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./V
1:t..1A 3D5
CIVil ENGINEERING STUDIES STRUCTURAL RESEARCH SERIES NO. 305
FLEXURAL ANALYSIS F ELASTIC-PLASTIC
REel ANGULAR PLATES
by L. A.lOPEZ
and A. H.-S. ANG
Issued as a Technical
Report of a Research
Program Carried out
under
NATIONAL SCIENCE FOUNDATION
Research Grant Nos. NSF-GP984
and NSF-GK589
UNIVERSITY OF ILLINOIS
URBANA, ILLINOIS ~_f MAY 1966
Metz Reference Room _ Civil Engineering Departmen~ BI06 C. E. Building University of Illinois Urbana _ Tll; YIn; ~ h 1 ~()1
FLEXURAL ANALYSIS OF ELASTIC-PLASTIC
R~CTANGULAR PLATES
by
L. A. Lop~z ~nd.A. H.~S. Ang
I~sued as a Technical Report of a
Research Program Carried out under
.NATIONAL SCIENCE FOUNDATION
Research Grant Nos. NSF-GP984
and NSF.". GK589 .
UNIVERSITY OF ILLINOIS
URBANA, I LL I NO IS
May 1966
ACKNOWLEDGMENT
The results reported herein were obtained as part of a research
program on the development of mathematical models for sol id media and
structures being conducted in the Department of Civil Englneering, University
of Illinois, and sponsored'by the National Science Foundation under
Research Grants NSF-GP984 and NSF-GKS89. This report is based on a doctoral
thesis by Leonard A. Lopez, submitted to the Graduate College of the
University of III inois in partial fulfillment of the requirements for the
degree of Doctor of Philosophy. The work was done under the immediate
direction of Dr. A. H.-S. Ang, Professor of Civil Engineering.
The authors wish to acknowledge the suggestions of Dr. J. W. Mel in,
Assistant Professor of Civil Engineering, related to the development of
computer programs and to Dr. J, H. Rainer, Assistant Professor of Civil
Engineering, for fruitful discussions during the course of the investigation.
Also gratefully acknowledged is the cooperation of the Department of Computer
Science in the use of the 7094-1401 system (partially supported by National
Science Foundation Grant NSF-GP700),
- i i i-
/
TABLE OF CONTENTS
List of Figures .... , .
I.
II.
INTRODUCTION
1.1 1.2 1.3 1.4 1.5
1.6
Object and Scope . . . . . . Notation. . . ........... . Limitations in the Theory of Plates, . Simpl ifying Assumptions in the Theory of Plates The Differential Equations of Equil ibrium .
1 .5. 1 1.5.2
Small Deflection Theory ..... . Large Deflection Theory.
Boundary Conditions ...... . •• ;t ,
1.6.1 1 .6.2
Small Deflection Theory .. Large Deflection Theory .....
1 .7A Numerical Approach to the Problem.
THE MATHEMATICAL MODEL
2.3 2.4 2.5
Mathematical Criteria for the Model The Flexural Model.
2.2.1 2.2.2
Desctiption of the Model Field Equations of the Model.
• 61 • II
Simpl ifications for the Small Deflection Theory Boundary Conditions ......... . An Incremental Form of the Field Equations ....
I I I. CONSTITUTIVE RELATIONS
IV
3.1 3.2 3.3 3.4 3.5 3.6 3.7
Gene ra 1 Rema rks . . . . . . . . The Prandt1-Reuss Sol id . Notation .... Elastic Stress-Strain Law. Plastic Stress-Strain Law. An Increm~nta1 Form of the Constitutive Relation Incremental Moment-Curvature Relations for Small Deflection Problems.
THE NUMERICAL TECHNIQUE ...
4.1 4.2
Introductory Remarks The Numerical Problem.
-iv-
Page
vi
1 2 4 5 6
6 9
11
11 12
13
14
14 15
15 16
21 22 25
29
29 29 30 31 32 35
38
41
41 41
v
VI
4,3
4.4 4.5 4.6 4.7 4.8
I I I
TABLE OF CONTENTS (Cont1d)
Details of the Solution Process ....... , , . , , , .. .
I 4.3,1 Small Deflection Problems 4.~~2 Large Deflection Problems 4,3.3 Forced Convergence of Iterative Scheme ...
solution of the Equations YiJld Surface Correction
I
Initiation of Yielding ..... ! ••
Unlload ing .. • .... . Undoupl ing of the Difference Equations
I NUMERICAU RESULTS.
• I
5.1 5.2 5.3 5.4
!
I Problems Considered.
I
Sq4are , Uniformly Loaded Plate with SqJare~ Uniformly Loaded Plate with
I Sq~are, Uniformly Loaded Plate with Su~ported and One Edge Free . . . . <::'l'ld~""A lin i -F" .... rnl \I I "!!lrlorl D1 "'+-0 ••• : +-1-. v';"""- ....... , VIIIIVIIIII Y L.UY'-4Y'-4 .. IOL.\,...o VVI LII
(LJrge Deflection Theory) . 5.6 coirectness of Solutions
SUMMARY 1ND CONCLUSIONS, . , , .
5.5
I
Four Simple Suppor~s. Four Fixed Supports. Three Edges Simply
III • • .. .. • • " • ~
Fou r Ro 11 er Supports . ,
REFERENCES
APPENDIX A
FIGURES .. , .. .. • " " • • " • • .. II • • .. • III • • • • .. ~ • II • II • ..
-v-
Page
42
42 43 44
45 47 49 50 51
54
54 55 57
58
58 60
62
63
65
68
Figure
2a
2b
3
4
5
6
7a
7b
8
9
10
11
12a
12b
13
14a
14b
15
16
1 7
18
LIST OF FIGURES
Sandwich Plate Configuration ...
Resultant Moments and Shear Forces Acting on an Infinitesimal Element. . . . . . . . . . . . . . . ....... .
Resultant Membrane Forces Acting on an Infinitesimal Element ...
The Flexural Model
Shears and Membrane Forces Act i ng at Node "0" ..
Sign Convention for Moments at Node lid' . ...
..:...4 -4 Element for Determining the Shears Q and Q , and the In-Plane Equil ibrium Equations x ~ . .
Modtftcations Along an Edge ..
Statics Along a Free Edge ...
Flow Diagram for Small Deflection Problems.
Flow Diagram for Large Deflection Problems ...
Computation of Moments and Deflections for Small Deflection Problems.. . .....
. ' ...
Computation of Strains, Stresses, Moments, and Membrane Forces -- Large Deflection Problems
General Form of the Stiffness Matrix.
El imination of the Elements in the ith Equation ..
Plastic Moment Corrections ..
Unsymmetrical Boundary Placement.
Symmetrical Boundary Placement.
Symmetrical Boundary Placement for a Square Plate ..
Coordinates Used in the Demonstration Problems ...
Load-Deflection Diagram at the Center of the Simply Supported Plate .....
Progression of Yielded Regions -- Square Plate with 4 Simple Supports . . . . . ...
-vi-
Page
68
69
70
71
72
73
74
75
75
76
77
78
79
80
80
01 01
82
82
83
84
85
86
Figure
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
LIST OF FIGURES (Cont1d)
Redistribution of Moments Due to Plastic Flow --4 Simple Supports ............... .
Redistribution of Moments Due to Plastic Flow --4 Simple Supports. . . . . . . . . . .....
Typical Displacement Contours 4 Simple Supports
Residual Moment Patterns -- 4 Simple Supports
Yield Criterion Used to Determine the Upper and Lower Bounds ..
Load-Deflection Diagram at the Center of the Plate 4 Fixed Supports . . . . ....
Progression of Yielded Regions -- Square Plate with 4 Fixed Supports . . . . . . . . . . . . . . . . . . . . ,
Redistribution of Moments Due to Plastic Flow -- 4 Fixed Supports ..
Redistribution of Moments Due to Plastic Flow -- 4 Fixed Supports
Typical Displacement Contours -- 4 Fixed Supports
Load-Deflection Diagram at the Mid-Point of the Free Edge
Progression of Yielded Regions -- Square Plate with 3 Simple Supports and 1 Free Edge ... 0 •
Redistribution of Moments Due to Plastic Flow -- 3 Simple Supports and 1 Free Edge . . . . . . . . . . . .
Redistribution of Moments Due to Plastic Flow 3 Simple Supports and 1 Free Edge
Typical Displacement Contours -- 3 Simple Supports, 1 Free Edge . . . . . . . . . . . . . . . . .
Qual itative Comparison of the Large Deflection Solution with Levy's Solution ................ .
Comparison of Elastic-Plastic Solutions Given by Small and Large Deflection Theories ........ .
Progression of Yielded Regions -- Square Plate with 4 Simple Supports Using large Deformation Theory and r = 10 .....
- vi i-
Page
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
F j gu re
37
38
39
40
LIST OF FIGURES (Cont1d)
Redistribution of Moments Due to Plastic Flow Large Deflection Solution ..... 0
Redistribution of Membrane Forces with Plastic Flow Large Deflection Solution .......... .
Redistribution of In-Plane Displacements a ~=O; ~=O Due to Plastic Flow -- Large Deflection Solution ..
Load-Deflection Diagram for In-Plane Displacements at ~=O; ~~O -- Large'Deflection Solution' .....
41 Ratio of Membrane Force to Moment at s=1/2; ~~1/24 --
42
43
44
45
46
47
48
Large Deflection Solution ....
Convergence of Model Solutions with Mesh Size 4 Fixed Supports ............. .
Convergence of Model Solutions with Mesh Size 4 Simple Supports ...... n ••
Load-Deflection Diagram at s=.5; ~=O, for 3 Different Mesh Sizes -- 4 Fixed Support~ ....
Variation in the Distribution of Moments with Mesh Size; q=qm -- 4 Fixed Supports ........ .
Sch:matic Diagram of the I'True' Flexural Model
Elevation of Section X-X after Deformation -- for Determining the Normal Strains . . . . . ...
Definition of Shear Strain in the Model
-vi i i...,
Page
105
106
107
107
108
109
110
111
112
113
114
1 15
I e I NTRODUCTI ON
1 Q 1 Db i ect and Scope
The existing theory of plates is noticeably deficient due to its
inabil ity to cope with the behavior of plates which have been subjected to
loads higher than their elastic-l imit load. This difficulty has been partially
overcome through the use of 1 imit theorems which enable one to reasonably
predict upper and lower bounds for the load carrying capacity of a plate, However?
since these theorems are based purely upon kinematical and statical considerations,
they still cannot be used to determine the behavior of a plate; i.e., its load-
deformation characteristics.
Several authors have found analytical solutions for circularly-
symmetric elastic-plastic plates on the basis of the Tresca yield criterion
(1'), C2), (]"), (4.')'i\·. However, each of these problems involves only one
independent spatial coordinate, the radius. This fact, combined with the
geometric simp1 icity of the Tresca yield criterion, allows one to directly
integrate the governing differential equations and hence obtain the above
mentioned solutions. Rectangular plates require two spatial coordinates x and
y; the corresponding problem then is consider-ably more difficult than the
circularly symmetric problem; analytic solution of the field equations has not
been possible so far.
The object of this work is to develop a numerical technique by which
one can obtain the behavior of a rectangular plate subjected to loads ranging
from zero to the ultimate capacity of the plate. Use is made of a mathematically
~,~
"Numbers in parentheses refer to entries in the bibl iography.
-1-
consistent lumped parameter model in which stresses~ strains, anp displacements
are defined at discrete points. Thus, the continuous problem is replaced by
one with a finite number of degrees of freedom. The field equations are then
formulated directly from the model and solved for the unknown displacements from
which all other quantities can be determined. In order to make the problem
manageable the analysis has been restricted to sandwich plates where the material
in the outer sheets follows the stress~strain law for a Prandtl-Reuss sol id.
The primary emphasis of the investigation is the development of a
technique for the numerical analysis of elastic-plastic bending of plates based
- on the small deflection theory of plates. However, the approach is appl icable
t9 large deflection theory; the equations for both large and small deflection
assumptions are presented, although the results for the large-deflection the~ry
are 1 imited.
1 .2 Notat i on
e spherical strain
~ spherical strain rate
E YoungBs Modulus
G shear modulus
h half-thickness of the sandwich plate
J2 deviatoric stress invariant
k y i e 1 d 1 i mit ins imp 1 e shea r
K bulk modulus
M M M x? y' xy
6M 6M .6M x' y' xy
M M M x? y' xy
N , N , N x Y xy
6N,~,6N x Y xy
moments per unit width
incremental changes in the moments per unit width
total moments at a node
membrane forces per unit width
incremental changes in membrane forces per unit width
N x' N y'J N xy
total membrane forces at a node
Q ~ Q x y
shear forces per unit width
Qx' Qy total shear forces at a shear point
q external load per unit area
6q incremental change in the external load
5 X'
5 y' 5 Z
deviatoric stresses
5 x? 5 y' 5 z
deviatoric st~ess rates
s spherical stress
s spherical stress rate
t thickness of the sheets in the sandwich plate.
u'J v in-plane displacements in the sheets of the sandwich plate
U, V, W displacements at the middle surface
6U, 6V, ~ incremental changes in the displacements of the middle surface
V.i positive scalar
x, y, z spacial coordinates
13'J X
f3 y 'J (3 xy
components of strain at the middle surface
.613 x 'J t:f3 y' fi3 xy incremental change in components of strain
€ x' E y
normal strains 0 . € x' E
y normal strain rates
6E x' 6E y incremental changes in normal strains
r dimensionless parameter
Yxy'J' y , r ·xz yz shear strains
Yxy ' y xz'
y yz shear strain rates
6Y , xy 61 , xz 61 yz incremental changes in shear strains
A. mesh size
I e , e , e x y xy
curvatures
v Po i sson·is rat i 0
-4-
6.(; x'
/;:,8 , 6.6 incremental changes in curvatures y xy
<P yield function
CT x' (J , (J" no rma 1 stresses
y z
CT , 0- , 0- normal stress rates x y Z
LtJ x' /YJ y' fff incremental changes in normal stresses z
'T xy' 'T , '[ shear stresses
xz yz
'I , 'I xz' 'I shear stress rates xy yz
6.'[ , L'I xz' 6'I incremental changes in shear stresses xy yz
1.3 Limitations in the Theory of Plates
The behavior of a plate subjected to lateral loads is greatly
influenced by the ratio of its length to its thickness. Consequently, for
practical reasons, the theory of plates consists of three parts:
1. small deflection theory of thin plates.
