Truss Cable FEM

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American Institute of Aeronautics and Astronautics

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Finite Element Analysis of Cable-truss Structures

Lin Liao1

AEROS, Montebello, CA, 90640

Baisong Du2

School of Civil Engineering & Architecture, Chongqing Jiaotong University, Chongqing, China, 400074

[Abstract] Finite element techniques are applied to analyze truss structures containing

pretensioned cables. This paper focuses on the study of the influence of cable property and

pretension on trusses and the variation of cable tension in undeformed and deformed

configurations. This work shows that cable pretension, elastic modulus, and cross-sectional

area have significant effect on truss performances. Stability and optimization associated with

structural design of trusses are briefly discussed. The paper presents some case studies of

cable-truss structures, in which commercial FE packages ANSYS and NASTRAN have been

utilized.

Nomenclature

E = elastic modulus

= Poissonsratio

A = cross-sectional area of

X = X coordinate

Y = Y coordinate

Z = Z coordinate

Ux = X component of displacement

Uy = Y component of displacement

Uz = Z component of displacement

Fx = applied force in X directionFy = applied force in Y direction

Fz = applied force in Z direction

Ci = the ithcable tension force

Ni = the ithnode

Elemi = the ithtruss element

I. Introduction

Truss structures have attracted tremendous interests due to their extensive applications in the construction of

infrastructures and space structures. Considerable efforts have been placed on the development and analysis of truss

bridges composed of concrete and steel. Research works have been focused on material characteristics, truss joint

design, and processing and construction of structural components in the field of civil engineering. Researchers have

carried out investigations of composite trusses for aerospace applications [1-3]. These space truss structures are

distinctively different from civil structures regarding materials, strength, stiffness, and weight. In recent years, the

overall performances and characteristics of composite truss structures have been studied [4, 5].

Cables serve as essential and important members of truss structures. Cables can only withstand tension forces and

are utilized to maintain stability and strength of truss systems. In this paper, cable is used to denote the structural

members that only support tension loads, which can also be called rope, wire rope, string, chain and etc. Cable

tension has been considered as a critical factor in optimal design of trusses and prediction of truss performances. The

1PhD, Aeronautical Engineer, AEROS, Montebello, CA, AIAA Member.2PhD, Associate Professor, School of Civil Engineering & Architecture, Chongqing Jiaotong University, China.

51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference
18th2 - 15 April 2010, Orlando, Florida

AIAA 2010-260

Copyright 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


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arrangement of cable configuration and determination of cable pretension are closely associated with load

conditions, application environment, and structural characteristics. Since cables are tension-only structural members,

nonlinear analysis was applied to study the erection of space trusses by cable tensioning technique [6]. A two-step

optimum method was developed by Zhang and Nai to optimize cable pretension, cross-sectional area, diameter, and

thickness [7]. Minimum cost was chosen as optimal objective, and stability and displacements were applied as

constraints. Multi-objective optimization technique was proposed for a cable-truss deployable space antenna [8]. In

the approach truss thicknesses and cable tensions were design variables, and minimum weight and surface accuracy

were the objective functions.

Besides the aforementioned investigations on cable-truss optimization, the researches primarily concentrated on

cable tension or prestress have been reported in the literature too. The effects of prestressed Kevlar/FRP cables in

composite structural systems were studied [9]. Experimental study and numerical analysis have been carried out on

composite space trusses with prestressed cables made of steel and compression members made of concrete [10, 11].

Zhang and his co-workers conducted research on the impact factors of cable tension of crescent-shaped multi-rib

concrete filled steel tube truss arch bridges [12]. The impact effects were caused by road surface roughness, vehicle

speed, and structural damping ratio. Li developed a method to determine cable prestress of truss-string structures

[13]. Cable prestress was realized by increasing temperature loads in the model of construction state and cable stress

and controlling point displacement were obtained correspondingly for this state . Liu introduced an approach for the

analysis of internal forces of cable-stayed space grids with multi-step-loading [14]. The stayed cables were

simulated as equivalent linearization elements. A method for pretension design of cable-net structures of axis-

symmetric parabolic antenna was proposed based on force balance equations of cable nodes and structural

characteristics [15].Based on the authors literature survey, the interaction between cables and truss members and the effects of cable

property and configurations on truss behavior have not been studied. This paper is aimed to conduct comprehensive

study of truss structures containing cables and truss members. First, finite element analysis (FEA) of cable-truss

structural systems is introduced. Hence, the effect of cable property and pretension on deformation of trusses having

diverse configurations will be covered. A variety of truss structures contained pretension cables will be modeled and

analyzed using commercial finite element (FE) packages ANSYS/NASTRAN. The authors discuss design

considerations of cable-truss structures. In particular, the stability and optimization problems will be addressed. This

paper is concluded with a summary in the last section.

