- Email: [email protected]
Nonlinear dynamic analysis of lattice structures
Pergamon
00457949(94)EO125-L
NONLINEAR
DYNAMIC ANALYSIS STRUCTURES
Computm & Srrucrurrs Vol. 52. No. I, pp. 9-15, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Brhain. All rights reserved co45-7949/94-~7.00 + 0.00
OF LATTICE
K. ZHU, F. G. A. AL-BERMANI and S. KITIFWRNCHAI Department of Civil Engineering, The University of Queensland, St Lucia 4072, Australia (Receiued 6 April 1993)
Abstract-A computational procedure for predicting the geometric and material nonlinear dynamic response of space trusses is presented. An updated Lagrangian method, based on the incremental formulation of the equation of motion, is employed. The constitutive law is cast into the Ramberg-Osgood polynomial model. Numerical studies of two- and three-dimensional truss structures under various dynamic loading are made. The structure dynamic response including different types of nonlinearity is compared with the linear response and with the results obtained from static analysis.
loading has attracted a good deal of interest to date. In contrast, the nonlinear structural response under dynamic loading has not received much attention. Lattice structures usually have higher natural frequencies than comparable solid structures, because of their relatively high stiffness-to-mass ratio. The nonlinearity of lattice structures under dynamic loading can stem from various origins: (i) geometrical-due to the variations in the geometrical properties of the structure as the load progresses; (ii) material-due to the inherent nonlinear behaviour of the materials under load; (iii) inertia-depending on the dynamic motion and the structural deformations; and (iv) dampingdepending on the structural joints and material. Kassimali and Bidhendi [l] have investigated the stability of truss structures under dynamic loading. The dynamic response of trusses with both geometric and material nonlinearities was presented by Noor and Peters [2] using a mixed formulation. More recently, a flexibility method was presented by Abu-Saba et al. [3] for predicting the dynamic characteristics of truss-beam structures. In this investigation, only those nonlinearities which affect the tangent stiffness of the structure have been accounted for (i.e. geometric and material nonlinearities). An incremental formulation of the equation of motion has been used, based on an updated Lagrangian frame of reference. The incremental equation of motion is discretized both in physical coordinates and time. Various implicit time integration schemes are used to obtain the transient structural response. During this incremental process, the effects of both the geometrical variations and material nonlinearity are accounted for through the tangent stiffness operator. An incremental/iterative solution scheme is employed and a residual force convergence criterion is used to terminate a loading
NOTATION
c
co,c, c, 1
D ; k 4, k,, k, KT
M n R r, i, F AR Ar, Ai, ti AR, :r T” V W
damping matrix of structure initial undeformed, known deformed and neighbouring configuration, respectively constitutive matrix instantaneous elastic modulus natural frequency current structural resistance parameter defining stress-strain curve element linear, geometric and tangent stiffness matrices tangent stiffness matrix of structure mass matrix of structure parameter defining stress-strain curve residual force total displacement, velocity and acceleration, respectively incremental applied nodal force incremental displacement, velocity and acceleration, respectively residual force increment time time increment natural period volume of element virtual work
Greek symbols B? Y c co CL. CN
0 00 T
coefficients used in Newmark method strain yield strain linear and nonlinear strain components of Green-Lagrange strain tensor stress yield stress Cauchy stress tensor
I. INTRODUCTION
in a structural system can have a profound effect on the transient structural response. The nonlinear behaviour of structures under static Nonlinearities
9
K. ZHU et al.
10
/ ,/ 2,/ :_$iz$ “”
the Cartesian component of the Cauchy stress tensor, D is the constitutive matrix, W is the virtual work and V is the element volume. The first integral in eqn (2) yields the element linear stiffness matrix, k,, while the second integral yields the element geometric stiffness matrix, k,. The element stiffness matrix, k,, is obtained by augmenting k, by k,. The geometric nonlinearity is accounted for through the geometric stiffness matrix and through continuous updating of the structure geometry. The material nonlinearity is incorporated by specifying a certain constitutive model consistent with the incremental formulation. A Ramberg-Gsgood type of constitutive model is used in this work, i.e.
CP /---
---
Cl
’
co
w
-x
/
i! /
Fig. 1. Updated Lagrangian configurations ement.
u -=6
of a truss el-
cycle. Results for several examples of truss structures under various types of dynamic loading are presented and compared.
