ANALYSIS OF SHELL-TYPE STRUCTURES SUBJECTED TO p/ 9% TIME-DEPENDENT MECHANICAL AND THERMAL LOADING 

A SEMI-ANNUAL STATUS REPORT 

SUBMITTED TO 

NASA-LEWIS RESEARCH CENTER 

CLEVELAND, OHIO 

G.J. Simitses 

R.L. Carlson 

R. Riff 

Schoo 

Georg 

of Aerospace Engineering 

a Institute of Technology 

At1 anta, Georgia 

ma themat 

solution 

The 

'4 April 1987 

(NASA GRANT: NAG 3-534) 

INTRODUCTION 

The objective of the present research is to develop a general mathematical model and solution methodologies for analyzing structural response of 

thin, metallic shell-type structures under large transient, cyclic or static 

thermomechanical loads. Among the system responses, which are associated 

with these load conditions, are thermal buckling, creep buckling and ratchetting. Thus, geometric as well as material-type nonlinearities (of high 

order) can be anticipated and must be considered in the development of the 

cal model. Furthermore, this must also be accommodated in the 

procedures. 

SUMMARY OF PROGRESS 

progress update has been elaborated upon in an interim scientific 

report submitted to the sponsor during the summer of 1986. A second report, 

describing the devel oped finite element, the computer code, and several 

(NASA-CR-IS0349) ANALYSIS OF ZHELL-TYPE N87- 197 56 

S'IRUC'IORES SUEJ€CTED TC TIHE-CEEEICENT 

EECHANICAL ANC 'lEEfiHAL LOBClh'G Semiannual Status Report (Georgia Xast. cf Tech.) 22 p Unclas 

CSCL 20K 63/39 43650 
appl i cati ons to cy1 i ndri cal and spheri cal she1 1 configurations, is being 

prepared. It is expected that this second scientific report will be submitted to the sponsor during the summer of 1987. 

A complete true ab-inito rate theory of kinematics and kinetics for 

continuum and curved thin structures, without any restriction on the magnitude of the strains or the deformation, was formulated. The time dependence 

and large strain behavior are incorporated through the introduction of the 

time rates of the metric and curvature in two coordinate systems; a fixed 

(spatial) one and a convected (material ) coordinate system. The relations 

between the time derivative and the covariant derivative (gradient) have been 

developed for curved space and motion, so that the velocity components supply 

the connection between the equations of motion and the time rate of change of 

the metric and curvature tensors. 

The metric tensor (time rate of change) in the convected material 

coordinate system is linearly decomposed into elastic and plastic parts. In 

this formulation, a yield function is assumed, which is dependent on the rate 

of change of stress, metric, temperature, and a set of internal variables. 

Moreover, a hypoelastic law was chosen to describe the thermoelastic part of 

the deformation. 

A time and temperature dependent viscoplastic model was formulated in 

this convected material system to account for finite strains and rotations. 

The history and temperature dependence were incorporated through the introduction of internal variables. The choice of these variables, as well as 

their evolution, was motivated by phenomenological thermodynamic considerations. 

The nonisothermal elastic-viscoplastic deformation process was described 

completely by "thermodynamic state" equations. Most investigators (in the 

area of viscoplasticity) employ plastic strains as state variables. Our 

study shows that, in general, use of plastic strains as state variables may 

lead to inconsistencies with regard to thermodynamic considerations. 

Furthermore, the approach and formulation employed by all previous investigators lead to the condition that all plastic work is completely dissipated. 

This, however, is in contradiction with experimental evidence, from which it 

emerges that part of the plastic work is used for producing residual stresses 

in the lattice, which, when phenomenologically considered, causes hardening. 
Both limitations are not present in our formulation, because of the inclusion 

of the "thermodynamic state'' equations. 

The obtained complete rate field equations consist of the principles of 

the rate of the virtual power and the rate of conservation of energy, of the 

constitutive relations, and of boundary and initial conditions. These 

formulations provide a sound basis for the formulation of the adopted finite 

element solution procedures. 

One of the most challenging aspects of finite strain formulations is to 

locate analytical solutions with which to compare the proposed formulation. 

Typically, as a first problem, a large strain uniaxial test case was analyzed. The case considered examines the rate-dependent plastic response of a 

bar to a deformation history that includes segments of loading, unloading, 

and re1 oadi ng, each occurri ng at varyi ng strain and temperature rates. 

Moreover, it was shown that proposed formulation generates no strain energy 

under a pure rigid body rotation. These are surely important demonstrations 

but they only represent a partial test because the principal stretch directions remain constant. Finally, a problem which was discussed by Nagtegaal 

and de Jong, and others too, as a problem which demonstrates limitations of 

the consitutive models in many strain formulations, is the Couette flow 

problem. 

test problems show that: 

she 

A 

This problem is solved as a third example. The results of these 

- The formulation can accommodate very large strains and rotations. 

- The formulation does not display the oscillatory behavior in the 

streses of the Couette flow problem. 

- The model incorporates the simplifications associated with rateinsensitive elastic response without losing the ability to model 

rate temperature dependent yield strength and plasticity. 

The problem of buckling of shallow arches under transient thermomechanical load was investigated next. 

The quasi-linear nature of the principle of the rate of virtual power 

suggests the adoption of an incremental approach to numerical integration 

with respect to time. The availability of the field formulation provides 

assurance of the completeness of the incremental equations and allows the use 

of any convenient procedure for spatial integration over the domain V. In 

the present instance, the choice has been made in favor of a simple first 
order expansion in time for the construction of incremental solutions from 

the results of finite element spatial integration of the governing equations. 

The procedure employed permits the rates of the field formulation to be 

interpreted as increments in the numerical solution. This is particularly 

convenient for the construction of incremental boundary condition histories. 

Even under the condition of static external loads and slowly growing 

creep effects, the presence of snap-through buckling makes the inertia 

effects significant. In dynamic analyses, the applied body forces include 

inertia forces. Assuming that the mass of the body considered is preserved, 

the mass matrix can be evaluated prior to the time integration using the 

i ni ti a1 configuration. 

Finite element sol uti on of any boundary-Val ue problem involves the 

solution of the equilibrium equations (global) together with the constitutive 

equations (local). Both sets of equations are solved sinultaneously in a 

step by step manner. The incremental form of the global and local equations 

can be achieved by taking the integration over the incremental time step 

At=tj+l-tj. The rectangular rule has been applied to execute the resulting 

time integration. 

Clearly, the numerical solution involves iteration. A simplified 

version of the Riks-Wempner constant-arc-length method has been utilized. 

This iteration procedure which is a generalization of the displacement 

control method also allows to trace the nonlinear response beyond bifurcation 

points. In contrast to the conventional Newton-Raphson techniques, the 

iteration of the method takes place in the velocity and load rate space. The 

load step of the first solution in each increment is limited by controlling 

the length ds of the tangent. Either the length is kept constant in each 

step or it is adapted to the characteristics of the solution. In each step 

the triangular-sized stiffness matrix has to be checked for negative diagonal 

terms, indicating that a critical point is reached. 

The analysis was performed with the aid of 24 paralinear isoparametric 

elements. The paralinear isoparametric element is intended for the analysis 

of oriented structures where the geometry is such that the thickness is small 

compared to other dimensions. The characteristics of the element are defined 

by the geometry and interpolation functions, which are linear in the thickness direction and parabolic in the longitudinal direction. Consequently, 
the element allows for shear strain energy since normals to a mid-surface 

before deformation remain straight, but not necessarily normal to the midsurface after deformation. 

The devel oped sol ution scheme is capable of predicting response which 

includes pre- and post-buckling with thermal creep and plastic effects. The 

solution procedure was demonstrated through several examples which include 

both creep and snap-through behavior. 

The last set of problems which are under investigation consists of creep 

To develop geometrically nonlinear, doubly-curved finite shell elements 

the basic equations of nonlinear shell theories have to be transferred into 

the finite element model. As these equations are in general written in 

tensor notation, their implementation into the finite element matrix formulation requires considerable effort. 

The nonlinear element matrices were derived directly from the incrementally formulated nonlinear shell equations, by using a tensor-oriented 

procedure. A modified version of the classical thin shell theory based on 

the Kirchoff-Love hypotheses, capable of large strains and rotations, is 

presently employed. For this formulation, we are using five "natural" 

degrees of freedom per mid-surface shell node: three incremental velocities 

and two rates of rotations about the material coordinates a and a . 

This element was introduced to the solution procedure described earlier. 

Few examples of creep and thermal buckling of a cylindrical panel and spherical cap are currently under investigation. 

In connection with the progress to date, two papers were published in 

the AIAA Journal and one in the "NASA-U. of Akron Sponsored Meeting'' Proceedings. Copies of these papers have been mailed to the sponsor. Moreover, 

two papers appeared in the Proceedings of the 28th AIAA/ASME/ASCE/AHS SDM 

Conference. 

or thermal buckling with plastic effects of shells of revolution. 

1 2 

Copies of these two papers are attached herewith. 

FUTURE TASKS 

With regard to additional future tasks, one must include the development 

of a finite element accommodating the more general shell theory formulations 

(A,B, & C), and the incorporation of this into a code. Moreover, threedimensional effects, especially those associated with normal stresses, will 
be incorporated, either through modification of the developed shell theory or 

through local-global finite element procedures in which the local part will 

be based on three-dimensional analysis. 