II.

Finite Element Analysis of Cable-truss Structures

Cable-truss structures having simple configurations and fewer members can be solved analytically. As for

complex truss systems we have to resort to FEA for efficient and accurate solutions. The construction of FE model

for cable-truss structures is similar to conventional approach of FEA. Usually, 1D element is used to model truss

rods and cables since they can be considered as slender structures. Based on the characteristics of structural

members (whether the members can support bending loads), they can be simulated as beams (compression-tension-

bending-twisting element), rods (compression-tension element), and cables (tension-only element). Equivalent

material properties could be applied for composite truss beams in the analysis of complex truss systems. Cables

serve as crucial members for generating determined truss structural systems. Cable pretension can be prescribed by

initial strains or prestress and is introduced in different ways in various FE packages. In this study the authors only

discuss the modeling of pretensioned cables in ANSYS and NASTRAN. It is noted that other FE software such as

ABAQUS, COMSOL can also deal with tension cables with specific build-in elements. Different FE codes have

their unique way of handling cable characteristics.

Thermal loads in terms of temperature differential can be applied to cable elements to achieve pretension effect

while truss elements are not subjected to thermal loads. According to the equation of thermal effect, F=EAT,

thermal loads can be converted to cable pretension. Generally, thermal expansion coefficients of cables could be

selected arbitrarily and dont affect the result of simulation. The exclusion of thermal loads of truss elements can berealized by setting either temperature differential or thermal expansion coefficients zero. CROD element in

NASTRAN can be used to simulate cables. NASTRAN linear solver SOL101 can be employed for this kind of

analyses. However, compressive forces could be generated for cable elements in deformed states using linear solver.

In such cases, a second analysis is needed by deleting cable elements having compressive forces. Alternatively,

NASTRAN nonlinear solver SOL106 is applicable to this problem. Cables are assigned nonlinear material property:

stiffnesses vary in tension and compression range. Resultant cable forces will be equal to or larger than zero using

this solver.


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The second method is to employ tension-only elements if they are available in existing FE software package.

According to the authors knowledge, ANSYS offers the straightforward solver for cables analysis. ANSYS

LINK10 elements are capable of dealing with only tension or only compression by assigning certain parameters for

cable elements. Since this kind of FE models involves tension-only elements, nonlinear iterative solver of ANSYS is

applied and sub step settings need to be specified for convergent study. This approach does not lead to compressive

forces in cable elements. However, the solution accuracy and convergence need to be prudently checked.

In this paper, both ANSYS and NASTRAN have been implemented into the analysis of cable-truss structures.

There are diverse types of cable-truss structures and the current work is focused on specific structures with the

assumptions listed below

1. Truss members are assumed to be perfectly connected at joints.

2. Cables are connected to truss rods at joints. There is no friction sliding of cables with respect to truss joints.

3. Small strain and displacement are considered. Geometrical nonlinearity is not accounted in this study.

4. Material nonlinearity of trusses is not considered.

5. Loads are applied at joints except gravity loads. Truss members only support tension/compression loads. All

trusses are axially loaded at the ends (truss joints) and have uniform property along the length of each member. It

should be noted that truss members can be designed to beam-type structures, which hold tension, compression, and

bending loads. This type of structure is beyond the scope of this work.

III.

Effect of Cable Property on Truss Performance

In this section, the effect of cable property on truss performances will be studied using ANSYS. Parameters usedto specify cables in FE models include elastic modulus, cross-sectional areas (diameters), and density.

An example of 3D Truss A (see Figure 1) is given

below to illustrate the influence of cable material

property. Blue lines represent truss elements and green

lines represent cable elements. Totally, there are nine

truss members and six cables. Nodal coordinates of all

truss joints in this example and subsequent examples are

given in Table 1. Original cable and truss properties areprovided in Table 2. The four nodes 14 of Truss A are

constrained. The forces applied at Node 5 and 6 are Fz=-

50 lbs. The deformation of Node 5 and Node 6 is the

same for this truss under the prescribed loading. Table 3

presents the variation of total displacement (magnitude of

total displacement, absolute value) of Node 5 or Node 6

with respect to elastic modulus when cable pretension is

kept constant (300 lbs). It can be seen that truss maximum

displacement decreases along with the increase of cable

elastic modulus. Under the condition of constant pretension, stiff cables generate less deformation than flexible

cables. Although cables with high stiffness can reduce truss deformation, there are drawbacks of using stiff cables.