:jl+k(;,.‘I
in which u0 and t0 are the yield stress and yield strain of the material, n is taken as 7 and k = 3/7. This model is shown in Fig. 2. 3. SOLUTION
2. INCREMENTAL
METHOD
FORMULATION
A three-dimensional truss element is shown in Fig. 1. Using an updated Lagrangian frame of reference, the element deformation can be traced using three different configurations: C,, C, and C, as shown in Fig. 1. Configuration C, at time t = 0 is the initial undeformed configuration, configuration C, at time t is a known deformed configuration and configuration C, at time t + At is a neighbouring unknown configuration. Using a tensor definition for the stresses and strains that follows the particle in its deformation [4], the incremental equation of motion of the structure expressed in configuration C, may be written MAf + CAi + K,Ar = AR
In order to obtain the structural response, eqn (1) has to be integrated in time. The time domain is discretized and the structural response is obtained using step-by-step integration coupled with a Newton-Raphson iterative procedure at each time step. The integration algorithm used is the constantaverage acceleration algorithm (y = 0.5, j3 = 0.25) [5]. This algorithm is unconditionally stable for linear analysis. To maintain stability in nonlinear analysis, however, a small time step has to be used. If the structure history is known up to time t and it is required to compute the structural response at time t + At, then at the first iteration the incremental acceleration and velocity may be written
(1)
Af=&Ar--&2f
in which Ar, Ai and dp are vectors of increments of nodal displacement, velocity and acceleration, respectively, M is the mass matrix, C is the damping matrix, K, is the tangent stiffness matrix, and AR is the vector of increments of applied nodal forces. The mass matrix for the structure, M, is assembled from the element lumped mass matrices, while the damping matrix, C, can be established for the structure if the structure damping characteristics are known. The tangent stiffness matrix of the system, K,, is assembled from the element stiffness matrix, k,, which can be derived using the virtual displacements principle, i.e. DE,be,dV + s”
C3)
m5c,,,dV = W
sY
(2)
in which tL and cN are the linear and nonlinear components of the Green-Lagrange strain tensor, z is
2 Ai=-Ar-2i. At
ov
I
0.5
I
1.0
I
1.5
I
2.0
Strain ratio, E/E~ Fig. 2. Ramberg-Osgood
constitutive model.
I
2.5
11
Nonlinear dynamic analysis of lattice structures Substituting
eqns (4) and (5) into eqn (1) yields
1
-$M+-&CfK,
Ar
=AR+[X+;M]i+2Mi.
(6)
Equation (6) may be solved for Ar and then substituted into eqns (4) and (5) to determine M and Ai. At the end of the first iteration the displacement, velocity and acceleration are updated by adding their incremental components. The current structural resistance, F, is F=F+K&
(7)
and the residual force increment, AR,, is AR,=R-@S?+Ci+Fj.
(8)
For the second and subsequent iterations, the system is solved under the residual force obtained from eqn (8) -$M+;C+&
1
Ar=AR,
(9)
and the corrections in the incremental velocity and acceleration are obtained using Ar from eqn (9) Al=$Ar At = d Ar.
(11)
At the end of each iteration the displacement, velocity, acceleration and the resistance are updated and a new residual force increment is found using eqn (8). The iterative process continues until a certain convergence criterion is satisfied. In this work a residual force convergence criterion using the Euclidean norm measure with a tolerance of 1% has been used. To account for material nonlinearity, the elasticity modulus has to be updated continuously. Differentiating eqn (3), the instantaneous elastic modulus, E,, will be
4. NUMERICAL
been examined extensively in the literature. The maximum deflection of the truss under three types of dynamic loading was presented by Kassimali and Bidhendi [I]. In this study the maximum deflection and the transient response of the truss under step, triangular and sinusoidal loads have been studied. The damping properties of the truss have been ignored. The maximum deflection response is compared with Kassimali and Bidhendi [I]. The natural period, T,, of the vertical vibration of the truss in its undeformed configuration has been found to be 0.139 set t& = 7.2 Hz). A time step of T,/50 was used in the analysis. Figure 3(a) shows the maximum deflection response of the truss under varying step load. The deflection chosen in this figure corresponds to the first peak value obtained in the nonlinear transient response. As seen in Fig. 3(a) the truss will snap at a step load of approximately 1200 kips, compared with a snap-through load of 1661 kips under static load. The nonlinear transient response under a step forcing function of 800 kips is shown in Fig. 3(b). The maximum deflection response of the truss under a varying triangular load with an impulse duration of 0.3 set is shown in Fig. 4(a). The deflection chosen is the maximum deflection obtained during the impulse duration. From Fig. 4(a) it is seen that the truss will snap at a forcing function of
Displacement, (a) Maximum vertical
A(h)
displacement
a! A
EXAMPLES
Several examples of truss structures under dynamic load are presented in this section using the formulation and solution method described earlier. The rest&s obtained from the analyses are compared with available numerical or analytical results wherever possible. 4.1. Ehtic
geometric nonlinear problems
4.1.1. ~~-member toggle truss. Figure 3(a) shows a shallow two-member toggle truss. The snapthrough behaviour of the truss under static load has
Time, t (sac) ib) Transient raspcnse
for step load = 800kipa
Fig. 3. Dynamic response of a toggle truss under a step forcing function.