2. large deflection theory of thin plates.
3. theory of thick plates.
Parts 1 and 2 are considered in the present work; part 3 is not.
The need to differentiate between a small and large deflection
theory is readily seen by considering a simpler, but analogous problem.
A beam is pinned at both ends and subjected to an increasing lateral load.
Initially the deflections are small and the lateral load is, carried primarily
in flexure. However, with increasing load the deflections become large enough
so that the beam will stretch and bend simultaneously. As a result of the
deflections, an increasin9 proportion of the load will be carried by the induced
axial forces. Timoshenko (5) has shown that in elastic plates where the ratio
of the center deflection to the thickness is less than .5, the plate behaves
similarly to a beam subjected to low inten~ities of external load in that
-5-
the load is carried primarily in flexure and that the membrane (axial) forces
can be neglected. These problems fall within the realm of the small deflection
theory. However, if the ratio of the center deflection to the thickness becomes
greater than .5~ the effect of deflections becomes increasingly significant, and
the large deflection theory~ which takes into consideration both bending and
membrane action, must be used.
1.4 Simplifying Assumptions in the Theory of Plates
The analysis of a plate is a complex three dimensiona~ problem.
However, by making the fol1owijng simpl ifying assumptions it can be reduced to
one in two dimensions:
1. plane sections normal to the middle surface before bending
remain plane and normal during bending; this is known as the
iK:i Jr·chh6ff.:. as:5;uf11pti on.
2. the transverse displacements do not vary throughout the
thickness of the plate.
3. stresses normal to the plane of the plate are neg1 igible.
The above assumptions were originally made by investigators who
were concerned only with the elastic theory of plates. The introduction of
plasticity compl Yeates the problem considerably since yielding may occur
and propagate through the thickness as well as in the plane of the plate.
Consequently, in order to continue to treat a plate as a two-space problem~
an additional assumption has been made. Fig. 1 shows a schematic representation
of a sandwich plate. ~t is assumed that the thin sheets are subjected to a
general ized plane stress condition; i.e., all stress components 1 ie within the
plane of the sheets and are distributed uniformly across the thickness of the
sheets. Shear forces resulting from the bending of the plate are carried by
-6-
the shear core (s~e Fig. 1). The resultant forces and moments per unit width
acting on a cross-section of the plate are related to the stresses by
M = (rr - rr xt) ht N == (rrxb + crxt ) t x xb x
M == (0' - rr ) ht N == (rryb + rryt ) t (1) Y yb yt y
M = (1;' - 'T ) ht N = (,It' + 1" ) t xy xyb xyt xy xyb xyt
where the subscripts band t, respectively, refer to the bottom and top sheets
of the plate; h is the half-thickness of the plate, and t is the thickness
of the sheets.
lp~ The Differential Equations of Eguil ibrium
1 .5.1 Sma 11 D~f 1 ect i on Theory
The equil ibrium equation is derived by consi~ering the statics of
an infinitesimal element within the ~ontinuum. Fig. 2a is a schematic
representation of such an element with the resultant shear forces and
moments acting on it (axial forces are neglected in the small deflection
theory). Summation of forces in the z direction yields
(2)
while summation of moments about the x and y axes, respectively, leqds to
(3 )
oM OM ~ ;. -.-Ei. -Q = 0 ox dy x
Differentiating Eqs. (3) and using the result in Eqo (2) yields th~ following
equi'l.lbrIumeql.iatiort in':ier.ms of moments:
-7-
- q
In the small displac~ment theory of elasticity (6) the strains
are approximated by
E X
_ dU - ..,...-
dX E
Y :::; dV
dy 'Y xy
dU dV :::;-+-dy dX
From the assumption outl ineq in Section 1.3, the in~plane displacements in
the thin sheets of a sandwich plate are related to the transverse displacement
by
- dW u :::; +h -
dX v :::; +h dW
dy
where the minus sign refers to the bottom sheet and the plus sign to the
top sheet. Hence, using Eq. (6) in Eq! (5) leads to
2 :::; +2h d W
dX dy
where again~ the minus sign refers to the bottom sheet and the plus sign to
the top sheet.
For a 1 inearly elastic isotropic material under plane stress
conditions, the stress-strain relation of Hooke is:
E (€ E ) (J :::; + v x l-v
2 x y
::: E (E V--:E ) (J 2 + y
l-v Y x
'f :::; 'G 'Y xy xy
(4)
(5)
(6)
( 7)
(8)
-8-
G is related to E by
G == E
z (l+v)
where E is Youngls modulus? and v is Poissonls ratio. Substitution of Eqs. (7)
and (8) into Eq. (1) results in the elastic moment-curvature relationships
for a sandwich plate.
M x
M Y
M xy
==
=
2Eh2t 2 d2W (9 W + v'2)
l-v 2 dX2 dy
2Eh2t (J2W d2W (-. + ;V -.-)
l-v 2 dy2 dX2
2Eh2t d2W =;----. l+v dx dy
Hence, comb in i ng Eqs. (4) and (9) y i e 1 ds in the fo 11 ow i n9 fou rth order
partial differential equation in W:
(9)
4 \]w= 9 ( 10)
A solution of Eqg (10) satisfying appropriate boundary conditions constitutes
the complete solution to an elastic plate bending problem under the above
stated assumptions of small deflections.
For an elastic-plastic material Eqs. (8) through (10) are no longer
val id; however, Eqs. (4) and (7) are still appl icable since these equations
are, respectively, statical and kinematical relationships and thus~ do not
depend upon material behavior.
-9-
1.5.2 large~D~flection Th~ory
If the center displacements become large with respect to the
thickness of the plate, it becomes necessary to include the effect of the
membrane forces in the equil ibrium equation. Fig. 2b shows the plan and
elevation of an infinitesimal element of the plate with just the membrane
forces acting on it; these forces must be added to those already shown in
Fig. 2a.Eqq (3) is not affected by the membrane forces provided the
deflections remain small compared to the lateral dimensions of the plate.
However, Eq. (2) must be modified to include the effects of their vertical
components. Thus~ Eqo (2) becomes
Substitution of Eq. (3) into the above leads to the equil ibrium equation for
the large deflection theory.
( 11)
In additol1 to satisfying Eq. (ll)~ the forces N ~ N ? and N x y xy
must also satl~fy in-plane equil ibrium. With reference to Fig. 2b summation of
forces::in the x and y directions lead~ respectively? to the following additional
equil ibrium requirements:
dN dN x +~::;; 0
dX dy
dN dN --::L. + -.E.. = 0 dy dX
( 12)
-10-
In the general theory of elasticity, the three components of
strains corresponding to those given in EqQ (5) are,
E X
In the small deflection theory of plates (see Section 1.5.1) it
was assumed that the second order terms in the strain-displacement relations
could be f:1egl'ected~:;hence the,a9o"e equatJons,:redute to Eqi,.·: -(5).:;.; In the
large deflection theory it is assumed that these terms are no longer
neg1 igible; however, the first and second terms in the bracket are usually
small compared to the last term. Thus, in the large deflection theory the
strains are given by the following (5):
E ;;: du + {' dW )2 x dx 2\ dX
E =: dV +l(,dW ).2 y dy 2\- dy -
dU dv . "Ow dW r :::;;-+-+--xy dy dX dx dy
The in-plane displacements are related to the transverse displace-
ments by
v = V +h "Ow "Oy
(13)
( 14)
-11-
where U and V are the in-plane displacements at the middle surface of the
plate. Using Eqo (14) in Eqo (13) leads to the strain displacement relations
for the large deflection theory of plates.
E . = dV +h d2~ + 1 dW )2 Y dy dy 2\ dy
(15 )
" =: dU + dV ;. 2h (J2W + dW .dW xy dy dX dX dy dX dy
If the material is 1 inearly elastic and isotropic Eqs. (1), (8),
(11), (12) 9 and (15) can be combined to form a set of three non-l inear
partial differential equations in U, V7 and W. A solution to these equations
satisfying the appropriate boundary conditions constitutes the complete
solution to a plate problem under the above stated assumptions.
A 1 imited number of numerical and series so:hjtions to the elastic
problem are in existence (7)~ (5).
1.6 Boundary Conditions
1.6.1 Small Deflection T~
I n order to obta ina compl ete so 1 ut i on to Eq. (10) it is necessary
to prescribe a set of boundary conditions which describe the constraints on
the edges of the plate. The three conditions most commonly encountered in
practice are:
1. simple ·support.
2. fixed support.
3. free edge.
Simple Support ~ A simple support along the edge y = a is characterized as
-12-
having zero moments normal to the edge and zero displacements along the
edge. Hence:
WI,. = 0 y=a (16 )
Fixed Support - A fixed support along the edge y=a is characterized as
having zero displacements and rotations along the edge. Hence:
w Iv=a = 0 'Ow = 0
'0 ; y y=a
(1 7)
Free Edge - The free edge condition is not as evident as the two preceding
conditions. Along such an edge it is normally expected that
Qy I y=a
= 0 MI = 0 MI· = 0 ( 18) y -'J=a xy y=a
However, Kirchoff (8) has shown that as a consequence of the assumptions in
the ordinary plate theory (see Section 1.3), only two independent conditions
can exist along a free edge. These are:
(19 )
These expressions can be derived from a variational procedure or from a
purely physical interpretation of the free edge (9). The latter method
was used to develop a similar boundary condition for the model presented
herein.
1.6.2 Large Deflection Theory
For large deflection problems it is necessary to prescribe
in-plane boundary conditions in addition to the type described in Section
1.6.1. The large deflection problem considered in this thesis is 1 imited
to the following boundary conditions:
· = 13-
w 1 - 0 y=a U I' y=a
= 0 Ny.lv=a = a (20)
This boundary condition corresponds to that of a plate supported
by a roller which is restrained to move in a direction normal to the edge;
no translatory motion parallel to the edge is allowed.
1.7 A Numerical Approach to the Problem
The consideration and inclusion of non-l inear material behavior
into flexural problems in plates results in partial differential equations
which are not amenable to analytic solutions; consequently, a numerical
technique of solution is necessary.
A method that has been appl ied successfully for the determination
of approximate solutions of continuum problems is that of digital simulation,
where the continuum is represented by a lumped parameter model (10), (11),
(12), (13), (14). The field equations are then formulated directly from
the model and solved for the unknown displacements~ from which all other
quantities can be computed. A lumped parameter model for treating f]exural
problems in plates is presented herein; both smal1 and large deflection
problems are considered. It should be emphasized that with this model
solutions are not restricted to a particular stress-strain relation.
However j for demonstration purposes 9 the problems treated herein are
confined to materials exhibiting elastic=perfectly plastic behavior.
I I. THE MATHEMATICAL MODEL
2.1 Mathematical Crit~ria for the Model
Currently there are three fundamental types of models being used
to solve problems in continuum mechanics:
1. lattice models.
2. finite element models.
3. mathematically consistent lumped-parameter models.
Historically, lattice models (15) were among the first to be
used for treating 1 inear problems in continuum mechanics. In this method
the continuum is replaced with a network of elastic bars whose load
deformation characteristics are, at discrete points 7 the same as those
for the continuum. However~ since the stress and strain tensors are not
defined expl icitly in these models, they are not amenable to solving
problems with non-1 inear material behavior.
In the finite element method (16) 7 (17) the continuum is divided
into discrete sol id elements which are then interconnected at their corners.
The stress and strain tensors are defined in these models; hence they can
be used to treat problems with non-1 inear material behavior. At the present
time there are a 1 imited number of solutions to non-1 inear problems available
which were obtained by the finite element method; however~ the displacement
functions for the individual elements which were used to obtain these
solutions are 1 inear (homogeneous states of stress and strain in each
element~· This type of displacement function is not appl icable to plates.
The third approach, that of using mathematically consistent
lumped parameter models, was first suggested by Newmark and formally
-14-
-15-
developed by Ang (11). Each physical quantit~ in these models is point-wise
compatible with a corresponding quantity in the continuous system. Thus 7
the model is a physical discretization of the continuous system.
In his models, Ang has establ ished the criterion of mathematical
consistency; i.e., the field equations of the model must be consistently a
finite difference form of the continuum equations. By virtue of the above
criterion~ the model is also a physical representation of the finite
difference equations for the continuum. Hence, the model solutions are
subject to the 1 imitations of the finite difference method. At the same
time, as more proofs similar to those given in Ref. (18) concerning the
convergence of finite difference solutions to those of the continuum~
problems become available, they can be immediately appl ied to existing
model solutions.
Proof of convergence of the method presented herein would
follow all ied work in finite difference theory. Such studies are beyond the
sc6pe of the present work; in the absence of these theoretical results?
however, the convergence of the solutions presented here'i'fiJcan be
judged, at leait partially, on the grounds that the formulation is physically
meaningful; furthermore, sequences of solutions corresponding to decreaSing
mesh sizes of the model would suffice to give rel iable indications that
convergence of the model solutions is more than plausible.
2.2 The Flexural Model
2.2.1 Description of the Model
The model described herein satisfies both the Kirchoff assumptions
for the theory of plates in finite difference form~ and the mathematical
-16-
consistency criterion establ ished in Section 2.1.
Fig, 3 is a schematic representation of the mathematical model.
The continuous plate has been replaced by a network of nodes (denoted by
arabic letters for purposes of description) interconnected by bars which
are infinitely rigid in flexure. Moments and membrane forces (stresses) 1
strains, and transverse displacements are defined at each node. Vertical shear
forces and in-plane displacements, such as those denoted by U and V in Fig. 3a,
are defined at the mid-points of the bars. Torsional elements emanate from
each node and are attached to the rigid bars a distance ~/2 away in each
direction. However, for clarity only one such set of elements is shown
in Fig. 3a; where the torsional elements emanating from node !lO" are
connected to the bars h-f, f-e~ e-g 9 and g-h. Hence, a rotation in any
one, or any combination of these bars will induce a twisting moment at
node DROID. The nodes are defined to have a cross-section identjca'i to that
of the plate it represents. Thus, the nodes shown in Fig. 3b are used to
schematically represent a sandwich plate. Note that there are two
independent sets of stresses and strains at each node; one set at z = -h
repres~nting the top sheet of the sandwich plate 9 and another set at
z = +h representing the bottom sheet of the sandwich plate.
