buckling and post-buckling behavior of a large aspect
TRANSCRIPT
BUCKLING AND POST-BUCKLING BEHAVIOR OF
A LARGE ASPECT RATIO ORTHOTROPIC PLATEUNDER COMBINED LOADINGS.
Fred Lewis Ames
^AVAL POSTGBADUATB SCHOW
ioHTEBK, CAT IF. 93940
BUCKLING AND POST-BUCKLING BEHAVIOR
OF A LARGE ASPECT RATIO ORTHOTROPIC PLATE
UNDER COMBINED LOADINGS
by
FRED LEWIS MIES
LIEUTENANT, UNITED STATES COAST GUARD
B.S./ United States Coast Guard Academy
(1968)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF OCEAN ENGINEER
AND THE DEGREE OF
MASTER OF SCIENCE IN MECHANICAL ENGINEERING
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June, 1973
^^.
LIBRARYNAVAL POSTGRADUATE SCHOODMONTEREY, CALIF. 93940
2
BUCKLING AND POST-BUCKLING BEHAVIOUR
OF A LARGE ASPECT RATIO ORTHOTROPIC PLATE
UNDER COMBINED LOADINGS
by
Fred L. AmesLieutenant, United States Coast Guard
Submitted to the Departments of Ocean Engineering and Me-chanicc'il Engineering on 11 May 1973 in partial fullfillmentof the requirements for the degrees of Ocean Engineer andMaster of Science in Mechanical Engineering.
ABSTRACT
Orthotropic plate theory has been increasingly used tomodel plate-stiffener combinations typical of those whichare used in ship hulls. The behaviour of a thin plate withlarge deflections is described by two nonlinear partialdifferential equations of equilibrium and compatibility.The orthotropic form of these equations is derived andsolved for a large aspect ratio plate with long edges simplysupported and short edges fixed as boundary conditions.Buckling and post-buckling regions are investigated undercombined loadings of lateral pressure, inplane edge com-pression and edge shear. Results are presented for virtualaspect ratios 1/1.5 to 1/6 and both isotropic and ortho-tropic plate properties in the form of design and behaviourcharts.
Thesis Supervisor: Alaa E. Mansour
Title: Associate Professor of Ocean Engineering
Thesis Reader: Thomas J. Lardner
Title: Associate Professor of Mechanical Engineering
Acknowledgement
The author would like to express his appreciation to
Professor Alaa Mansour for his support and guidance in this
research. Gratitude is extended to Commander Joseph L.
Coburn,, U. S. Coast Guard Headquarters Research and Develop-
ment Branch, for providing the funding of the necessary
computer services.
Special thanks is given to my wife Holly for her con-
tinued understanding and long hours spent deciphering and
typing the rough draft. In addition, the fine typing of
the final draft by Ms. Cathy Bayer is greatly appreciated.
Table of Contents
Page
Abstract 2
Acknowledgement 3
Table of Contents 4
List of Figures 7
List of Tables 9
Nomenclature 10
1. Introduction 14
2. Formulation of the Problem 17
2.1 Orthotropic Material Properties 17
2.2 The Rectangular Orthotropic Plate with
Large Deflection 17
2.3 Bending and Membrane Stresses 21
2.4 Load Boundary Conditions 22
2.5 Support Conditions 23
3. Theoretical Analysis 24
3.1 Analysis Procedure 24
3.2 Solution 25
3.3 Bending and Membrane Stresses 29
3.4 Effective Width 33
3.5 Principal Stresses 35
4. Numerical Solution. 37
4.1 Analytic Solution 37
4.2 Computer Solution 39
5
Page
5. Results 41
5.1 Design Charts 41
5.1.1 Charts of Deflection 41
5.1.2 Charts of Effective Width 41
5.1.3 Charts of Bending Moment 41
5.2 Behaviour Charts 4 2
5.2.1 Charts of Deflection 42