K. ZHU et a/.
12
Pigs 5(a) and 5(b), the time needed to reach a stable response in Fig. 5(a) is nearly four times that in Fig. 5(b). Further, from Figs S(a) and 5(b) it is clear that higher order modes contribute to the response when AA is negative (i.e. upward). This is because the toggle stiffens as member forces become tensile. I
OV
10
I
I
(a) Maximum vertical
I
I
30 20 Displacement,
40 Atin)
50
displacement
I
60
at A
1 -30w40 Time, (bf Transient
response
t (see) for triangular
loads
Fig. 4. Dynamic response of a toggle truss under a triangular forcing function.
4.1.2. Geodesic truss dome. A geodesic truss dome is shown in Fig. 6(a). The snap-through behaviour of this dome under static load was studied by Papadrakakis [6]. The snap-through load for the dome under static load is about 3 kips. The natural period of vertical vibration of the dome in its initial configuration is 0.00728 set Cr;,= 137.4 Hz). The dynamic response of the dome under a triangular forcing function is investigated here. Two triangular functions with different impulse durations have been used. The maximum deflection obtained during the impulse duration is presented against the forcing amplitude in Fig. 6(a). For the impulse duration chosen, the snap-through load is higher than that for a static load. As the forcing duration is increased, the snap load decreases. Figure 6(b) shows the transient response of the dome under two different triangular forcing functions with the same amplitude but different impulse duration. It is clear from this figure that varying the impulse duration may alter the nonlinear transient response significantly. 4.2. Geometric and material nonlinear problems
approximately 1600 kips which is higher than the buckling load obtained using a step forcing function, However, as the impulse duration is increased it is expected that the critical load under the triangular forcing function will approach that of the step function. The nonlinear transient response under two triangular forcing functions of the same duration is shown in Fig. 4(b). The response during the impulse duration only is shown. From this figure it is clear that a quite different response is obtained when varying the forcing function magnitude. Figures S(a) and 5(b) show the nonlinear transient response of the truss under sinusoidal loading of the same period but different amplitude. Because a nonlinear analysis is used, the truss natural frequency changes as the stiffness changes. The response shown in Figs 5(a) and 5(b) is of an oscillatory nature starting with increasing amplitude followed by resonance. The response then stabilizes as the amplitude starts decreasing. From Figs 5(a) and 5(b) it can also be seen that the resonance starts to appear when the downward deflection at A is close to 40 in. It can be shown with simple calculations that at this deflection the toggle frequency is very close to the forcing frequency (1.666 Hz). The response stabilizes when the toggle becomes nearly horizontal (A, r 50 in). Due to the differences in the forcing function amplitude in
4.2.1. Two-bay cantiIeoer truss. Figure 7 shows a two-bay plane truss. The dynamic response of this 1401
,
,
2
(
4 Time,
,
,
,
6 tkec)
8
,
10
, ,
12
Fig. 5. Dynamic response of a toggle truss under a sinusoidal forcing function.
Nonlinear dynamic analysis of lattice structures 0
I
-
I
assumes a more prominent role than the geometric nonlinearity in this example. In this analysis a time step of 1 x 10m4set has been used. The material constitutive law has been taken according to eqns (3) and (12). Both the linear and nonlinear responses are shown in Fig. 7 and are compared with Noor and Peters [2]. Figure 7 shows the time histories of the maximum deflection and maximum member force of the truss. It is clear from this figure that the nonlinear and linear responses are distinctly different. The results obtained are in good agreement with Noor and Peters [2]. The nonlinear response of the same truss under a sinusoidal forcing function of the same amplitude as the step function has also been investigated. The loading/unloading of the truss members was described through eqns (3) and (12), again neglecting the hysteresis characteristics of the material. The linear and nonlinear time histories of the truss response under the sinusoidal forcing function are shown in Fig. 8.