Finally, another important extension is to replace the material constitution (initially homogeneous and isotropic-metal) by a more general one that 

allows the analysis of layered fiber-reinforced composites. 
A COLLECTION 

OF 

TECHNICAL PAPERS 

PART 1 

Structures and Materials 

(Sessions #I-3, 6-9, 12-15, 18-21, 24, 

27-29, 32-35) 

AIAAIASMEIASCEIAHS 

28TH STRUCTURES, STRUCTURAL DYNAMICS 

AND MATERIALS CONFERENCE 

April 6-8, 1987/Monterey, California 
87-0804 

SOLUTION METHODS FOR ONE-DIMENSIONAL VISCOELASTIC PROBLEHS 

by 

John M. Stubstad* and George J. Simitses** 

School of Engineering Science and Mechanics 

Atlanta, Georgia 30332 

Georgia Institute of Technology 

Abstract 

A recently developed differential methodology for solution 

of one-dimensional nonlinear viscoelastic problems is presented. 

lysing the example of an eccentrically loaded cantilever beamcolumn, the results from the differentid formulation are compared to results generated using a previously published integral 

solution technique. It is shown that the results obtained Gom 

these distinct methodologies exhibit a surprising high degree of 

correlation with one another. A discussion of the various factors 

affecting the numerical accuracy and rate of convergence of these 

two proredures is also included. Finally, the inlluences of some 

"higher order" effects, such as straining along the centroidal axis 

are discussed. 

I. htroductioa 

A number of methods are available to solve viscoelastic 

problems in which the behavior of the material may be adequately characterized by a,linear viscoelastic operator and where 

the deformation of the body is suRciently small to allow the 

use of a linear kinematic formulation [1,2]. Commonly, integral 

transform methods, separation of variables, series expansions 

or other such techniques provide methodologies wherein exact 

clored form solutions may be derived. When exact solutions 

cannot be obtained, approximate techniques, such BS one proposed by Schapery (31, may provide an alternate approach. 

The inclusion of nonlinear effects in the analysis significantly reducer the mathematical tractability of the problem. 

These nonlinear influences can be induced by geometric factors 

resulting from the magnitude of the deformation or Gom ~OSS 

rotation of cross sections. Alternatively, nonlinearities in the 

material response may need be included to provide an accurate 

model for material behavior. 

~ 

*Currently with the Aerospace Corporati< . El Segundo, California; Member ASME. 

**Professor of Aerospace Engineering. 

Associate Fellow AIM; Member ASMECopyighI 0 American Institute of AcronanUn and 

Astronautics. Inc.. 1987. All rights resewed. 

Independent of whether these nonlinearities are produced 

by geometric or material effects, they invariably result in an 

overall formulation governed by nonlinear equations. Thus, the 

solution methods mentioned above, applicable to linear problems, cannot be employed. As such, approximate solution methods have been developed and are routinely employed to malyze 

such problems [4]. 

One of these methods is to idealize the problem is such a 

manner so as to generate inherent simplifications to the governing relations. A classical example of this technique was the utilization of an ideal"'1" cross sectional geometry in early column 

creep buckling studies 151. With this approximation, the equations governing equilibrium of the column were reduced to much 

simpler forms involving the "average" stresses in the flanges of 

the ideal beam-column. 

Another approach used extensively was to restrict considerations to only certain types of time dependent material behavior [6]. In some cases, this involved retaining only secondary 

creep behavior in the material model. Alternatively, and especially when "power law" type constitutve laws were used, 

the constants or exponents of the law were restricted to spe 

cial values for which closed form solution was possible [i]. In 

a few cases, this simplification of the material model, as well 

as the aforementioned geometric simplification technique, were 

employed simultaneously to enable solution. A survey of most 

of these methods has been provided by Hoff [SI. 

An exact numerical solution technique for geometrically 

nonlinear viscoelastic problems has been presented by Rogers 

and Lee [9]. A recent paper by the authors [lo] provides a 

method for bounding the solution for such prohlems. In both 

of these techniques, the solution to the problem is formulated 

in terms of an integral equation which is nonlinear with respect to both time and space. From this, solutions may then 

be readily obtained using relatively standard numerical techniques. Generally, both the exact and bounding technique can 

be employed for problems wherein the response of the material may be adequately characterized using a linear viscoelastic 

model, but where the resulting time dependent deformation of 

the body warrants the use of a nonlinear kinematic formulation. 

... .... .. ......... <...a, 

.... . .~ ............ .. - ............ __ ... . ~~ 
Since the integral solution technique can be applied only 

when the response of the material either is or may be reasonably approximated as being linear, problems involving nonlinear 

viscoelastic material behavior cannot be addressed using these 

methods. rnfortunatdy, many materials, and especially the elevaied temperature behavior of most metals, require such nonlinear characterizations. Consequently, an alternate solution procedure for one-dimensional problems involving nonlinear kinematic 

and nonlinear material effects has been developed. This method, 

hereinafter referred to as the differential formulation, is based 

on the direct solution of the nonlinear differential equations of 

equilibrium. 

Similar to the integral method, the overall procedure is 

predicated on the assumption of a quasistatic response. This 

asumption effectively "decouples" the temporal and spatial dependence of the problem in such a manner so as to allow the 

pnerd solution to be treated as the combination of the sohtions to a nonlinear "houndary value" problem and a nonlinear 

-initial value' problem. The first of these, the equations characterizing the time dependent states of quasistatic equilibrium, 

are solved through the use of a Newton type method Ill]. In 

contrast, the "initial value" problem, resulting from the nonlinear constitutive law. governs the manner by which the body 

progresses from one state of quasistatic equilibrium to the SUCreeding one. Sumerical solutions for this part of the problem are 

eerierated using a fourth order Runge-Kutta method. Using this 

technique. pro1)lems involving the nonlinear thermoviscoelastic 

bhavior of thin structural members have recently been examined 12,. 

In addit ion to presenting the differential formulation technique. a comparison of results obtained through the use of the 

integral and differential formulations is provided. The prohleiti 

of an eccentrically loaded viscoelastic cantilever beani-column is 

emploved as the vehicle through which the comparison is performed. Because of the inherent limitation of the integral technique. this comparison is restricted to the consideration of a 

linear viscoelastic material. The specific case considered is that 

of the three parameter viscoelastic solid which has been examined in a number of st.udies [10,13,14]. The results obtained 

from these.two distinct methods of solution exhibit a surprisingly high degree of correlation with one another thereby establishing a high level of confidence in the vdidity of the methods. 

Finally. the differential formulation is employed to examine the 

influences of some of the "higher order" effects for the class of 

prohlenis under consideration. 

11. Integral Formulation 

Since the integral solution is available in literature [9,10], 

only a brief outline of its development is presented herein. The 

459 

solution is based on the assumption that the beam-colunn is 

thin and composed of a linearly viscoelastic material. Its geomerty in the deformed configuration is illustrated in Fig. l. 

Note that the eccentric load is assumed to be applied quasistatically and its direction does not vary with time. 

Reference line extensional strains are assumed to be negligibly smalL Thus the coordinate s' is employed to specify position in both the initial and deformed configurations. A nondimensional coordinate, s, is defined by dividing 6' by the length 

of the beam, L. Assuming a linear distribution of the strains 

through the depth, bending thus occuring within a BernoulliEuler context, results in a moment-curvature relationship given 

by 

k(S',t) = (!) /' -aD J(1 - T)[-)] dT ' (1) 

where s(J',~) denotes the curvature and M(s',T) the bending 

moment at location s'. I is the moment of inertia of the beam 

and J(1) the creep compliance of the material. For the eccentric 

load, R(t), the nioment at position s' is given by 

Figure 1. Beam-Column Geometry for Integral Formulation 

From kinematic considerations, it is noted that 

aw,q K(S',t) =- 

- az(S', 1) = cos qq s', 1) ad' 

as# 

as, 

= sin &(st, I) 

The boundary conditions for the problem are 

(3) 

4(04- = 0 

(4) M( 1 ;y = aR( t) cos d[L, t) 

where a(t) = 4(L,t) for a "rigid" extension. 

Substituting Eqns. (2) and (3) into Eqn. (1) followed by 

differentiatlon with respect to 8 yields the governing differential equation. Using the methodology detailed in 191- it can be 
shown that, for quiescent initial conditions, the solution for this aU ad 

equation is given by 55 N= -F(1 i- + ~((acosd(L) + W(J) - r(L))) (110) 

and 

~(s,r) = (I) L2 [J(o)R(t)@(s,t) + 1' J'(t - r)R(r)Ws,t)dr] 

hl = F(a cos o(L) x(5) - w( L 1) (1Ihl 

where N and M, the force and tiioment resultants. respectively. 

are defined as 

(5) 

where 

and 

N = lnlld.4 and hI = (1%. h) 

Based upon an additive decomposition for total strain. et. 

such that et = e, + cc where ec and c, represent the elastic and 

creep strain components, respectively, yields, after substitution 

into Eqns. (11) and (12). 

g(5,u) = u, 0 5 u 5 s 

=s. S<USl 

(7) 

and where the prime, ( 0, in Eqn. (5) denotes differentiation 

with respect to the argument of the function. 

Note = -F( lt x+z " (a cos b( t 1- w(s)- x( L )) ) - S, ( 13a ) that Eqn. (5) represents a time convolution of anonlinear Fredhoh type integral equation, Eqn. (6). Numerical solutions for this set of equations are accomplished using Picard's 

method of successive substitutions [15]. The spatial integrals 

are approximated using Newton-Coates formulae; a fixed step 

and 

(136) EIz 86 = F(~casp(L) -i W(S) - ~(t)) - 11, 

trapezoidal rule is used for the time convolution. -. 

where A represents the area of the cross section. I denotes its 

111. Differential Formulation : moment of inertia and the "pseudo-resultants" Xc and >Ic are 

defined by 

Similar to the integral formulation, the differential formulation for this prohlem is based on the assumption that bending 

of the hem occurs in accordance with the Euler-Bernoulli hy- K, = Ec,dA and XI, = qEt, d.4 ll4O.b). 

potheses. The functions u(s,/) and w(s,f) are employed to denote, respectively, the axial and transverse deflection of points 

on the centroidd axis. From this, the extensional strain dong 

the centroidd axis, eo, is approximately given by 

Numerical solutions for Eqns. (14) are cotiipured using a 

modified Newton type nrethod suggested by Thurston .13. To 

illustrate this method consider a nonlinear differential term of 

the form du"dw" where du and dx are differentials of the functions u and w and m and n represent integer exponcnt5. t\scunling that close trial solutions it and 1 are available which differ 

from the exact solution hy the small quantities 4w and lu so 

that = fit Au and H, = + As, then 

eo 55 - al law2 (8) 

Sote that the term :($)' has been neglected as small in comparison to 5. If, in addition, both the strain at the centroidal 

axis and 2 are small in comparison to 1 then it is relatively 

as + Z(Z) 

Thus, for a linear variation of strain through the cross section, 

this yields 

where j( ) denotes a nonlinear function of fi and in. and 0': 

indicates terms of order AuAw and higher. If the trial solution 

is indeed close to the true solution, then the corrections. lu 

(10) and As, will be small. Consequently. the quadratic and higher 

~,~~l~~i~~ the Principle of Virtual Work followed hy integrating by parts and then manipulating algehraicdly eventually 

yields the equilibrium equations 

order terms in the corrections sill he negli-:' le in coniparison 

to the linear terms and therefore may be neglected. Thus. under 

these conditions, the right-hand side of Eqn. (15) may be closelv 

approximated by the linearized form consisting of just the first 

. 

three terms on the right-hand side. 