Stiff cables tend to lose tension forces in contrast with cables with low elastic modulus, which will be studied in

details in Section V.

Next, we study the effect of cable geometry on truss deformation. Table 4 shows the variation of Node 5 and

Node 6 displacement due to cross-sectional areas when cable pretension (100 lbs) and elastic modulus (1E6 psi) are

constants. The increase of cable cross-sectional areas results in the decrease of displacements. Industrial cables most

often consist of a few strands of small fibers or metal ropes (such as 6, 7, or 8 strands). The bulk area provided in

cable specification is usually larger than the actual cross-sectional area. Real cross-sectional area used in FE models

needs to be calculated by taking out the gaps between strands of ropes. Additionally, nonstructural massesassociated with cable adjustment and installations need to be accounted in the construction of FE models for

structural systems. Some FE programs provide the option of nonstructural masses.

Table 1. Nodal coordinates of Truss A-H.

N1 N2 N3 N4 N5 N6 N7 N8 N9 N10

AX (in) -10 10 10 -10 0 0

Y (in) -10 -10 10 10 -10 10

Figure 1. Schematic of Truss A.


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Z (in) 0 0 0 0 10 10

B

X (in) 250 125 -125 -250 -125 125 0 375 375 0

Y (in) 0 216.5 216.5 0 -216.5 -216.5 -433 -216.5 216.5 433

Z (in) 90 90 90 90 90 90 0 0 0 0

C

X (in) -100 100 -100 100 -100 100 -100 100 -250 250

Y (in) 0 0 100 100 0 0 100 100 0 0

Z (in) 200 200 200 200 0 0 0 0 100 100

D

X (in) -19.02 -11.76 11.76 19.02 0 -9.51 -5.88 5.88 9.51 0

Y (in) 6.18 -16.18 -16.18 6.18 20 3.59 -7.59 -7.59 3.59 10.5

Z (in) 0 0 0 0 0 8 8 8 8 8

E

X (in) 0 30 60 90 120 120 90 60 30 0

Y (in) 0 0 0 0 0 30 30 30 30 30

Z (in) 0 0 0 0 0 0 0 0 0 0

F

X (in) 0 34.64 69.28 103.92 13.92 43.92 73.92 103.92 34.64 43.92

Y (in) 0 0 0 0 51.96 34.64 17.32 0 0 34.64

Z (in) 0 0 0 0 0 0 0 60 20 20

G

X (in) 0 34.64 69.28 103.92 143.92 34.64 69.28 103.92 0 34.64

Y (in) 0 0 0 0 0 0 0 0 30 30

Z (in) 0 0 0 0 0 20 40 60 0 0

HX (in) 0 40 40 0 -60 -40 0 0 40 40Y (in) 0 0 30 30 0 30 0 30 0 30

Z (in) 0 0 0 0 40 40 40 40 40 40

N11 N12 N13 N14 N15 N16

B

X (in) -375 -375 0

Y (in) 216.5 -216.5 0

Z (in) 0 0 140

E

X (in) 30 60 90 90 60 30

Y (in) 0 0 0 30 30 30

Z (in) 30 30 30 30 30 30

F

X (in) 69.28 73.92

Y (in) 0 17.32

Z (in) 40 40

G

X (in) 69.28 103.92 143.92 34.64 69.28 103.92

Y (in) 30 30 30 30 30 30

Z (in) 0 0 0 20 40 60

Table 2. Material property.

E (psi) A (sq in) initial strain

truss 1.5E7 1.0E-1 0

cable 1E6 1.0E-2 1.0E-2

Table 3. Variation of nodal displacements with respect to cable material properties.

E (psi) 1E6 3E6 4E6 8E6 12E6

Initial strain 3.000E-2 1.000E-2 0.750E-2 0.375E-2 0.250E-2

Displacement (in) 0.4269E-02 0.4229E-02 0.4209E-02 0.4131E-02 0.4056E-02

Table 4. Variation of nodal displacements with respect to cable cross-sectional areas.

A (sq in) 1E-2 2E-2 4E-2 5E-2 8E-2

Initial strain 1.000E-2 0.500E-2 0.250E-2 0.200E-2 0.125E-2

Displacement (in) 0.4229E-02 0.4169E-02 0.4056E-02 0.4001E-02 0.3847E-02


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IV.