1
This paper
0
Displacement. (a) Maximum vertical
A (in)
displacement
at A
I
I
0.0025
(b) Transient
I
I
0.005 Time, t (sac)
0.0075’,
response
13
4.2.2. Shallow truss arch. Figure 9 shows a shallow truss arch with a rise-to-span ratio of about 2%. Both the static and dynamic responses under a vertical load applied at the crest have been investigated. A step load function was used for the dynamic load. The static and dynamic responses considering the geometric nonlinearity only (GNL) and both geometric and material nonlinearities (GMNL) are presented in Fig. 10. It is clear from this figure that the GMNL behaviour is different from the GNL behaviour particularly for the dynamic response.
cI.0 1
for triangular’loads
Fig. 6. Dynamic response of a geodesic truss dome.
truss under the step loading shown in the figure was studied by Noor and Peters [2] using mixed formulation. Both geometric and material nonlinearities have been considered. The material nonlinearity
5. CONCLUSION The nonlinear behaviour of truss structures under dynamic loading has been investigated in this paper.
ao= 280N/mm2
~,=0.4052xlO
t
p/-ES_ A -2q
0.6 0.5 -
z 2
1
I
P
I
x104
I
-
This paper
---
Noor 8 Peters (1980)
1
\
Peters
-I
0.4Geometric 8
E z 0.3 -
material nonlinear analysis (GMNL)
: =$ 0.2 E 0.1 -
/
Time, t (set9
Linear elastic CL)
1
Time, t bed
Fig. 7. Linear and nonlinear dynamic responses of a two-bay canteliver truss under a step forcing function.
K. ZHU et al.
14
z ^_
GMNLfi
I_
-“-20eltY3 Time,
Geometric
201
6 material
’
0
I
I
1
2
4
6
IL 8
/
:103
l( IX
time, tbec)
tkec)
Fig. 8. Linear and nonlinear dynamic responses of a two-bay cantiliver truss under a sinusoidal forcing function.
Both geometric and material nonlinearities have been considered in the analysis. Structural responses under different types of forcing functions with varying amplitudes and durations have been presented and compared. Further, the structural responses accounting for different nonlinearities have been compared, and a comparison with the structural response under static load has been made. It has been shown that the critical load using step forcing functions is lower than the critical load obtained from static analysis. For the other types of forcing functions used, the critical load is always higher than that obtained using a step forcing function and can be higher than the static buckling load depending on the impulse function duration as compared to the natural period of the structure. However, as the impulse duration increases, the
2 /‘@‘Eryy; 1
3
8
8
4
critical load will approach that determined using a step forcing function. For the same forcing function, varying either the impulse duration or amplitude can completely alter the structural response. Since the structure stiffness is changing continuously, resonance may appear at a certain stage during the response and higher order modes may contribute to the response when the structure stiffens. In comparing the linear and nonlinear responses in Figs 7 and 8 it is clear that the inclusion of nonlinearities can change the response completely. Further, as shown in Fig. 10 the structural response using GNL and GMNL are quite different. Therefore it can be concluded from these figures that nonlinearities have a profound effect on the dynamic response of the truss structure.
10fA
7
5 34.3m
34.3m
E = 7.17~10” N/m’ A - 1.6x10-’ rd (cbm6) - 1.3x104 In’ (web) P - 2766 ks 4 = 2.8X10’ N/d f - 0.403x10-y
I
I
I
Fig. 9. Shallow truss arch.
I
-I
Nonlinear dynamic analysis of lattice structures QNL=Qeometric
Nonlinear
GMNL=Geometric
I
STATIC
Displacement.BA (a)
Static
Analysis
8 Material
I
15
3.0 -
Nonlinear
Analysis
DYNAMIC
(m)
Time, t (set)
response
(b) Dynamic
response
Fig. 10. Static and dynamic responses of a shallow truss arch.