5.2.2 Charts of Maximum and Minimum
Total Principal Stresses 42
6. Discussion of Results 43
6.1 Orthotropic Properties 43
6.2 Comparison with Existing Solutions 43
6.3 Design Charts 44
6.3.1 Charts of Deflection 44
6.3.2 Charts of Effective Width 45
6.3.3 Charts of Bending Moment 4 6
6.4 Behaviour Charts 4 6
6.5 Examples Demonstrating Use of the Charts. ... 49
6.5.1 Design Example 49
6.5.2 Behaviour Example 50
7. Conclusions and Recommendations 52
8. References 54
9. Appendices 101
Appendix A. Details of Derivation of Basic
Equations of Large Deflections 102
Appendix B. Effective Width 115
6
Page
Appendix C. Details of Solution 118
Appendix D. Computer Programs 126
List of Figures
Figure Title Page
1 Illustration of Coordinate System ggand Inplane Edge Loadings
Design Charts of Deflection vs. InplaneLoad N*
X
2,3 p = 0.667 67
8,9 P = 0.50 70
14,15 P = 0.25 73
20,21 P = 0.167 76
Design Charts of Effective Width vs.Inplane Load N*^ X
4,5 P = 0.667 68
10,11 P = 0.50 71
16,17 P = 0.25 74
22,23 P = 0.167 77
Design Charts of Bending Moment vs.Inplane Load N*^ X
6,7 P = 0.667 69
12,13 P = 0.50 72
18,19 P = 0.25 75
24,25 P = 0.167 78
Behaviour Charts of Deflection
26,27 p = 0.667, N* = 79
36,37 p = 0.80, N* = 84
46,47 P = 0.667 89
8
Figure Title Page
Behaviour Charts of Maximum andMinimum Total Principal Stresses
28-31 p = 0.667, N* = 0, X = 80,81
32-35 p = 0.667, N* = 0, y = 82,83
38-41 p = 0.80, N* = 0, X = 85,86
42-45 p = 0.80, N* = 0, y = 87,88
48-51 p = 0.667, X = 90,91
52-55 p = 0.667, y = 92,93
56-60 Deflection and Principal Stresses, 94,95,96Top and Bottom of Plate
61-63 Design Charts of Deflection, Effective 97,98Width and Bending Moment vs. InplaneLoad N*, p = 1.0
y
64 Plate Deflection Comparison with Levy 99Solution [5]
65 Effective Width Comparison with Levy 100Solution [5]
List of Tables
Table Title Page
1 Coefficients C 56pq
2 Rigidity and Compliance Coefficients 60
3 Integrals 61
4 Computer Symbols 65
10
NOMENCLATURE
y= ordinate in long direction
X = ordinate in short direction
a = plate length in x-direction
b = plate breadth in y-direction
3 = — = aspect ratio
u, V, w = displacements of a point in x-, y-, and z-
directions respectively
e , e / e = middle-plane strainsX y xy ^
D , D = flexural rigidity of orthotropic plate in x-
or y-directions respectively
D = effective torsional rigidity of orthotropic
plate
V , V = Poisson's ratio of orthotropic plate in x- or
y-directions respectively
E , E = modulus of elasticity in x- or y-directionsX y
respectively
G = modulus of elasticity in shear
N # N / N = middle-plane loads per unit length
M , M = bending moment in orthotropic plate acting
around a line perpendicular to x- or y-axis
respectively, per unit width
M = twisting moment in orthotropic plate
b = nondimensional deflection coefficientmn
11
F =
J =
Airy's stress function
1
hE
h = plate thickness, orthotropic
J =hE
X
xy Gh X y y x
2D =Dv +Dv +4Cxy X y y X
C = 9hl12
e = i2^(i-v v„)1,2
X y
P , PX y
q
s
= total loads in x- and y-directions respectively
= uniform lateral load, per unit area of plate
= constant inplane shear load, per unit length
P = KY _ virtual aspect ratio of orthotropic plateX
n = xy _
Id dvl X !
^= torsion coefficient of orthotropic plate
xy
J JM X y
N a'
N*X 2t^
IT D= nondimensional in-plane load x-direction
12
N b^_ yN* = —
i
= nondimensional in-plane load
y-direction
qb"* . .
