1

Nonlinear Dynamic Analysis of

Reticulated Space Truss Structures

Chung-Yue Wang1 , Ren-Zuo Wang1 , Ching-Chiang Chuang2, Tong-Yue Wu3

1 Department of Civil Engineering, National Central University, Chungli, 320 Taiwan, ROC

2 Department of Civil Engineering, Chung Yuan Christian University, Chungli, 320 Taiwan, ROC

3 Department of Mechanical Engineering, Chung Yuan Christian University, Chungli, 320 Taiwan,

ROC

Abstract In this paper, a simpler formulation for the nonlinear motion analysis of reticulated

space truss structures is developed by applying a new concept of computational

mechanics, named the vector form intrinsic finite element (VFIFE or V-5) method.

The V-5 method models the analyzed domain to be composed by finite particles and

the Newton’s second law is applied to describe each particle’s motion. By tracing

the motions of all the mass particles in the space, it can simulate the large geometrical

and material nonlinear changes during the motion of structure without using

geometrical stiffness matrix and iterations. The analysis procedure is vastly simple,

accurate, and versatile. The formulation of VFIFE type space truss element includes

a new description of the kinematics that can handle large rotation and large

deformation, and includes a set of deformation coordinates for each time increment

used to describe the shape functions and internal nodal forces. A convected material

frame and an explicit time integration scheme for the solution procedures are also

adopted. Numerical examples are presented to demonstrate capabilities and

accuracy of the V-5 method on the nonlinear dynamic stability analysis of space truss

structures.

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Keywords: space truss, nonlinear, dynamic, stability, particle, convected material

frame, deformation coordinates

1. Introduction

The nonlinear behavior of a reticulated space structure has been an important

problem in classical structural analysis. This is, in part, because most of the space

structures are quite flexible. Thus, the understanding of changes in the structural

geometry due to external loading is a critical aspect of the design process. In

practice truss structures are usually constructed by using a large number of

components and joints, and the structural geometry can be complex and innovating.

It is thus necessary to use numerical procedures for the analysis. A literature survey

shows that, during the past three decades, a large variety of formulations and

algorithms have been proposed for the study of reticulated trusses [1-20]. Some are

quite elaborate and complex. In general, several techniques seem have adopted the

stiffness or flexibility method as the basis of formulation for the classical matrix

analysis of structures. Modifications are introduced to account for the geometrical

changes. A common example is the addition of geometrical stiffness matrices. For

the calculation process, the basic algorithm is essentially linear. An iterative

procedure is employed to perturb the added nonlinear and geometrical terms. For

post-buckling and snap-through responses, special algorithms are required to impose

displacement inputs and to trace the full load-displacement path.

Among the earlier reports, Noor [3, 6] used a mixed formulation, with unknowns

consisting of both the force and displacement parameters, to analyze the geometrical

and material nonlinear problems of space structures. However, the mixed method

increases the computation effort required for the matrix operation, so a reduction

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method is applied to reduce the amount of computation. Jagannathan et al. [7] and

Wood et al. [24] used the stiffness method and the finite element method, respectively,

to study snap-through instability problems containing large deformation and large

rotation. In their analyses, Lagrangian coordinates and incremental moving

coordinates were adopted to describe the large displacement and rigid body rotations.

Jagannathan, Epstein and Christiano [8] used the Newton-Raphson method to analyze

the post-buckling behaviors of reticulated space structures. Furthermore, Rothert,

Dickel and Renner [9] compared the accuracy of four different types of modified

Newton-Raphson method for the analysis of the post-buckling behaviors of reticulated

space structures. These modified Newton-Raphson methods combine the

conventional incremental and iteration techniques with a predictor-corrector principle.

Papadrakakis [10] proposed a two-vector iteration method, that is, a combination of,

the dynamic relaxation method and the conjugate gradient method, to correct errors

due to unbalanced forces, to obtain the post-buckling analysis response curves.

Meek and Tan [11] applied the arc length method, and Watson and Holzer [25] used

the Crisfield’s method, to study the post-bucking behavior of a space truss. Holzer et

al. [17] has checked whether the location of the combined loading in the

loading-interaction diagram exceeds the stability boundary or not, to define the force

equilibrium path of a lattice structure. Leu and Yang [18, 37] have pointed out that

an unbalanced force will cause a fictitious response during rigid body motion, and

higher order nonlinear stiffness matrix, together with geometric stiffness matrix,

should be included to remove computational errors. Similar analyses have been also

conducted by Leu and Yang [26], Mallett [27] and Rajasekaran et al. [28]. Levy et al.

[19] have developed a so called exact geometrical nonlinear displacement method to

study the buckling problems of a truss with an arbitrary shape and applied loading.

Krishnamoorthy et al. [20] have done post-buckling analysis of structures using

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three-parameter constrained solution techniques to get the correct load-deflection

path.

As discussed previously, the nonlinear behavior of structures under static loading

has attracted a good deal of interest to date. In contrast, the nonlinear structural

response under dynamic loading has not received much attention. Kassimali et al.

[41] had studied the stability of truss under dynamic load based on an Eulerian

formulation accounting for arbitrarily large displacements. Zhu et al. [42] stated that

the nonlinearity of reticulated structures under dynamic loading can stem from

various origins: (i) geometrical; (ii) material; (iii) inertia; (iv) damping characters of

the structure system. Due to the complexity of the problem, the nonlinear response

and stability of these space structures under dynamic loads have thus far received only

limited attention [38-44]. It attracts the authors of this article to investigate this topic

by the newly proposed VFIFE method [30-32].