Acknowledgemenrs-This project has been supported by funds made available by the Australian Research Council (ARC) and from the Australian International Development Assistance Bureau (AIDAB). The authors wish to thank Mr Warren Traves for proof-reading the manuscript. REFERENCES
1. A. Kassimali and E. Bidhendi, Stability of trusses under dynamic loads. Compur. Srrucr. 29, 381-392 (1988). 2. A. K. Noor and J. M. Peters, Nonlinear dynamic analysis of space trusses. Compur. Merh. Appl. Mech. Engng 21, 131-151 (1980).
3. E. G. Abu-Saba, W. M. McGinley and R. C. Montgomery, Dynamic analysis of truss-beam system. AXE J. Aerospace Engng 4, 341-354 (1991). 4. F. G. A. Al-Bermani and S. Kitipomchai, Nonlinear analysis of thin-walled structures using least element/member. AXE J. Srrucr. Engng 116, 215-234 (1990).
5. N. M. Newmark, A method of computation of structural dynamics. AXE J. Engng Mech. Div. 85, 67-94 (1959).
6. M. Papadrakakis, Inelastic post-buckling analysis of trusses. ASCE J. Srrucr. Engng 109, 2129-2147 (1983).
00457949(94)EO125-L
NONLINEAR
DYNAMIC ANALYSIS STRUCTURES
Computm & Srrucrurrs Vol. 52. No. I, pp. 9-15, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Brhain. All rights reserved co45-7949/94-~7.00 + 0.00
OF LATTICE
K. ZHU, F. G. A. AL-BERMANI and S. KITIFWRNCHAI Department of Civil Engineering, The University of Queensland, St Lucia 4072, Australia (Receiued 6 April 1993)
Abstract-A computational procedure for predicting the geometric and material nonlinear dynamic response of space trusses is presented. An updated Lagrangian method, based on the incremental formulation of the equation of motion, is employed. The constitutive law is cast into the Ramberg-Osgood polynomial model. Numerical studies of two- and three-dimensional truss structures under various dynamic loading are made. The structure dynamic response including different types of nonlinearity is compared with the linear response and with the results obtained from static analysis.
loading has attracted a good deal of interest to date. In contrast, the nonlinear structural response under dynamic loading has not received much attention. Lattice structures usually have higher natural frequencies than comparable solid structures, because of their relatively high stiffness-to-mass ratio. The nonlinearity of lattice structures under dynamic loading can stem from various origins: (i) geometrical-due to the variations in the geometrical properties of the structure as the load progresses; (ii) material-due to the inherent nonlinear behaviour of the materials under load; (iii) inertia-depending on the dynamic motion and the structural deformations; and (iv) dampingdepending on the structural joints and material. Kassimali and Bidhendi [l] have investigated the stability of truss structures under dynamic loading. The dynamic response of trusses with both geometric and material nonlinearities was presented by Noor and Peters [2] using a mixed formulation. More recently, a flexibility method was presented by Abu-Saba et al. [3] for predicting the dynamic characteristics of truss-beam structures. In this investigation, only those nonlinearities which affect the tangent stiffness of the structure have been accounted for (i.e. geometric and material nonlinearities). An incremental formulation of the equation of motion has been used, based on an updated Lagrangian frame of reference. The incremental equation of motion is discretized both in physical coordinates and time. Various implicit time integration schemes are used to obtain the transient structural response. During this incremental process, the effects of both the geometrical variations and material nonlinearity are accounted for through the tangent stiffness operator. An incremental/iterative solution scheme is employed and a residual force convergence criterion is used to terminate a loading
NOTATION
c
co,c, c, 1
D ; k 4, k,, k, KT
M n R r, i, F AR Ar, Ai, ti AR, :r T” V W
damping matrix of structure initial undeformed, known deformed and neighbouring configuration, respectively constitutive matrix instantaneous elastic modulus natural frequency current structural resistance parameter defining stress-strain curve element linear, geometric and tangent stiffness matrices tangent stiffness matrix of structure mass matrix of structure parameter defining stress-strain curve residual force total displacement, velocity and acceleration, respectively incremental applied nodal force incremental displacement, velocity and acceleration, respectively residual force increment time time increment natural period volume of element virtual work
Greek symbols B? Y c co CL. CN
0 00 T
coefficients used in Newmark method strain yield strain linear and nonlinear strain components of Green-Lagrange strain tensor stress yield stress Cauchy stress tensor
I. INTRODUCTION
in a structural system can have a profound effect on the transient structural response. The nonlinear behaviour of structures under static Nonlinearities