Q* = —^ = nondimensional lateral load7T'*hD
y
SabS* = = nondimensional edge shear load
TT^Dxy
b , a = effective width, x- or y-directionse e r j:
respectively
o , o ', o* , a* = membrane stresses in x- and y-
directions respectively; nondimensional
T ; T* = shear stress in xy plane; nondimen-xy xy J t- '
sional
^bx' ^by' ^bx' ^by ^ bending stresses in x- and y-
directions respectively; nondimensional
T, '"^h
~ twisting stress in xy plane; non-
dimensional
o. , a. , T^ = total stressestx ty txy
a* , ai / tJ = nondimensional total stresses
ai'f of^' "^t't~ nondimensional total stresses
cr, « ; a? p = maximum and minimum principal stresses,
plane stress; nondimensional
a r a = edge membrane stress in the x- or y-e ^e
direction respectively
13
M*, M* = nondimensional benclinq momentsX y -^
per unit length
M* ,M* = nondimensional bending moments pero ^o
unit length due to curvature in x-
direction only (M* ) and y-directiono
only (M* )
^o
C ,6 , A / X , Y = coefficientspq pq pq q q
a ; a* = Von Mises combined stress; non-o o
dimensional
1,1 = moments of inertia of the stiffenersX y
with effective plating in the x- and
y-directions respectively
1,1 = moments of inertia of the effectivepx' py
plating alone in the x- and y-
directions respectively
S , S = spacings of the stiffeners extending
in the x- and y-directions respec-
tively
h , h = equivalent thickness of the plate
and the stiffeners (diffused) in the
x- and y-directions respectively
h = thickness of plate aloneP
14
1. Introduction
Of major concern to the naval engineer is the reaction
of ship bottom plates to a combination of inservice loadings.
In gene::al, the plate-stiffener combinations will be sub-
jected to inplane compression or tension edge loads due to
ship hogging or sagging, lateral hydrostatic loads and in-
plane edge shear loads. With increasing compression and
shear loads, a critical state will be reached where buckling
will occur. Plate deflection will exceed plate thickness,
however, the plate may still carry considerable loads. High
slenderness ratio (b/h) plates are encountered in modern
longitudinally framed ship designs. The yielding load may
be considerably higher than the buckling load and investi-
gation of the plate-stiffener residual strength in the post-
buckling region is of great importance.
Orthotropic plate theory has been increasingly applied
to ship structures. With this analysis, the plate-stiffener
combination is modeled as an equivalent flat plate with
elastic properties that are different in two perpendicular
directions. Schade [12, 13] and Mansour [9, 10]*, among
others, have worked with this concept.
The small deflection theory is useful only in the pre-
buckling region where deflections of a plate are small in
*Numbers in brackets designate references in Chapter 8.
15
comparison with its thickness, ^.he assumption of no deforma-
tion in the middle plane of the plate is made in this case.
w 1If deflections are not small, r- > :r, the strain in the middle
h 2
plane is no longer negligible and must be considered. Von
Karman's equations describe the behavior of an isotropic
plate under combined lateral and inplane loads. The non-
linear equations describe the behavior of the plate in both
small and large deflection regions.
Considerable application has been made of von Karman's
equations. Levy [4, 6] investigated square and rectangular
plates simply supported and subjected to both inplane edge
loading and norroal pressure. Coan [3] included the effects
of small initial curvature for this case. Levy [5], using
two fixed and two simply supported ends, solved the equations
with edge loading only for an aspect ratio of 4 , and studied
pure shear for the simply supported square plate [7, 8].
The square plate with variations of the boundary conditions,
subjected to edge loading only, was studied by Yamaki [17]
.
Payer [11] applied von Karman's equations to deep web frames
and included uniform edge shear in addition to compressive
edge stress and normal pressure. Shultz [15] investigated
wide plates having aspect ratios of 3 = 1.5 to 8 for con-
ditions of a transversely framed ship. All of these
analyses apply only to isotropic plates. Mansour [9, 10]
extended von Karman's equations to consider slightly rec-
tangular orthotropic plates under various boundary, loading
and initial deflection conditions.