Although procedures based on matrix formulations and perturbation concepts [45,

46] seem to yield excellent results, there are some drawbacks when the proposed

procedures are implemented for engineering practices. Firstly, large geometrical

changes are often combined with material property changes, such as yielding and

damage. In fact, the formation of local yielding zones could be the primary source

of global shape changes of structure. Furthermore, for a large engineering structure,

yielding may occur simultaneously at multiple locations. If iterative procedures are

used to perturb both the geometrical and material nonlinearities, the accuracy and

stability of the algorithm can not be assured. Secondly, for a large structure,

geometrical changes may be caused by the buckling and yielding of a large number of

members. If a special technique is required for each member, to describe a

displacement controlled procedure the algorithm can become quite complex. Thirdly,

iterative procedures for nonlinear behavior are often limited to specific types of

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loading conditions. The restriction on applying loading conditions (type, direction,

magnitude) into the system can be a problem when a realistic simulation needs to be

performed. Another fundamental consideration in developing a practical algorithm

is the choice of a vector or matrix operation. Classical structural analysis for linearly

elastic material and static loading adopts a matrix formulation. However, vector

formulations as Hallquist [29] have been shown to be the preferable choice for the

study of the structure response under transient loading, shocks, and impacts with

complicated material properties.

This paper reports on a recent attempt to develop a simpler formulation for the

analysis of nonlinear dynamic behaviors of 3D reticulated structure. A recently

proposed approach by Ting et al. [30-32] named the vector form intrinsic finite

element (VFIFE, or V-5) for the analysis of large geometrical changes in continuous

media is discussed. The concept is adopted into the present paper for 3D truss

members. The formulation does not follow traditional approaches. Instead, a

convected reference frame, fictitious reversed rigid body motion and updated

deformation coordinate system are used to separate the rigid body motion and pure

deformation of the system. Then the internal force is calculated from the

deformation of element and applied to the mass particle to constrain its motion.

After combining with explicit time integration scheme, the proposed method can

effectively simulate the dynamic behaviors of space truss structures having large

deformation. This leads to a straightforward incremental algorithm. Nonlinear

dynamic force-displacement responses can be accurately calculated without using any

iteration, which also permits force or displacement inputs, without the need for special

handling.

Results for several examples of space truss structures under various types of

dynamic loading that had been discussed by previous researchers are compared and

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extended analyses were conducted to demonstrate the capability, accuracy and

stability of this newly proposed computation method.

2. Rotation and Deformation

The basic modeling assumptions for the vector form intrinsic finite element (V-5)

method for truss structures are essentially the same as those in classical structural

analysis. A truss is constructed by means of prismatic members and pin-connected

joints. Members are subjected to axial forces only. Joints have work equivalent

masses, and can be modeled as mass points. Motions of the joints can be described

by the principles of virtual work or equations of motion for particles. Members have

no mass, and are thus in static equilibrium. Therefore, the structural configuration of

a space truss can be described by the positions of the joints.

As shown in Fig. 1a. the position of a truss member is defined by the locations of

its attached end joints, denoted as nodes (1, 2). At time t the nodal positions are

( 1x , 2x ) and the length and cross-sectional area are l and A . At time tt ∆+ ,

following a time increment, t∆ , the corresponding values become ( 1x′ , 2x′ ) , l ′ and

A′ . Thus, relative positions between the nodes are

122 xxx −=d (1)

122 xxx ′−′=′d (2)

2x′= dl (3)

2xdl =′ (4)

The nodal displacements are

222 xxu −′= (5)

111 xxu −′= (6)

If the motion of the end node 1 is defined as the rigid body translation of a truss

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member 1u during the time increment, the relative displacement of an arbitrary point

on the member is then the displacement due to member rotation and the deformation;

see Fig. 1b.

xxuuu ddd −′=−= 1 (7)

For large geometrical changes, displacements due to rotation and deformation

can both be finite. Superposition does not apply. More importantly, there is no

prior knowledge as to how the applied load induces each displacement component.

Rigid body rotation and deformation of the truss member can not be calculated

independently. Therefore, the traditional co-rotational description of the frame

kinematics is no longer valid.

Although finite rotation and deformation can not be exactly separated there is

still a need to calculate deformation. Firstly, internal stresses are related to

deformation only. Secondly, the deformation component of the displacement is

usually much smaller than the rotational component. If these components are not

separated, at least approximately, the accuracy of the stress calculation may quickly

be lost and the algorithm diverges. For the purpose of evaluating deformation and

internal stresses, the approach of V-5 suggests the following.

Consider the state of the truss member (1, 2) at tt ∆+ , after it has completed its

motion for the time increment t∆ . The member is then subjected to a fictitious

reversed rotation described by a rotational vector θ or a rotational matrix R . The

fictitious motion induces a relative displacement vector rdu at an arbitrary point.

The deformation vector of the arbitrary point A is then defined as

rd ddd uuu −= (8)

A schematic diagram of the rotation and deformation at tt ∆+ is shown in Fig. 2.