Cable Pretension and Truss Deformation

Cables are usually loaded with pretension when they are assembled with truss members. The mutual

reinforcement between truss members and tensioned cables enhances overall structural strength. Cable pretension is

closely linked to truss deflection and element forces of truss members. The study of pretension and structural

strength is significant and informative for truss structural design.

A. Effect of cable pretension on deformationTwo numerical examples are given below to study the influence of cable pretension on deflection. The first

example is a 3D dome-shape truss with pretensioned cables (see Figure 2). The properties of truss elements are

E=2.5E7 psi; =0.33; A=4.0 in2; and the properties of cables are

E=3E6 psi; =0.25; A=0.05 in2. An external force of 500 lbs is

applied at Node 7 along positive Z direction. The six bottom

nodes (Node 8Node 13) are constrained.

Table 5. Nodal displacements of Truss B.

Table 5 shows the variation of the Node 7 displacement in Z-direction with respect to cable pretension. When

Truss B is subjected an external load in positive Z direction, a small value of cable pretension 150 lbs causes a

positive displacement Uz at Node 7. The absolute value of displacements decreases when pretension increases from

150 lbs to 300 lbs. The increase of cable pretension up to 450 lbs leads to a negative displacement Uz at Node 7.

When the cable pretension is larger than 450 lbs, the displacement continues to increase. Apparently, Truss B has a

minimum deformation (the absolute value of displacements) when cables are given a pretension force around 300

lbs.

The second example is Truss C shown in Figure 3.

The properties of truss element are E=2.5E7 psi; =0.25;

A=5.0 in2; and the properties of cables are E=3E6 psi;=0.25; A=0.02 in2. The bottom six nodes (Node 12, 5

6, 910) are restrained. Fy=200 lbs forces are applied to

Nodes 3, 4, 7, 8. The displacements of Node 3 and Node

8 in three directions are listed in Table 6. It can be seen

that the absolute values of Ux & Uz displacements

increase due to the increase of pretension. It is interesting

to know that the Y displacement changes from positive to

negative. It shows that the top portion of truss tends to

deflect upward when pretension is much smaller than

applied loads; Node 3 and Node 8 deflects downward when pretension is close to or larger than applied loads.

Appropriate selection of cable pretension can minimize the deformation due to external loads.

Table 6. Displacements of Truss C.

Pretension (lbs)Displacement of Node 3 (in) Displacement of Node 8 (in)

Ux Uy Uz Ux Uy Uz

60 -0.5895E-04 0.9042E-04 -0.8556E-04 0.5895E-04 0.9042E-04 0.8556E-04

120 -0.1363E-03 0.3316E-04 -0.1588E-03 0.1363E-03 0.3316E-04 0.1588E-03

180 -0.2137E-03 -0.2409E-04 -0.2320E-03 0.2137E-03 -0.2409E-04 0.2320E-03

240 -0.2911E-03 -0.8135E-04 -0.3053E-03 0.2911E-03 -0.8135E-04 0.3053E-03

Pretension (lbs) Displacement Uz (in)

150 4.094E-3

300 -9.862E-4

450 -6.066E-3

600 -1.115E-2750 -1.623E-2

Figure 2. Schematic of Truss B.

Figure 3. Schematic of Truss C.


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The above examples show that truss deformation is not linearly proportional to cable pretension. Final

configurations of deformed trusses are influenced by cable pretension. To minimize truss deformation, it is

important to select suitable cable pretension. Determination of cable pretension is relevant to loading conditions and

cable property. Both theoretical prediction and practical experiences are used to specify cable pretension. Usually,

skillful workers are able to adjust cable pretension by observing the structural behavior.

B. Effect of cable pretension on truss element forces

Cable pretension not only has effect on truss deformation but also affect truss element forces. The study of the

variation of truss element forces due to cable pretension is useful for design and analysis of truss systems, since

element force is one of the key considerations of truss geometrical and strength design. Table 7 displays the

variation of element forces of Truss A subjected to different cable pretension. The data shows that truss element

forces are significantly increased when cable pretensions become large. We increase cable pretension in order to

reduce structural deformation, and the strength of individual truss members need to taken account at the same time.

Too much pretension might lead to excessive truss element forces, and thus enhances the demand of strong truss

materials.

Table 7. Variation of element forces with respect to cable pretension.