~,~
2.2.2 Field Equations of the Model"
The strains E and r a at a distance z from the middle y" xy#
j'~
See Appendix A for a detailed derivation of the strains.
-17-
surface at a typical node 110 " are given by (see Fig. 3a)p
u - U4 Ca -2W 0 + Wb ) 0 2 E ~ - z ,
1\,2 x I\,
reWa - W 2
CWo: Wb )2 ] 1 0 ) +1+ + A.
V1
.... V /W - 2W + W . ) 0 3 \ c o G:
E =: .... Z Y A. A.
2
[ (We: Wo 3 (W - W .2
] + 1 + ,,0 d ) "4 A.
-where, for a sandwich plate~ z is restricted to z = +h. Eqs. (21) were
derived directly from the physical model. The forces acting at a typical
node DIOID are shown in Fig. 4; the equil ibrium equation in the z direction
for the node is given by
+ ,2No xy (
',W .... W ,e f . f..
W g - W h ). + NO C' We"" 2W 0 + W d ); ~ 0 A. .' Y f..
(21 )
(22)
-18-
where the barred quantities represent the total forces at the node in the
specified directions. Each interior node (exceptions occur at the
bounda ry) is defined as having an effective width, ~/2. Hence, the total
forces in Eq. (22) are related to the corresponding forces per unit width
by
MO ::: MO
~/2 NO ::: NO '~/2 x x x x
-0 MO
~/2 NO NO ~/2 M ::: ::::
y y y y
MO ::: MO
~/2 NO ::: NO ~/2 xy xy xy xy (23)
-1 Ql ~/2 -2 Q2 ~/2 Qy
::: Qx :::
y X
-3 Q3 '~/2 -4 Q4 /\/2 Qy
::: Qx :::
y X
-0 Q is the total external load appl ied to node "d'~ and is defined by
-0 0 2 0 Q ::: q ~ /2, where q is the average load intensity over the area
bounded by the points, e, f, g, and h in Fig. 3a (see Section 4.8 for a
detailed explanatio\1 of this definition). Substituting Eqs. (23) into
Eq. (22) leads to the following:
(24)
g h + ~o. co. W - w) (. w - 2W + Wd ) 1 ~ 2. Y :\ ~ 2 , _
Fig. 6 is an enlarged view of the bars h-g and o-b in Fig. 3a with the
resultant membrane forces, shears, and moments from the nodes, 0, h, b,
-19-
and g, acting on the ends of the bars. Summation of moments about the y
and x axes, respectively, and simpl ifying through the use of Eqs. (23),
results in the following expressions for Q4 and Q4: x y
S im! 1 ar
into Eq.
~ ;:
Q4 = Y
expressions
(24) lead
Ma _ x
MC _
+ y
MO _ Mb Mg - Mh
x x + Xv Xv I\. A-
Mg - Mh MO - Mb V V + Xv Xv
I\. A-
can be written for 2 Qx'
1 Qy'
to the equil ibrium equation
2Mo + Mb Me _ Mf x x + 2 ( xy Xv -
A-2 2 0A-
and Q3, which, if substituted y
in the z direction at 110".
Mg _ Mh
) Xv ~V
1-..2
2Mo + Md (W - 2W + W " b \ y v + NO ' a 0
1-..2 x A-2 /
oW - W W - W h) D (We - 2W D + W d ) _ + 2ND ( e f 9 0
A-2 ° + N - -q xy I-.. 2 y I-.. 2
It is also necessary to satisfy the equil ibrium of in-plane
forces. With refe~ence to Fig. 6, summation of forces in the X and y
directions, respectively, at point 4, and simpl ifying, leads to
NO _ Nb Ng - Nh x x xy Xv 0 + =
A- I\.
Ng - Nh NO Nb
Y V + xy xy = 0 A- I-..
(25)
(26)
(27)
-20-
Eqs. (21), (26)? and (27), are, respectively, central finite difference
expressions of the continuum equations (15), (11), and (12).
The above equations can also be expressed in terms of displacements
for plates of 1 inear Hookean material. To do this, it is necessary to
first relate the moments and membrane forces to the stresses. In general,
the desired relations are
~ jh M =I
h
(J' z dz N =lh
(J' dz x x x x
!h h M -I~ (J' z dz N =lh cr dz (28) ~ "
y y y Y ,..h
h h
M;(y=l 'T z dz N =1 'T dz xy xy xy
-h -h
However, for the sandwich plate being considered in this thesis, Eqs. (28)
red u ce to Eq s . ( 1 ) •
For a 1 inear and isotropic Hookean material, subjected to a state
of plane stress, the stress-strain relations are given by Eqso (8). Using
Eqs. (21) in (8) and the resulting expressiol1s in Eqs. (1) leads to a set
of six equations relating the moments and membrane forces at 1I0 Ul to the
displacements of the surrounding points~ Hence 7
2 2Et Go· 2Eh t eO eO ) NO ( f30 + f30 ) M = -'--..... + v =-.-x 1 2 x Y x l-v
2 x v y -v
(29)
MO 2Eh2t eO + v eO. ) NO _ 2Et ·0 f30 =-- f3y + v Y l-v
2 x Y Y ---2
(< l-v
\... 2Eh2t NO =: 2Et. 0 eO ) 0
M = f3xy xy l+v
xy xy l+v
..
-21-
where
eO :::; _ (Wa - 2Wo + Wb
) x . A. 2
_ (We - 2W o + Wd
) eO :::;
~2 y
(W - W W - W
) eO e f 9 h :::; - ')- ? xy ~L. ",'"
and
[ (Wa - W 2 2
~o U2 - U4 1
0 ) + (Wo ~ Wb ) ] = +4 x ~ '"
V1
- V 2'
- W 2
~o 3 +l [(We ~ WO) + (Wo d '\ J :::;
Y A 4
A )
U1
... U V - V4 (Wa - Wb ) (We -Wd) . 0 3 +2 ~xy ::::; +
"- .'" 2"- .. 2",
Expressions similar to Eqs. (29) through (31 ) cc;m be written for nodes
a, b, c, d, e, f? g, and h, which, if substituted into Eqs. (26) and (27)
lead to a set of non-1 inear difference equations in terms of the displace-
ments U, V, and Wof the various nodes.
2.3 Simpl ifications for the Small Deflection Theory
In Section.l.3 it was pointed out that in many engineering
problems, one is justified in neglecting the strains of the middle surface
of the plate. In the model, this assumption is identical ito setting ~ , x
(30)
(31 )
, \
-22~
f3 and f3 to zero at all points. Eqs. (21) and (26) reduce to y xy
and
+
o E
Y eo
=: Z y
o = .... q
Eqs. (25) and (29) remain val id in simpl ified forms, while EqD (27) is
satisfied identically.
2.4 Boundary Conditions
Due to the discrete nature of the model, the introduction of an
edge or any other type of discontinuity will, in general~ necessitate
the derivation of special equations at nodes in the region of the
discontinuity. The purpose of this section is to present the physical
interpretation of the model at an edge so that one can derive these speciai
equations directly from the model.
The introduction of an edge is characterized by cutting the nodes
along the edge in~half and severing the bars which cross it. However)) in
doing so, one is faced with the same inconsistencies which appear in the
continuum theory of plates (see Section 1.6). Consequently~ the model
has been modified or PfinishedDl along the edge as shown in Fig. 7a. With
(32)
this modification the twisting moments along the edge are converted to shears
internally, and in a manner identical to that described by Kelvin and
-23-
Although it is possible to prescribe a wide range of boundary
conditions on the model~ only the four which were used during this investi-
gation will be presented here. The first three are for problems using the
small deflection model (~ =:~ =~ == 0) the fourth is for the large X y xy ,
deflection model.
Simple Support: Deflections and moments are zero along the edge. Hence,
with reference to Fig. 7a
Wb :::;: W =:: W =: = 0 1 a
Mb .=: Ma =: == 0
y y
Fixed Support: The deflections and slopes normal to the support are zero
along the edge. Hence? with reference to Fig. 7a
W =: C
• 0 I' :=;:: 0
:::; 0 (zero displacement)
(average slope is zero at a~ b, .. o)
(average slope is zero at c, •.. )
Free Edge: External forces and moments are zero along the edg~. Hence~
with reference to Fig. 7a
b Mc3 0 M = = :::;
y y
Vb =: V1 ..... Va =: =: 0
R R R
b 1 a where VR, VR' and VR are reactive forces along the edge which can be
determined from statics; e.g.~ 1
the force VR is. determined as follows:
(34)
(3s)
(36)
-24-
Fig. 7b is a free body of the bars a-b and c-l of Fig. 7a, showing
1 only the moments and s~ar forces that are necessary for determining VRo
Summation of forces in the z direction leads to
This shows that the treatment of a free edge condition as defined in the
model is mathematically consistent wi th the last of Eq .. (19) .. Ql can be y
determined by summing moments about the x axis. Hence
and the boundary condition is
Similar expressions can be found for the reactions at nodes band a.
Large Deflection Problem: The boundary condition used simulates a plate
supported on rollers which are free to move only normal to the edge.
Consequently~ in addition to satisfying Eq. (34) the following additional
boundary conditions are specified for the large deflection formulation.
For the edge y~a
N
!v=a = 0
y u Iy=a = a
wh i 161 for the edge x=b
N ! .- 0 ·X ::x=:b V'I x=:::b
- 0
(37)
-25-
2.5 An Incr~mental Form of the Field Equations
In the preceding sections the field equations for the model were
presented in terms of tota'l displacements, moments, and membrane forces.
However, the numerical procedure which is presented in Chapter IV requires
that the governing equations be in an incremental form; i.e., where the
load is changed by only a small amount, say 6~, at anyone time. This
restriction is dictated by the incremental stress-strain relations of the
incremental theory of plasticity. TL .. _ ..... L_ .-1 __ '! ....a_-1 __ .. _ ..... ~ ~ __ I flU:::', LIlt:; Ut:;:;'IBtU t:;t.tUClLIUII;:) are
6(2° ::::: Z 6eo + 6.(30 x x X
6EO = :z; 66° + tt30
y -y Y (39)
ti° 2z b.eo ° :;: + tt3xy 'Y xy xy
where
_ (~a - 2~ + ~b) b.eo := 0
x t-. .z
_ (~c - 2~ + ~d '\ b.eo .= ° 'y ;...2 ) (40)
A6W ~&I tM-l:M '\ b.eo := ~C e f_ s. h
. xy ,'2 : 2 ) ;... ;...
and
-26-
)J q q
r (Wo w '-'(( &/0 - !M .. )+(Wo - Wd )l tMo - 0Wd ) ](41) I 1 .c c ,.. - L '\ )\ .\-A. 2 A. A. .f...
q q
in which the parenthetical quantities with a subscript Ilqll are evaluated
prior to the addition of the load .6.q; i .e. ~ if the external load is being
increased from q to q+.6.q, then the terms in question are evaluated before
changing the external load. In deriving Eqs. (41), terms which involve
the square of the incremental displacements were omitted. As a consequence?
the quantities .6.f3~? .6.~~, and ~~ybecome 1 inear functions of the incremental
displacements.
Eqs. (26) and (27) can then be wr i tten as fo 11 ows:
-27-
(42)
and
6N 0 - fflb ffl9 .. 6N h
x x + _x....,...y"-----.-_x~y = 0
A.
, (43)
,6N9 _,6Nh 6N 0 .. 6N b
Y. Y + ~x_.:..y.r.....,.,....-.--.-_x..L-.y = 0
where 1 a9~in, the quantities with a subscript "qll are to be evaluated
prior to the addition of ~q.
The incremental moments and membrane forces are related to ,the
incremental stresses as follows:
.6M ::= (LYrxb
- ro ) ht 6N :::: (roxb
+ Mxt
) ~ x ' xt x
6M :;;:: (ro b- ro t) ht .6.N ::= (ro + ro ) t (44) y y y y yb . yt
LM :;::: (~'rxyb - ~'r ) ht .6N = (~'r + ~ ) t xy , xyt xy . xyb xyt
and Eq9 (29) becomes
.-28-
f}10 2Eh2t ( ~eo + v ~eo ) ,6,N0 2Et ( fijo + v 1$0 ) = :::;:~ x l ... v 2 x Y x l-v x Y
6Mo 2Eh2t ~eo -+ v ~eo 6N
0 2Et ( 6f30 + v 6f30 ) (45) = =-.-
y .. ' 2 y x y 2 y x l-v l ... v
MO 2Eh2t ( 8eo L),N0 2Et ( .6(3~y ) = =; ------xy 1 v xy xy 1 v
For the small deflection theory? EqsQ (39) and (42), respectively, reduce
to Eqs G (46) and (47):
° . ° !:::,E = Z 6.8 x x
° ° !:::,E = Z 68 Y . y (46)
II L CONSTI rUTI VE REL,ATIONS
3.1 General Remarks
In general 9 most engineering materials can be classified within
one of the fol lowing four categories:
1. elastic -- 1 inear and non1 inear
2. visco-elastic -- 1 inear and non1 inear
3. inviscid plastic ~- inelastic-time independent
4. visco-pl~stic -- i~elastic-time dependent
where each of the above may be subdivided into several smaller groups. The
theory of perfect-plasticity, from which the Prandtl-Reuss Sol id is derived,
1 jes within one of the subdivisions of category 3.
3.2 The Prandtl-Reuss Sol id
The behavior of a Prandtl-Reuss Sol id is characterized as being
elastic-perfectly plastic. In addition to satisfying the
postulate of stabil ity (19), the following are assumed:
The yi-eld function, ~? which defines the initiation of yielding in
a materlal is given as a function of the independent stresses; in a general
six dimensional stress space, ~ = ~ (crx ' cry' CJ"z' '"[xy' 'Ixz' 'T yz ). If <P < k2
,
where k is the yield 1 imit of the material in simple shear, the material
behavior is 1 inearly elastic and is governed by Hooke l sLaw. 2 I f cD = k-, the
material will undergo plastic flow and the plastic stress-strain law, known
as the Prandtl-Reuss flow rule, is used. The condition cD > k2 is not
permissible. I~ after plastic flow has occured at a point, the state -of
stress becomes such that cD < k2
, the material is said to have unloaded from
a prior plastic state; its behavior during unloading is incrementally elastic;
i.e., incremental stress is proportional to incremental strain.
in the following sections the general ized three dimensional stress-
strain relations for a Prandtl-Reuss sol id are reduced to those for a plane
state of stress (the conditions that exist in the thin sheets of the sandwich
plate). Since it is not the purpose of this investigation to develop new
concepts in plasticity, no discussion pertaining to the val idity of the
basic equations for specific materials is given. It suffices only to say
that the resultin~ one-dimensional moment-curvature relation is also elastic-
perfectly plastic which resembles closely that of an under-reinforced concrete
beam; the r~sults.·presented her~1n are directly appl icable to sandwich plates
of structural steel.