The relationships of the rotational vector θ , rotation matrix R and relative

displacement rdu can be obtained through some algebraic manipulations [33]. The

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derivations are summarized in the following. Referring to Fig. 3, consider a plane of

rotation OAB with a normal vector θe , and a set of cylindrical coordinates with the

unit vectors ( re , se , θe ) as shown. The vector rdu can be written as

=rdu AD DB+uuuv uuuv

sr rr ee θθ sin)1(cos +−= (9)

Note that;

αθ sinxxe ′=′×= ddr (10)

)(sin

1As eee ×= θα

(11)

xxe′′

=dd

A (12)

θeee ×= sr (13)

Representing re and se in terms of θe and x′d , we get

)]()[cos1( xeeu ′××−= dd rθθθ

)(sin xe ′×+ dθθ (14)

A cross product of vectors can be written alternatively in matrix form as

xAxde ′=′× dθθ (15)

in which

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−=

00

0

θθ

θθ

θθ

θ

lmln

mnA (16)

with

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

θ

θ

θ

θ

nml

e (17)

rdu May be written as

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xRu ′= dd rθ (18)

and θθθ θθ AAR sin)cos1( 2 +−= (19)

From Fig. 3, we see that

=′′xd O B O A AB′ ′= +uuuv uuuv uuuv

rdd ux +′=

xRI ′+= d)( θ (20)

The rotational matrix is then

θRIR += (21)

The deformation vector of an arbitrary point can be written as

x-xRIu ddd d ′−= )( θ (22)

For finite incremental displacements, the above deformation vector is clearly not the

pure deformation. The magnitude and direction of the vector also depend on the

choice of rigid body rotation vector. Although it is possible to derive the exact

formulation, it is interesting to note that an exact measure is not needed in the present

numerical approach. As long as the assumed rotation is a close approximation of the

true rotation, this deformation vector should have the same order of magnitude as the

true deformation of the member. Then, as shown in following sections for the

equations of motion and numerical verifications, it is sufficient to obtain a convergent

solution. The advantage of not using the complicated exact formulation is that the

resulting algorithm is simple and flexible.

In the numerical analysis, a change of member orientation during the time

increment is taken as the rotation vector. That is, the angle between the relative

displacement vectors 2xd and 2x′d can be calculated as follows:

θθ eθ = (23)

)(sin 221 xddx ′×= −θ (24)

10

22

22

xxxxe′×′×

=dddd

θ (25)

3. Equations of Motion

The motion of each joint mass is assumed to satisfy the principle of virtual work

associated with incremental displacements

0=+− αα δδ WU , n,...,3,2,1=α (26)

where α is an arbitrary joint, and n is the total number of joints in the structure.

It is straightforward to evaluate the virtual work that occurs due to the applied joint

force and the inertia of the joint mass. We have

αααααα δδδ dMdPd &&TTW −= (27)

where { }zyxT ddd αααα =d are the incremental joint displacements; αP is the

total applied force vector acting on the joint at time tt ∆+ ; αM is the joint mass

value. The internal virtual work of the joint αδU is the sum of the work that occurs

due to the deformation of the connecting members, and can be expressed by the

following equation

∑=e

eUU αα δδ (28)

where e is the number of elements connected to the joint. The internal virtual

work of the member referred to is the deformed state at time tt ∆+ and is associated

with an incremental virtual displacement )( udδ . The deformed state is the truss

position (1′ , 2′ ) , which is shown in Fig. 1a. Since the member has no mass, and the

internal forces are assumed to be in static equilibrium, the virtual work that occurs

due to a fictitious translation 1u and a rigid body rotation θ are both equal to zero.

Thus, the internal work may instead be referred to as the fictitious state of the

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deformation, the truss position (1′′ , 2 ′′ ) shown in Fig. 2, and is associated with a

virtual deformation vector )( dduδ .

To describe the deformation of a member, a deformation coordinate system x

( x , y , z ) is assumed for the time tt ∆+ . The x axis is parallel to the deformation

vector of the end node 2, that is dd 2u . The projection of members (1, 2) and (1′′ ,

2 ′′ ) are used to evaluate the deformation and the stress distribution. A schematic

diagram of the deformation coordinates and the projection values is shown in Fig. 4.

Using the prismatic assumption of the truss member,

1ˆˆˆˆˆ eu ∆=∆lx , where (29)

u∆ is the deformation vector at a point on the member projection on x ; 1e a unit

directional vector parallel to the deformation coordinate axis x ; ∆ is the

projection of the incremental deformation dd 2u of node 2 on the x axis. The

corresponding axial strain along the projected member is then

1ˆˆˆ

ˆˆˆ euε

lx∆

=∂∆∂

=∆ (30)

Where l is the projection length of the truss element at time t on the deformation

coordinate x . The virtual work of the member, at time tt ∆+ , due to the virtual

incremental deformation )ˆ( u∆δ , can be calculated as

xdAUl T ˆˆ)ˆ(ˆ

0∫ ′∆= sεδδ

∫ ′∆=

lxdA

l

ˆ

0ˆˆ

ˆˆ

sδ (31)

sss ˆˆˆ ∆+= t , where (32)

ts and s are the total stress vectors at time t and tt ∆+ ; 1ˆˆˆ es s∆=∆ is the

incremental stress vector; A′ is the cross-sectional area at time tt ∆+ . Define a

set of equivalent internal forces acting on the end nodes

1ˆˆˆ ef ii f= , 2,1=i (33)

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such that

2211ˆ)ˆ(ˆ)ˆ( fufu TTU ∆+∆= δδδ

2211ˆˆˆˆ( f)uδ(f)u ∆+∆= δ (34)

where 1u∆ and 2u∆ are the incremental deformation vectors of the end nodes in

the time increment t∆ .

Since 0ˆ1 =∆u , ∆=∆ ˆˆ2u and (35)

2ˆˆ fU ∆= δδ (36)

the nodal force at node 2 can be obtained as

xdAssl

fl

t ˆ)ˆˆ(ˆ1ˆ ˆ

02 ′∆+= ∫ (37)

For a member with a uniform cross section and of uniform material,

∆′

+= ˆˆ

ˆˆ22 l

AEff t , where (38)

E is the tangent modulus at the stress value ts , and tf 2ˆ is the internal nodal force

of node 2 at time t . The equilibrium requirement of the truss member yields

21ˆˆ ff −= (39)

Using the total incremental displacement of the nodes, including the components due

to rigid body motion, we get

21 UUU δδδ +=

2211 fufu TT δδ += , where (40)

if and iu are nodal forces and displacements expressed in global coordinates,

rd dd 2212 uuuu ++= (41)

1ˆˆ ef ii f= )2,1( =i (42)

12 ˆˆ eu ∆=dd (43)

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1Uδ and 2Uδ are each summed into the connecting joints, and 1u and 2u and

equal to the corresponding joint displacements. Now, returning to an arbitrary joint

α , the internal virtual work of the joint is the sum of the contributions by all the

connecting members.