Pretension (lbs) Elem 5 (lbs) Elem 6 (lbs) Elem 7 (lbs) Elem 8 (lbs) Elem 9 (lbs)

100 -92.76 -92.76 -92.76 -92.76 -162.37

200 -150.22 -150.22 -150.22 -150.22 -324.87

300 -207.67 -207.67 -207.67 -207.67 -487.38

400 -265.12 -265.12 -265.12 -265.12 -649.88

In this section, influence of pretension on structural performance of cable-truss structures is addressed. Cable

pretension can be favorable or unfavorable for the overall structure, depending on load conditions and structural

configurations. Although increase of pretension helps to reduce truss deformation, it also increase truss member

forces. The results indicate that appropriate and optimal positioning and pretensioned loading of cables have

significant influence on the performance of trusses. At the same time of increasing cable pretension, we should

consider the stiffness of truss members.

V. Variation of Tension Forces in Undeformed and Deformed Configurations

Before the structures are subjected to external loads, each cable is loaded with the same pretension in undeformed

configurations. Constant cable pretension can lead to variable tension forces in cables in deformed truss structures.Cable tension after deformation might be larger or smaller than

pretension. Cables might lose all preloaded tension in truss deformed

states. A few examples are presented below to study the change of

cable tension subjected to loads.

Truss D (see Figure 4) consists of 15 truss members and 15 cables

(the number of cables is shown by green circles). The properties of

truss element are E=2.5E7 psi; =0.25; A=0.15 in2; and the properties

of cables are E=3E5psi; =0.25; A=0.04909 in2. A force Fz=-1000 lbs

is applied at each of the five middle nodes (Nodes 6-10). Five bottom

nodes (Nodes 1-5) are constrained. All cables are assigned a pretension

of 147.26 lbs. Table 8 shows that 15 cables are still subjected to tension

loads and cable tensions in deformed configuration vary from

pretension. It can be seen that cable tension forces range from 85.22 lbs

to 196.28 lbs, which is around 58% to 133% of pretension. The appliedloads cause reduction of tension in some cables as well as increase of

tension in some cables.

Table 8. Cable tension in deformed configuration of Truss D.

Cable C1 C2 C3 C4 C5 C6 C7 C8

Tension (lbs) 176.89 107.35 176.89 107.35 196.28 85.22 142.79 142.79

Cable C9 C10 C11 C12 C13 C14 C15

Tension (lbs) 196.28 85.22 141.22 141.22 142.04 141.68 141.68

Figure 4. Schematic of Truss D.


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As displayed in Figure 5, Truss E consists of 30 truss members and 30 cables. Elastic modulus for truss members

and cables are 6E5 psi and 1E5 psi, respectively. Poissons ratio of both materials is 0.33. Cross-sectional areas of

truss rods and cables are 0.8 in2 and 0.1 in2. Truss E is supported at six nodes: Nodes 2-4, 7-9. The prescribed loads

are Fz=-500 lbs at each of the six nodes 11-16, and Fz=500 lbs at Node 1 and Node 10. A pretension of 120 lbs is

applied to each cable. The cable tension in the deformed state is listed in Table 9. It can be seen that substantial

changes of cable tension occur after deformation. Some cables lose tension while some cables gain double value of

pretension. The cables 23, 25, 27, and 29 lose tension and compressive forces are generated for the first run.

NASTRAN input file was modified by deleting these cables and rerun again.

The undeformed and deformed configurations are displayed in Figure 5. Blue lines represent undeformed state

and black lines represent deformed state. The deformation of rectangular frame formed by Nodes 2,3,11,12 causes

Cable 24 gains much larger tension than pretension, and Cable 23 completely lose tension. Similarly, the same

effects occur to the other three pairs of cables: Cables 25-26, Cables 27-28, Cables 29-30. It can be understood that

one of the two cross-cables becomes shorter and the other becomes longer when the rectangular frame was altered to

a parallelogram. To identify which cable in the structure will lose tension under specific load cases is a complicated

problem and need high fidelity prediction of deformed state and element forces. Generally, cables with high

modulus are easier to lose tension than those with low modulus. Stiff cables might completely lose pretension in

deformed configurations. Although Flexible cables are good for maintaining tension, they are are not helpful for

deformation minimization compared with cables with high modulus for the same configuration and load conditions,

as discussed in Section IV.

Table 9. Cable tension in deformed configuration of Truss E.