3.3 Notation
It is convenient to define the following quantities prior to'
deriving the constitutive relations.
The spherical components of stress and strain are, respectively,
1 s = - (rr + rr + rr ) 3 x y z
e= 1 (E + E + E ) 3 x y z
(48)
The deviatoric components of a stress and strain tensor are, respectively,
s = rr - s e = E - e x x x x
s = rr - s e = E - e (49) y y y y
s = rr - s e = E - e z z -z z
In addition, each of the above quantities may appear in rate form; e.g.,
3.4 Elastic Str~ss-Strain Law
In the elastic range (<1> < k2) stress is 1 inearly related to strain
by~
E ~i[ 0- ~ v (0- + 0- ) 1 "xy = T /G x x Y z xy
E ~ iT 0- - v (0- + 0- ) ] "xz = '! /G y y x z xz
E =H 0- - v (0- + (5) J; ", = T /G z z x yz yz
where E~ v, and G, respec;tively, are Young1s modulus, Poisson~;s ratio, and
the shear modulus, .G is related to E by,
E G==---2 (1 +v)
From Eq. (Sl) it fo 11 ows that
5 ::;;: 3 K e
where K is the bulk modulus, and is given by,
(so)
(s1)
(S3)
E K = 3(1-2v) (S4)
For a state of plane stress, o-t == 'fxz =: 'fyz = 0 0 Using this
information in Eqo (51) and inverting leads to the stress-strain relations
for the elastic range as follows:
E + v IT == E E X
l~v 2 x Y
E + v (55) cr == E E Y l-v
2 Y x
'[ :;: G r xy xy
or 9 in rate form
• I; E + v E ) IT :::; .....----x 1 2 x Y -v
E ~ E ) (56) (!J =; -. ··-2 + v
y l-v Y x
:r ..., G r xy xy
where the derivitives are taken with respect to the load, q. Since Eqs. (55)
and (56) can be u$ed interchangeably in the elastic range, and since Eq. (56)
must be u$ed during unloading (Section 3.2), the latter form will be used
throughout.
3.5 Plastic Str~s~-Strain Law
For a Prandtl-Reuss sol id, the yield criterion is given as a
function of the second invariant of the d~viatoric stresses as follows:
(57)
which reduces for plane stress condition 7 to
(58)
~.33'~
where k is the yield 1 i mit in simple shea r. The deviatoric stress rates
a six dimensional state of stress are given by (20)
5 := 2G ( e w t G ( t W - 2k2
s ::::
- 2k2 't
x x X xy xy xy
5 2G ( e w ) T G ( Yxz w
:=
- 2k2 s :::: - .......-....- 't
Y Y Y xz 2k2 xz
5 2G ( e w t G (-:'1'
w ) ::::
- 2k2 s ::::
- 2k2 . 't
z ·z Z yz ,·yz yz
where W, a positive scalar defined by Eq. (60), may be interpreted as the
rate of work of the deviatoric stresses in distorting the material"
W :::: S e +s e + s e + 't t + 't t +.'1' t x x y y z z xy xy xz xz yz yz
For a state of plane stress Eqg (60) reduces to
w:::: cr E +cr E + 't r - 3se x x y y xy xy
or, by using the rate form of Eqo (53))1
W• :::: cr E" + ~ EO + ~ ~ 3ss x xUy y 'xy'xy- K
For plane stress the spherical stress is given by
1 (cr + cr ) S :::: -
3 x y
Hence, using Eq. (63) in Eqs. (49) and inverting the result lead to
(j =: 2 s + S x x y
cr == 2 s + S Y Y x
for
(59)
(60)
(61)
(62)
(64)
-34-
which, in rate form, are
cr =: 2 s + s x x y
(J = 2 s + s y y x
Substituting Eqs. (49) and (59) into (65) leads to
. ,
cr = 2G ( 2 E + E - ~ - ~ (J )
x x y K 2k2 x
(J =: 2G ( 2 y
E + E Y X
, . s w
... = "" - (J" K 2k2 Y
Hence, using Eqs. (50) and (62) in (66) results in the following plastic
stress-strain relations:
[ ( 4k2 _ (J"2 4G 2 + ( 2k2 - 4G 2
(J = c1
't E (j' IT 't E X x 3K xy x x y 3K xy y
( - 2G ) Yxy ] + (J" 'f +- (J" - (J" 'f X xy 3K y x xy
C 1 { 2 4G 2 ( 4k2 2 4G 2
(J" = (2k - (J" (J" 't E; + - (J" - - 'f E Y X Y 3K xy x y 3K xy y
+ ( - (J" 'f + 2G (J" - (J ) 't ) Yxy ] x xy 3K x Y xy
where
G c
1 =:
k2 2G 2k2 1.5
2 ) + 3K - s
Hence s, which is given by Eq. (50) , is
(65)
(66)
(67)
(68)
5
and W,
W
~ ;1 [ ( _ 6k2
-2 cr - crcr x x y
- (cr + cr ) 1" 'Xy] x y xy
which is defined by Eqo
c1
k2
=:
~:'-
+ 1" xy
[ + 2G ( cr ( cr x 3K x
+4G ).t, oJ 3K 'xy
) E x
(61)
- 0" y
Using Eqo (70) in the expression for
t = c1 [ "xy ( - 0-
+ 2G 0-
xy x 3K Y
+ '! ( - -0- +2G 0-
xy y 3K x
+ ( k2 1"2 + 4G
xy 3K k2
~35~-
+ ( 6k2 - 2 ) E cr - cr cr
y x y y
becomes
2G ) ) E + ( cry + 3K cr - () ) ) E
x y x y
t leads to xy
- 0- ) ) E x x
- 0- ) ) E y y
2 3 2 ) ) t xy J = 1" - "4 s xy
Eqs. (67) and (71) are the constituti've relations for the material in the
plastic range and remain val id as long as
w > 0
3.6 An Incremental Form of the Constitutive Relation
Eqs. (56), (67), and (71)? wh ich express the st ress rates in terms
of strain rat~s~ can be used to compute incre~ental changes in stres$ for a
corresponding incremental chang~ in strain.
(70)
(71)
(72)
~:36~
The independent parameters in a plate problem are the spacial
coordinates x, y, and z, and the load 9 q. Hence,
0" ::; O"(x,y,z,q)
E = E(X,y,Z,q)
and the differentials of cr and E, respectively, are
dO" dO" = "-. - dx + dO" dy + Ocr dz + dO" dq dX dy dz dq
dE d' dE dE dE ::;~ dx +~ dy + ~ dz +- dq
dX dy dZ dq
For a fixed point in space dx = dy = dz =i o. Consequently,
dO" dO" = -- dg = rr dq
dq
dE dE = --. dq::; ~ dq dq
Hence, for a finite increment in load, ~q, the incremental changes in the
stress and strain components resulting from an incremental change in load
become,
fu = 0- 6q l::.E = E l::.q x x x x
!.j:f := 0- .6q l::.E := E l::.q Y Y Y Y
61: := Txy.6q 6r :; r L\q xy xy xy
Substitution of Eqso (56) ~ (67), (70), and (71) into the above leads to the
following incremental stress-strain relations~
Elastic Stress-Strain Law:
60" X
E =--2
l-v 6E + V 6E
X Y )
(73)
!:::rr = Y
E 2
l-v 6.E + V 6.E ) Y . x
6.1' ::: G 6.Y xy xy
Plastic Stress-Strain Relations:
&J [ ( 4k2 2 4G 2 4E + ( 2k2 -= c1
- 0" 1:" X x 3K xy x
( - 2G 6y ] + 0" 'r + --,... ( 0" - 0" l' X xy 3K Y x xy xy
txs [2 4G 2 ) 6.E + ( 4k2 ::;; c1 (2k - 0" 0" + 3K 1:"
Y X Y xy x
2G 6Y
XY ] + ( - cr l' +--. (J' - cr l' Y xy 3K x Y xy
61:" ::: 9 1 [ Txy ( - 0" + ,2G
0" - 0" ) ) 6.E xy x 3K Y x x
+ l' - 0" + 2G
0" - cr ) ) 6.E xy y 3K x y y
+ ( k2 1'2 + 4G 2 2 3 2 ) k = l' - 1+ s xy 3K xy
W becomes 6w/~, and is given by
+4G t2 0" 0"1;" x y 3K xy
2 4G 2 - cr l'
Y 3K xy
) f::..y xy ]
k2
6.E: ~w c 1 [2G x 2G - - -- (0" + -r-- (0" - O"y ) ) /\" + ( O"y + 3 K (0" - Cf' .6q - G .; x 3K x '-"-I y X
+ l' (1 + 4G xy 3K ~J'" 6.q
)
6.E Y
6E Y
6.E ) ) --Y..
6.q
Strict;ly speaking, the stresses which appear in the coefficients
of the incremental strains in Eqs. (75) and (76) should be evaluated at a Ai
load q such that
(74)
(75)
(76)
":38· ...
However, for small ~q the total stresses at the load q may be used; this
simpl ifies the problem considerably and the resulting error is expected to
be small and tolerable. Then the coefficients in the plastic stress-
strain relat10ns can be evaluated by using the stresses which exist in the
structure just prior to the addition of the load, ~q.
3.7 Incremental Moment-Curvature Relations For Small Deflection Problems
The small..,.deflection strains of Eqs. (46) can be used in the
preceding expressions for the incremental stresses; the resulting expressions
then can be used in Eqs. (44) thus leading to a set of simpl ified moment-
curvature relations.
Let M o
M
2kht
M +M x y
Then the elastic incremental moment-curvature relation~ becomes
.6M 2Eh2
t = x l-v
2
2 .6M
2Eh t =
y l-v
2
(~6 + v 66 ) x Y
~f) Y
66 xy
+ v ~e x
and the plastic incremental moment-curvature relation is given by,
(77)
(78)
M. = c _.:-' [ ( 4M2 .. M2 ,_4G M2 69 + ·x 2 0 x 3K xy x
2M2 - M M - 4G M2 69 o x Y 3K xy y
+ 2 ( - M M + lQ (M - M ) M ) 68 J x xy 3K Y x ~y xy-
-,6M = c [. 1 ( 2M2 - M M 4G M2 ) 68 + Y 2·· . 0 x y 3K '~y x
+2 ( - M M' + 2G y xy 3K M - M ) M ) 68 ] x Y xy xy
M ( - 2G ) ) 68 :;:; C [ M M + --..... ( M - M xy 2 xy x 3Ky x x
+ M' ( - M ;.. 2G ( M '- M ) ) 68 xy y 3K x y y
+ 2 ( M2 ... M2 +- 4G ( M2 _ M2 - ~ .H2)
0 xy 3K 0 xy
where c2
i$ given by,
The yield criterion reduces appropriately to
and ~/ 6q becomes
68 J' xy
(80)
(81)
~:40~
t:,e x
~q
t:,e t:,e "'l' + (M + 2 G (M _. M ) ) ..:.J... + 2 M (1 +4G ) w..2ri '. I
Y 3 K Y x .6q xy 3 K 6q _.i
(82)
IV THE NUMERICAL TECHNIQUE
4.1 I ntroductory Remarks
The essential elements for the solution of elastic-plastic plate
problems are formulated in Chapters II and II I. The determination of
complete solutions in terms of displacements, moments, and membrane forces,
requires numerical techniques of calculation; these techniques are presented
in the following sections.
The primary objective of the present work is to develop a method
for treating elastic-plastic plate problems in the context of the small
deflection theory; however, the extension of the method to large deflection
problems is investigated for one specific problem.
4.2 The Numerical Problem
Using Eqs. (40) and (46) in ~qs. (78) and (79), respectively,
1 inear expressions relating incremental change.s in transverse displacem~nts
to incremental changes in the elastic and plastic moments can be obtained.
Consequently, the quantities in Eq. (47) can be transformed into 1 inear
functions of 6.W provided it is known, a priori, which nodes are elastic and
which are plastic. The basic numerical problem, therefore, is the generation
and solution of a set of 1 inear algebraic equations for each additional
increment of load, 6.q.
For large deflection problems, using Eqs. (1), (39), (40), and
(41) in Eq. (74) or (75), a set of 1 inear expressions relating incremental
changes in elastic or plastic moments and membrane forces to 6.U, 6.V, and ~
can be ob t a i ned. E q. ( 43), the ref 0 r e , i s 1 i n ea r i n 6.U, 6. V, a n <:I 6.W. I n
-41-
-42-
addition, some of the terms in Eq. (42) are also 1 inear in 6U? 6V, and 6W;
howeveri terms such as
appearing in Eq. (42) will be quadratic in t6.Ul' 6V, and &I since both 6N~
and 680 are functions of the unknown displacements. x
From Eq. (42) it can be seen tha t if the va 1 ues of 6N~, 6N~ and
~o are assumed or known, the resulting equation is 1 inear. The numerical xy
solution of Eqs. (42) and (43) is started by assuming values of 6N , .6N ?
X Y and ~ ,at each node; correct values of these incremental membrane forces
xy
are then obtained by iteration. Hence, for large deflection problems, the
solution tec~nique remains incrementally 1 inear.
4.3 Details of the Solution Process
4.3.1 Small Deflection Problems
The solution procedure for small deflection problems is summarized
graphically by the flow diagram shown in Fig. 8.
Initially, all moments~ displacements, and app1 ied loads, are
prescribed to be zero; all nodes are elastic. A matrix containing the
coefficients of the moment~curvature relation at each node is generated; if
a node is ~lastit,:the elastic coefficients are used. When a node turns
plastic, the corresponding coefficients are functions of the existing
moments at the node. The appl led load is incremented in finite increments
and the equil ibrium equations are solved for e~ch load level.