16
A large aspect ratio orthotropic plate with the shoD't
edges fixed and the long edges simply supported is considered
in this paper. The plate-stiffener combination is subjected
to loads of inplane edge compression, uniform edge shear and
uniform pressure normal to the plate. Figure 1 illustrates
the loadings and coordinate system used.
The orthotropic form of von Karman's equations are
solved similar to the theoretical analysis of Mansour [9, 10].
The IBM System 37 model 165 is employed to produce the
numerical results. "Design" charts of deflection, effective
width and bending moment are given for orthotropic plate
virtual aspect ratios of 1/1.5 to 1/6 with inplane edge
compressive and lateral loadings. In addition, "behaviour"
plots of plate centerline deflection and total principal
stresses are given for virtual aspect ratios of 1/1.5 and
1/1.25 for various combinations of inplane edge compression,
edge shear and lateral loadings.
17
2. Formulation of the Problem
2. 1 Orthotropic Material Properties
An orthotropic material has different elastic proper-
ties in two perpendicular directions. For plane stress in
the xy plane, the stress-strain relations are
E
^x = l-v^'v ^S."^ ^y^y^
X y -^-^
Eo = -. ^— (e + V e ) (1)y i~^x^y y X X
xy 'xy
from energy symetry
E V = E VX y y X
which gives 4 independent elastic constants.
2. 2 The Rectangular Orthotropic Plate v/ith Large Deflection
(Figure 1)
w 1For thin plates with large deflection, r- > y, a satis-
factory approximate theory makes the following assumptions:
1. Points initially on the normal to the middle plane
of the plate remain on the normal after bending.
(Disregard of shear deformation.)
2. The normal stresses in the direction transverse to
the plate can be disregarded. (Plane stress where
a = 0.)z
3. Hooke's Law relating stress-strain applies.
For the large deflection theory, deformation in the
middle plane of the plate must be considered. These strain
components therefore include the effect of deflection and
are approximately
X ~ ax 2^dx'
e ~ 9X + 1(9^)2 (2)y 3y 2^3y' ^"^^
^ 9u 9v; 3w ^ 9w^xy ~
8y 9x 9x * ay
Equilibrium of forces and moments on an element of a
plate produces the equilibrium equations
aN aN
ax ay
aN aNz + -^ =
(3)
ay ax
and
a^M a^M a^M^ - 2. ^y +
ax^ ^^3^ ay^
_(5 + N ^+ 2N ^-t- N ^) (4)Xg^2 xyaxay y ^^^^
19
Introducing the familiar Airy's stress function F which
satisfies equations (3)
N = l!Z; N = ^ ; N = -v^ (5)
^ 8y^ ^ 8x2xy 8x8y
From substitution of large deflection strains (2) into
the stress-strain relations (1) and using the definition of
bending and twisting moments, the moment-curvature relations
for an orthotropic material are derived
M = -D (^ + V ^)^ ax^ ^ 8y2
M = -D (^ + V i^) (6)y ^\y' ^ 8x2
M = 2CB^w
xy 8x8y
where the rigidity coefficients are defined as
E h^D -X - 12(l-v^Vy)
E h-
D = JL
y - 12(l-v V )J ^ X y
and
Gh^C
12
Substitution of equations (6) and equations (5) into equation
(4) , the "equilibrium" equation is obtained as
20
D i^:^ + 2D 9'^ -f D i>
(7)
Q + •*2 • — Z • + •
rv 2 ^2 3x3y 8x8y ^2 ^2
where
2D =Dv +Dv +4Cxy X y y X
To obtain the "compatibility" equation, the strain equations
(2) are first differentiated and combined to eliminate dis-
placements u and V
^'^x _ ^'^xy ^^'^y
_( a^w^2 _ 3lw ^ aH/^g^
3y2 axay g^2 axay ^^2 9^2
Differentiation of equations (1) , substitution into equation
(8) along with (5) and using the equilibrium equations (3)
,
the "compatibility" equation is obtained as
J i!l ^. 2J 9'^ + J 3^""ax** ""^ax^ay^ ^ay"*
a^w.j a^w a^w O)= (~:;^)axay .^2ax^ ay 2
where
J =^ ; J = ^
and
X E„h ' y E^h
2J = -prr- - V J - V Jxy Gh X y y x
21
Equations (7) and (9) are fourth-order non-linear
partial-differential equations that describe both small and
large deflections of an orthotropic plate. This includes the
buckling and post-buckling behavior of the stiffened plate.