Cable C1 C2 C3 C4 C5 C6 C7 C8

Tension (lbs) 139.75 139.75 120 120 120 120 111.24 111.24

Cable C9 C10 C11 C12 C13 C14 C15 C16

Tension (lbs) 75.58 75.58 91.777 91.77 55.21 55.21 106.97 106.97

Cable C17 C18 C19 C20 C21 C22 C23 C24

Tension (lbs) 110.54 110.54 67.77 67.77 68.25 68.25 0 362.28

Cable C25 C26 C27 C28 C29 C30

Tension (lbs) 0 344.83 0 362.28 0 344.83

Next, Truss F is given as another example (see Figure 6). The properties of truss rods are E=8E5 psi; =0.3;

A=0.8 in2 and those of cables are E=1E6 psi; =0.3; A=0.1 in2. The displacements of Nodes 1, 4, and 5 are

restrained. Fx=-400 lbs is applied to Nodes 9, 10, 11, and 12. The pretension of each cable is 100 lbs. Undeformedand deformed configurations are shown in Figure 6. The resulting cable tensions in the deformed state are listed in

Table 10. Cable tensions vary from 0 to 359.01lbs. The four cables (5, 8, 10, 15) dont sustainany loads in deformed

truss, which be explained similarly to that of truss E.

Although 4 out of 18 cables are completely released from tension loads for this specific load case, it is not always

like this for other load cases. To investigate the influence of loading conditions on cable behavior, we rerun the

program and change the loads at Nodes 9, 10, 11, and 12 to Fx=-200 lbs, and the same pretension for each cable is

used. The resulting tensions of cables are displayed in Table 11, which are significantly different from the data in

Table 10. When the external loads are decreased, tension forces are produced in all cables. It is reasonable to deduce

Figure 5. Schematic of Truss E (left: schematic, right: undeformed and deformed configurations).


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that cables might regain tension or lose tension if the structure is transferred from one load case to another load case.

Diverse loading conditions lead to considerable variation of cable tension with respect to pretension. As far as

structural design is considered, we need to solve for a variety of load cases and find the maximum possible cable

tension, which will be used in the selection of cables. When we design truss structures sensitive to cable tension, it is

necessary to install devices for cable tension adjustment and structural health monitoring.

Table 10. Cable tension in deformed configuration of Truss F.

Cable C1 C2 C3 C4 C5 C6 C7 C8 C9

Tension (lbs) 26.18 147.11 85.59 76.44 0 214.56 248.60 0 137.53

Cable C10 C11 C12 C13 C14 C15 C16 C17 C18

Tension (lbs) 0 94.04 48.06 121.68 86.75 0 359.01 4.14 255.12

Table 11. Cable tension in deformed configuration of Truss F.

Cable C1 C2 C3 C4 C5 C6 C7 C8 C9

Tension (lbs) 66.61 110.72 85.24 84.23 7.64 132.29 135.24 8.51 97.77

Cable C10 C11 C12 C13 C14 C15 C16 C17 C18

Tension (lbs) 48.72 84.41 65.91 100.20 89.11 29.96 196.89 47.28 167.55

In this section, a few numerical examples are presented to show that variation of cable tension in contrast topretension. Final tensions of cables are mostly determined by structural deformation and applied loads. Whether

cables are stretched or released pertains to external loads and vary case by case. At one hand the presence of cables

contributes to the strength and integrity of truss structures, while at the other hand cables allow for structural

recovery flexibility: cables accommodates diverse loading by changing tension forces. The breaking strength of

cable materials must meet the requirement of maximum possible tension forces with the consideration of a safety

factor.

VI.

Discussion of Design Considerations

The objective of truss design is to achieve stable, strong, and light weight structures. For cable-truss structures,

there are some crucial design considerations associated with unique characteristics of cables. Some examples are

presented to address these factors.

Truss G is shown in Figure 7. The properties of truss rods are E=8E5 psi; =0.3; A=0.8in2and those of cables are

E=1E6 psi; =0.3; A=0.1 in2. The displacements of Nodes 1-5, and 9-13 are restrained. Fz=-100 lbs is applied toNodes 7, 8, 15, and 16. The pretension of each cable is 100 lbs. Three design architectures are presented in Figure 8.

Design A is not stable due to the lack of connection between truss members along Y direction. If six cables are

included, we can obtain a stable structure: Design B. Four truss members are replaced by two cables in Design C.

The design architectures B & C are stables because the placement of cross cables between two XZ planes in the

truss. Generally speaking, a parallelogram is not stable, and a triangular shape whether formed by truss rods only or

both truss rods and cables generates a stable structural system, and the loads at joints can be transferred loads among

more than three members. The displacements for two design architectures B&C are compared in Table 12.