)(∑==i

iTTU fdfd αααα δδδ (44)

αf is the sum of all global internal force components and

{ } ∑==i

iT

zyx fff ffα (45)

Hence, from the principle of virtual work for joint α can then be written as

0=−+− ααααααα δδδ dMdPdfd &&TTT (46)

This leads to the following equations of motion for the joint :

αααα fPdM −=&& (47)

α

α⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

−−−

=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

zz

yy

xx

z

y

x

fPfPfP

ddd

M&&

&&

&&

(48)

Various techniques may be used to calculate the solution of a standard equation of

motion. An explicit time integration procedure is adopted for the numerical

examples. Discussions concerning the stability criterion, size of the time step, and

the procedures to obtain quasi-static solutions are well-documented in textbooks and

literatures. A central difference, similar to that used by Rice and Ting [34], is

adopted here. For each equation of motion, the difference equations produce

)( 3212 mccc

net

tttttFddd +∆−∆=∆ ∆+∆+ (49)

mc

=α , tt

a∆

+∆

=2

121

α (50)

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1a

c = , 222t

c∆

= (51)

14

ttI mm ddF &&& α+= (52)

F P f Fnet Iα α= − − (53)

Thus, the equations of motion formulated at time tt ∆+ yield the displacements at

tt ∆+ 2 . The algorithm requires special attention at the initial time step

20001 )(

21)( tt ∆∆+∆∆−∆=∆ dddd &&& (54)

It is interesting to note that the internal nodal forces at time t+Δt do not act in

the axial direction of the truss member; see Fig. 5. This is due to the following two

reasons: Firstly, the assumed rotation is in general, not the true rotation. The error

leads to deviations of the force components. The resulting transverse components

form a couple, which provides a correction to the assumed rotation vector. In an

implicit algorithm for solutions, the correction is carried out by an iterative process.

In an explicit procedure, it is performed by subsequent time steps. Analogous to a

predictor-corrector scheme, the assumed rotation vector plays the role of a predictor in

the present algorithm, and the force couple behaves as a corrector. Secondly, in the

V-5 formulation, the rigid body motion of a member is defined by the motions of the

joint masses. The internal force is related only to the deformation and the member

stress at time t. Thus, the resulting nodal forces are not axial forces. This is

different from the traditional nonlinear formulation of a reticulated truss member.

By using a Lagrangian or Almansi strain measure to include both the effects of the

rigid body rotation and the deformation, the total internal force for a nonlinear

formulation should be in the axial direction. The physical interpretation of the nodal

force components related to rotation and deformation, is frequently discussed in the

literatures by Leu and Yang [18] and Yang and Chiou [37].

In the present formulation with the correction mechanism and a separate

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treatment of rigid body motion, a nonlinear strain measure is not assumed.

Furthermore, the formulation for member rotation does not have to exact. This leads

to a simple and flexible algorithm.

4. Numerical Examples

Three examples are presented in this section. The first one dealing with a

buckling of a space truss structure for which detailed results are available shows the

validity of the present formulation. It also implies that the capability of applying a

dynamic formulated method to solve a static problem. The following examples

dealing with the space structure under various types of excitation are aimed at gaining

hindsight into the behavior of the dynamic instability of space truss structures.

4.1 Snap-though of a space truss dome.

Figure 6 shows a space dome structure constructed using 168 truss members.

This problem has been studied by Leu and Yang [36] as an illustration of the

importance of rigid body motion in large dome deflection. The truss members are

made of the same material. The Young’s modulus is 81004.2 ×=E 2/ mKN and the

cross sectional area is 410431.50 −×=A 2m . The calculation results by using the V-5

method are compared with the ones in reference [36] and good agreement for all cases

are shown in Figs. 7 and 8. For the V-5 calculations on this static problem, the

displacement rate is arbitrarily set at 45 m/s, and a constant time step of s8100.1 −×

is used.

4.2 Two-member toggle truss

Figure 9(a) shows a two-member toggle truss. The snap through behavior of

the two-member truss under dynamic load has been examined by Kassimali et al. [41]

and Zhu et al. [43]. In this study the maximum deflection and the transient response

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of the truss under step, triangular and sinusoidal loads as shown in Fig. 9(b) have been

studied. The damping properties of the truss have been ignored. The truss

members are made of the same material. The Young’s modulus E is 27 /100.3 ink× ,

the cross sectional area A is 1 2in and the density ρ is 24 sec10339.7 −× − ksi . The

length L of each member is in100 . Two types of structure of inclination

angle, o30=α and o75=α were considered. For the steep two—member truss

with o75=α , an additional horizontal eccentric load PeP 01.0= as shown in Fig.

9(a) is considered into the analysis.

Figure 10 shows the dynamic response of a toggle truss ( o30=α ) under a step

forcing function with P = 800 kips. Figure 11 shows the maximum deflection

response of the truss joint under static and step loads. As seen in Fig. 11 the truss

will snap-through at a step load of approximately 1310 kips, compared with a

snap-through load of 1659 kips under static load. The deflection chosen in this

figure corresponds to the first peak value obtained in the nonlinear transient response.