Substitution of the isotropic material properties
EE =E =E;v =v =v; G = Tm—\X y X y 2(l+v)
result in the familiar von Karman's equations.
Solution of these equations with 16 boundary conditions
results in the two functions F and w. The boundary con-
ditions are eight support conditions and eight edge load/
displacement conditions.
2 . 3 Bending and Membrane Stress
The membrane stresses are determined from equations (5)
X 1 8^Fas N.. -. ^ 2
X h h 3^2
a =!z 1 alF(10)
^ h ^ ax^
xy ^ _ 1 9^Fxy h h 9x8y
Noting that the maximum normal stress acts on those sections
parallel to the xz or yz planes, and using equations (1)
,
(2) and (6) the bending and shear stresses are obtained as
bx Cl-v V ) DX y X
E M
^ X y,l y
22
E M
M
bxy C
The maximum bending and shear stresses occuring at z = ±y
gives
M M Ma, = ±6— ; a, = ±6-^ ; x, = +6-^^ (11)bx ^2 'by ^2 ' bxy ^ ^2
where M and M and M are given in equations (6) . Thex y xy ^ ^
total stresses are the sums
a. = a ± a,tx X bx
"ty = "y * "by <12)
txy xy bxy
2. 4 Load Boundary Conditions
The following most general conditions state that the
edges are subjected to an inplane average compressive load
per unit length in the y-direction and x-direction of
magnitudes N and N respectively. Additionally, all edges
are subjected to a constant inplane shear load per unit
length of magnitude S. Referring to Figure 1,
23
at X = ±2
N = S , —-^ = -Sxy ' dxdy
Inplane load resultant P is (13a)
^x = : ,. H ^^ = "^x^-b/2 9y
at y = ±2
N = S , TT^—^ = -Syx ' dxdy
Inplane load resultant P is (13b)
P = / ^-^ dx = -N ay -a/2 8X2 y
2.5 Support Conditions
Edges simply supported at x = t-y
Deflection equals zero: w =
External moments equal zero: (14a)
9^+ V ^ =Bx^ ^ 8y2
Edges clamped at y = ± y
Deflection equals zero: w =
Slope equals zero: |— = (14b)
24
3. Theoretical Analysis
3.1 Analysis Procedure
The solution method used is identical to that of
Mansour [9, 10] which is an extension of Levy [4], Coan [3]
and Yamaki [17]. Briefly, the outline of the procedure is:
1. Express the deflection of the plate satisfying the
support boundary conditions, in a double trigono-
metric series choosing only a finite number of
terms, w will be a function of unJcnown nondimen-
sional coefficients bmn
2. Substitution of this expression into the "compati-
bility" equation (9) results in a fourth order
partial-differential equation with the Airy stress
function F and quadratic functions of the coeffi-
cient b expressed as coefficients Cmn ^ pq
3. A solution for the stress function is assumed which
satisfies equation (9) and the load boundary con-
ditions. F will be expressed as functions of
unknown coefficients 6 which in turn are functionspq
of coefficients Cpq
4. To determine the unknown coefficients b , Galerkin'smn
Method is applied to the "equilibrium" equation (7)
25
with substitution for w and F by their appropriate
expressions. Using the orthogonality properties of
the trigonometric functions, a set of simultaneous,
non-linear algebraic equations involving cubic
products of the coefficients b results.^ mn
3. 2 Solution
The deflection of the plate surface can be expressed in
the form
w = h E Z b f (x) g (y) (15)m n mn m ^n -^ ^^-'^
Satisfying the boundary conditions (14) / the deflection terms
take the form
J. I V miT , _ pf (x) = COS—X ; m=l, 3, 5....m^ ' a '
/ \ / T \ n+1 , 2mT T o ->
gj^(y) = (-1) + cos-^-y ; n = 1, 2, 3
Differentiation of expression (15) and substitution into
the "compatibility" equation (9)
J i!£ + 2j^"^
+ J i^
= h'ai;
b„ (!!^) (2niL)sin^x sinSgly}^m n mn a b a b
- h={E E b„r (-!)"+!+ cos2gi:y]{!HL)acosi^x}m n mn b a a
•{E Z b (-T—)^ cos—-X cos-r—y}m n mn b a b
26
{Z Z E E 4b b. .mnii sm—^x sm-r—-y2, 2 . . mn 11 -^ a b -^
a b m n 1 3-^
. ilT . 2111 „., , 2-2 IHTT• Sin—X sm-^—y - Z E Z E 4b b. .m^i'^cos—^x
a b ^ ^ ^ . j mn 13 J
r/ , V n+1
,
2n7T , iiT 2i7r , ,^,.• I (-1) + cos-^y] cos—^x cos-^y} (16)
m, i = 1/ 3/ 5....