For each additional load increment, the corresponding incremental
moments are computed as shown in Fig. 11; these are added to the prior
moments to obtain the total moments in the structure. If necessary, the
-43-
plastic moments are corrected as described in Section 4.5. The corrected
moments are then used to determine if new yielding, unloading, or an
('overshoot", as described in Sections 4.6 and 1+.7, has occurred. If an
"overshoot!! occurs, a smaller increment in load 6q is determined by new .
interpolation. The incremental moments and displacements are then scaled
by a factor of 6Qnew/6q; these are then added to the moments which existed
prior to the addition of this load increment. The new moments are then
used to check for new yielding, unloading, or an "overshoo~'. The cycle
is repeated until an acceptable value of 6q is found. After a correct new
6q and the corresponding incremefltal moments and displacements are com-new
puted, they are added to the corresponding quantities which existed in the
structure prior to the additIon of 6q ,thus forming a new set of total new
lnads, moments, and displacements .. These quantities are used during the next
load increment. If new yielding or unloading has occurred~ these are noted
appropriately. A new coefficient matrix is generated based on the new
set of moments and information on the yield regions. The above process,
except for initial ization, is repeated for each new load increment until
a desired level of loading is reached, or the approximate 1 imit load of the
plate is obtained.
4.3.2 Large Deflection Problems
Fig. 9 is a graphical summary of the computational procedure used
to solve the large deflection problem. In general, the procedure IS the same
as that used for solving smal-1 deflection problems; however, in this case,
stresses and strains take the place of moments and curvatures, and, Eqs. (42)
and (43) are used instead of Eq. (47). Fig. 11 summarizes how the incremental
stresses, strains, moments, and membrane forces, are computed from the
-44=
incremental displacements.
The iterative scheme for the solution of the non=l inear equations
is initiated by setting the values of ~U? ~V, 6W? ~ , 6N ? and 6N to x Y xy
zero at all nodes. The resulting equations are then solved for ~U, ~V, and
~W. !f the difference in the incremental displacements from two successive
iterations is significant, the new values of ~U, ~V, and 6W are used to
compute the corresponding values of 6N ,6N , and DN at all nodes; the x y xy
cycle is repeated until the uncremental displacements from two successive
iterations agree to within a specified tolerance. The convergence of this
iterative calculation can be improved as described below.
4.3,3 Forced Convergen~e of iterative Scheme
The rate of convergence of the iterative scheme for solving the
non-l inear equation tends to decrease rapidly as the appl ied load approaches
its 1 imiting value. The following technique can be used to force a more
rapid convergence to the solution:
Assume that three iterations have been performed, and that the
resulting solution vectors are, respectively?
1 1 1 al
a2 a3
Al =: A2 =: A3 = 'N 'N 'N a1
a2
a3
Let the true solution vector be A, Defining the rate of convergence for o
the ith element in the solution vector by
change in error in one iteration r = ------~-~----------------------~~-----error before the iteration is performed i=1,2.,.,.N
and assuming that the rate of convergence is constant within the three
(83 )
iterations, the following relation can be given:
i i i i a1
- a2
a2 - a
3 i ., r ::::: :::::
i i i i I=1,2,."N
a1 - a a
2 - a 0 0
Solving the above for i yields a 0
i I (a~) 2 a1a
3 ~
i a :::::
i 0 i 2 i a1
- a2 + a
3
Actually, the rate of convergence is not a constant; therefore, the above
expression represents only an improved approximation to the solution vector
rather than the solution vector itself. Nevertheless, using A in place of o
A3 should result in an improved rate of convergence.
4.4 Solution of the Equations
There are two fundamental numerical methods for solving 1 inear
algebraic equations; the method of iteration or relaxation, and el imination.
Allen and Southwell (21) and Ang and Harper (12), were successful
in applying relaxation techniques to plane strain problems of plastic flow
in sol ids. The main advantages of the method are that it requires a minimum
amount of coding and computer storage, and that solutions can be obtained
reasonably rapidly for well-conditioned equations. However, during the
latter stages of extensive plastic flow, the diagonal elements of the stDff~
(84)
(85)
ness matrix become small compared to the off-diagonal elements. Thus, as the
appl ied load approaches the 1 imit load, the equ]l ibrium equations tend to be
severely ill-conditioned; iterative or relaxation methods then become ex-
tremely inefficient at best.
=46-
On the other hand, the amount of calculation required in the
solution of a set of 1 inear algebraic equations by Gaussian el imination
does not depend on how well the equations are conditioned, but only
on the number of equations and the average band width of the associated
stiffness matrix. Thus, in terms of total computation time, and in terms of
accuracy, the e1 imination method was found to be superior to the method
of relaxation for the problems considered herein. However, the classical
Gaussian e1 imination method requires large amounts of computer storage.
This problem may be alleviated somewhat through the use of auxil iary
storage such as magnetic tape units; such an alternative, however, usually
leads to much longer computation times.
A modified Gaussian e1 imination scheme which is particularly
useful In solving problems involving banded stIffness matriciescan be
used. Fig. 12a is a schematic representation of the stiffness matrix
corresponding to the equil ibrium equations of the model. All non-zero
elements 1 Ie within the shaded area which has an average band width B.
Flg. 12b shows the same matrix after the terms to the left of the diagonal
i nth e fir s t i - 1 row s are eli min ate d . I tiS e v ide n t from the f', g u ret hat
. d t l' 0 h 1 0 h 0 th 0 0 In or er 0 e Imlnate tee ement a .. In tel row 9 It IS necessary to IJ
use t-he 0 0 1 t 0 t-h . th remainIng non-zero e.emen 5 In L ,e J ro"v. Furthermore? i-j ::: B/2.
The minimum number of elements which can be used to e1 iminate all of the
elements to the left of the diagonal in the ,th row is, therefore, approxi-
2 mately (B/2) (crosshatched area in Fig. 12b). However, solving the equations
while keeping only this minimum number of elements in the computer memory
would require that one rely heavily on auxiliary storage units. in order to
minimize the computation time, the entire shaded area in Fig. 12b can be kept
in memory in addition to the program itself, thus, el iminating the need fbr
-47-
auxi1 iary storage. The computer storage required for the e1 imination
process is then approximately B~N; for 300 equations with an average band
width of 100, the required storage is 15,000 locations, a figure well
within the capacity of large computers.
The data processing involved in the solution technique is to
generate one equation, e1 iminate the non-zero elements to the left of the
diagonal, and store the remaining non-zero elements for later use. Repeating
the process N times and then performing a simple back substitution leads
to the desired solution vector.
4.5 Yield Surface Correction
In deriving Eq. (79) it is assumed that J 2 =: 0; i ,e., the moment
rate vector 1 ies within the tangent plane to the yield surface. However,
with reference to Fig. 13 it is evident that a finite incremental moment
vector cannot remain on the yield surface. The state of moments at lid',
in Fig. l~ must be modified to conform with the assumptions of perfect
plasticity. This can be accompl ished by adding a correction vector to the
state of moments a "2'; in effect, this determines the corresponding state
of moments which is on the yield surface.
The derivation of the correction vector is performed in a nune-
dimensional deviatoric moment space. The results are special ized to the
plane stress condition that exists in the sandwich plate. Define first the
following quantities:
;0 = the uncorrected deviator!c moment vector
-4
m =: the corrected deviatoric moment vector
-?
CB =: the correction vector defined by
-?
m n;1 + CB (86)
-48-
where
n;'1 ~ (m' ml , m', m' ml m' , m' , m' ml
) x' y z XV' yx' xz zx yz' zy
~
(m , mzy ) m == m , m , m , m yx' m m , m yz' x y z xy xz zx
in which m , m , m , x y z
m', ml, ml x y z'
are, respectively, the components
of the corrected and uncorrected deviatoric moment vectors.
Then,
where
J' == the second invariant of the deviatoric moments - computed Z
from the uncorrected moments.
Jz == the second invariant of the deviatoric moments - computed
from the corrected moments.
z JZ == J2 + s J == MZ and sZ is the error in JZI 2 0'
-7
CB == the correction vector, and is defined to be normal to the yield
surface in the nine=dimensional deviatoric moment space, and has
length c; hence
-? \7J 2
CB == C\\7Jz\
JZ is given by
1 2 . 2
m2 + Z Z 2 Z Z 2 J == 2" (m + m + m + m + m + m + m + m ) 2 x Y z xy yx yz zy xz zx
(87)
(88)
(90)
Therefore, \7J2 -7 = m, and the length of the gradient vector to the yield surface
is
J'-' 2
== 2M o
(91 )
Substituting Eqs, (89) (90), and (91) into Eq. (86) leads to
(92)
Since J 2 and JZ are quadratic functions of the deviatoric moments,
JI = 2
(1 -
=49-
Using Ecjs. (87) and (88) in Eq9 (93) yields
c=J~
The negative sign is readily seen to be the desired solution for c.
Therefore, from Eq. (92),
2 ~I = (1 + -~ -) ~
2M2
Let
Then
~2 6 = 2M2
o
~
m =
o
and the corrected moments are given by
W W M = --'-V __
Y 1 + 6 M -~ xy - 1 + 0
For large deflection problems, Eq. (97) is appl icab1e if the
moments are replaced with the corresponding stresses.
4.6 Initiation of Yielding
After computing and correcting the moments, each node which was
elastic prior to the addition of the incremental load, 6q, must be
re-examined on the basis of Eq. (81) for possible yielding as a result of
the additional load. For the points in question:
if J2< M!, the point remains elastic
if J 2 = M!, yielding has occured.
(93)
(94)
(95)
(96)
(97)
H . f . J M2 2 h 2. h • f· d 0wever, , at some pOint, 2 = 0 + 6 , were 6 IS greater t an a speci Ie
-50-
tolerance ~2, the point is said to have rloversho/t'l yield; i.e., the
additional increment of appl ied load is too large and has caused the state
of stress at a node, which was previously elastic, to move significantly
outside of the yield surface. This violates the assumption of perfect
plasticity. In such cases, a 1 inear interpolation is performed to determine
a smaller load increment, 6q ,which will cause the point to fallon the new
yield surface rather than outside of it. This is determined as follows:
6qnew
where the subscripts q and q+6q refer, respectively, to the valu~s of J2
before and after the addition of 6q. This interpolation must be performed
at each point where J 2 - M! > ~2 and then the smallest value of ,6qnew
selected for use. After determining the minimum value of 6q ,the new
incremental displacement and moment vectors are multipl ied by the ratio
6q /6q. The total moments are then recomputed, corrected, and then new
rechecked for yielding.
For large deflection problems, yielding is checked by using Eq. (58)
instead of Eq. (81). In the event of an 'Iovershoot", one must generate
and solve the equil ibrium equations again since, in large deflection
problems, the incremental displacement vector is not 1 inear with respect to
the incremental load vector as it is in small deflection problems.
4.7 Unloading
If
6w ( A < 0 see Eq. uq -
(82) )
the point is said to have unloaded. During the unloading stage, the elastic
-51-
stress-strain relation is used.
For large deflection problems, 6W/6q is computed by Eq. (76).
4.8 Uncoupl ing of the Difference Equations
The equations of equil ibrium of the model can be represented
in matrix notation and partitioned as follows:
where {~Wl} and (~2} are, respectively, the incremental displacement
vectors of the shaded and unshaded nodes in Fig. 14a and {~qlJ and {6q2}
are the incremental load vectors appl ied at the corresponding nodes; kll
,
k12' ... are the sub-rnatric;es.~of the stiffness matrix. If the stiffness
matrix in Eq. (98) is generated numerically under the assumption that the
plate is completely elastic, it can be shown that
Hence, expanding Eq. (98) leads to two independent sets of linear algebraic
equa t ions.
If the model is loaded in a checker board pattern; e.g.,
the result will show that one-half of the nodes will assume a deflected
position while the other half remains undeflected. This means that in the
elastic range, the model can be decoupled into two independent network
systems; indeed for 1 i'nearly elastic problems only one of the systems need
(98)
(99)
=52~
be used. For elastic-plastic problems, the complete model must be used.
However, in order to avoid or minimize the decoupl Ing of the two network
systems, the loading oH the model must be properly appl ied. This is
achieved by imagining that the model is loaded indi rectly through a
system of stringers as shown in Fig. 14a. The nodes a, b, c, d, and o?
are interconnected by ~tringers. The external load is then appl led to the
middle of each stringer. Hence~ the externa"j ly appl jed load at node 110" is
given by
Qo 1
(Ql + Q2 + Q3
+ (24) :::: -2
where , 2 r..., )...,2 r...,2 r...,2
Ql :::: qf 1+ Q
2 = q2 "4 Q3 ~ q3 1+ Q
4 =:
q4 "4
and Ql' Q2' Q3' and Q4' are the average load intensities under each
stringer; e.g., Q4 is the average load intensity over the shaded area in
Fig. 14a.
The above discus s ion 1 ead i ng to Eq. (99) is based on the
assumption that the plate is completely elastic. ~n the plastic range
k12 and k2l are not, in general, null matrices. Ho'wever 7 in many cases
the non-zero terms in these sub-matrices are not numerically significant
when compared to the diagonal terms of kjj
and k22" Hence 7 the undesirable
situation described above will also occur in the plastic range unless the
above loading concept is used.
~AAot~er proble~, which presents itself when the appl led load
approaches its limiting value? and which is directly related to the
decoupl ing phenomenon, is that of "plastic separation". However, unl ike
the former problem, "plastic separatiod ' is dependent upon mesh size and
vanishes in the 1 imiting case. The following illustrative example is
used to describe the problem.
-53-
Assume that the model shown in Fig. l4a represents a uniformly
loaded plate with fixed support conditions at the left and right ydges
and has free edge conditions along the two remaining edges. As the level
of the app1 ied load is increased, the four shaded nodes which 1 ie along
the fixed boundaries will yield. Increasing the load further will cause
the shaded nodes along the center 1 ine to yield. Since the terms which
couple the equil ibrium equations for the shaded and unshaded areas are not
numerically ~ignificant, the two systems act independently of each other.
Due to the pattern of yielding, the shaded system of nodes will deflect
at a much higher rate than the unshaded system since the latter remains
completely elastic, and separation of the two sets of nodes will occur.
Although this phenomena vanishes a ~ ~ 0, it is quite pronounced for
practical values of~. The following procedure can be used to e1 iminate the
problem.
If, instead of the model shown in Fig. 14a, the arrangement
shown in Fig. 14b is used, the effect of plastic separation is e1 iminated
completely since both systems of nodes are forced, by symmetry, to yield
simultaneously,
Fig. 15 shows how the boundaries were placed in obtaining
solutions to the problems presented in Chapter V.