ORIGINAL PAGE IS 

OF POOR QUALITY 460 
the process repeated until convergence is obtained. 

Table 1. Differential Elastic Solution Versus Number 

Sumerical of Nodes for PIP, = 0.75 and a/L = 0.05 solution of Eqns. (13) also requires evaluation 

of P/P, = 0.75. The Euler load, P,, here is based on a perfect 

geometry and use of the instantaneous compliance of the mate- 

- rial, J(0). Table 2 provides a similar comparison for the same 

loading and eccentricity ratio but for the integral formulation. 

of the "pseudo-resultants" N, and M, at each point of the finitedifference mesh. This is accomplished by evaluating the accumulated creep strain at select number of points across the cross 

. section and then using a three point Newton-Coates quadrature 

Transverse Deflection for Various Numbers of Elements (in) 

Number of Elements 

formula repetitively to approximate the area integral. Evaluaticn of the accumulated creep strain at each of these points is 

acconiplished through the use of a fourth order Runge-Kutta inregration routine to integrate the constitutive law expressed in 

a differential format. 

IV. Example Problem 

The specific example considered is that of a 12 in. long 

s = s'/L 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

1.0 

40 

0.000000 

0.024653 

0.098189 

0.219074 

0.384931 

0.592512 

0.837815 

1.116191 

1.422456 

1.750997 

2.095877 

20 

0.000000 

0.024705 

0.098307 

0.219332 

0.385376 

0.593187 

0.838755 

1.117426 

1.424007 

1.752879 

2.098098 

% Diff 

0.130 

0.120 

0.118 

0.116 

0.114 

0.112 

0.111 

0.109 

0.107 

0.106 

beam-column. For simplicity, a square cross section of dimension s = S'IL 40 10 % Diff 

0.5 in. has been assumed. It is also assumed that the beam- 0.0' 0.000000 0.000000 

0.1 0.024673 0.024809 0.551 

0.2 0.098189 0.585 column is fabricated from a material which can be modeled as 

a three parameter viscoelastic solid. The creep compliance for 0.3 0.219074 0.220289 0.555 

0.4 0.384931 0.387110 0.566 

this model. illustrated in Fig. 2, is given by 0.5 0.592512 0.595755 0.547 

0.098763 

0.6 0.837815 0.842435 0.551 

0.7 1.116191 1.122177 0.536 

J(r)/J(O) = 1 + !Ez/El]e-''+* (16) 0.8 1.422456 1.430090 0.535 

0.9 1.750997 1.760165 0.524 

where r, = ,'El. For these computations, the numerical values 1.0 2.095877 2.106817 0.522 

for the pararricters have been selected SO that T. = 1. Thus, (% differences are with respect to 40 elenlent solution) 

A five point grid in the transverse direction is used for 

computation of the "pseudo-resultants" in the differential formulation. The points are equidistantly spaced with the first 

and last located at the extreme fibers and the central point POsitioned on the centroidal axis. 

Since the solutions of both the' differential and integral fotmulations are only satisfied at a discrete number of points over 

the length of the beam-column, the fist question that must be 

481 

element model results. Additionally, the relative magnitude of 

the errors between the 10 and 40 element models of the differential formulation are virtually identical to those exhibited by 

the integral solution methodology. Although not specifically indicated on these Tables, it should also be noted that the angles 

of rotation are not always small. For example, at this load and 

eccentricity, the initial elastic end rotation of the beam-column 

exceeds 17 deg. Of course, smaller rotations are exhibited at the 

lower loads and lower eccentricities. 

ORIGINAL PAGE IS 

M POOR QUALfTY 

-. . - .. ~ 

- . . - . . . .. . . .... . _____- , . -I---- .. . 

-~ . 
Table 2. Integral Elastic Solution Versus Number 

of Nodes for P/P, = 0.55 and a/L = 0.05 

Table 3. Integral and Differential ($) Elastic 

Solutions for Various Loads a/L = 0.05 

~~ 

Transverse Deflection from Various Solutions (in) 

Transverse Deflection for Various Numbers of Elements (in) Exact 

S = J'/L 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

1.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

1.0 

s = s'/L 

Number of Elements 

40 % Diff 

0.000000 

0.024616 

0.097967 

0.218586 

0.384079 

0.591200 

0.835951 

1.113686 

1.419227 

1.i46972 

2.091003 

40 

0.000000 

0.024616 

0.097967 

0.218586 

0.384079 

0.591200 

0.835951 

1.113686 

1.419227 

1.746972 

2.091003 

20 

0.000000 

0.024652 

0.098125 

0.218890 

0.384691 

0.592089 

0.837267 

1.1 15374 

1.421422 

1.749594 

2.094171 

10 

0.000000 

0.024781 

0.098753 

0.220123 

0.386547 

0.595443 

0.841727 

1.120936 

1.429034 

1.758422 

2.105685 

0.146 

0.161 

0.139 

0.159 

0.150 

0.155 

0.152 

0.155 

0.150 . 0.152 

% Diff 

0.650 

0.802 

0.703 

0.643 

0.718 

0.691 

0.651 

0.691 

0.655 

0.702 

sin d Angle 

s =d/L Integral a/& 2 Diff (de:) 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

1.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

for PIP, = 0.25 

0.000000 0.000000 

0.002615 0.00261G 0.04 

0.010450 0.010448 -0.02 

0.023444 0.023445 0.00 

0.041 520 0.04 1531 0.03 

0.064590 0.064589 0.00 

0.092475 0.092483 0.01 

0.125010 0.125031 0.02 

0.203256 0.203259 0.01 

0.248513 0.218497 -0.01 

0.162039- 0.162044 0.00 

for PIP, = 0.50 

0.000000 0.000000 

0.008283 0.008285 O.O? 

0.033056 0.033044 -0.04 

0.053940 0.073948 0.01 

0.130441 0.130516 0.06 

0.202009 0.201999 0.00 

0.287513 0.285559 0.02 

0.385945 0.386070 0.03 

0.496355 0.496382 0.01 

0.00 

0.23 

0.50 

0.i4 

0.91 

1.22 

1 .11 

1.66 

1 .s7 

?.25 

3.07 , 

0.00 

0.59 

1.57 

2.33 

3.07 

3.56 

4.40 

5.01 

_. 3 .32 

6.01 

1.0 0.746172 0.546650 -0.01 6.31 

(% differences are with respect to 40 element solution) 0.9 0.616919 0.615049 0.02 

Therefore, it is concluded that both formulations exhibit 

the same low level of sensitivity to the number of elements used 

in the analysis. Concurrently, these results also indicate that 

a 10 element model cin be used with either method without 

generating si6;Lificant errors in the analysis. It should be noted 

that all of the diflerential formulation results presented above 

are based on the use of an exact expression for evaluating the 

angle of rotation. 

A direct comparison between the resutts generated by the 

two formulations is provided in Tahles 3 and 4. Again, the comparison is based on the 0.50 eccentricit; ratio which inherently 

produces larger angles of rotation. The angle of rotation of the 

cross section, determined tom the integral technique, are also 

provided. 

Table 3 is based on the use of an exact expression for 

evaluation of sin 4 in the solution of the differential methodology. The "approximate" sin4 results, provided in Table 4, on 

the other hand, are based on use of the common approximation sin4 2 -aW/aJ, to calculate the angle of rotation. Note 

that, other than this particular change, these two differentid 

formulations are, otherwise, completely identical. Both include 

the influences of axjal i~s well as transverse deflection on the behavior of the beam-column. The influence of deleting the axial 

deformation on the accuracy of the predicted results is discussed 

later. ' 

for P/Pe = 0.55 

0.0 0.000000 0.000000 0.00 

0.1 0.024781 0.024809 0.11 7.37 

0.2 0.098753 0.098i63 0.01 4.6Q 

0.3 0.220123 0.220289 0.06 6.91 

0.4 0.386547 0.385110 0.15 9.03 

0.5 0.595443 0.595T55 0.05 10.9; 

0.6 0.841727 0.842435 0.08 12.69 

0.7 1.120936 1.122157 0.11 14.23 

0.8 1.429033 1.430090 0.07 13.41 

a.9 1.73842? 1.760165 0.10 1G.41 

1.0 2.105685 2.106815 0.05 1 5.04 

(% differences are with respect to integral solution) 

(cornparisons hased on results froiii 10 rlcliirric 111411hsl~) 

As the data of these Tables indicate. there are no cicnificant differences in predicted transverse deflections. Even for an 

end rotation angle of 17 deg, the differences between the various 

results is well below that required in engineering computations. 