which can be expressed in the form
J i!£ + 2J -A!IL + J i!Z
EEC cos^^i^^x cos^^y (17)h^TT** V ^ A^ 2077 2qTrEEC cos—^£—^x cos-^—
^
a2j32 P q pq a b
P/ q ~ 0/ 1/ 2....
where C are quadratic functions of the nondimensionalPSl
coefficients bmn
A particular solution for (17) is assumed in the form
F = h^ E Z ({) cos^^x cos^y (18)p p q pq a b -^
substitution of (18) into equation (17) defines coefficient
6 aspq
27
D 2 /-I
^ = P3(19)
where
The stress function F to satisfy the load boundary conditions
(13) and equation (9) is
N NF = - -^x^ - -~Y^ - Sxy + F (20)
The method of B.G. Galerkin is applied to the "equilib-
rium" equation (7) to determine the unknown coefficients b^ mn
Galerkin 's condition requires that the following equation be
satisfied by all functions f (x) g (y)
a/2 b/2 ^ It ^ ! ^j,
/ / [D 1^ + 2D _AJ!L_ + d ^ "^
^ax"* ^^9x^8y^ ^ay**
8^F 9^w_^^ a^F 9^^ 9^F 9 ^w, ,^^." ^ " 77 • 7T "" ^91^ * 93^ - 77 * 7T^ ^^^^
9y 9x -^ -^ 9x 9y
f^(x) g^(y) dxdy =
where f (x) g (y) is given be equation (15) . Equation (21)
may also be obtained by applying the principle of virtual
work.
Substitution of w and F, expressions (15) and (20)
respectively, into equation (21) and using the orthogonality
28
properties of the trigonometric functions, there results
,k
Y TT^D y IT D^ y -^ X
Zf g b^^{2(-l)^"^^r'* + [rS2p2n(2s) 2r2+ p'*(2s)'*]6j
eJ n f g rn'^ ' - . .- .-h ,,v--y - h x--/ j u^^g
2.r2(-l)^-'^^)} + i; I°f E 4in(i)^A (-1) s+l^(-l!^) b2r5 m r n s tt' mr v tt^d
^^^ X ^ xy
odd 2
+ I Z Z 2mn(i) ^A (-I- + -!_) §£_ (_SafcL.) bm r , IT mr n+s n-s GJ 2rs ninn+s y 7T D^ xy
+ Z E E E b {2(-l)^'^^n^rM(j) ^ + (}) + <^ ]m n r s mn ^m+r ^m-r ^r-m2 fH 2 'J^ 2 '^
+ 2(-l)^"^-'-m^s^[(}) ^ + (j) + (}) ]^m+r ^m-r ^r-m^/S <),S -),S
+ (ms-nr)^I4) ^^ +4> +4> •-(|) +cl) ]^m+r , ^m-r ^m-r ^r-m ^r-m—2"fH+s —2—/ n-s —J—/S-n —2~>n-s —2~> s-n
+ (ms+nr)^I(}), +
<i> , + <^ +<t> ]>m+r ^m+r ^m-r ^r-m
—J-,n-s —2—/ s-n —2—/ n+s —2~7n+s
r s ej.. rfT _it,
3r+l ^ Q -,it= Z Z (.l)^+-2-^ :^ (-2^—) (22)
^ y
m, r = 1, 3/ 5....
