparent bending stiffness of the joint. Although the joints used 

in this study were preloaded to the same nominal value using a calibrated torque 

wrench, any differences between the individual joints were averaged when 

determining the joint bending stiffness. 

The analytical procedure developed in the appendix treated the section of the 

beam containing the two joints and the node as having an effective bending 

stiffness distributed over its length. Use of this analysis for joints which are 

thought to have a high bending stiffness should yield accurate results. Since all 

of the deformation observed during the joint bending tests occurred at the 

jointlreceptacle interface, modeling the joint as a torsional spring located at the 

interface and representing the other components as rigid links may give a more 

accurate estimate of the joint bending stiffness than the model used. Such a rigid 

link analysis would probably be more applicable to joint designs which have a 

low bending stiffness. 

CONCLUSIONS 

In this study, a technique to estimate the bending stiffness of large space structure 

truss joints was developed. To demonstrate this technique, the bending stiffness 

of one erectable joint concept was studied for three applied bending moments at 

seven joint orientation angles. The joint bending stiffness was found to be nonlinear and to vary significantly at a given joint orientation angle. The joint 

bending stiffness ranged between 6 percent of the strut bending stiffness for an 
7 

applied moment of 113 in-lb and 17 percent of the strut bending stiffness for a 32 

in-lb moment (at orientations of 90 and 180 degrees, respectively). Because the 

joints are used in a structurally redundant truss, the bending loads in the truss 

members are much smaller than their axial loads. Consequently, it is expected 

that the bending loads that the joints will encounter in service are within the range 

of applied loading studied here. The bending stiffness of a space truss joint will 

be highly dependent on the design of the particular joint concept. With this in 

mind, the techniques developed in this study may serve as a guide for 

determining the bending stiffness of other joint designs. 
a 

REFERENCES 

1. Schneider, M. E. and Keinholz, D. A.: Characterization of Joints Using the 

Force-State Mapping Technique, CSA Engineering Technical Report No. 85-1 2- 

02, December 1985 

2. Keinholz, D. A. and Allen, B. R.: Force-State Testing of a Quick-Connect Joint, 

CSA Engineering Technical Report No. 87-02-01, February 1987 

3. Razzaq, Z.; et al.: Stability, Vibration and Passive Damping of Partially 

Restrained Imperfect Columns, NASA TM 85697, October 1983 

4. Pappa, R. S. and Juang, J. N.: Some Experiences with the Eigensystem 

Realization Algorithm, Sound and Vibration, January 1 988 

5. Warnaar, D. B. and McGowan, P. E.: Effects of Local Vibrations on the 

Dynamics of Space Truss Structures, Published in the proceedings of 

AIAA/ASME/ASCE/AHS 28th Structures, Structural Dynamics and Materials 

Conference and AlAA Dynamics Specialist Conference, Monterey, CA, April 6-8, 

1987, Paper No. AlAA 87-0941 
9 

APPENDIX 

The derivation of an expression for the lateral deflection of a discontinuous crosssection beam in three-point bending is shown below. Additional information is 

given in figure 3. 

2 

dx 4EI(x) dy = -px +ci 

solve for 0 s x s a 

solve for a I x I L/2 

find C2 by solving at x = L/2 
10 

equate slopes at x = a and solve for C1 

substitute (A3) and (A4) into (Al) and (A2) 

-Pa -Pa + PL2 2 

=-+c, =- 

p12 

x-a 4EIs 4EIj 16EIj 

16e1j 

integrating again, 

- -Px + PL2x +c4, as~s% (AB) 12EIj I~EIJ 

solve for C3 at x = 0 

ylx=o = O+O+O+C3 = 0; C3 = 0 (A91 

equate deflections at x = a to find C4 
11 

C4= ($-$)d, El El 6 (A10) 

substitute (A9) and (A10) into (A7) and (A8) 

solving (A12) at x = L/2, 
ORIGINAL PAGE 

BLACK AND WHITE PHOTOGRAPH 

12 

Figure 1 : Mechanically Preloaded Joint 

Figure 2: Joint Internal Mechanism 
13 

X 

(a) Symmetric Problem 

Boundary Conditions 

y=Oatx=O Continuity of 

*=Oatx=L/2 Deflect ion at 

dx x=a 

Slope and 

(b) Reduced Problem and Boundary Conditions 

- 72 P 

/2 

(c) Free-Body Diagrams 

- 72 P 

/2 

Figure 3: Analysis Schematic for a Discontinuous Cross-Section 

Beam in Three-Point Bending 
ORIGINAL PAGE 

BLACK AND WHITE PHOTOGRAPH 

Figure 4: Three-Point Bending Test 

14 

Figure 5: Joint Orientation Angle 
6- 

5- 

Y 

U 

.- 

n Unloading curve 

0 Experimental 

data point 

0 .10 .20 .30 .40 SO .60 .70 .80 

Mid-span deflection (in.) 

0 .10 .20 .30 .40 SO .60 .70 .80 

Mid-span deflection (in.) 

Figure 6: Mid-Span Load-Deflection Response 

Joint Orientation Angle = 180 degrees 

Joint orientation Joint angle (degrees) 

angle 

.15 

.10 

.05 c Mid-span applied ii 3 

moment (in.-lbs) 

o Experimental 

data point. 

.oo .25 .50 .75 

Mid-span deflection (in.) 

Figure 7: Normalized Joint Bending Stiffness 
Report Documentation Page 

1. Report No. 2. Government Accession NO. I 3. Recipient's Catalog No. 

NASA TM-101565 

4. Title and Subtitle 

Characterization of the Bending Stiffness of Large 

Space Structure Joints 

7. Authork) 

K. 

NASA 

9. Performing Organization 

Chauncey 

Langley 

Wu 

Research 

Name and Address 

1 

Center 

Hampton, VA 23665-5225 

5. Report Date 

May 1989 

6. Performing Organization Code 

8. Performing Organization Report No. ' 

10. Work Unit No. 

506-43-41-02 

11. Contract or Grant No. 

13. Type of Report and Period Covered 

17. Key Words (Suggested by Authork)) 

12. Sponsoring Agency Name and Address 

Technical Memorandum 

18. Distribution Statement 

National Aeronautics and Space Administration 

Washington, DC 20546-0001 

19. Security Classif. (of this report) 

Unclassified Unclassified 16 

20. Security Classif. (of this paG) 21: No. of pages 

14. Sponsoring Agency Code 

22. Price 

A0 3 

I 

15. Supplementary Notes 

16. Abstract 

A technique for estimating the bending stiffness of large space structure joints 

is developed and demonstrated for an erectable joint concept, 

deflection data from a three-point bending test is used as input to solve a 

closed-form expression for the joint bending stiffness which was derived from 

linear beam theory. Potential error sources in both the experimental and analytical procedures are identified and discussed. The bending stiffness of a mechanically preloaded erectable joint is studied at three applied moments and seven 

joint orientations. Using this technique, the joint bending stiffness was 

bounded between 6 and 17 percent of the bending stiffness of the graphite/epoxy 

st rut member. 

Experimental loadLarge space structures 

Joint bending stiffness 

Truss member end fixity 

Unclassified - Unlimited