In this problem, the structure responses under larger loads were further investigated.

Figure 12 shows the transient responses of the vertical displacement of the node 2 for

various magnitude of step load. The occurrence of the snap-through can be easily

identified in Fig. 12 from the bifurcation of the vibration modes at the load is 1311

kips which is a little bit different from the values predicted by Kassimali et al [41] and

Zhu et al. [43] as shown in Fig. 11.

Figure 13 shows the transient response of the vertical displacement of the node 2

for triangular forcing functions of P = 200 kips and P = 800 kips, sec3.0=dT . From

Fig. 14, it is found that the dynamic snap-through load is increased as the impulse

duration is decreased. It is expected that the critical load under the triangular forcing

function will approach that of step function as the impulse duration is increased.

Figure 15 shows the transient responses of the vertical displacement of the node 2 for

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various forcing amplitude P of triangular forcing functions but the same duration

sec3.0=dT . From this figure the dynamic snap-through load is easily identified as

1621 kips.

For the sinusoidal loading case, some representative load-deflection curves are

presented in Fig. 16. It seems that, unlike the load-deformation curves with steadily

decreasing slopes presented previously for the step and triangular loadings, the

snap-through load for the two-member toggle truss under sinusoidal loading is a

function of the loading frequency. The response curves obtained by the V-5 methods

have some variations from the ones presented by Kassimali et al [41]. In this figure,

the response curves of the truss after the snap-through and the deflection of very large

deformation states were also predicted by the V-5 method.

Numerical solutions generated for the steep two-member truss ( 075=α ) under

various static and step forcing functions with lateral load are summarized in Figs

17-18. For this structure, the primary loading is similar to that of the shallow truss

(i.e. a vertical load, P, applied symmetrically at the free joint), except that a small

lateral load eP, is now introduced to simulate a condition of imperfection with a

parameter e. The predictions by the V-5 method compared with the ones presented

by Kassimali [41] are shown in Fig. 17. The snap-through load of 4466 kips for this

case can be identified from the transient response curves of different load level P in

Fig. 18.

4.3 Dynamic stability of geodesic truss dome

The dynamic stability of geodesic dome under a triangular forcing function as

shown in Fig. 19 is investigated here. Two triangular forcing functions as shown in

Fig. 9(b) with different impulse durations have been used. The maximum deflection

obtained during the impulse duration is presented against the forcing amplitude in Fig.

18

20. Same conclusion is obtained as previous researchers that the snap-through load

is decrease as the forcing duration is increased. Figure 21 shows the transient

response of the dome under two different triangular forcing functions with the same

amplitude but different impulse duration.

To demonstrate the capability of the V-5 method on the analysis of very large

motion of structure, the problem presented in Figs 20-21 was further investigated with

very high forcing amplitude. As seen from Fig. 22, it is very interesting to find that

there are two snap-through load levels for this geodesic truss dome, this first one is at

the forcing amplitude P equal to 6.5 kips and the second one is at the forcing

amplitude equal to 50 kips for the forcing duration of 0.005 sec. To understand the

mode instability behavior at these two bifurcation points, the transient response of the

vertical displacement of the nodal point 1 of the dome are shown in Figs. 23-24. The

nonlinear responses can also be qualitatively analyzed by studying the trajectories in

the phase plane. As shown in Figs. 25-27, the mode of instability is seen as the

shifting of the repelling range of the trajectories. Highly nonlinear motion is seen

for the forcing amplitude is equal to 50 kips from Fig. 27. .

Since the V-5 method does not need to solve any matrix equation, this character

allows the V-5 method be capable to analyze the dynamic behavior of structure with

members of very much difference in their material properties. To verify this, the

elastic modulus E of the member “a” in the geodesic truss is changed to ksi9100.1 ×

and keep other members of the same stiffness ( ksi4100.1 × ) as previously. The

Transient response of a geodesic truss dome with and without a member of high

stiffness difference is shown in Fig. 28. It is clear from this figure that varying the

stiffness of a member may alter the nonlinear transient response significantly. For

the structure without stiffness difference, the forcing amplitude of 6.5 kips is a

bifurcation load. However, the increase of the stiffness value of a member causes

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the 6.5 kips not a bifurcation forcing amplitude.

5. Conclusions

A vastly simple numerical procedure is developed in this paper for motion analyses of

the nonlinear response and stability of reticulated space truss structures subjected to

large geometrical changes and complicated excitations.

Different from conventional matrix form structure analysis methods, the vector

type motion equation of each mass particle makes the analysis procedure of the

proposed method dramatically simple. Due to the inherited predictor-corrector

mechanism, iterations are not required as conventional methods in nonlinear motion

analysis. In addition, due to the nature of discrete independent particle point, it is

not required to set essential boundary conditions of the system. It is very easy to

prescribe the displacement and forcing conditions on each particle during the

procedure of analysis.

Through the numerical analyses of a few benchmark problems of features as

large rotation and dynamic instability, the newly proposed method demonstrates its

accuracy and superior capability on the nonlinear motion analysis of space truss

structure. As well, the vector form nature of the V-5 method allows it to be linked

with parallel computation techniques to study the large scale problems that have

complicated geometrical variations and loading histories. It is believed that the V-5

method can be a very effective tool for engineers on the structure analysis.

Acknowledgements

The authors wish to express their gratitude to Professor E. C. Ting for his help, advice

20

and encouragement during this work.