Figure 6. Schematic of Truss F (left: schematic, right: undeformed and deformed configuration).


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The weight efficiency is always a big concern

of designing space truss structures that are

mainly used in the field of aerospace engineering.

It is known that linear weight density (weight per

unit length) of truss members is usually much

larger than that of cables. To reduce the weight

of overall system, truss rods can be replaced by

cables under the conditions that same

performances can be achieved. Design C (shown

in Figure 8) reduces 4 numbers of truss rods by

using 8 cables. The weights of B and C are 107.9

lbs and 94.3 lbs, respectively. The structural

behavior of these two configurations subjected to

same loading conditions is compared in Table 12&13. As shown in Table 12, the displacements of Nodes 6, 7, 8 for

Design C are larger than those of Design B. Truss element forces are provided in Table 13. It can be seen that

weight reduction is achieved by substituting truss rods with tension-only cables with the increase of deformation and

element forces as trade offs.

Substituting more truss rods with cables is beneficial for weight reduction. The key issue is to obtain an optimal

solution with the combined consideration of weight, deformation, and truss strength. The objective is to accomplish

sufficient weight reduction while maintain the acceptable margin of deformation and strength. When we replace

truss members with cables, the problem of cable losing tension should be accounted. Unlike truss rods, cables cannot support compressive loads. Thus, we need to make sure that the replacement of truss rods will not affect

structural integrity for all load cases in application.

Table 12. Comparison of displacements of two design architectures of Truss G (in).

Design Node 6 Node 7 Node 8

Ux Uy Uz Ux Uy Uz Ux Uy Uz

B-2.46E-3 1.79E-3 -2.30E-3 -4.05E-3 1.20E-3 -8.08E-3 -4.97E-3 8.87E-4 -1.11E-2

SUM: 3.81E-03 SUM: 9.12E-03 SUM: 1.22E-02

C2.18E-3 1.74E-3 -4.51E-3 2.59E-3 1.00E-3 -1.72E-2 -6.02E-3 8.52E-4 -1.36E-2

SUM: 5.30E-03 SUM: 1.74E-02 SUM: 1.49E-02

Table 13. Comparison of element forces of two design architectures of Truss G (lbs).

Design Elem 1 Elem 2 Elem 3 Elem 4 Elem 5 Elem 6

B -52.55 -68.23 -36.64 -57.19 -52.55 -68.23

C -5.91 -95.56 -90.70 -70.70 -5.91 -95.56

Elem 7 Elem 8 Elem 9 Elem 10 Elem 11 Elem 12

B -68.23 -68.23 -76.79 -51.38 -37.86 -73.74

C -90.70 -70.70 -74.09 -42.78 -36.36 -144.35

Truss H is used as the last example (Figure 9). The properties of truss rods are E=1E6 psi; =0.3; A=0.8 in2and

those of cables are E=1E6 psi; =0.3; A=0.1in2. Fz=-100 lbs is applied to Node 5 and 6. A pretension of 100 lbs is

applied to each cable. All the DOFs of Nodes 1-4 are fixed. The undeformed (yellow color) and deformed (black

color) configurations are shown on the right side of Figure 9. The variation of design architectures for Truss H is

Figure 8. Schematic of unstable and stable configurations (from left to right: Design A, B, C)

Figure 7. Schematic of Truss G.


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shown in Figures 10&11. Design A and B are stable while design C can not hold the prescribed loads. NASTRAN

program shows that singularity occurs at Y translational DOF of Node 5. A portion of the truss structure is free to

move in Y direction. The two parallelogram frames in Design C formed by Nodes 1-4-6-5, Nodes 5-7-8- 6 dont

have any cross-cables for interconnection. Either Cables 1-2 or Cables 3-4 provides links for structural members of

these two frames. Figure 11 displays the functions of Cables 5-10 in truss configurations D, E, F. The removal of

Cable 5-8 can still ensure the structure is stable (Design E). However, if all six Cables 5-10 are eliminated (design

F), the truss loses its stability. Usually, triangular configuration can generate steady truss and some kinds of

interconnection is needed for parallelogram configurations. To find out an optimal configuration for cable

positioning and arrangement is not a straightforward direct solution, and it involves a trial and test iterative analysis

procedure.

