A theoretical analysis is presented for estimating the in-space large displacement elastic stability behavior of structures
subjected to either proportional or non-proportional dynamic loads. The analysis adopts the beam-column approach, which models
the structure’s members as beam-column elements. The formulation of the beam-column element is based on the Eulerian approach
allowing for the influence of the axial force on the bending and the torsion stiffness. Also, change in member chord length due to the
axial deformation and flexural bowing are taken into account. Newmark- β method is used as a time integration technique to plot the
time-deformation curves. Damping effect is expressed using Raleigh damping matrix. Several examples have been solved to insure
the accuracy of the present analysis.

A dynamic load may cause instability of a structure, even if the
structure remains stable under a static load of the same magnitude
as the dynamic load .Such phenomenon may occur within the elastic
range. So, the maximum deformation response abruptly increases
at some point with respect to the magnitude of loading. When this
kind of instability problem to be solved ,geometrical non-linearity
must be considered in the dynamic analysis as that in static
instability analysis. To days, studies conducted on dynamic analysis
of frames with geometric non-linearity are mainly concerned with
plane frames. Such studies dealing with three dimensional frames
are seldom found in the literature. The objective of this paper is
to extend the large deformation static analysis of space structures
presented previously by Oran [1] and then by kassimali and
abbsania [2] to deal with the time dependent loading. Members
are assumed to be prismatic connected by frictionless hinges for
trusses and rigid joints for frames.

Basic Member Force Deformation Relations

The basic member force-deformation relations consistent with
the conventional beam column theory [1] Figure 1 are:

Figure 1: Google Earth Imagery Showing River Ethiope.

Both stability and bowing are dependent on the axial force
parameter (qn) and their explicit transcendental and series
expressions are given by Oran [3,1] and kassimali and abbsania [2].

The torsional factor (Ct) in equation [4] considers the influence
of axial force on member torsional stiffness [1]

Member End Effects

Consider an arbitrary prismatic member of a space frame,
and let {F} and {F’} denote member end forces in the global and
local coordinates respectively (Figures 2a & 2b), the relationship
between {F} and {F’} can be written as:

{F}=[R]{¯F} (11)

Figure 2:

Member end effects,

A. Member forces in global coordinates,

B. Member forces in local intermediate coordinates,

C. Relative member deformations and corresponding
forces.

[R]: is the (12x12) orthogonal matrix [5] and defined as:

[r]: is the (3x3) member orientation matrix, refers to the
deformed configuration of the member. This matrix refers to the
deformed configuration and must be updated according to the N-R
type technique [6]. Similarly the member forces {F}are related to
the forces {S} associated with the relative member deformations
(Figures 2b & 2c) by:

{F}=[B]{S} (13)

[B]: is the instantaneous static matrix defined as:

It must be used for the first increment and for the first iteration
only. This matrix can be derived by taking in consideration that the
transformation from the local to the global coordinates to take place
by rotating the member through the angles (β, and ) about the Z-,
Y- and X-axis, (Figures 3a & 3b).The final form of this matrix is [5]:

Figure 3:

Transformation of coordinate system.

A. Rotation of a space frame moment about x-axis.

B. Rotation of axes for arbitrary space frame member.

C and D Rotation of axis for vertical space frame.

When the member is vertical, the initial member orientation
matrix [r_{0}] will be :

CY = 1 for γ=900 and CY =-1 for γ=2700

Current Member Orientation Matrix

Three alternate methods are available in the literature that
are used to determine the current member orientation matrix for
the deformed configuration. Two of these are presented by Oran
[1] and the third one was presented by Kassimali and Abbasania
[2]. One of the procedures of Oran was utilized in the present
analysis. This method is based on the variation of the principal
directions from one section to another, then the incremental joint
orientation matrix is calculated in each increment, and the total
joint orientation matrix is updated for the next increment. The
current member orientation matrix, then, is obtained by averaging
the orientation of the two end sections of the member [1]. Thus, the
current member orientation matrix can be expressed as:

[r]=1/2{[D^{(1)}]+[D^{(2)}]} (18)

Where [D(1)],[D(2)]: are the current section orientation
matrices of the two ends of the member.