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Struct. Div., ASCE, 97(9), pp. 2189-2202 (1971). 2. Baron, F. and Venkatesan, M. S., “Nonlinear analysis of cable and truss

structures,” J. Struct. Div., ASCE, 97(2), pp. 679-710 (1971). 3. Noor, A. K., “Nonlinear analysis of space trusses,” J. Struct. Div., ASCE, 100(3),

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body rotation,” J. Struct. Div., ASCE, 101(11), pp. 2472-2476 (1975). 8. Jagannathan, D. S., Epstein, H. I. and Christiano, P., “Nonlinear analysis of

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21

17. Holzer, S. M., Plaut, R. H., Somers, A. E. and White, S. W., “Stability of lattice structures under combined loads,” J. Eng. Mech. Div., ASCE, 106(2), pp. 289-305. (1980).

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22. Richard, R. M. and Goldberg, J. E., “Analysis of nonlinear structures : force method,” J. Struct. Div., ASCE, 91(9), pp. 33-48 (1965).

23. Hensley, R. C. and Azar, J. J., “Computer analysis of nonlinear truss structures,” J. Struct. Div., ASCE, 94(9), pp. 1427-1439, (1968).

24. Wood, R. D. and Zienkiewicz, O. C., “Geometrically nonlinear finite element analysis of beam, frames, arches and axisymmetric shell,” Comput. Struct., 7, pp. 725-735 (1977).

25. Watson, L. T. and Holzer, S. M., “Quadratic convergence of Crisfield’s method,” Comput. Struct., 17(1), pp. 69-72 (1983).

26. Leu, L. J. and Yang, Y. B., Discussion of “Post-buckling analysis of steel space trusses,” by Hill , C. D., Blandford, G. E. and Wang, S. T. (Proc. No. 23407), J. Struct. Eng., ASCE, 117(12), pp. 3824-3828 (1991).

27. Mallett, R. H. and Marcal, P. V., “Finite element analysis of nonlinear structures,” J. Struct. Eng., ASCE, 94(9), pp. 2081-2105 (1968).

28. Rajasekaran, S. and Murray, D. W., “Incremental finite element matrices,” J. Struct. Eng., ASCE, 99(12), pp. 2423-2438, (1973).

29. Hallquist, J. O., LLS-DYNA Theoretical Manual, Livermore Software Technology Corporation, (1998).

30. Ting, E. C., Shih, C. and Wang, Y. K., “Fundamentals of a vector form intrinsic finite element: Part I. basic procedure and a plane frame element,” J. Mech., 20(2), pp. 113-122 (2004).

31. Ting, E. C., Shih, C. and Wang, Y. K., “Fundamentals of a vector form intrinsic finite element: Part II. plane solid elements,” J. Mech., 20(2), pp. 123-132 (2004).

32. Shih, C., Wang, Y. K. and Ting, E. C., “Fundamentals of a vector form intrinsic finite element: Part III. Convected material frame and examples,” J. Mech., 20(2), pp. 133-143 (2004).

22

33. Goldstein, H., Classical Mechanics, MA: Addison-Wesley Publishing, (1959). 34. Rice, D. L. and Ting, E. C., “Large displacement transient analysis of flexible

structures,” Int. J. Num. Meth. Eng., 36, pp. 1541-1562 (1993). 35. Crisfield, M. A., Non-linear finite element analysis of solids and structures, John

Wiley & Sons, England (1991). 36. Yang, Y. B., Yang, C. T., Chang, T. P. and Chang, P. K., “Effect of member

buckling and yielding on ultimate strengths of space trusses,” Eng. Struct., 19(2), pp. 179-191 (1997).

37. Yang, Y. B. and Chiou, H. T. “Rigid body motion test for nonlinear analysis with beam elements,” J. Engrg. Mech., ASCE, 113(9), pp. 1404-1419 (1987).

38. Abrate, S. and Sun, C. T., “Dynamic analysis of geometrically nonlinear truss structures,” Comput. Struct., 17(4), pp. 491-497 (1983).

39. Noor, A. K. and Peters, J. M., “Nonlinear dynamic analysis of space trusses,” Comput. Meth. Appl. Mech. Engng., 21, pp. 131-151 (1980).

40. Coan, C. H. and Plaut, R. H., “Dynamic stability of a lattice dome,’’ Earthquake. Engng. Struct. Dynam., 11, pp. 269-274 (1983).

41. Kassimali, A. and Bidhendi, E., “Stability of trusses under dynamic loads,” Comput. Struct., 29(3), pp. 381-392 (1988).

42. Sllaats, P. M., Jough, J. de. and Sauren, A. A. H., “Model reduction tools for nonlinear structural dynamics,” Comput. Struct., 54(6), pp. 1155-1171 (1995).

43. Zhu, K., Al-Bermani, F. G. A. and Kitipornchai, S., “Nonlinear dynamic analysis of lattice structures,” Comput. Struct., 52(1), pp. 9-15 (1994).

44. Tada, M. and Suito, A., “Static and dynamic post-buckling behavior of truss structures,” Eng. Struct., 20(4-6), pp. 384-389 (1998).

45. Walker, A. C. and Hall, D. G., “An analysis of Large deflections of beams using the Rayleigh-Ritz finite element method,” Aeronautical Quarterly., pp. 357-367 (1968).

46. Walker, A. C., “A nonlinear finite element analysis of shallow circular arches,” Int. J. Solids Struct., 5, pp. 97-107 (1969).

23

Figures:

1x

1x′ 1u

2u

x

y

ztat

tttat ∆+=

1 2

1′

2′

2x

2x′

u

2xd ′

2dx

x

x′

Figure 1a. Motion of a truss member in space.

2du

1′

2′

1 2dx

xd ′du

Figure 1b. Relative displacements.

24

θe

2xd ′

dx A

durdu

ddu

2du

d2du

r2du-

1 1′

2′

2 ′′

2

θ-

xd ′

1′′

l′

l′2xd ′′

l

2dx

xd ′′

A′

A ′′

Figure 2. Fictitious reverse rotation.