It should be noted that this high level of correlation continues 

to exist for even greater angles of rotation. The reason why 

this high correlation exists is that, under compressive loadins. 

the derivative of the axial displacement is negative. Since. the 

square of the slope of the transverse deflection is always positive. 

these algebraically combine so that the sum is lower in niagnitude than either of the individual terms. This serves to reducc. 

the magnitude of the centroidal axis swain. Because the differG62:GINAL PAGE IS 

462 ' WPOORQUALrrY 
Table 4. Integral and Differentid (2) Elastic The net result of these changes is to create a differential model where the centroidal axis effectively is inextensional. Solutions for Various Loads a/L = 0.05 I 

While it might be anticipated that this would improve the correlation between the merentih and integral results, such is not 

the case. When the angle of rotation is very small, as those 

Transverse Deflection from \%trious Solutions (in) 

0.o 

0.1 

0.2 

0.3 

0.4 

0.5 

! 0.6 

0.7 

I 

0.6 

0.9 

1.0 

i 

I 

I 

0.0 

0.1 

I 0.2 

1 0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

1 .0 

I 

I 

i 0.0 

0.1 

I 0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.S 

0.5 

1 .o 

i 

Approx. 

sin 6 

Integral a/& % Diff 

0.000000 0.000000 

for P/P, = 0.25 

0.002615 0.002616 0.00 

0.010450 0.010448 -0.02 

0.023444 0.023435 0.00 

0.041520 0.041531 0.03 

0.064590 0.064589 0.00 

0.092355 0.092483 0.01 

0.125010 0.125031 0.02 

0.162039 0.162044 0.00 

0.203256 0.203279 0.01 

0.248513 0.248497 -0.01 

f0l 

0.000000 

0.008283 

0.033056 

0.053940 

0.130441 

0.202009 

0.285513 

0.385945 

0.496355 

0.616919 

0.546572 

- P f P, = 0.50 

0.000000 

0.009286 0.04 

0.033044 -0.04 

0.053949 0.01 

0.130516 0.06 

0.202000 0.00 

0.285560 0.02 

0.3860il 0.03 

0.496383 0.01 

0.617050 0.02 

0.746672 -0.01 

for P/P, = 0.75 

0.000000 0.000000 

0.024781 0.024810 0.12 

0.098753 0.098567 0.01 

0.220123 0.220299 0.08 

0.386547 0.385128 0.15 

0.595443 0.595782 0.06 

0.841727 0.842473 0.09 

1.120936 1.122228 0.12 

1.429034 1.430155 0.08 

1.758422 1.560244 0.10 

2.105685 2.106911 0.06 

Angle 

(deg) 

0.00 

0.25 

0.50 

0.74 

0.98 

1.22 

1.44 

1.66 

1.87 

2.07 

2.25 

0.00 

0.79 

1.57 

2.33 

3.07 

3.76 

4.40 

5.01 

5.52 

6.01 

6.38 

0.00 

2.37 

4.68 

6.91 

9.03 

10.97 

12.68 

14.23 

15.41 

16.44 

li.04 

which result from a low level of loading and a small eccentricity, 

all the formulations provide virtually identical predictions. HowTable 5. Integral and Modified Differentid ( f ) 

Elastic Solutions for Various Loads a/L = 0.05 

Transverse Deflection from Various Solutions (in) 

Modified 

(w/o 4 Angle s = s'/L Integral B/Bt % Diff (deg) 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

1.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

1.0 

for P/P, = 0.25 

0.000000 0.000000 

0.002615 0.002616 0.00 

0.010450 0.010446 -0.04 

0.023444 0.023444 0.00 

0.041520 0.041527 0.02 

0.064590 0.064586 -0.01 

0.092475 0.092478 0.00 

0.125010 0.125030 0.02 

0.162039 0.162043 0.00 

0.203256 0.203287 0.02 

0.248513 0.248508 0.00 

f01 

0.000000 

0.008283 

0.033056 

0.073940 

0.130441 

0.202009 

0.287513 

0.385945 

0.496357 

0.616919 

0.746772 

r P/P, = 0.50 

0.000000 

0.008294 

0.033073 

0.074032 

0.130665 

0.202274 

0.287975 

0.386711 

0.497264 

0.618270 

0.748235 

for PIP, = 0.75 

0.13 

0.05 

0.12 

0.17 

0.13 

0.16 

0.20 

0.18 

0.22 

0.20 

0.00 

0.25 

0.50 

0.74 

0.98 

1.22 

1.44 

1.66 - 

1.87 

2.07 

2.25 

0.00 

0.79 

1.57 

2.33 

3.07 

3.76 

4.40 

5.01 

5.52 

6.01 

6.38 

0.00 

3.07 2.37 

2.98 4.68 

3.15 6.91 

3.30 9.03 

3.32 10.97 

3.46 12.68 

I"; differences are with respect to integral solution) 0.0 0.000000 o.~odooo 

1 roliiparisons based on results from 10 element.models) 0.1 0.024781 0.025543 

0.2 0.098753 0.101698 

0.3 0.220123 0.227053 

0.4 0.386547 0.399299 

0.5 0.595443 0.615237 

0.6 0.841727 0.870876 I cnce hetaren the exact and approximate expressions for sin 4 is 

rt.1att.d to the "'1 + 2c0 in the denominator of the exact expression. reducing the magnitude of the centroidd strain inherently 

tnlpro\er tl-e accuracy of the approxhation. 

This influence is best illustrated by the data of Table 5. 

Here. the integral solution is compared to an approximate sohtion for which the effect of the centroidd axis strain terms has 

heen suppressed. This wa accomplished by first eliminating 

6' the (arvjas)* terms from the governing equations. The Ed 

niodulus-area product H'= then artificially increased through 

multiplication by a factor of 1000. This second change reduces 

the magnitude of the axial deflection by approximately the same 

i 

I 

i 

1 

i 

0.7 1.120936 1.161483 3.62 14.23 

0.8 1.429034 1.481683 3.68 15.41 

0.9 1.758422 1.825548 3.82 16.44 

1.0 2.105685 2.186516 3.85 17.04 

(% differences are with respect to integral solution) 

(comparisons based on results from 10 element models) 

ever, a3 the angle of rotation increases, as those which occw at 

the higher loadings and the greater load eccentricity, the modified dflerential formulation predictions begin to diverge kom 

those of the other two. The difference between the predicted 

results is most notable for the highest load and largest load ec- I factor. centricity example. 

463 GRZcjli\lAL PAGE IS 

OF POOR QUALI'IY 
This behavior results from the way the end of the beamcolumn deflects. In the modified model, the end of the beamcolumn basically moves only in the vertical direction (see Fig. 1). 

The standard model, however, allows the end to move both vertically and horizontally. Thus, for a given vertical deflection, the 

leftward movement predicted by the standard model increases 

the angle of rotation. This reduces the moment by reducing the 

moment arm. Thus, to support a given load, the standard for. 

mulation beam-column model need not deflect as much as the 

modified model must. 

Comparisons of the viscoelastic predictions provided by 

the integral and the exact differential are provided in Table 6. 

A comparison with the approximate sin4 differential solution 

is not included because the differences between the exact and 

approximate results again are so small as to be negligible. These 

computations are based on the use of an ideal three element 

"limited" material model illustrated in Fig. 2. 

As demonstrated by the data of this Table, the high correlation between the elastic solutions provided by the integral 

and differential solution methods carries over directly into the 

viscoelastic analysis. 

The final item meriting discussion is the length of the time 

increment used in each of the formulations. Unlike the prior results, some differences do exist between the maximum allowable 

time step increments for the integral and differential formulations. Additionally, the allowable time step increment for the 

integral formulation exhibits a higher dependence on the Actual 

angle of rotation than does the differential formulation. 

In general, a relatively small time step increment must be 

used with the integral solution methodology. For example, the 

results presented above are based on the use of a 0.01 time step. 

As the length of this time step is increased, the accuracy of the 

solution decreases and tends to underpredict the deflection. This 

convergence "from below" is not surprising since the convolution 

integral is approximated as the sum of a finite number of terms. 

In contrast, much larger time step increments can be used 

with the differential formulation. This is principally attributed 

to the high accuracy provided by the RungcKutta integration 

routine. Most of the results provided above were developed using 

a 0.10 time step. Additionally, the allowable length for this time 

step increment tended to be rather insensitive to the angle of 

rotation. The allowable time step for the integral formulation, 

on the other hand, exhibited a high level of sensitivity to the 

angle of rotation. Larger angles of rotation required significantly 

shorter time steps for accurate results to be obtained. 

These factors combine in a rather interesting manner with 

regard to which method of analysis is computationally more efficient. Typically, for the analysis of short periods of viscoelastic deformation the integral solution method was two to three 

times faster than the differential method. This is attributed to 

Table 6. Integral and Differentid Viscoelastic 

Solutions for Three Time Constants 

P/P, = 0.50, and a/L = 0.05 

Transverse Deflection (in) 

s = s'/L 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

1.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

1.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

1.0 

Exact 

sin 6 

Integral a/& % Diff 

at 

0.000000 

0.008283 

0.033056 

0.073940 

0.130441 

0.202009 

0.28 75 13 

0.385945 

0.496355 

0.616919 

0.546772 

time = 0.0 

0.000000 

0.008285 0.02 

0.033044 -0.04 

0.073948 0.01 

0.130516 0.06 

0.201999 0.02 

0.287559 0.02 

0.386050 0.03 

0.496382 0.01 

0.617049 0.02 

0.546G70 -0.01 

at time = 1.0 

0.015083 0.015069 0.04 

0.060151 0.060128 -0.04 

0.134319 0.134344 0.02 

0.236430 0.236612 0.08 

0.365200 0.365193 0.00 

0.518035 0.518161 0.02 

0.692634 0.692946 0.05 

0.886869 0.886967 0.01 

1.096666 1.097006 0.03 

0.000000 0.000000 

1.320154 1.320004 -0.01 

at time = 2.0 

0.000000 0.000000 

0.019127 0.019138 0.06 

0.056254 0.076232 -0.03 

0.150149 0.170202 0.03 

0.299200 0.299484 0.09 

0.461621 0.461664 0.01 

0.653844 0.654093 0.04 

0.852716 0.873246 0.06 

1.115343 1.115622 0.02 

1.356225 1.356851 0.05 

1.652861 1.652902 0.00 

Angle 

I deg I 

0.00 

0.59 

1.5; 

2.33 

3.05 

3.56 

4.40 

5.01 

5.52 

6.01 

6.35 

0.00 

1.44 

2.36 

4.22 

5.54 

6. 76 

5.86 

P.57 

9. ;o 

10.44 

10.95 

0.00 

1.33 

3.62 

5.3 

i.00 

8.52 

9.R6 

11.13 

12.12 

12.99 

13.55 

(% differences are with respect to integral wlution) 

(comparisons based on results from 10 eleriirnr ntodels ! 

two factors. The first is the comparitively sIo\v Runge-Iiurta 

integration procedure used in the differential formulation. For a 

short period of viscoelastic deformation, the cdrularion of the 

convolution integral of the integral formulation. requiring siniple summation of a limited number of terms. can be performed 

much more rapidly. 