Vo NUMERICAL RESULTS
5.1 Problems Considered
Four Problems of uniformly loaded square plates are presented
with the following boundary conditions:
10 All four edges are simply supported s
2. All four edges are fixed supports.
3~ Three of the edges are simply supported and one edge is free.
4s All four edges are roller supportss
The first three solutions are based on the small deflection theory;
the fourth is a large deflection problem. These are presented for the
prupose of illustration the results obtainable by the proposed method, and
also to give an indication of the rel iabil ity of the methods
The results are presented in non=dimensional forms using the
following 1 ist of parameters.
For Small Deflection Problems
~ == x/a M == M /M W :;;;; EhW/ka2 == W/2h r x x 0
yla M M /M 2 T1 = =: q == qa 1M
y y 0 0
M =: 2kht M =: M /M r== (k/2E) (a/h)2 0 xy xy 0
where: a =: span length of the plate; k ~ the yield strength of the
material in simple shear; h = the half-thickness of the plate; t = the
thickness of the thin sheets of the sandwich plate.
For Large Deflection Problems
s == x/a
1) = y/a
r =: (k/2E) (a/h)2
2 q :;;;; qa /M
o
=54-
=55-
M :::;: M 1M N :;:;: N IN U ::;;: Ua/2h2 x x 0 x x 0
M :::: M 1M N :::;: N IN V :::; Va/2h2 y y 0 y y 0
M ::::: M 1M N :::: N IN W :::; W/2h xy xy 0 xy xy 0
where: N :::;: 2kt and all other quantities are defined above. o
In addition the appl ied loads are non-dimensional ized in terms of
one of the following:
q - the dimensionless load level at which yielding is first el -
initiated.
q :::: the dimensionless load level at which the pattern of m
yielding forms a "pseudo-mechanism ' . Increasing the
appl led load by a small amount beyond qm usually results
in large deflections; hence, this may be considered the
state of impending collapse.
q+ = the dimensionless load at which computation was terminated. I
All results are presented with reference to the coordnnate system shown in
Fig. 16. Poisson's ratio is taken to be .3 for all problems.
5.2 Square, Uniformly loade~ Plate with Four Simple Supports
The model used in obtaining the solution to this problem IS
that shown in Fig. 15; however, the mesh size used is ~ :::: 1/24. The
appl ied load is uniformly distributed over the plate, and the boundary
cond i t ions a re those given by Eq. (34).
With reference to Fig. 15, it is evident that the deflections
and moments at the center of the plate are not defined in the model. For
the purpose of comparision with elastic results tabulated by Timoshenko (5)
-56-
and Levy (7), these quantities are approx~mated by passing a parabola
through the two nodes nearest to the center of the plate such that the
slope at the center point is zero.
Fig. 17 is the load-deflection diagram for the center of the
plate; the appl led load has been increased from zero to qf and then
decreased to zero again. Comparison of the elastic center deflection
and moments with those tabulated by Timoshenko (5) indicates that the
error in the center deflection is approximately +.25%~ while the error in
the center moments is much smaller.
Applying a load greater than q ~ qel ~ 30.84 causes plastic flow
to be initiated in the plate; however, it is evident from Fig. 17 that this
flow does not appreciably affect the stiffness of the plate until the load
reachesq =1.32q 1" Any small increase in load beyond this level results m e
in large deflections of the plate; for instance~ increasing the load by
5% (from qm to qf) results in an increase in center deflection of 160%.
Thus, qf can be considered as the carry~ng capacity of the plate under the
assumption of small deflections.
Fig. 18 shows the extent of plastic flow in the plate at a number
of load levels between qel an qf. Figs. 19 and 20 illustrate the redistri
butions of moments as a result of plastic flow; while Fig. 21 shows the
deflection configurations. The center of the plate is subject to a
spherical state of moments. Consequently, after yielding occurs there, the
val ues of M and M rema in at M = M =- i3. x y x y V
Upon reaching q == qf~ unloading was initiated by applying negative
load increments to the system. When unloading was initiated, all plastic
nodes returned to the elastic range, Thus, the unloading path shown in
Fig. 17 is a straigrt 1 ine parallel to the orignnal elastic loading curve.
-57-
Fig. 22 shows the residual moments and deflections at a number of cross
sections in the plate.
There are no upper and lower bound solutions presently available
for plates which are governed by the Mises yield criterion. However, there
are solutions for plates which are governed by the Tresca yield criterion.
Consequently, the following indirect procedure can be used to determine the
upper and lower bounds such as those shown in Fig. 17.
Fig. 23 shows the first quadrant of the princi.pal moment space
(all principal moments lie within this quadrant for the plate in question)
where the Mises yield criterion plots as an ell ipse with major and minor
axes MI = M2 and Ml = , M2
, respectively. If a Tresca yield criterion is
circumscribed about the Mises criterion (see Fig. 23), then an upper bound
for the Tresca condition is also an upper bound for the Mises condition.
Similarly, if a Tresca criterion is inscribed within the Mises criterion,
then a lower bound for one condition is also a lower bound for the other
condtion. Hence, the Tresca yield criteria shown in Fig. 23 were used to
determine the upper and lower bound solutions shown in Fig, 17.
5.3 Square, Uniformly Loaded Plate. with Four Fixed Supports
The method of treating this problem is identical to the method
used for the simply supported plate in Section 5.2. The boundary conditions
are given by Eq. (35). Fig. 24 shows the load-deflection diagram for the
center of the plate. The appl ted load is increased monotonically from zero
Comparisons of the center deflection and of the moments at the
center of the fixed support with the corresponding quantities tabulated by
Timoshenko (5) indicate that the deflection is in error by approximately +.6%
while the moments are in error by -2,5%. The upper bound solution shown
in Fig. 24 was obtained by the method described in Section 5.2. No lower
bound solution is presently available.
Applying loads greater than qel = 38.73 causes plastic flow to
occur along the edges of the plate; however, the load-deflection characteris-
tic of the plate is not appreciably affected until the center yields at
q = 1,66 qel' Further loading beyond q ~ 1.66 qel causes a I'pseudo
mechanism! to form at g = 1,98 gel; qf of this plate is approximately
equal to 2,05 qel d Fig 25 shows the extent of plastic flow in the plate
corresponding to different levels of the applied load, while Figs. 26
through 28 show the redistribution of the moments and deflections as a
result of plastic flow.
5.4 Square, Uniformly Loaded Plate with Three Edges Simply Supported and One Edge Free
In this problem the edges ~ = t .5 and S = 0 are simply supported
while the edge S = 1.0 is free. The load is uniformly distributed over the
entire plate; a mesh length of ~ = 1/16 is used.
Fig. 29 is the load-deflection diagram at the center of the free
edge. Comparison of the elastic deflections and moments at this point with
those tabulated by Timoshenko (5) indicate that the elastic deflections are
in error by approximately 2%, and the the error in the moments is neg1 igible.
Figs. 30 through 33 demonstrate how the moments and deflections
redistribute as a result of plastic flow.
5.5 Square, Uniformly Loaded Plate. with Four Roller Supports (Large Deflection Theory)
The solutions to each of the three preceding problems indicate
-59-
that the deflections in an elastic-plastic plate can become large with
respect to the corresponding deflections in a completely elastic plate
subjected to the same level of appl ied load. Hence, it is reasonable to
expect that membrane action will effect th~ elastic-plastic problem more
than the corresponding elastic problem.
The following soluton of a simply supported plate on roller
supports is obtained through the use of the large deflection formulation as
described in the preceding chapters; the boundary conditions are those given
in Eq. (38), and a mesh length of ~ := 1/24 is used.
A similar problem was previously presented by Levy (7). However,
whereas tre problem considered herein assumes that the roller supports are
confined to move normal to the edge~ this constraint was not imposed in
Levy's problem. Also, Poisson ' s ratio is taken to be 0.3 for the problem
presented herein as opposed t~ 0.316 used by Levy. Since these differences
are minor, the two solutions can be expected to be comparable. Fig. 34 is
a comparison of these two elastic solutions. The load-deflection diagram
for the elastic-plastic problem of the present study is shown in Fig. 35.
Solutions for three values of the parameter f= 1,2, and 4, are shown where
r is a dimensionless parameter defined in Section 5.1, r= 1 to f=4 give
results which are almost identical to those given by the small deflection
theory; this means that with r::s 4, the membrane forces have on'ly a minor
effect on the solution,
Fig. 35 also shows that, for r =: 10, the load-carrying capacity
of the plate given by the small deflection solution is overly conservative.
The calculations were performed up to qf =: 525. Beyond this load level
considerable difficulty was encountered 'in'the computations, hence, the
ultimate capacity was not determined. However, at this load level the
-60~
results are sufficient to show that the contribution of the membrane forces
is becoming increasingly important.
Fig. 36 shows the progression of yielding in the plate at four
different levels of appl ied load; while in Figs. 37 through 39 are shown
the redistribution of the moments and membrane forces? and of the in-plane
displacements during increasing plastic flow. Note, that during plastic
flow it is possible for the moments or the membrane forces to decrease
in magnitude at certain regions of the plate although loads are being
increased. Fig. 40 sh~ws the variation of the in-plane displacements at
the edge of the plate with load, while in Fig. 41 is shown the variation
of the ratio of membrane forces to moments at the center of the plate
with increasing load.
5.6 Correctness of Solutions
A formal discourse on the correctness of the approximate solutions
presented herein for elastic-plastic problems of plate bending requires
further mathematical study of the underlying quasi-l inear or non-l inear
equations. No such rigorous study was made In the present Investigation.
In the absence of a formal proof of the correctness of the solution method,
the rel iabil ity of the numerical solutions may? nevertheless, be shown on
the basis of heuristic or physical reasoning.
The convergence of solutions determined with the model can be
verified, at least partially, by considering the convergence of various
physical quantities obtained with the model using monotonically decreasing
mesh sizes. Figs. 42 and 43 show? respectively~ a number of solutions from
the fixed and simply supported plate problems presented in Sections 5.2
and 5.3, all plotted against ~2 ~t can be observed that, extrapolating
-61-
the elastic quantities to the limiting mesh length, ~ = 0," leads to the values
tabulated by Timoshenko. Fig. 44 shows the load-deflection diagram at the
center of the fixed plate for three different mesh sizes; the difference
between the curves appears to be decreasing with decreasing mesh size,
indicating, therefore, that the method of solution is convergent for this
case. A similar plot for the simply supported plate is given in Fig. 17
which shows that the three curves (corresponding to three mesh sizes) are
coincident with each other. Fig. 45 shows the moments in the fixed plate
at an app1 jed load of ~ = ~. Again, it appears that the curves are . m
approaching a 1 imiting value. Similar plots for the simply supported plate
are shown in Figs. 19 and 20 which represent moments for three mesh sizes.
Also, comparison of upper and lower (where possible) bound solutions with
the model solutions, indicates that the model does yield reasonable results,
at least with respect to the load carrying capacity of the plate.
All the results, therefore, show that a sequence of solutions
determined with decreasing mesh lengths of the model tend 'to a unique
solution. Although no formal proof is available, results such as those
shown above, supported by bounds of 1 imit :analysis and by 'the fact that
the problems were formulated on physically meaningful terms leave 1 ittle
doubt that the solutions determined with the proposed methbd are correct
consistent with the mesh size.
VI SUMMARY AND .CONCLUSIONS
A numerical technique has been presented from which approximate
solutions to elastic-plastic plate problems can be obtained. By using a
lumped-parameter model the continuous plate is replaced by one with a
finite number of degrees of freedom. The field equations are then derived
directly from the model and shown to be a""finite difference equivalent of
the corresponding continuum equations. Hence, a problem f,ormulated through
the model can be shown formally to tend to the corresponding problem of a
continuum.
In order to make the problem mathematically tractable, the plates
under consideration are assumed to be a sandwich construction, consisting of
a shear core between two thin sheets; the material comprising the thin sheets
of the sandwich plate is elastic-perfectly plastic and its behavior is
described by the Prandtl-Reuss stress-strain relation. The shear core is
assumed to be rigid in shear and incapable of developing flexural stresses.
The Prandtl-Reuss equation in plane stress condition is used for the thin
sheets.
Four problems are investigated including a problem containing
both geom~tric and material non-1 inearities. The techniques used in
solving these problems are discussed in detail.
In the absence of fofmal proofs, the problem of icon vergence has
been studied on a -h~~ristic basis. The results of this iQvestigation show
that a sequence of solutions determined with decreasing m~sh lengths of the
model tend to a unique solution. Furthermore, the val idity of these
solutions is strongly supported by the upper and lower (w~ere available)
bound solutions of 1 imit analysis.
-62-
REFERENCES
1. French, F. W., dlE1astic Plastic Analysis of Centrally Clamped Annular Plates Under Uniform Loads," in SR-62, a report distributed by The Mitre Corporation, Contract Number AF33(600)39852, Project 607, December, 1962.
2. Haythornthwaite, R. M., liThe Deflections of Plates in the ElasticPlastic Range:' Proceedings of the Second National Congress of Appl ied Mechanics, Ann Arbor, 1954, pp. 521-526, 1955.
3. Hopkins, H. G., and Prager, W., liThe Load Carrying Capacities of Circular Plates," Journal of Mechanics and Physics of Sol ids, v. 2, pp. 1-13, 1953.
4. Tekinalp, B., "Elastic-P1astic Bending of a Built-In Circular Plate Under a Uniformly Distributed Load;' Journal of Mechanics and Physics of Sol ids, v. 5, pp. 135-142, 1957.
5. Timoshenko, 5., and Woinowski-Krieger, 5., Theory of Plates and Shells', McGraw-Hill Co., New York, 1959.
6. Timoshenko, 5., and Goodier, J. N., Theory of Elasticity, McGrawHill Co., New York, 1951.
7. Levy,S., IILarge Deflections of Rectangular Plates,1I NACA, T. N., 846, 1942.
II
8. Kirchhoff, G., "Vor1esungen uber Mathematische Physik,1I Mechanik, 1877, p . . 450.
9. Kelvin, and Tait, liTreatise of Natural Philosophy," v. 1? part 2, p . 1 88, 1 883 .
10. Ang, A. H.-S., and Newmark, N. M., "A Numerical Procedure for the Analysis of Continuous P1ates," Second Conference on Electronic Computation; ASCE, September, 1960.