The dynamic equation of motion [7,8] for nonlinear systems
can be written as

[M]{X¢¢}+[C]{X¢}+[T]{X} = [F ( t )] (21)

The consistent mass method has been adopted in the present
study. In this method, mass coefficients corresponding to the nodal
coordinates can be evaluated depending on the principle of virtual
work. Accordingly, the mass coefficients can be given in a general
form as:

Damping matrix has been represented using the so-called
Raylieh type of damping which may be expressed as:

The nonlinear dynamic response of structures is investigated in
this work using the step -by -step numerical integration procedure
with a Newton type of iteration performed within each time step
to satisfy the equation of motion .The Newmark -β method with
β=1/4 and γ=1/2 is used

Assuming that the displacement, velocity and acceleration
vectors are known at time to=0

Then, the general procedure of analysis is as follows [9,10]

a) Calculate the following constants

b) form the matrices [T],[M] and [c] for each member ,then,
form the overall matrices by the assemblage of the element
stiffness matrices

c) From the effective stiffness matrix, initially assuming
linear behaviour

[T_{eff}]=a_{1}[M]+a_{2}[C]+[T] (24)

[C] =b_{1} [M]+b_{1}[T] (25)

d) From the effective load vector

e) Solve for the increment displacements (for time t + Δt )

f) iterate for dynamic equilibrium

a. i=i+1

b. evaluate the ( i-1 )th approximation of acceleration,
velocity and displacement vectors

c. evaluate the (i-1)th unbalanced joint forces

d. compute the correction of the displacement increment
{ΔX}

e. evaluate the corrected displacement vector

f. check for convergence

g. if “No” ,go to 6.a else return to step 3 to proceed to the next
time step

The response due to a harmonically varying vertical load
at the crown of the dome shown in Figure 4 has been studied by
Remesth [11] using the finite element approach with two elements
per member . Newmark-method with ( β=1/4 and γ=1/2 ) has
been used. The peak value of the concentrated load is chosen to be
(34.4MN) with time step size of (0.0025sec) and natural frequency
of the applied load is taken to be (0.15sec).Damping is included
in the analysis and it is determined from a model damping ratio
(5%) for the mode with the longest natural period initially equal
to (0.175sec). This example is analyzed in the present work with
and without damping using Newmark method with the same load
properties, each member is modeled with one element and the
vertical response of the loaded joint for both studies is illustrated
in Figure 5. It is clear that a good agreement may be obtained using
the present analysis including damping effects.

Figure 4:

Framed Dome.

A. Top view.

B. Elevation.

C. Load –time diagram.

Figure 5: Dynamic response of the framed dome.

Geodesic Dome under Triangular Load

The geodesic dome shown in Figure 6 is to be analyzed under
triangular dynamic load applied at the central point [6] Kassimali
and Bedhendi [12] analyzed this structure using a stiffness method
based on an Eulerian formulation accounting for arbitrarily large
displacements. Two time durations (Td=0.005sec, Td=0.01 sec)
were used. It was found that for short impulse duration, the critical
load decreases steadily as duration increases and it approaches
that of the static load case for long time durations. The problem
is solved in this study under the same loading conditions and the
results are shown in Figure 7 which indicates the efficiency of the
computer program developed in the present study to predict the
large displacement behaviour of space frames under dynamic loads.

Figure 6: Geodesic dome.

Figure 7: Load deflection curves for the geodesic dome.

Space Truss with Nine Prismatic Elements

The space truss shown in Figure 8 has been analyzed by Weaver
[5] using linear analysis .The same structure is considered here
to show the importance of studying large deformations for such
structures under dynamic loads. The variation of axial force with
time for members [6,13] for both studies are shown in Figure
9 & 10. It is obvious that at time(t=19 msec),the linear analysis
underestimates the axial forces for the two members by (15%) and
(30%) respectively [14-17].

Figure 8: Space truss with nine prismatic elements

Figure 9: Variation of axial force with time for element.

Figure 10: Variation of axial force with time for element.

The present work shows the importance of considering the large
deformations behaviour in the analysis of space structure under
dynamic loading within elastic range. The beam column which have
been utilized previously for the analysis of plane structures under
static and dynamic loads and also for space structures under static
loading is extended in this work to deal with the analysis of space
structures under dynamic loading.

Although of its simplicity, Rayleigh model may be adopted to
represent the damping effects. Also it is found that the method
of updating the member orientation matrix by averaging the end
section orientation matrices is adequate enough to be adopted
in the analysis of space structures subjected to time dependent
loading (Appendixs 2 & 3).