The second factor is that the fixed point iteration scheme 

of the integral formulation, although requiring more iterations 

than the Newton method, is also performed more rapidly since it 

is simply an algebraic operation. The h'erton method. in contrast, requires inversion of the matrix premultiplying the wctor of trial function corrections and then numerical evaluation 

through solution of the system of equations. Even for just a 10 

464 ORIGINAL PAGE Is 

Of POOR QlJAlm 

..... 

...... . ....... ........ - 

~ ~~~ ~ 

~-. .. . -.-. .- - 
riPrlrenr he;lnl. thj, process is slow in conlparison to the fixed 

point it vaticin. 

~J~,,~~~~. ai tile length of the period of viscoelastic deb 

iilcp-aspS. this relative speed relationship reverses. Eventually. t he difierential formulation begins to generate solutions 

"inre rapiillv ihan the integral method. In the example problem 

2e.rriheil ahove. this generally occurred approximately between 

the second and third time constants. The reason for this change 

ic directl! related to the conlputation of the convolution inteCAI of :he integral technique. As time increases, the nuniher 

of ternis in the summation increases linearly. This in turn, inrreacer the nuxuher of additions which nust be performed and 

:herefore 1inc.arly increases the time need for each computation. 

In cont:ast. the speed of the Runge-h'utta integration routine is 

virtually independent of time. Thus. the continually increasing 

roiiiput at ion effort required in the integral technique eventually 

osrecd. t trat YI~W,~ for the differential technique thereby reversing 

1 ti,, r+,litt ivi. *ped rdationbliip. 

V. Conclusions 

D;d nn the results reported herein and elsewhere 1121 

j* I- ricrIudwi chiit the differential foriiiulation procedure pre- 

-vn:vd c;m rtiiplowd for the analysis of quasistatic nonlinear 

i,!i~.-(~iiiienci~~yial v3roeiastic problems. This conclusion is based 

4i:ertiv OIL :he high level of correlation between results developed ri.ir~q rhir formulation technique to those obtained with 

;he p:~viousl~ published integral method for solution of such 

?rnbi-n:-. .Additionally. it is noted that both of these methcd- px!.ihit exreptionally similar accuracy characteristics with 

rrghrd to tile niinher of elements employed in the approximnaiiorr. For hoth. a relatively low number of elements can be used 

wi:hnEr Pngendering any significant errors. 

A cknorledgement 

The research described above has been performed under 

S.45.4 Grant So. 3-53-1. The financial support provided by 

S.AS.4 ic gratefully acknowledged hy the authors. The authors 

xiGh rn extend their thanks and appreciation to Dr. C. Chamis 

$0: r!ir SAS.A.Lewis Research Center for his support and for the 

in.irhlr into the prohleni which he provided during our many 

rerfin;ral dizcussions. 

3. Schapery, R.A., "Approximate Methods of Transform Inversion for Viscoelastic Stress Analysis," Proceedings of 

4th US National Congress on Applied Mechanics,Vol. 2, 

1962, pp. 1055-1085. 

4. Kraus, H., Creep Analysis, John Wiley 6. Sons, New York, 

1980. 

5. Libove, C., "Creep Buckling of Columns," -1 of th 

-s, Vol. 19, No, 7, 1952, pp. 459-467. 

6. Zyczkowski, M., "Geomertically Non-linear Buckling of 

Bars," IUTAM Colloquium on Creep in Structures, Stanford Vniversity, CA, Academic Press, New York, 1962, 

pp. 305-325. 

7. Patel, S. A., "Buckling of Columns in the Presence of 

Creep," The Aeronautical Quarterly, Vol. 7, No. 2, 1956, 

8. Hoff, N. J., "A Survey of the Theories of Creep Buckling," 

Proceedings of the 3rd US National Congress of Applied 

Mechanics, 1958, pp. 29-49. 

pp. 25-134. 

9. Rogers, T. G. and Lee, E. H.,"On the Finite Deflection 

of a Viscoelastic Cantilever," Proceedings of the 4th US 

National Congress on Applied Mechanics, Vol. 2. 1962, 

pp. 955-987. 

10. Stubstad, J. M. and Simitses, G. J., "Bounding Solutions 

of Geometrically Nonlinear Viscoelastic Problems," 8198 

Journal, Vol. 24, No. 11,1986, pp. 1843-1850. ' 

11. Thurston, C., "Newton's Method Applied to Problems 

in Nonlinear Mechanics," Journal of Applied Mechanics, 

Vol. 32, 1965, pp. 383-388. 

12. Stubst ad, J. M., "Nonlinear Thermoviscoelastic Analysis 

of Metallic PLane Curved Beanis," Doctoral Thesis submitted in partial fulfillment of the requirements for the 

degree of Doctor of Philosophy, Georgia Institute of Technology, Atlanta, GA, 1986. 

13. Vinogradov, A. M., "Nonlinear Effects in Creep Buckling 

Analysis of Columns," Journal of Engineering Mechanics, 

Vol. 111, No. 6, 1985, pp. 757-765. 

14. Vinogradov, A. M. and Wijeweera, H., "Theoretical and 

Experimental Studies on Creep Buckling," Proceedings 

of the 26th AIAAJASMEJASCEJAHS Structures, Structural Dynamics and Materials Conference, 1985, Vol. I., 

pp. 160-164. 

15. Triconli, F. G., Integral Equations, Dover, N.Y., 1985. 

Peferences 

1. Gristensen. R. .\I.. Theor;von'iscoelasticity,Introduction 

Academic Press. Sew York, 1982. 

2. Hahotnov. Y. S..Elements of Hereditary Solid Mechanics 

>I!: Puhlirations, Moscow, I'SSR, 1980. 

ORlGlMkL PAGE 6 

OF POOR QUALITY 

I 

465 
NON-ISOTIIERXIAL ELASTO\'ISCOPLASTI(' SNAP-THROUGH AND CREEP BIlCKLING 

OF SHALLOW ARCHES 

C;. J. Sinutses and R. Riff'* 

Scliool of Aerospace Engineering 

Georgia Institute of Technology 

Atlanta. Georgia 30332 

Abstract 

The prol,leiii of Iiuckhg of sl~allow arches uiidrr transient 

theriiie~iiierh;ctiical I~ads is investigatetl. The analysis is lmed on 

noiiliiiiw geoiiietrir ad ronstitiilive relations, uicl is expressed 

in a riltr forni. The tiiaterial ronstitiitive eqirations are capable 

of rqirotliicing all non-isot hernial. elasto-visroplast ic cliararteristic\. The <olut ion sclieriie is rapal)le of predicting response 

rliir11 iiirliicie~ prr and posfl~ucklir~g with creep and plastir efFv-i.. TIir whil iiin ))roceditre is cleiiioiistratetl through several 

ex;tiiipIi-~ wliirli include I~o11i rreep anel siiap-~ hroiigli Iwhavior. 

L Intrcrcl~~rt ioi) 

Ela-rir wap-through of low arches under quasi-static loads 

)ins Iwrn tlre subject of several investigations. The significanre of 

the prd1Ieiii. in so far a5 it illustrates certain iniportant features 

in iiiorc roiiiplicated buckling probleiiis of plates and shells, was 

pointed OUI I,! Marguerre 11). who constructed a siniplified merhiiniral iiiodel to deiiionstrate these features. Tiriioshenko 121 

oI>raii~vil an approlriniate solution to the prohleiii of a low arch 

under a uiiiforndy distributed load. Biezeno 13) considered the 

prihhii of a low paraholic arch loaded laterally at the iiid-point 

{vir !I a ro~irentratetl load. He alto investigated snap-through 

hiirLlii~g of a slrallow spherical cap, pinned along its circular 

Iioiinil~~rv. itiitler the action of a concentrated load applied dong 

!lie ai* of rot ational nynuiietry. He presented his approxiinate 

vJii1 ifm~ iiieans of load-deflection curves atid eclua~ions from 

c.Iiicli tlir critical loacl roulcl 1)c rnlrulatrtl. 

in IY:,~. Fring aiid KapIan 141 investigslecl the prohletii of 

lo\$ pinrird arrlier of various initial shaprs and spatid distributioils of the 1;cteral load. Their results slior that a very shallow 

arch sriapc -titrough syiiuiietricdly. wherea.. a higher arch huckIc.2 a~~iiuiietrically. They also ran a limited nuiiiber of experi111enrn1 te\ts. and their experitiienlal data are in good ;yreenient 

\r.ith ~lrrir tiieoretiral results. Ahorit the saiiie tiiiie. HofT iuid 

Uriicr 5'. in invrstigating the 1iossil)ility of snap-tlirough buckling of a low pilined arch with a half-sine-wave initial shape 

tiritlcr a half-rine-wave distrit)rited clynaiiur load (suddenlv applied with iiitinite duration). ohained results for the quasi-st.at.ir 

loa11 raw which are identical to those obtained by Fung iuid KapI~:i for rl~ carlie problciii. 