11. Ang, A. H.-S.; "Numerical Approach to Wave Motions in Non-l inear Sol ids,11 Conference on Matrix Methods in Structural Mechanics, WrightPatterson Air Force Base, October, 1965.
12. Ang, A. H.-S., and HarpE!r, G. N., II Analysis of Contained Plastic Flow in Plane Sol ids," Journal of Engineering Mechanics, ASCE, October, 1964.
13. Ang, A. H.-S., and Rainer, J. H., "Mode1 for Wave Motions in AxiSymmetric Sol ids;1 Journal of Engineering Mechanics, ASCE, Apri 1, 1964.
-63-
-64-
1 4 . S c h nob ric h, W. C., II A Ph Y sic a 1 An a log u e for t he N u me ric a 1 A n a 1 y sis of Cyl indrical Shells," Ph.D. Thesis, University of 111 inois, 1961.
15. Hrennikoff, A., "Solution of Problems In ElastIcity by Framework Method," Journal of Appl jed Mechanics, December, 1941.
16. Argyris, J. H., "Energy Theorems and Structural Analysis,11 Butterworth, London, 1960.
17. Clough, R. W., liThe Finite Element Method in Plane Stress,1I Proceedings of the Second Conference on Electronic Computation, ASCE, 1960.
II
1 8 . Co u ran t, R., F r i e d ric h s, K., and L ewy, H., I I U b e r die Par tie 1 1 en Differenzeng1eichungen der Mathematischen Physic," Mathematische Annalen, v. 100,1930.
19. Drucker, D. C., "Plasticity,I' Structural Mechanics,. J. N. Goodier and N. J. Hoff, Ed., Proceedings of the First Symposium on Naval Structural Mechanics, Pergamon Press, New York, pp. 407-455, 1960.
20. Prager, W., and Hodge, P. G., Theory of Perfectly Plastic Sol ids, John Wiley and Sons Inc., New York~ 1963.
21. Allen, D. N .. de G., and Southwell, R. V., IIRelaxation Methods App1 ied to Engineering Problems, XiV Plastic Straining in TwoDimensional Stress Systems;' Philosophical Transactions of the Royal Society of london, Series (A), v. 242, 1950.
22. Shull, H. E., and Hu, L. W., I'Load Carrying Capacities of Simply Supported Rectangular Plates;' Journal of App1 led Mechanics, ASME, pp. 617-621 ~ December, 1963.
APPENDIX A
Al Sandwich Plates
The schematic representation of the model as given in Chapter I I
is an oversimpl ification of the fitrue'l flexural model in that it does not
explain the relationships between the model and Kirchhoff l s assumption.
A more complete representation of the model is shown in Fig. 46. The middle
surface consists of flexurally rigid bars interconnected by shear hinges,
Transverse displacements are defined at the shear hinges while in-plane
displacements are defined at the mid-points of the rigid bars. Thus, the
middle surface described above is similar to the model as given in
Chapter II.
The vertical rod i-r is constrained so that, under deformation,
it rem~ins normal to a-o and e-f. Similarly k-s remains normal to
o-b and gOh. Hence, with reference to Fig. 47
u = U - h i 2 (
Wa
(A 1)
Similar expressions for nl"\ i ni- I .... tJv I I J (.. "''' (Al) is exactly a
central finite difference analogue of Eq. (14) evaluated at z = +h.
Kirchhoff's assumption, therefore, is implicit in Eq. (Al). Resistance
to deformations is defined at the stress points. With reference to
Figs. 46a and 46b, two stress points are positioned at each node;
one at z = +h, and the other at z = -h. The ends of the stress points
are then attached to the vertical rods such as i-r and s-k. In addition,
-65-
... 66-
the center of each stress point is attached to the corresponding shear
hinge.
The normal strain in an element with undeformed length ~ is
given by
fY... (A2) A.
where ~ is the deformation of the element. Applying Eq. (A2) to the
stress points at Ildl and simpl ifying through the use of equations similar
to Eq. (AI) leads to the first two expressions in ~q. (21).
By virtue of the mathematical consistency establ ished in Section
2.1, Eq. (21) must be a finite difference form of Eq. (15). It is evident
that'the first two terms in the expressions for E and E are central x y
finite difference analogues of the corresponding terms in Eq. (15);
however, the mathematical consistency is not SQ evident in the non-l inear
terms of Eq. (21). The reason for the ambiguity lies in the fact that,
in the model, there is no unique definition for the slope at 110'1. Conse-
quently, the terms in Eq. (21) are the forward and backward difference
expressions for the slope at lid'. In the limit as A.~, the value of these
expressions tends to the value of the slope at IId l ,
In the general the6ry of elasticity, the shearing strain at a
point is defined to be the cosine of the' angle between two 1 ines which
were normal to each other prior to deformation. Hen~e, with reference
to Fig. 48 a, the she a r i n g s t r a ina til jl 1 i s give n by
j 'Yxy :::: cos (Xl (A3)
Fig 48b shows that, in the model, there are two possib,le angles from which
one can measure the shear strain. In the model, the shear strain is defined
by
(A4)
in which cos al and cos a2 are, respectively,
v,, . V • I J m)
(t-.j2) 2
V'l ' V. k J J
(t-.j2) 2 (A5)
where the vectors are defined in Fig. 48. Using Eqs. (Al) and (A5) in
Ed. (A4) leads to the expression for shear strain given in Eq. (21). Again,
the 1 inear terms are exactly the central finite difference expressions for
the corresponding terms in Eq. (15); the non'" 1 inear terms are central
difference expressions taken over two mesh lengths,
A2 Plates with Sol id Cros~-5ections
For plates with sol id cross-sections, additional stress points
parallel to those shown in Fig. 46 must be inserted; the governing
equations for this case can then be derived accordingly.
-68-
Thin Sheets
Shear Core
FIG. SANDWICH PLATE CONFIGURATION
/"
I I I I I I I I I
A / ........
~ y z . x
1
FIG. 2 a : RESULTANT MOMENTS AND SHEAR FORCES ACTING ON AN INFINITESIMAL ELEMENT
- }o-
f\ Ny
I
jNxy
V Nxy + ciNxy ax
Nx 1
Nxy~
L Nxy
+ IdNXY
dy
Ny + aNy
a;-
1
PLAN
ELEVATION
FIG. 2 b · RESULTANT MEMBRANE FORCES ACTING ON AN INFINITESIMAL ELEMENT
- 71-
I II II
h f )(
II 3 III <
'-... ~ Torsional < Element
'-... ~
<
r '-...
b J\ J\ J\ b A A A
I 4 / I I II \ \ \ 2 III a
~ v v v ~v V v \
" i'-. ~
"
"i'-. I>
"" e 9 J> r-
III I ---.jJooU • l v
c II II III
( a) PLAN VI EW
Node (after deformation)
Node (undeformed) Rigid Bars
(b) ELEVAT ION VIEW
FIG. 3· THE FLEXURAL MODEL
-4 Ox
· ... 72- .
z
FIG. 4: SHEARS AND MEMBRANE FORCES ACTING AT .NODE 11 0 "
x
-73-
FIG.5: SIGN CONVENTION FOR MOMENTS AT NODE 11 0 "
! \ .,.
-7'-1--
FIG. 6 ELEMENT FOR DETERMINING THE SHEARS Q~ -4 AND Q y ,AND THE IN -rPLANE EQUILIBRIUM
EQUATIONS
-75-
( React ion)
x
Over A/4 Half)
Severed At The Boundary
(Reaction)
b Tee Transmits Twist, Mxy 1 To Point I As A Shear
(Reaction)
FIG. 70 . MODIFICATIONS ALONG AN EDGE.
. M b • A/4 xy I V ~ . A/2 (Reaction)
FIG.7b STATICS ALONG A FREE EDGE
z
'1111!111
yes
~.., ,., - /0-
IStartl
Read da ta; I n i ti ali ze ; Compute constants
....
Generate the coefficients in the moment-curvature relation at each node
Increment the load by ~q
Generate and solve theequil ibrium equations for the incremental changes in displacement
Compute the changes in the curvatures and moments at all nodes
....
Compute the total moments and displacements at all nodes
Correct the moments at all plastic nodes
Determine if new yielding, unloading, or an "overshoot" has occurred. Interpolate if necessary
f""""'---<\ I n te r po 1 ate on ~q ? >------1 no
Scale the incremental moments and displacements by.6q /~q new
Replace prior moments _ and displacements w1th
,.....,.,----1 new va1ues. Generate
I a new yield table
Output I
I End l~ ______ y~e_s ______ ~ '--_______ --J
last load? \}--_n_o _____ --.:.... _____ ~_
FIG,8· FLOW DIAGRAM FOR SMALL DEFLECTION PROBLEMS'
-77-
l Startl
Read data; ~ n J t i ali ze ; Compute constants
J
Generate the coefficients ,~
in the stress-strain relation at each node
1ncrement the 1 Dad by .6q
~J I
Set the Incremental stressesl and displacements to zero
Generate and solve the equil ibrlum equations for the incremental changes in displacement
Compute the incremental strains and stresses
Compare the lncjemental displacements { wIth those from the previous Iteration
n°i""\Wlthin yes
tolerance ?"
~ Rep~ace the previous Compute the total incremental stresses I--+- stresses and with the new ones displacements
Icorrect the stressesl ·at all plastic nodes
Determine if new yielding, unloading, or an overshoot has occurred. Interpolate If necessary
t yes Interpolate on 6q ?
Replace prIor stresses and displacements wi th the new val ues. Compute the moments ahd membrane forces. Generate a new yield table.
I Output I
r End I ~ 'Les ILast load ? I no ...
. I ...
FIG.9· FLOW DIAGRAM· FOR LARGE DEFLECTION PROBLEMS
Elastic or unloaded node
-78-Given: LM at all nodes
Go to the first node
Compute 6 9x ' 6Ay ' 6$xy'
at the node Eq. (40)
. Check the yield table
Plastic node
I
Compute ~x, ~y' ~xy . Compute ~x, ~y' DMxy at the 'node 1----I:.~IIIIIIfI_-_I at the node
Eq. (78) Eq. (79)
1
Compute the total mo~ ments and displacements at the node
'M x
M y
M xy
'. W
......... ---1 Go tot he ~n_o---.----.-_....--( next node
1181 (Mx) q . +.6M x
a (M ) +d'1 y q Y a (M ) + t}1 .
xy q. xy
• (W) ... tM q .
Last node ?
remainder of program
FIG.IO: COMPUTATION OF MOMENTS AND DEFLECTIONS FOR SMALL DEFLECTION PROBLEMS
E 1 ast f.c or unloaded node
-79-
IG i ven: LW, r[~N, lM 1 at all nodes
ISet i ilia lJ I J
IGO to the first I node
J I
Compute b"fi • , /:),E t D
XI Y and ~ 1 at the node xy Eqs. (39). (40), (41)
. t Che·ck the yield table
J ::: 1 is associated with the stress pointS in the bottom sheet of the sandwich plate; i = 2 is associated with the top sheet
Plastic node
Compu,t~ llCJ xi' tiJ Y i ' : '.C~p~ te ~(j xi' /::.rr y i' I.
:a nd li:r;~ y i at t he nod~ 1--_----;~ .... 4r__----':--'___'i. a'ri(j·~T xy' at t he node
··;.,.Eq. (74) Eq. (75)
i
....
compute the total stresses at the ·n'ode
'<T xi =. :(,0-xi) q +.6cr x i cr·
1 ~ ( ... 6- ,) +00 i
y ::' yl q Y :~. t == (r i) + b,,'t' , ·xy xy q XYI
Go to:. ;the ~ __ n_o ___ ---< Last node ? nex.t ·n·ode
yes
........ ---(, i = 2:·rl-4----:;..n~o---___< ; ( I == 2 ) ?
yes
Compute the moments and .memb ra-ne:; forces at a 1 r ·nodes
Eq. (1)
; rema·j nde·r~ of program
FIG. II' COMPUTATION OF STRAINS, ST:RESSI;S,., MOMENTS, . AND MEMBRAN·E FORCES - LARGE ·DEFLECTION ·PROBLEMS
FIG. 120
FIG.12b
N
-80-
......... 8 ••• 0 •••• , • ., · ....... " ... . • • • a ••••••• 0 • · .............. . • • • • • • • • • • • 0 •••
•• • • • • • • • • • • • • • e I ................. · ................ . •••• 9 ••••••••••• Q •• .................... ••• ~ •••••••• 4 ••••••••
• CD •••••••• ., • " ••••••••• III •• 0 •••••• •• e· • ., •••••• II •••••••••••••• 0 ••••••• 0 ................. " .....
OGelDeGm •• O •• S0oeeeeees ••• CD ••••••••• " •••••••••
4'1 •• 0 ••••• 0080.0 •• · •••••• e ••.••••••••• , ••••••••••
• • • •••••••••••••• ct •••• If " ••• 0 •••••••• d •••••••••
eeeeeeeoe ••• IIG •• O •••• OD ••• OD.O· ••••••••••••••••
0 ••• "0 •••• 0.01380 ••••• '. ·.·.·a·.O •. 1lI0.·.·.·.·o·a·.·.·.·.·.·.·.·.·.·o eoo •••••••••••••••••
• ••••••••••••••••• '111 ••• 80 •••••••••••••• 4 ••••• & .......................
•••••• 0 ••••• 00 ••••••••• • • • • Q \9 •••• , ••••••••••••
• 0 ••• 0 ••••••••••• , ••••• lit •••••• 8 ••••••••••••••• ••..•..... ~ .••....•....
••• 00 •••••••••••••••• 0 .................... " . '" . o ••••••••••••••••••••••
• • • • • • • 0 ••••••••••••••• • • • • lit • 0 •••• 0 ........... . •• G ••••••••••••••••••••
••••••••••••• 8 ••••••• , •••• 8 •••••••••••••• 0 •
• • • • • • • • • • • • • • • • • • • t
••••••••••••••• III ••• • • • " ............. III •
• • • • • • • G •••••••••
• • " •••••• 0 •••••• . . . . . . . . . . . . . . . . •• o~ •••• 6 •• o ••
• • o •• 111 •••••••• • • • " ••••••• 0 •
• G ••• III " •• 0
GENERAL FORM .OF THE STIFFNESS MATRIX.
j th
N
r Firs1 Element, ali) l To Be Eliminated In Row
ELIMINATION OF THE ELEMENTS IN THE i TH EQUATION.