9
K. ZHU et al.
10
/ ,/ 2,/ :_$iz$ “”
the Cartesian component of the Cauchy stress tensor, D is the constitutive matrix, W is the virtual work and V is the element volume. The first integral in eqn (2) yields the element linear stiffness matrix, k,, while the second integral yields the element geometric stiffness matrix, k,. The element stiffness matrix, k,, is obtained by augmenting k, by k,. The geometric nonlinearity is accounted for through the geometric stiffness matrix and through continuous updating of the structure geometry. The material nonlinearity is incorporated by specifying a certain constitutive model consistent with the incremental formulation. A Ramberg-Gsgood type of constitutive model is used in this work, i.e.
CP /---
---
Cl
’
co
w
-x
/
i! /
Fig. 1. Updated Lagrangian configurations ement.
u -=6
of a truss el-
cycle. Results for several examples of truss structures under various types of dynamic loading are presented and compared.
:jl+k(;,.‘I
in which u0 and t0 are the yield stress and yield strain of the material, n is taken as 7 and k = 3/7. This model is shown in Fig. 2. 3. SOLUTION
2. INCREMENTAL
METHOD
FORMULATION
A three-dimensional truss element is shown in Fig. 1. Using an updated Lagrangian frame of reference, the element deformation can be traced using three different configurations: C,, C, and C, as shown in Fig. 1. Configuration C, at time t = 0 is the initial undeformed configuration, configuration C, at time t is a known deformed configuration and configuration C, at time t + At is a neighbouring unknown configuration. Using a tensor definition for the stresses and strains that follows the particle in its deformation [4], the incremental equation of motion of the structure expressed in configuration C, may be written MAf + CAi + K,Ar = AR
In order to obtain the structural response, eqn (1) has to be integrated in time. The time domain is discretized and the structural response is obtained using step-by-step integration coupled with a Newton-Raphson iterative procedure at each time step. The integration algorithm used is the constantaverage acceleration algorithm (y = 0.5, j3 = 0.25) [5]. This algorithm is unconditionally stable for linear analysis. To maintain stability in nonlinear analysis, however, a small time step has to be used. If the structure history is known up to time t and it is required to compute the structural response at time t + At, then at the first iteration the incremental acceleration and velocity may be written
(1)
Af=&Ar--&2f
in which Ar, Ai and dp are vectors of increments of nodal displacement, velocity and acceleration, respectively, M is the mass matrix, C is the damping matrix, K, is the tangent stiffness matrix, and AR is the vector of increments of applied nodal forces. The mass matrix for the structure, M, is assembled from the element lumped mass matrices, while the damping matrix, C, can be established for the structure if the structure damping characteristics are known. The tangent stiffness matrix of the system, K,, is assembled from the element stiffness matrix, k,, which can be derived using the virtual displacements principle, i.e. DE,be,dV + s”
C3)
m5c,,,dV = W
sY
(2)
in which tL and cN are the linear and nonlinear components of the Green-Lagrange strain tensor, z is
2 Ai=-Ar-2i. At
ov
I
0.5
I
1.0
I
1.5
I
2.0
Strain ratio, E/E~ Fig. 2. Ramberg-Osgood
constitutive model.
I
2.5
11
Nonlinear dynamic analysis of lattice structures Substituting
eqns (4) and (5) into eqn (1) yields
1
-$M+-&CfK,
Ar
=AR+[X+;M]i+2Mi.
(6)
Equation (6) may be solved for Ar and then substituted into eqns (4) and (5) to determine M and Ai. At the end of the first iteration the displacement, velocity and acceleration are updated by adding their incremental components. The current structural resistance, F, is F=F+K&
(7)
and the residual force increment, AR,, is AR,=R-@S?+Ci+Fj.
(8)
For the second and subsequent iterations, the system is solved under the residual force obtained from eqn (8) -$M+;C+&
1
Ar=AR,
(9)
and the corrections in the incremental velocity and acceleration are obtained using Ar from eqn (9) Al=$Ar At = d Ar.