Cop?ripht I American Insliiutr of Acronsulia rod 

.4rtronaulicr. Inc.. 1987. AU rights rcurved. 468 

In 19G2, Gjelsvik and Bodner [G] obtained mi approxiiiiate 

solution to the problerri of a lor circular arch with a ronrentrat.ed load at tlie iilid point of tlie arch and claiiiped houndary ronditions. They also report ed on experiiiieiital tests. and 

tliere is good agreeiiient hetween their theoretical prerlictiorrr 

and their exj)eriiiientaI resull s. Schrever iuid hlawr IS] ohtainecl 

mi exact solution lo their prol)leni (and for the load race of 

uniforiii pressure). and they showed that for the coricmtrated 

load caw.t he arrh snaps-I hrough syiiuiietricallr regarcllesr of 

the value of the rise paraiiieter. hlasiir and Lo !R, prc*\eiitecl 

a general eliscussion of tlir I)eliavior of the rliallr~r cirrrilar arch 

regartling l~irckling~ post -1~iickIiiig and iinyierfert ion wn4 ivit v. 

Snapping of ICIW pinned arches rcsling on ai elastir foiiiitl;itioii 

ha\ Iwen investigated . .. I)? Siiiutse* 19;. This SWI~III exliiliit* dl 

forin* of rxperiiirnt.dly ol).;ervetl I)urkling pli~nc~riie~ra (siiioot 11 

and violent) and of theoret ically predicted respoiisec (liiiut point. 

bifurcation with stable hrancliing and bifurcation with unstalile 

I)ranrhing), and it is presented ritli sufficient detail in Ref. 10. 

Experiixiental results have also heen reported hy Roorda :1 I]. 

The eflerls of inelastir iimterial behavior found their rav into 

the literature since the 1960's. Onat and Shu j12: ron4clered the 

behavior to be one of rigid-perfectly plastic. Froiiiriosi. Augucti 

wid Spararjo 1131 disrussed the rollapse of arches under repeared 

loads with inelastic nmterial behavior. Studies of inelast ir snapthrough buckling of shallow arches were also reported by Lee 

and Murphy [Id]. In addition Aiigusti !15) investigated plastic 

buckling of a model of a three hingrcl cucli in 1968. and a inore 

coiiiplete analysis of the same model was provided I)! Barteritian {I(;j in 13il. Finally. the reatlrr rho is intermled in the 

1111 iiiiatc s~rtvigtli of parahidir stprl nrclies wit I1 11r;iriiig rvst rin 

is refrrrctl to Koiiiatsu [I 71. who considers inchst ir iii-IiIaiie iujd 

orit-of-plarir i~istal)ilities and provicles clesigri foriiiiilir for each 

rase. 

The elastic resportse of arches under sudden (tlvnmilir ) application of the exteriial loads 11a. been reported Iiv Hoff ruicl 

Bruce 15). Hsii (211. [E]. arid Lock [XI. For a more roiiiplete 

biBliograpliy we Hef. 24. 

('reel) Inickling of sliallow arclie5 hac Iwen inve~tigatecl hv 

Huang and Nachlm 1181. Krajcinovir (19;. and Botros axid Bienek ;'LO]. 

As far as the aittliors know no work hac I)een reported ori the 

non-isotheniial elastovisro-plastic hehavior of cliallor arches. 

The piirpose of this paper is to deiiionstrate the erect of liighlv 

nonlinear iiiaterial behavior on the snap tliroiigli and rreep I)uckling of sIia1low arches. 

~- . . .. 

L 
11. Elasto-Theriiio-~iscoplastic Constitutive Relations 

The prediction of buckling loads and post buckling behavior 

of ctructural coliiponcnts. like shallow arche% made of a redistic 

r:rrtaibc Illaterial and subjected to non-isotherlid theriiioruerliaiiirill lciadb hac increased in importance hi recent yews. 

1.nairr ttii, kind of severe loading conditions, the structural 

be!iavior i. highly nonlinear due to the combined action of geoiiirtrird and physical nonlineluities. On one side, finite deformation in a st reaced structure introduces nonlinear geonietric 

erect.. Orl thr other side. physical nonlinearities arise even in 

.iuall st rain regimes, whereby inelastic phenoniena play a particularly important role. From a theoreticd standpoint, nonlinear rori'titutire equations should he applied only in connection 

with riorilinear transforniat ion iiieasures (implying both deformation and rotations). However, in aliiiost dl of the works in 

rhis area 251. the two identified sources of nonlinearities are 

AIHAI, separated. The separation yields, at one end of the spectruii~. pru!,lrlns of large response, while at the other end, probhi!, of vi.riniG d;or non-isotherniirl behavior in the presence 

qtf \III;III ,triiiri. 

Tire cla.-ical theories, in which the inaterial response is characti-riird ;,. ;t rotiiliinatiori of distinct elastic. thennal, tiirie indc 

pibtiqiwt itac.I;r\tic (plastic) and time dependent inelastic (creep) 

i:$.faoriilar iori roiiiponents cannot explain some phenoitlena, that 

rail t,i,-~rvctd in coiiiplex therino-iiiechanicd loading historiei. Thib is particularly true when high-temperature noni.othz:uld processes iiiust be taken into account. There is a 

SizeaIDle hody of literature (251, [261 on phenomenological constirritiv.r rquatioris for the rate and temperature dependent plastic 

c1rfori:i;t: ion behavior of metallic iimrerials. However. diiiost all 

of thew ricxv "unified" theories are based on sinal1 strain theorip. irrd wvt-ral wfGr froill .soiiie theniiodynaiilir inconsistencies. 

If1 it ;J~-VII~J+ paper 1271. the authors have presented a CO111- 

plrrr -+,i id ctiiictit utive relations for nonisothermal. large strain, 

+Iarir, ..I-( oplactic heharior of nietals. It was shown there 1271 

that the iriptric tensor in the convected (itlaterial) coordinate 

rv\tm1 ran he linearly decoiiiposed into elastic and (visco) phiir parrs. So a yield function was assumed, which is dependent 

on the iat~ of change of stress on the nietric, on the temperature 

ani1 on R <et of internal variables. hIoreover, a hypoelastic law 

~ir- cli~wn to describe the theriiio-elastic part of the defoniia1 IOII. 

.A t iiiir and temperature dependent viscoplasticity iiiodel W~U 

furtiiulated iii thic convected niaterial systeiii to account for finite 

st rains and rotations. The history and temperature dependence 

rere inrorporatecl through the introduction of internal variables. 

The rhrr of these varialiles. as well ac their evolution, WIS 1110- 

t I\ at ed In therriiodynaiiuc considerations. 

errierges tliat part of the plastic work IS used lor produciig iebiuud stresses in the lattice, which, when phenoiiienologically considered. causes hardening. Both liiuut at ions were excluded fro111 

this 1271 forinulation. Accuracy of the foriiiulation was checked 

on a wide range of exiuiiples [28]. 

The constitutive relation will he rephrased here in some different form. For brevity we coinpile only soiiie notations and 

fundanieiital relations which are used in the forniulation of the 

intended Constitutive low. For details, see (271 and (281. 

Concerning the foriiiulation of constitutive laws it is advantageous to use a iiiaterid (co-iiioving) coordinate svsteiii. The 

transforillat ion froiH the uiiderforiiied state (iuetric g,&) to the 

deforiited state can he represented l~p the tensor: 

0 

I =ytr grk or (j-' I; = g'' ;,& (1) 

The total deformation rate is defined by 

represents the synuiietric part of (jvk or the covariant derivative with respect to tiiiie. also called the roiiverlcd derivation, 

which is due to Zareiiiha aid Jauiiiaun. 

The total deformation CNI be deroiiiposed according to 

. 0. (i!ip) 

fI =Y'm9mr LIS& =f;f; (4 1 

where the superscript ( ) relates to a fictitious configuration defined by a fictitious reversible process with frozen internal variables. The decoiiiposition of Eq. (4) leadr to a11 additive deconiposition of the deforiiiation rate 

(r) (11 

8; =8& t 8& (5) 

(7) (8) 

di is related to the reversible defonnations. whereas di. denotes 

the reiiiaining part of the deforiiiation rate. 

For the description of the siress state, we introduce the Kirchoff stress tensor s;, which is connected with the red Cauchy 

stress tensor u;, hy the relation: 

Assuiiung that the elastic beliavior is not affected essentially 

by inelastic defoniiations, the following hypoelastic increiiiental 

law was chosen [Zi] 

with 

The noni-orheririal elasto-viscoplastic defonuation process 

Wac dewihed coinpletely hy "thermodyn;unic state" equations. t; : weighted stress deviator 

Moct inr.esrigators :2j], [26] (in the area of viscoplasticity) ein- C : shear inodulus 

plov plastic strain$ as state variahles. The authors' previous K : bulk iiiodulus I 

*rudy .XI shows that, in general. use of plastic strains BS state o : coefficient of theriiial exi)wioion 

v.viab1- may lead IO inconsistencies with regard to thennody- I 

The following constitutive relations were established (2i) for narilic considerations. Furtheriiiore, the approach and fornula? ion prnployed in previous works leads to the condition that 

dl the nlactic work is completely dissipated. This, however, 

the 

yield -- 

is ir. contradiction with experiiiiental evidence, froiii which it F = (fi - c;g$,(t: - dg$) - k*(A,T) = f* - cz > 0 (8) 

467 

ORIGINAL PAGE is 

t 1 OF POOR QUALITV 
with 

evolution law for tlie internal variables: 

if 

then 

with 

if 

or 

then 

F<O 

IVit Iiiii tlic developed frame the elasto-viscoplastic behavior is governed by the scalar material functions e(~;. T. A,&). 

k'(.~.T).g(si..d,T.4).~(d.T1d~).and s(d,.T,&). These~uaterial fuiict ions can be determined froin a series of nionotonic 

and cyclic procr\ces with proportional and nonproportional pathc 

at dill-rent teiuperature levels 1281. 

Ill. <klieral Foriiiulation and Soh tion Scheme1 

Tlie rate forin of the constitutive equations suggests that a 

rate approarli I)c taken toward the entire problem so that flow 

i* viewed a% liictory dependent process rather than an event. 

For tldc piirpose. a coniplete true ah-initio rate theory of kineiliatics and kinetics for continuuiii and curved thin structures. 

nit hout any restriction on tlie magnitude of the trancfonnation 

was pretented in Ref. 28. It is iiiipleniented with respect to 

468 

a body- fixed system of concected coordinates. and it is valid 

for finite strains irnd finite rotations. The time dependence and 

large strain behavior are incorporated through the intrudurtion 

of the time rater of change of the metric and of the curvature. 