-81-
8
7
FIG.13 PLASTIC MOMENT CORRECTIONS
-82-
A
A
For Applying Load
B
8
FIG.14a UNSYMMETRICAL BOUNDARY PLACEMENT
A B
FIG.14b SYMMETRICAL BOUNDARY PLACEMENT
-83-
Are Halved Along The Support
Support
FIG. 15 : SYMMETRICAL BOUNDARY PLACEMENT FOR A SQUARE PLATE
~84-
. ·1 ~ -I
(0 ,0) (1,0)
(0, .5)
,
FIG. 16: COORDINATES USED IN THE DEMONSTRATION PROBLEMS
50 Upper Bound F rom Ref. 22 ///////////////
q
10
Displacement; Vi = .307
.1 .2 .3 .4 .5 .6 .7
w
FIG. 17: LOAD-DEFLECTION DIAGRAM AT THE CENTER OF THE SIMPLY SUPPORTED PLATE
D' eb' tn, i
~ Indicates Region Of Plastic Flow
• I I _
q=1.14Qel q=1. 17Q el q=1.20Qel q = 1. 25Q e'l
q = 1.:31 qel q = 1.32 qel = q m q = 1.35 qel q = 1.39 Ci el = q f
FIG.18 PROGRESSION OF YIELDED REGIONS-SQUARE PLATE WITH 4 SIMPLE SUPPORTS
I co (J'\ I
M .. ••. Y
+1
-I~:;;;""""-~-....I
I /
/ /
Twist ing Moments Along TJ =- -.5
---, '\
+1 \ \ \
1.0
\ \ \
oL---------------~----------------~--~-o .5 1.0
Moments Across 7} =- - .25
+2.0 /, \ \ \ \
+ 1.0 \
Moments Along TJ=- 0
FIG. 19: REDISTRIBUTION OF MOMENTS DUE TO PLASTIC FLOW -- 4 SIMPLE SUPPORTS
-88-
2.0
0.5
Moments Along TJ~-.25
2'O~ /-, IJ
...,., I
-Mx qel
1.0
o~·----------------~--------------~----.-o 0.5
Moments Along "I} = 0 .
FIG. 20: REDISTRIBUTION OF MOMENTS DUE TO PLASTIC FLOW -- 4 SIMPLE SUPPORTS
w
Deflections (Elastic) Along 'fJ= 0
w
.6
Deflections Along '1J = 0
I
FIG. 21: TYPICAL DISPLACEMENT CONTOURS --, I
4 SIMPLE SUPPORTS
1.0
-.5
Twisting Moments Along 7] = -.5
Moments Across .." = ,.... .25
+.5
-My 0
..... 5
Moments Across ..,,=0
I
My +.:t ~. ~ ~ III e 0 "- LO .;.)
Moments Along ",=-.25
.5 1.0
-.5
Moments Along 7] = 0
FIG. 22: RESIDUAL MOMENT PATTERNS--4 SIMPLE SUPPORTS
-91-
Circumscribed Tresca Criterion
Mises Criterion
Inscribed Tresca Cr iter i on
(0,0) (..f3 ,0)' (2,0)
FIG~ .23: YIELD CRITERION USED' TO DETERMINE THE UPPER AND LOWER BOUNDS
100 1//////// flit/Upper Bound From Limit Analysis/////I/t //1/11/11//1
80 --0 0 0 0 0 \
qm=76.e> Ci f =79.4
q 60 Yields
40
20
OL'--------------~--------------~-----------o .1 .2 .3 .4 .5 .6 .7
w
FIG. 24: LOAD-DEFLECTION DIAGRAM AT THE CENTER OF THE PLATE --4 FIXED SUPPORTS
I \..0 N
~
~ ~
Q=1.66QeJ
q = 1.97 Ci e!
,~
~ Indicates Region Of Plastic Flow
Q=1. 82Q e! Q=1. 9Q el q=1. 93Q el
Q=1.98Qel=qm q = 1.99q el q = 2.05 qet = Qf
FIG.25 . PROGRESSION OF YIELDED REGIONS-SQUARE PLATE WITH 4 FI.XED SUPPORTS
i I..D VJ
+1
-·M Y 0
-I
+1
-
Moments
/ .....................
/ /
/ I
I.
/ /
/ I I I I
Moments
/
-94-
,5 1,0
Across TJ=-·5
.5
Across TJ =-,25
My O~~~---------+----------~~~--~ .5
Moment 5 Across 7J = 0
-I
FIG. 26: REDISTRIBUTION OF MOMENTS DUE TO PLASTIC FLOW .... - 4 FIXED SUPPORTS
-M)(
Moments Along "1 =-.5
+1
-M)( O~~~~--------+---------~~--~~~
+ 1, -2
-2
~ I
/
fl I
I I
.5
Moments Along TJ = - .25
.5
Moments Along 7J = 0
FIG. 27: REDISTRIBUTION OF MOMENTS DUE TO PLASTIC FLOW -- 4 FIXED SUPPORTS
0
.01
,.02 W
.03
.04
.05
Deflections (Elastic) Along 7J = 0
w
Def lect ions Along 7J = 0
FIG. 28: TYPICAL DISPLACEMENT CONTOURS --4 FIXED SUPPORTS
30
-q
20
10
00
FIG. 29
I'////////////Upper Bound From Limi-t Theorems///////// q =28,3
-qm = 23.87
-q f = 24.80 .
qel = 15.52
L_ L I _ .1 .2 .3 .4 .5 .6 .7 .8
w
LOAD - DEFLECTION DIAGRAM AT THE MID-POINT OF THE FREE EDGE
I
\..D -......J
I
~ Indicates Region Of Plastic Flow
Free Edge~
t
q ::: 1.03 (lei q ::: I. 23 (l eJ q = 1. 35Qel q = 1. 48Q el
q ::: 1.52 qel q ::: 1.54 qel::: qrn Ci ::: 1.58Qel q ::: 1.61 qel =qf
FIG.30 PROGRESSION OF YIELDED REGI()NS-SQUARE PLATE WITH 3 SIMPLE· SUPPORTS AND I FREE EDGE
I
1...0 OJ
I
-99-
0.5 I.()
. ----- -.;-.., ,.,...... . ......... -1.0 t------- ................ _-"".. ........
Twisting Moments Along 7J = -.5
2.0
-My 1.0
0.5 1.0
Moments Across 7J = - .25
My 1.0
----------------~----------~--~--~"' 0.5
Moments Across 7J = 0
FIG. 31: REDISTRIBUTION OF MOMENTS DUE TO PLASTIC FLOW-3 SIMPLE SUPPORTS AND I FREE EDGE
-100-
2.0
1.0
OLV--______ ~----~----------------~~.~ ~ - 0.5 a + 0.5
Moments Along ~ = 1.0
+1.0 ~ .
. I
-1.0
Twisting Moments Along e = 1.0
FIG. 32 REDISTRIBUTION OF MOMENTS DUE TO PLASTIC FLOW 3 SIMPLE SUPPORTS AND I FREE EDGE
-101-
o 0.5 1.0 o------------------~----------------~----------
.2
.W
.4
.6
Deflections Along 7] = 0 .
-0.5 o 0.5 o------------------~----------------~---------
.2
w .4
.6
Deflections Along t = 1.0
FIG. 33 : TYPICAL DISPLACEMENT CONTOURS - 3 SIMPLE SUPPORTS II I FREE EDGE
-q
1600
1200
800
400
. * - Points Taken From Levy's Solution
(11=.316)
1.0
-102-
2.0
w
S rna II Deflec tion Solution
Large Deflection Solution
3.0
FIG. 34: QUALITATIVE COMPARISON OF THE LARGE DEFLECTION SOLUTION WITH· LEVY·S SOLUTION
q
600
500
400
300 ~
I 200
100
Large Deflection (EI a s tic) ,
Bottom Sheet Near Center Of Plate Yielded
/ . .('
// -
q el
q f = 525
r = 10,0
...--------_ ............ _-- ____ CiZ!SlD=- ____ _
Additional Yielding In The Bottom Sheet Near The Center Of The Plate qm
Largle Deflection Theory
_ ........... Small Deflection Theory
r = 4.0 (Small And Large Deflection Theory Yield The Same Results)
r = 1.0 (Small And Large Deflection Theory Yield The Some Results)
o g~----------~------------~----------~-=---, J .4 .8 .12 .18 w .20 24 .28 .. 32
FIG. 35 : COMPARISON OF ELASTIC--PL,ASTIC SOLUTIONS GI\/EN BY SMALL AN~b LARGE DEFLECTION THEORIES
.' 0 (.,) J
~ Indicates Regi~n Of Plastic Flow
q = qel ,q = q rn .q = I. 75 q el q=1. 94Q el'
Top Sheet
• q = q eJ q=qm q = 1.75 qel q = 1.94 qel
Bottom Sheet
FIG. 36: PROGRESSION OF YIELDED REGIONS-SQUARE PLATE WITH 4 SIMPLE SUPPORTS USING LARGE DEFORMATIC)N THEORY AND r:: 10
o -J::
I
-105-
+ 1.0
-Mxy o~--------------~~--------------~----~
- \.0
Twisting Moments Along 7J = -.5
-My 1.0
°oL---------------~----------------~----~e 1.0
My 1.0
Moment 5 Across "7 = 0
- q ,....... __ ---.. __ ....---~ ;; f
I I
I i
I
/ --- \-/ - ~
0.5 - 1.0
Moments Across "7 = -.25
FIG. 37 REDISTRIBUTION OF MOMENTS DUE TO PLASTIC FLOWLARGE DEFLECTION SOLUTION
-106-
0.4
N xy 0.0 ~-------~~---------t~--~-1.0
0.4
Twisting Force Along 'rJ = -.5
0.5
~~------------~----------------~-------.-t 0.5 1.0
Membrane Force Along TJ = 0
·0.2 "..- ...... / ......
/
/ /
/
o~~~----------~--------------~~----~~ a 0.5 1.0
Membrane Force Along TJ =-:25
FIG. 38: REDISTRIBUTION OF MEMBRANE FORCES WITH PLASTIC FLOW- LARGE DELECTION SOLUTION
12
\ 8 \
-107-
\ -
U (0,0) 4 \,\-/_Qf
-
\
\ \
-4
-8
-I~
\ \
\ \
\ \
\
FIG.39 REDISTRIBUTION OF IN-PLANE DISPLACEMENTS
AT e =0; "1 =0 DUE TO PLASTIC FLOW-LARGE DEFLECTION SOLUTION.
q 200
o~----~------~------~--------~~ o 4 8 12
-U(O,O)
FIG. 40 : LOAD- DEFLECTION DIAGRAM FOR IN-PLANE DISPLACEMENTS AT e=O; 7]=O-LARGE DEFLECTION SOLUTION
600
Elastic-Plastic
400 I I
/~Elastic -q
200
oL---------~--------~--------~--------~~ o 0.5 1.0
FIG. 41: RATIO· OF .MEMBRANE FORCE TO MOMENT AT e= 1/2; "1 :: 1/24 -- LARGE DEFLECTION SOLUTION
-q
50
45
40
37.7
35
Elastic Limit Load In A Fixed Plate
30b&------~----~------~------------o .004 .008 .012 .016
A2
Def I ect ion At The Center Of An Elastic Plate
.0014
.0013
w
.0012
.0011 '-T i moshenko
.0010 a".. --.....1...,--_""",&"" __ ",,:,,, __ ...1...._-o .004 .008 .012 .016
A2
-Mx
.055
.045
Moment At The Genter Of A Fixed Edge In :An Elastic Plate
Timoshenko
.040' - , o .004 .008 .012
A,2
~ -" (Jj B
.016
FIG. 42: CONVERGENCE OF - MODEL SOLUTIONS WITH ME"SH" SIZE -- 4 FIXED SUPPORTS
.00377
.00375
w .00373
Deflection At The Center Of An Elastic Plate
Timoshenko
.00369'h-----------------~----*-----o .004 .008 .012 .016
A2
M
QQ2.04788
.04784
Moment At The Center Of An Elastic Plclte
TimoshenkO-
:~
.04780~·----------~----·~------~----o .004 .008 .. 012 .016
A '2.~
41.0
40.9
q
40.8
40.7 ~. __ ....&... __ """"-__ -'-_ ......... .....1.
o .004 .008.012 .016
Xl.
FIG. 43 CONVERGENCE 'OF MODEL SOLUTIONS \NITH' MESH SIZE -~ 4 SIMPLE SUPPORTS
t
--o· I!
Upper Bound From Limit Analysis
80
I A = 8"
60 I II \
A = ...L 24
q I/~qel = 45.79 (x= t ) q - 39 el- .73 (A=J....) 16 40r ~ -qel = 38.73 (A= _I ) 24
J /
O~'------------~------------~----------~------------~------------~------------~----------~-----------o
FIG. 44
.1 .2 .3 .4 .5 .6 .7
w
LOAD-DEFLECTION DIAGRAM AT (=.5; "1=0, FOR 3 DIFFERENT MESH SIZES-4 FIXED SUPPORTS
~
1
+ 1.0
-My
0
-1.0
+2.0
-My +1.0
-1.0
-112-
r r; 0.5
. Moments Acro ss TJ::: - .25
~'l II
I J
0.5
Moments Across 7J::: 0
A = -'-8 A = -L
16 A = ..!.....
24
FIG. 45: VARIATION IN THE DISTRIBUTION OF MOMENTS WITH MESH SIZE; q:: qm-- 4 FIXED SUPPORTS
x
t
'\ '\1\.,/\
h b
1,'. ~
.h
/\ /\/\ v v
\VLV~, v, r--?V .
I~ II
4
k
-113-
d
3 (m)
I (f)
1--"> C / .~",.
(a) Plan
V\ :../'v"
0 1111
"v"v """v"
~ e. / >"
II
2
( b) Elevation (Section x-x)
'\V~
J\/'1jYY
a /"I .. .~
m )(
J\.II'Vv
FIG. 46 : SCHEMATIC DIAGRAM OF THE II TRUE " FLEXURAL MODEL
'".
Uj
FIG. 47 : ELEVATION OF SECTION X-X AFTER DEFORMATION-FOR DETERMINING THE NORMAL STRAINS
l
I
- 1 J ~)-
y = COS a
(a) Elasticity Definition
m
(b) Model Definition
FIG.48 : DEFINITION OF SHEAR STRAIN IN THE MODEL