θ

θe

A

B

re

θe

se

O

O′

r

αAe

rD

rdu

xd ′

θ

Figure 3. Rotational displacement.

25

l

∆ 1e x

1 1′′2 ′′

2ddu

x ′′x

xl

d2du

u∆

Figure 4. Deformation coordinates.

1

x

y2

2 ′′ 1e

1e x

1e2f

2f−1′

2′

d2du

1′′

1u

2u

tttimeat ∆+

Figure 5. Internal nodal forces.

26

1

2

(a) Top view

P

66.9 cm

116.7 cm

135.0 cm

181. cm 194.9 cm 204.1 cm 580. cm

(b) Side view

Fig 6. Reticulated space truss structure composed of 168 members.

27

0.00 10.00 20.00 30.00 40.00 50.00Central Disp. (cm)

-800.00

-400.00

0.00

400.00

800.00

1200.00

P (K

N)

Yang [36]

V - 5

Figure 7. Response curve of load versus vertical displacement at joint 1.

-2.00 0.00 2.00Vertical and Horizontal Displ. of Joint 2 (cm)

-800.00

-400.00

0.00

400.00

800.00

1200.00

P (K

N)

Yang [36] (Vertical)

VFIFE (Vertical)

Yang [36] (Horizontal)

VFIFE (Horizontal)

Figure 8. Response curves of load versus vertical and

horizontal displacements at joint 2.

28

eP

3

Y

X1

P

2

L

α

(a)

P

Load

Time

P

Load

Time

dT

P

Load

Time

dT (b)

Figure 9 (a) two-member toggle truss, (b) various loading types of P

29

0.00 0.10 0.20 0.30 0.40 0.50 0.60Time (sec)

0.00

4.00

8.00

12.00

16.00

Dis

plac

emen

t (in

)

ZhuVFIFE

Figure 10 Transient response of the vertical displacement of the node 2 for step load

P = 800 kips ( 030=α , e= 0) (1 kips=4.45 kN; 1 in. = 25.4 mm)

0.00 40.00 80.00 120.00 160.00Displacement (in)

-2000.00

-1000.00

0.00

1000.00

2000.00

Load

P (k

ips)

Zhu ( Step Load )

Kassimali ( Step Load )

VFIFE ( Step Load )

Kassimali ( Static Load )

VFIFE ( Static Load )

Figure 11 Maximum vertical displacements at node 2 under static and step loads ( 030=α , e= 0) (1 kips=4.45 kN; 1 in. = 25.4 mm)

30

0.00 0.40 0.80 1.20

Time (sec)

0.00

20.00

40.00

60.00

80.00

100.00

120.00

140.00

160.00

180.00

200.00

220.00

Dis

plac

emen

t (in

)

P = 1300 kips

P = 1310 kips

P = 1311 kips

P = 1315 kips

P = 2000 kips

Figure 12 Transient responses of the vertical displacement of the node 2 for various magnitudes of step load ( 030=α , e= 0) (1 kips=4.45 kN; 1 in. = 25.4 mm).

0.00 0.10 0.20 0.30Time (sec)

-8.00

-4.00

0.00

4.00

8.00

12.00

Dis

plac

emen

t (in

)

Zhu ( P=200kips )

VFIFE ( P=200 kips )

VFIFE ( P=800kips )

Figure 13 Transient response of the vertical displacement of the node 2 for triangular loads (P = 200 kips and P = 800 kips, sec3.0=dT , 030=α , e= 0)

(1 kips=4.45 kN; 1 in. = 25.4 mm)

31

0.00 40.00 80.00 120.00 160.00

Displacement (in)

0.00

1000.00

2000.00

3000.00

Load

P (k

ips)

Kassimali (Triangular Td=0.3 sec)

Zhu (Triangular Td=0.3 sec)

VFIFE (Triangular Td=0.3 sec)

Kassimali (Triangular Td=0.2 sec)

VFIFE (Triangular Td=0.2 sec)

Figure 14 Maximum vertical displacements at node 2 under various triangular

forcing functions ( 030=α , e= 0) (1 kips=4.45 kN; 1 in. = 25.4 mm)

0.00 0.20 0.40 0.60 0.80 1.00

Time (sec)

-50.00

0.00

50.00

100.00

150.00

200.00

Dis

plac

emen

t (in

)

P = 1600 kips

P = 1620 kips

P = 1621 kips

P = 2000 kips

Figure 15 Transient responses of the vertical displacement of the node 2 for various forcing amplitudes of triangular forcing functions ( sec3.0=dT , 030=α , e= 0)

(1 kips=4.45 kN; 1 in. = 25.4 mm)

32

0.00 40.00 80.00 120.00 160.00Displacement (in)

0.00

400.00

800.00

1200.00

1600.00

2000.00

Load

P (k

ips)

Kassimali (Sinusoidal Td=0.1 sec)

VFIFE (Sinusoidal Td=0.1 sec)

Kassimali (Sinusoidal Td=0.3 sec)

VFIFE (Sinusoidal Td=0.3 sec)

Kassimali (Sinusoidal Td=0.6 sec)

VFIFE (Sinusoidal Td=0.6 sec)

Figure 16 Maximum vertical displacements at node 2 of a shallow two-member

truss ( 030=α , e= 0) under sinusoidal forcing functions with various dT values.