The constitutive law may be appiied to the conservation of 

nioiiientum via an appropriate variational principle. The principle of virtual power (or of virtual velocities) reads 

where 613, are the virtual velocities. fJ the bodr force\ per iitiit 

iiiass and vTJ the surface tractionr. 'Total differentiation of Eq. 

(22) yields, 

At any instant. Eq. (23) 11iiist be satisfied. Tlie virt~~;rl 5-1.- 

lociry and its tiiiie derivative are then indepeiitlerit. Ilorrover. 

tlie last three terms of Eq. (23) are equivalent to Eq. i2i. 

Hence, tlie principle of the rate of virtual power ;ria! lw ditained in its concise foriii. For further classifirarion6. tlie total 

derivative of the stress components rill be reprevnted t)y 111e 

Jauiiian derivative. and the following integrals are defined hy 

Then. sul)slitutioii in Eq. (43) yields the final fimi IJf the priri 

ciple of the rate of virtual power. 

The quasi-linear nature of the priliciplc id tlie riitv Id vir~u;J 

power suggests the adoption of M increiiierital approacli to numerical integration with respect to time. The availahiliry of the 

field foriiiulation provides ammtnce of the coiiipletene.~ of the 

increriiental equations and allows the use of anv converiieril procedure for spatial integration over the domain V. In the prrbent 

instance the choice has heen iimde id favor of a \iriiple lir-t order 

expansion in time for the construction of incremental solution. 

froin the results of finite element spatial integration of the governing equations. 

The procedure eniployed pernuts the rates of thr field foriiiulation to he interpreted a5 increments in the nuliieriral hddution. 

This is particularly convenient for the construrtiuti of iricretiiental boundary condition histories. 

The finite element method for spatial discretization ha* Iwen , 

well documented (see. e.g. Zienkiewicz [?9; or Oh :3:) iuid 

will not be detailed here. Restricting attention to i~ single finite 

element B,", the velocity field is approxiuiaretl hv f,( I' 1. 

tli(rJ) fii(d) = (ra~)~tl;(~J)\-L3 0.3 = i......~ (2s) 

ORGlNAL PAGE i3 

OF POOR QUALITY 
! 

The algebraic rounterpart of Eq. (51 after the finite eleiiient 

eii.rr~~~iza;i(m ic rhe well-known stiffness expression 

iiiri, .iir i;llic(.lli ~tiffiiess iiiarrix YK;. the vector of the incre- 

:i+.~iikI .;*lorii ies {I.}. and the vector out-of-halance-force rates, 

exrrr:lai forrr rate6 {P} iiuiiur internal force rates {F}. Tlie hoIIK,~~;N~II. ra-e of Eq. (39) indicates either the non-uniqueness 

of i ttl. r~(jiiilit~riiiiii path at a stable or unstahlc bifurcation point 

cir it,* ~itiic~iir ibiit uiictalde situation at a liiilit point. Heiice tliis 

r:i:e:ioii riav lw evaluated by a deteniunant check or suppleiii-:it;trv iai:riivalue analysis for the load paraiiieter ~)arallel to 

r!,r lraktlir~; pr,o+*. 

E:*!, xi:,Glw the condition of static external loads ancl~slorly 

:r,,ir I:#? t ra+lj vffi.rt\. the prewwc of snap-tliroiigh I~li~klit1.g 

11d1.. I 111. iwr~ ii~ &rts signilirant. 111 dynaiiuc iuialyses, tlie 

qb;Jiid 1,1dv force< include inertia forces. Assuiiling that the 

111;c.. ,d rile. i~,ilv considered is preserved, the iililss iimtrix can 

1,s. r\:d:i;tivil prior to the tiiiie integration using the initial conrzi::afiam. 

Fi:iii+ viriiient solution of any boundary-value proldeiii in- 

:l,i..- ;h* -o!iifion of the eyuilihriuiii equation (gl01)al) together 

ivirli iiir r~~ii-:itritive equation (local). Both equations are solved 

-ir:iul:;c:~*~ .i~.ii. in a step hy step iiiaiiner. The increiiiental foriii 

(i r!l+ zlo~d ilrirl local equations can he achieved by taking the 

inrezrkiilch over the increiiiental tiiiie step At = fj+l - tj. The 

r-rra:;n!iiar rirk tias heen applied to execute the resultiiig tiiiie 

ii,i+.z::tii<qi. 

i6,;triv. r hr niiiiierical solution iiivolves iteration. A siiiipliri*4 i-r-iim 21: of Hiks Weiiipner constant-arc-length method 

!.+ I+CII iirilize~l. This iteration procedure which is a generaliz;tii(c of rhe ilicplareiiient control iiiethod also allows to trace 

the :io alineitr response beyond hifurcation points. In contrast 

:!I t!w rimwnrioii.4 Sewtoii-Raphson techniques, the iteration 

4.; rtw riirrhncl rakes place in the velocity and load rate spare. 

'i itmi I IT)^ ol' the first solution in each increiiient is liiiutecl 

1,: ro:~fr~~Iiri~~ ihc length ds of the tangent. Either the length is 

i..,;': riiii-raiii iii each step or it is adapted to the characteristics 

,#I :ill. .,,Iiir ion. Iri each step the triaiigular-sized stiffness iiiatrix 

!.A- I,, IW rtrrrkrcl for negative diagonal teriiis. indicating that a 

I r::ir;:l 1,<1ir11 i4 rrarlled. 

! 

11.. Shallow Circular C'laiilped Arch 

I T!)*. r tI*,orv arid roiiiputational procedures described in the 

l~r+-~liitc .-rrion. have heen applied to the creep buckling anal- 

:-i- <if a -iidlorv rirrular clariiped arch. The prohlein of the 

r:;tiiip*ii ;rrrIi Iwcitle being of wine practical interest contains a 

:.li::ii**r <If .i:iularities to that of the unifortidy loaded sphericd 

r+. Titr ~r~wl of result5 of the arch prol)letii serves as a good 

i:iiilrhi(,r IO rlw Iwhavior of the latter. 

I 

i 

The shallow circular claiiiped arch sitl~jected to a single central conceiitrrted load. as shown iii Fig. 1, is iuialyzecl. The 

niaterid rhosen for tlie nuiiierical experiiiientation is the carhon steel ('-45 (DIN 1520) with E = IO' psi, v = 0.3 and 

oy = 2.i x 10' pri at rooiii teiiiperature. The viscoplastic prop 

erties (the scalar iiiaterid functions) were obtained in Ref. 28. 

I 

Fig. 1 Claiiiped C'ircular Arch 

The analysis is perforriled with the aid of 24 paralinear isoparaiiietric eleiiients, Fig. 2. The paritlinear isopariuiiet ric element is intended for the analvsis of oriented structures where 

the geometry is such that the thickness in siiiall roiiipnred to 

other diiiiensions. The characteristics of tlie eleiiient arc defined 

hy tlie geotiietry iuicl interpolation hinctions. rliirli are linear 

in the tliickness direction and parabolic in tlie longitudinal direction (see Fig. 2). (:oiisequentlv, the eleiiieiit allows for shear 

straiii energy since iioriiials to a iilicl surface before cleforiiiat ion 

reiiiain straight, hut not necessarily noriiial to the slid stirface 

after deforiiiation. 

I Nooe I Ni I 

Fig. 2 Paralinear Isopariuiietric Eleiiient 

Tlie elastic behavior, corresponding to hoth axisymetric and 

asyiiietric response, is shown on Fig 3. These curves are in 

coiiiplete agreeiiient with those produced by.C;jelsvik and Bodner [GI, only heckuse the young's iiiodulus irnd Poisson*s ratio 

values used are virtually the saine (carbon steel C:-45 here, and 

469 ORiG!NAL SAGE IS 

OF POOR QUALITY 

I 
, 

2024-T-l aluiiunuiii alloy in Ref. G ). Note that these elastic 

response curves are hypothetical for our iimterial but true for 

the 2024-T4 alloy. The triie behavior for our riiaterid is elasto- 

. viscoplastic and it is labeled as such on Fig. 3. Note thqt this 

curve represents quasi-static (steady state) elastovisco- plastic 

response, as described by the chosen constitutive law. According 

to this. snapping occurs at a load of 26.20 lhs, prhilarily because 

of the low yield strength. Then, the post-liirut point behavior 

sehiiis to be pririiarily driven by viscoplastic behavior. 

1. ir 6 

Fig. 3 The Arch Response 

It is expected here that if loads up to approxiniately 11 Ihs 

are reached quasi-statically and left applied for a long time the 

priniary recponse will lie creep and the critical time to crwp 

will be extreiirely large. On the other hand, for loads between 

14 Ibs and 26.2 Ibs ( especially for the higher range ) the hehavior will he a co~~ibination of creep and snap-through buckling. 

This ir liest demonrtr~ted by the curve on Fig. 4. The applied 

load is reached quasi-statically in 13 nlinutes and then it is kept 

constant. The curve of Fig. 4 depicts the behavior of the arch 

in tcriiic of nudpoint deflection versus time. Creep. initially, is 

very slow. lhcn snap-through takes place iii 32 nuniites, curve 

IK'. atid tlien tlie crwp beliavior cotitiiiurz until a critical tinie 

to creep (creep buckling occurs) is reached after a totd tinie 

of 95 iiunutes. Note that for this loading condition the critical 

tinie to creep is 95 nunutes. Creep buckling and critical time to 

creep are Iiased on the phenoinenon that the deflection increased 

very rapidly. For loads higher than 26.2 Ibs, it is expected that 

snapping will occur early, during quasi- static loading and then 

the creep behavior will he similar to that shown on Fig 4, past 

point c. 