(11)
At the end of each iteration the displacement, velocity, acceleration and the resistance are updated and a new residual force increment is found using eqn (8). The iterative process continues until a certain convergence criterion is satisfied. In this work a residual force convergence criterion using the Euclidean norm measure with a tolerance of 1% has been used. To account for material nonlinearity, the elasticity modulus has to be updated continuously. Differentiating eqn (3), the instantaneous elastic modulus, E,, will be
4. NUMERICAL
been examined extensively in the literature. The maximum deflection of the truss under three types of dynamic loading was presented by Kassimali and Bidhendi [I]. In this study the maximum deflection and the transient response of the truss under step, triangular and sinusoidal loads have been studied. The damping properties of the truss have been ignored. The maximum deflection response is compared with Kassimali and Bidhendi [I]. The natural period, T,, of the vertical vibration of the truss in its undeformed configuration has been found to be 0.139 set t& = 7.2 Hz). A time step of T,/50 was used in the analysis. Figure 3(a) shows the maximum deflection response of the truss under varying step load. The deflection chosen in this figure corresponds to the first peak value obtained in the nonlinear transient response. As seen in Fig. 3(a) the truss will snap at a step load of approximately 1200 kips, compared with a snap-through load of 1661 kips under static load. The nonlinear transient response under a step forcing function of 800 kips is shown in Fig. 3(b). The maximum deflection response of the truss under a varying triangular load with an impulse duration of 0.3 set is shown in Fig. 4(a). The deflection chosen is the maximum deflection obtained during the impulse duration. From Fig. 4(a) it is seen that the truss will snap at a forcing function of
Displacement, (a) Maximum vertical
A(h)
displacement
a! A
EXAMPLES
Several examples of truss structures under dynamic load are presented in this section using the formulation and solution method described earlier. The rest&s obtained from the analyses are compared with available numerical or analytical results wherever possible. 4.1. Ehtic
geometric nonlinear problems
4.1.1. ~~-member toggle truss. Figure 3(a) shows a shallow two-member toggle truss. The snapthrough behaviour of the truss under static load has
Time, t (sac) ib) Transient raspcnse
for step load = 800kipa
Fig. 3. Dynamic response of a toggle truss under a step forcing function.
K. ZHU et a/.
12
Pigs 5(a) and 5(b), the time needed to reach a stable response in Fig. 5(a) is nearly four times that in Fig. 5(b). Further, from Figs S(a) and 5(b) it is clear that higher order modes contribute to the response when AA is negative (i.e. upward). This is because the toggle stiffens as member forces become tensile. I
OV
10
I
I
(a) Maximum vertical
I
I
30 20 Displacement,
40 Atin)
50
displacement
I
60
at A
1 -30w40 Time, (bf Transient
response
t (see) for triangular
loads
Fig. 4. Dynamic response of a toggle truss under a triangular forcing function.
4.1.2. Geodesic truss dome. A geodesic truss dome is shown in Fig. 6(a). The snap-through behaviour of this dome under static load was studied by Papadrakakis [6]. The snap-through load for the dome under static load is about 3 kips. The natural period of vertical vibration of the dome in its initial configuration is 0.00728 set Cr;,= 137.4 Hz). The dynamic response of the dome under a triangular forcing function is investigated here. Two triangular functions with different impulse durations have been used. The maximum deflection obtained during the impulse duration is presented against the forcing amplitude in Fig. 6(a). For the impulse duration chosen, the snap-through load is higher than that for a static load. As the forcing duration is increased, the snap load decreases. Figure 6(b) shows the transient response of the dome under two different triangular forcing functions with the same amplitude but different impulse duration. It is clear from this figure that varying the impulse duration may alter the nonlinear transient response significantly. 4.2. Geometric and material nonlinear problems
approximately 1600 kips which is higher than the buckling load obtained using a step forcing function, However, as the impulse duration is increased it is expected that the critical load under the triangular forcing function will approach that of the step function. The nonlinear transient response under two triangular forcing functions of the same duration is shown in Fig. 4(b). The response during the impulse duration only is shown. From this figure it is clear that a quite different response is obtained when varying the forcing function magnitude. Figures S(a) and 5(b) show the nonlinear transient response of the truss under sinusoidal loading of the same period but different amplitude. Because a nonlinear analysis is used, the truss natural frequency changes as the stiffness changes. The response shown in Figs 5(a) and 5(b) is of an oscillatory nature starting with increasing amplitude followed by resonance. The response then stabilizes as the amplitude starts decreasing. From Figs 5(a) and 5(b) it can also be seen that the resonance starts to appear when the downward deflection at A is close to 40 in. It can be shown with simple calculations that at this deflection the toggle frequency is very close to the forcing frequency (1.666 Hz). The response stabilizes when the toggle becomes nearly horizontal (A, r 50 in). Due to the differences in the forcing function amplitude in
4.2.1. Two-bay cantiIeoer truss. Figure 7 shows a two-bay plane truss. The dynamic response of this 1401
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Nonlinear dynamic analysis of lattice structures 0
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assumes a more prominent role than the geometric nonlinearity in this example. In this analysis a time step of 1 x 10m4set has been used. The material constitutive law has been taken according to eqns (3) and (12). Both the linear and nonlinear responses are shown in Fig. 7 and are compared with Noor and Peters [2]. Figure 7 shows the time histories of the maximum deflection and maximum member force of the truss. It is clear from this figure that the nonlinear and linear responses are distinctly different. The results obtained are in good agreement with Noor and Peters [2]. The nonlinear response of the same truss under a sinusoidal forcing function of the same amplitude as the step function has also been investigated. The loading/unloading of the truss members was described through eqns (3) and (12), again neglecting the hysteresis characteristics of the material. The linear and nonlinear time histories of the truss response under the sinusoidal forcing function are shown in Fig. 8.