Next, we study the influence of truss rod materials on structural performances. The model in Figure 9 is solved

for four kinds of truss materials, which have elastic modulus 1E6 psi, 5E6 psi, 1E7 psi, and 5E7 psi, respectively

(denoted by Mat 1-Mat 4 in Tables 14&15). Maximum displacements occur at Node 5. As shown in Table 14, the

three displacement components are reduced due to the increase of elastic modulus. The enhancement of truss rod

stiffness helps to reduce structural deformation. Table 15 shows that axial forces in truss element are increased at the

same time. Most of truss rods are subjected to compression loads, and elements 5, 6 have the maximum axial forces.

Because truss are constrained at Nodes 1-4, elements 1-4 does not support any loads. These four members are

redundant for this specific condition, and might be useful for other load cases. Efficient usage of truss elements and

elimination of redundant members can be a vital design consideration. This study demonstrates that implementation

Figure 9. Schematic of Truss H (left: schematic, right: undeformed and deformed configurations).

Figure 10. Schematic of Truss H (from left to right: Design A, Design B, Design C)

Figure 11. Schematic of Truss H (from left to right: Design D, Design E, Design F)


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of truss materials with high elastic modulus helps to minimize truss deformation. Alternatively, increasing cross-

sectional area of truss rods can be a simple way to reduce truss deformation.

Table 14. Comparison of maximum displacements of Truss H (in).

Displacement Mat 1 Mat 2 Mat 3 Mat 4

Ux 9.729E-2 8.861E-2 8.748E-2 8.657E-2Uy 2.105E-2 1.595E-2 1.524E-2 1.465E-2

Uz 1.867E-1 1.418E-1 1.357E-1 1.308E-1

Table 15. Comparison of axial forces of Truss H (lbs).

Elem 5 Elem 6 Elem 7 Elem 8 Elem 9 Elem 10 Elem 11

Mat 1 -250.95* -218.32 -92.58 106.09 -61.08 -107.95 -80.87

Mat 2 -272.59 -234.33 -103.15 112.37 -67.39 -117.29 -97.23

Mat 3 -275.56 -236.50 -104.63 113.21 -68.31 -118.63 -99.57

Mat 4 -277.99 -238.27 -105.84 113.88 -69.06 -119.72 -101.49

Elem 12 Elem 13 Elem 14 Elem 15 Elem 16 Elem 17

Mat 1 0.309 -210.36 -87.43 -197.95 -68.22 -147.97

Mat 2 -5.96 -226.73 -93.57 -214.73 -81.35 -162.57Mat 3 -6.88 -228.99 -94.44 -217.06 -83.22 -164.63

Mat 4 -7.65 -230.84 -95.14 -218.98 -84.77 -166.32

(* negative sign represents compressive forces, positive sign represents tensile forces)

In this section, the influence of cables on the stability of cable-truss structures is presented. Essentially, cables

play an important role in stability and integrity of truss systems. The issue of truss weight reduction is also

addressed. In order to reduce total weight of truss-cable structures, some truss members can be substituted with a

pair of cross-cables. The unique property of cables that they can only sustain tension loads should be carefully

considered when we replace truss members with cables. The enhancement of truss material property helps to reduce

truss deformation, but results in the increase of internal forces of truss rods.

VII.

Summary

In this paper, a variety of truss structures containing pretensioned cables are analyzed. Commercial FE packagesANSYS and NASTRAN are employed to study the static behavior of cable-truss structures. Nodal displacements,

truss element forces, and cable tension of various trusses are calculated. This paper provides some general guideline

for truss analysis and configuration design. Some findings have been obtained from this work:

1. Cable geometry and property have influence on truss performances. The increase of cable elastic modulus

results in the decrease of truss deformation. Stiff cables generate less deformation than flexible cables for the same

pretension. The increase of cable cross-sectional areas results in the decrease of truss displacements.

2. Cable pretension is of extreme importance for prediction of truss performance. Appropriate setting of cable

pretension is helpful for the minimization of truss deformation.

3. Cable tension of deformed trusses changes significantly in contrast with pretension. Cables losing tension is

raised as an issue in truss configuration design. Cables could completely lose tension in deformed configurations.

4. The paper discusses some design considerations and factors in the last section. The stability of truss structures

associated with cable placement is addressed by analyzing diverse cable positioning and configurations and by

comparing deformations and axial forces. The behavior of cable-truss structures is affected by the stiffness of trussmembers.

The paper addresses weight reduction issue in the last section. For general optimization of cable-truss structures,

comprehensive considerations of loading, constraint condition, weight, strength, material, and cost are needed.

Future work can also be directed toward the design of optimal architectures with local reinforcements and variable

geometry and materials.


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