(1 kips = 4.45 kN; 1 in. = 25.4 mm)

0.00 20.00 40.00 60.00 80.00 100.00 120.00 140.00

Maximum Horizontal Displacement (in)

0.00

2000.00

4000.00

6000.00

Load

P (k

ips)

Kassimali (Step Load)

VFIFE (Step Load)

Kassimali (Static Load)

VFIFE (Static Load)

Figure 17 Load-deflection curves for steep two-member truss ( 075=α ) under

various static and step forcing functions ( 01.0=e ). (1 kips=4.45 kN; 1 in. = 25.4 mm)

33

0.00 2.00 4.00 6.00Time (sec)

-200.00

-100.00

0.00

100.00

200.00

300.00

Dis

plac

emen

t (in

)

P = 4400 kips

P = 4465 kips

P = 4466 kips

Figure 18 Transient responses of the vertical displacement of the node 2 for various forcing levels of step forcing function ( 01.0=e ， 075=α )

(1 kips=4.45 kN; 1 in. = 25.4 mm)

34

21

FREE

FIXED

34

5

6 7

11

10

12 8

13

9

Z

Y

a

(a) top view

2.1 in

8.2 in

25.4 in 25.4 in

43.3 in 43.3 in

P

Z

X

(b) side view

Figure 19 Geodesic dome: dimensions, properties and loading (1 kips=4.45 kN; 1 in. = 25.4 mm)

35

0.00 0.40 0.80 1.20 1.60 2.00Displacement (in)

0.00

2.00

4.00

6.00

8.00

Load

P (k

ips)

Kassimali ( Td = 0.005 sec )

Zhu ( Td = 0.005 sec )

VFIFE ( Td = 0.005 sec )

Kassimali ( Td = 0.01 sec )

Zhu ( Td = 0.01 sec )

VFIFE ( Td = 0.01 sec )

Figure 20 Maximum vertical displacement at the top joint of a geodesic truss dome under triangular impulse with different durations. (1 kips=4.45 kN; 1 in. = 25.4 mm)

0.000 0.002 0.004 0.006 0.008 0.010Time (sec)

-0.40

-0.20

0.00

0.20

0.40

0.60

Dis

plac

emen

t (in

)

Zhu ( Td = 0.005 sec )

VFIFE ( Td = 0.005 sec )

Zhu ( Td = 0.01 sec )

VFIFE ( Td = 0.01 sec )

Figure 21 Transient response of a geodesic truss dome under triangular forcing

functions of the same forcing amplitude (P = 2kips) but different durations. (1 kips=4.45 kN; 1 in. = 25.4 mm)

36

0.00 5.00 10.00 15.00 20.00 25.00

Displacement (in)

0.00

20.00

40.00

60.00

80.00

Load

P (k

ips)

VFIFE ( Triangular Load )

Td = 0.01 sec

Td = 0.005 sec

Figure 22 Maximum vertical displacement at the top joint of a geodesic truss dome under triangular impulses with different durations and different forcing amplitudes.

(1 kips=4.45 kN; 1 in. = 25.4 mm)

0.00 0.04 0.08 0.12 0.16 0.20Time (sec)

-2.00

0.00

2.00

4.00

6.00

8.00

Dis

plac

emen

t (in

)

P = 6.49 kips

P = 6.5 kips

Figure 23. Bifurcation of the transient response of the top joint at the first critical

forcing amplitude of a geodesic truss dome under triangular forcing function ( sec005.0=dT ).(1 kips=4.45 kN; 1 in. = 25.4 mm)

37

0.00 0.04 0.08 0.12 0.16 0.20

Time (sec)

-5.00

0.00

5.00

10.00

15.00

20.00

25.00

Dis

plac

emen

t (in

)

VFIFE (Triangular Load)

P = 49 kips

P = 50 kips

Figure 24. Bifurcation of the transient response of the top joint at the second critical

forcing amplitude of a geodesic truss dome under triangular forcing function ( sec005.0=dT ).(1 kips=4.45 kN; 1 in. = 25.4 mm)

-3.00 -2.00 -1.00 0.00 1.00 2.00 3.00Displacement (in)

-1000.00

-500.00

0.00

500.00

1000.00

Velo

city

(in/

sec)

Figure 25. Phase diagram of the vertical motion of the node 1 (forcing amplitude

P= 6.49kips, sec005.0=dT ) (1 kips=4.45 kN; 1 in. = 25.4 mm).

38

-2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00Displacement (in)

-1000.00

-500.00

0.00

500.00

1000.00

Velo

city

(in/

sec)

Figure 26. Phase diagram of the vertical motion of the node 1 (forcing amplitude P= 6.5kips, sec005.0=dT ) (1 kips=4.45 kN; 1 in. = 25.4 mm).

-10.00 -5.00 0.00 5.00 10.00 15.00 20.00 25.00Displacement (in)

-8000.00

-6000.00

-4000.00

-2000.00

0.00

2000.00

4000.00

6000.00

8000.00

Velo

city

(in/

sec)

Figure 27. Phase diagram of the vertical motion of the node 1 (forcing amplitude P

= 50 kips, sec005.0=dT ) (1 kips = 4.45 kN; 1 in. = 25.4 mm).

39

0.00 0.00 0.01 0.01 0.02 0.02Time (sec)

-1.00

0.00

1.00

2.00

3.00

4.00

Dis

plac

emen

t (in

)

a member E=1.e4 ( P = 2 kips)

a member E=1.e9 ( P = 2kips )

a member E=1.e4 ( P = 6.49 kips )

a member E=1.e9 ( P = 6.49 kips )

a member E=1.e4 ( P = 6.5 kips )

a member E=1.e9 ( P = 6.5 kips )

Figure 28 Transient response of a geodesic truss dome with and without a member

of high stiffness difference, sec005.0=dT . (1 kips = 4.45 kN; 1 in. = 25.4 mm)