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4.2.2. Shallow truss arch. Figure 9 shows a shallow truss arch with a rise-to-span ratio of about 2%. Both the static and dynamic responses under a vertical load applied at the crest have been investigated. A step load function was used for the dynamic load. The static and dynamic responses considering the geometric nonlinearity only (GNL) and both geometric and material nonlinearities (GMNL) are presented in Fig. 10. It is clear from this figure that the GMNL behaviour is different from the GNL behaviour particularly for the dynamic response.
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Fig. 6. Dynamic response of a geodesic truss dome.
truss under the step loading shown in the figure was studied by Noor and Peters [2] using mixed formulation. Both geometric and material nonlinearities have been considered. The material nonlinearity
5. CONCLUSION The nonlinear behaviour of truss structures under dynamic loading has been investigated in this paper.
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Both geometric and material nonlinearities have been considered in the analysis. Structural responses under different types of forcing functions with varying amplitudes and durations have been presented and compared. Further, the structural responses accounting for different nonlinearities have been compared, and a comparison with the structural response under static load has been made. It has been shown that the critical load using step forcing functions is lower than the critical load obtained from static analysis. For the other types of forcing functions used, the critical load is always higher than that obtained using a step forcing function and can be higher than the static buckling load depending on the impulse function duration as compared to the natural period of the structure. However, as the impulse duration increases, the
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critical load will approach that determined using a step forcing function. For the same forcing function, varying either the impulse duration or amplitude can completely alter the structural response. Since the structure stiffness is changing continuously, resonance may appear at a certain stage during the response and higher order modes may contribute to the response when the structure stiffens. In comparing the linear and nonlinear responses in Figs 7 and 8 it is clear that the inclusion of nonlinearities can change the response completely. Further, as shown in Fig. 10 the structural response using GNL and GMNL are quite different. Therefore it can be concluded from these figures that nonlinearities have a profound effect on the dynamic response of the truss structure.
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Fig. 10. Static and dynamic responses of a shallow truss arch.
Acknowledgemenrs-This project has been supported by funds made available by the Australian Research Council (ARC) and from the Australian International Development Assistance Bureau (AIDAB). The authors wish to thank Mr Warren Traves for proof-reading the manuscript. REFERENCES
1. A. Kassimali and E. Bidhendi, Stability of trusses under dynamic loads. Compur. Srrucr. 29, 381-392 (1988). 2. A. K. Noor and J. M. Peters, Nonlinear dynamic analysis of space trusses. Compur. Merh. Appl. Mech. Engng 21, 131-151 (1980).
3. E. G. Abu-Saba, W. M. McGinley and R. C. Montgomery, Dynamic analysis of truss-beam system. AXE J. Aerospace Engng 4, 341-354 (1991). 4. F. G. A. Al-Bermani and S. Kitipomchai, Nonlinear analysis of thin-walled structures using least element/member. AXE J. Srrucr. Engng 116, 215-234 (1990).
5. N. M. Newmark, A method of computation of structural dynamics. AXE J. Engng Mech. Div. 85, 67-94 (1959).
6. M. Papadrakakis, Inelastic post-buckling analysis of trusses. ASCE J. Srrucr. Engng 109, 2129-2147 (1983).