The next exlunple considers the influence of cyclic loading on 

the response. Figure 5 illustrates the load deflection behavior 

of the arch under a cyclicnlly applied external load. The load 

is increa5ed. quasi-statically, froiii zero to 26 Ibs in 5 minutes, 

then it is held constant for 2.5 iiunutes, after that it is reduced 

to 20 lbs and held constant for 50 iilinutes, then raised to 25.5 

11,s for 2.5 iiunutes and finally it is reduced back to 20 111s and 

held constant. 

470 

as I. a 

Fig. 4 Deflection Tiirie History 

1 I I I J 

1 in d 

Fig. 5 Multi-Cycle Arch Response 

The steady state response under this type of loading exhibits 

several liiiut points, which limy imply that snapping is iriuninent shortly after the load reaches the value of 26 Ihc Il)erween 

points A and B on Fig 5). The dashed curve correc1)onds ro 

the hypothetical elastic static response and it is only shown for 

coiiiparison purposes. 

The last exluiiple presented in Fig 6. consider the influence 

of teiirperature on the arch behavior. The loading history is the 

sanie on the one shown on Fig. 4. The curve rorresporlding to 

T = 50°F was discussed previously (Fig. 4), and it is used here 

N a hksis for comparison. When the terriperature is raised to 

2OOoF (after this, the loading is applied), the tinie to snap is 

reduced to 2G ~iunutes, while the critical titiie to creep is not 

appreciable dected. On the other hand at the highest teiiiperature T = IOOO"F, for which results are shorn. The critical ti111c 

to creep is reduced to G2 iiunutes. and the steady state response 

does not show a clear snap-t hrough beliavior. 
. 

1. Marrgperre. K., "Die Durchschlagskriift eines Schwach 

Gekruiiuiiten Balken", -. C :esi, Vol 35, 

1938, pp 92-108. 

2. Tiiiioshenko, S.P.. "Buckling of Curved Bars with Sild1 

(Iurvature-. J. Appl. Mech.. Vol 2. No. 1, 1935. pp 15-23. 

3. Biezeno. C.B., "Das Durchschlagen cines Scliwach 

(;ekruiiuiiten Stabes". Zeitschrift Ange. Math. und Mekh 

Vol 18. 1938, 1)p 21-29. 

4. Fung. Y.<'., and Kaplan. A., "Buckling of Low Arches 

or Curved Bcaiiis of Siiidl Curvature", NACA TN 2840. 

Washington, D.C., 1952. 

tint ,hr 5. Holf. N.J.. mid Bruce, V.C& "Dynamic Analysis of tlie 

c ---- . - -___-___-- AA .d Buckling of Laterally Loaded Flat Arches"2. hlath. miel 

PI'?.. Vd 32. NO. 4. 1954, pp 256-2RR. 0.s , 1.1 L6 

6. Gjelsvik. A., nncl Uodner, SJ., "The Eiiergv C:rilerion and 

Siiai) Biirkling of Arches". J. Eng. Mecli. Dir. ASC'E. Vol. 

88. Eh15. 19(i2. pp 85-108. 

I'ic. The IiiIIiie~ice of Teiiiperature Raise 

i'. D iscussion 

5. Schreyer. H.L., aid Masiir. E.F., "Buckling of Slidor 

Archer". J. Eng. Mech. Div.. ASCE, Vol. 92, Eh14. 1966. 

I'P 1-15. 

8. Masur. E.F.. and Lo, D.L.C., "The Sliallow Arch-General 

Buckling. Post Buckling. and Iiiiperfection Analysis". 

J. Struct Me&. Vol 1, No 1, 1972, pp 91-105. 

9. Siiiutses. G.J., "Snapping of Low Pinned Arches on an 

Elastic Foundation"J. Appl. Mech., Vol. 40, No3., 1953, 

pp 541-544. 

10. Siiiutses. G.J., Elastic Stability of Structures, Prent.icelid1 Inc.: Ennlewooci - ClitTs. N.J., 1976; second printing - 

R.E. Krieger Puhlirhing Co.. Melbourne. F.L., 19AG. 

11. Roorda. J.. "Stal)ility of Structures with Siiiall Iiiiperfectionr". J. Eng. Mecli. Div., ASCE, Vol 91, EM1. 1965. pp 

85-108. 

I 

12. Onat, E.T.. and Shu. L.S.. "Finite Deforiiiation ofa Rigid 

Perfectly Plastic Archr, J. Appl. hfech.. Vol. 29. No. 3, I 

I 1962. pp 549-553. 

13. Franciosi. V.. Augusti, C;., and Sparrnrrio. R., "Collapse of 

Arche. Under Repeated Loading", J. Struct, Div., ASC'E. 

Vol. 90, ST1, 1964. pp 165-159. I 

'1'11.. iriflii~iir~ of rtirriiio~i~ie~lianical coiipliitg can heconic 

k1.w I;t:<e in .tal)iiity prol)leiiis. Such proresses are dwavr 

rmnwii4 with a rapid growth of inhoiiiogeneity of the st.ate. 

Tiiorwiqli investigation of such prohleiii~, however, iiiust be perfcr!tiwi with the necessary detail. 

14. Lee, L.H.N.. and Murphy. L.M., '*Iiielast.ic Buckling of 

SliaUow Arches". J. Eng. Mech. Div., Vol. 94, EMl. 1968. 

pp 225-239. 

i 15. Augusti. G., "Biickling of Inelastic Arches: Siiiiple Model- 

-. Vol 3. 1968, pp 102-105. !I 

Arkno\vleil~riiient 1G. Batteriiiiui. S.C.. *Plastic Stability of Archcs:Re- Consideration of a Model", 

45-456. 

J. of Tech, . Vol. 9. 1951. pp 

I T!ic rp-e;trrlr de,cril)ed 'tove ha5 been perforiiiecl under NASA 

(*:rnr Xn. 3-534. The financial support provided I)? NASA is 

;:~t+fiilIv arknowledged hy the authors. The authors dso wish 15. Kaiiurtsu, S., "Braced Steel Arches" Ch. 9 in Steel FfRiiied I to oxteiiil their thanks and appreciation to Dr. <:.C:. Chruiur of 

thr S.AS.i-l.rwi~ Research Center for his support and for the 

in*irfit\ into the prohleiii which he proridetl during our many 

ier ii riiral t1i.r 114onc. 

Structires; Stahi1it.y and Strength, ed. I)g R Narayanan, 

Elsevier Applied science Publishers Ltd, London. 1985. 

I ORIGINAL PAGE 1s 

471 OF poofz QUALtTY 
18. Htiang. N.C'.. and Narhhr. W.. "Dynaiilic Snap-Tlirough 

of imperfect Viscoelastic Shallow Arches". TR-AF- AFOSR 

I??R-ti. Oiiiverdty of ('aliroriiia at Ldolla, hlarch 1967. 

19. hrnjcinovic, D., "('reep nuckling of a Tied Arch3 in 

Staliilitv of Strtictctre* lhcler Static iuid Dynaiilic Loads. 

puldislird I)y ASC'E. sew York. 1975. 

20. Botros. F.R.. and Bienek. h1.P.. "Creep Buckling of Structure\". Procccclings oft he AIAA/ASArE/AS('E/A}rS 24th 

SDSI Conference (AIAA Papr No. 83-O8(i4). Mag 198J 

21. Hlu. ('.S., "On the Dynaulic Staldity of Elastic Bodies 

with Prescrihed Initial <'onditions*. Int'l J. Eng. Sci.. Vol. 

4. 1966. pp 1-21. 

22. Hsu. C.S.. "The Erects of Variou;.Paraileters on the Dynamic Stability of a Shallow Arch*. J. Appl. Mech., Vol. 

34. K0.2. 1965. pp 349-358. 

23. Lock. M.H.. "The Snaplhg of a Shallow Sinusoidd Arch 

l'ncler a Step Pressure LOMI". AIAA. 1'01. 4. No. i, 

19OG. pp 1249-1250. 

24. Sinlitqes. G.J.. "Suddenlv-Laded Structural ('onfiguralions". J. of Eng. hlech.. AWE, Vol. 110. No. 9. 1984. 

pp 1320-1334. 

25. \\aiLrr. K.P., "Research and Developiiient Pragranr for 

Nonlinear Structural hiodeiing with Aclvmced Time Tenlperat ure Deprnclent Const it it ti vc R elat ions', NASA Contract Report, NASA ('R 165533: 1981. 

2C. C'han. K.S.. Bother. S.R., Walker. K.P., and Lindholiri. 

I'.S.. "14 Siirvey of Unified Constitutiw Thebri-', NASA 

('ontract Report, NASA-23925, 1984. . 

27. Riff. R.. C:arlsoii. R.L., and Sinlitses. C.J.. "The Thenilcr 

tIyiiaiiiicaIIy (bnsistent Constitutive Equations Fo~ Nwisot lieriiial Large Strain. Elasto-Plastic. Creel) Behavior". 

AlAA/AShlE/ASCE/AHS 26th SDi'vl Conf.. Orlando. 

Floricla. April 1955. AIAA' Paper No. 85-0621-CP. 

in h(A&,!, 

28. siiilitses. G.J.. <:arlson. R.L.. and RifT, R.. "Forilnilatioll 

of the Nonlinear Aiialysis of Shell-Like Structcires. Subjrrtrd to Tiiiie-I)ependent !vlcclianical anid Theriiiat Loading" NASA Report, 1956. 

29. Zieiikiewicz, OX.. and ('heiiiig, Y.E., The Finite Eleiiieiit hlet liotl in Structural aid <'oat inutiiii hlechanics, 

Mc(;raw-Hill. 1967. 

30. Odrn. J.T.. me E~PIII~IIIS of Noiiliiirar C 'ont iiiiu). .. 

hWraw Hill. 1972. 

ORIGINAL PAGE IS 

OF POOR QUALrrV 

31. Raiiuii. E.. 'Strategies for Tracing the Nonlinear Rezpolise 

Kear Liiiut points'. Proceedings Europe-Il.S. Worksliol) 

OII "Noitliiiear Finite Ele~~rent Alialysis in Structural hiechanicc'. Boc11111ii. l!IRO. Spriiiger-1'erlay 19RI. 

472