29
where 6. . is the Kronecker delta, (|) . . = if i < or
j < 0, = if n = s and-" ' n-s
A = —,— if —^r- oddmr m+r 2
or
A = if —rr- oddmr iti-r 2
Equation (21) is identical, except for the addition of the
two shear and two N load terms, to that of Mansour [9] case
(i) and, if isotropic material properties are substituted,
reduces to Yamaki [17] case Illb for zero initial curvature.
Substitution for coefficients cf) in equation (22)
results in a set of simultaneous, non-linear algebraic
equations involving cubic products of the coefficients b .
Solution of these equations gives the coefficients b and^ -^ mn
hence defines the functions w and F.
3 . 3 Bending and Membrane Stresses
The membrane stresses are now determined from equations
(10) , (18) and (20) . These are
NX T_ r. V X /2q7Tx 2 2pTr 2q7T
a = - -u hZZd) (-r— ) cos-^x cos-g—
y
X h p q ^pq b a b ^
NY 1- V V J.
/2p7T. 2 2p7r 2qTra = - -tf
—
h E Z d) {-^—-) cos--^-—
X
cos-^—
y
y h p q ^pq a a b -^
_ S u V V A /2p'n'. ,2qTT. . 2p7T . 2qTr„ ,^-.
30
The meml^rane stresses will be represented in the most
general form, the nondimensional form used by Mansour [9]
.
Using equation (19)
a N*-a. X XX 2u 12(l-v V )
a^Jy
(24)C q^p"* o o
- Z Z ^-^ cos --X cos y- yP ^ 4(p'* + 2Yp2p2q2 + p'^q'*) ^ ^
cr.. N** = iL_ y^
(_zi2l)12(l-v^Vy)
b^JX
(25)
2p7T 2q7Tcos--^—-X COS-——
y
4(p'* + 2YP^P^q^ + p-^qM
Z E C p^ o opq^ 2p7T 2q7Tp q £-s cos--^—-X cos-——
y
and
T* = ^xy ^ _ S*nYxy ^2, 12(l-v V )r_jT_h_, X y
^abJ ^
xy
Z Z £-^ sin—^-—X sm—r—
y
p q It 2 a b4(2— + 2p^q^ + ^')
P/ <3 - , 1 , 2....
The bending and twisting stresses from equation (11) are
31
6DQ = + _r^ = ± - [- i!:^: - V
6D
'by= ± _Lm = ± __Z [-
2 Y
8x^
8^w
3y^
- V
a^wy ay^
S^wX
8x2
(26)
bxy- 12C a^w•*"
,2 8x8y
The normal bending stresses may be expressed as functions of
nondimensional bending moments M* and M* where^ x y
MM*X D D
M* + Vx_ "^
D
^ M*D y
(27)
MM*y hD
b^
M*y.
+ VX
D•
D XX o
and
M*X
M^o
h pnr^[
= - Da^w
X yX . 2 h nj~D^
8x \| X y
M
M*y. hD
b^
= - Da^w b'
y -N 2 hD^ 9y y
(28)
The bending stresses as functions of the nondimensional bending
moments are
32
bx
6M*X
D D'
'by
6M*(29)
hD
The nondimensional moment due to curvature in the x-direction
only (M* ) and that due to curvature in the y-direction only^o
(M* ) are determined from (15) and (28)^o
M*X
M*
„ „ 'n'^m^, m-n r, , . n+1
,
2n7T ,
E E ^b^^cos-—X [(-1) + cos^j—-y]m n 2 inn a b
Z Z 411 n'^b cos—X cos-T—-ym n mn a b -^
(30)
m = 1 , 3 / 5 . , . .
The bending stresses may be expressed as nondimensional bend-
ing stresses in a form compatible to equations (24) and (25)
Using equations (15) and (26) , the nondimensional bending
and twisting stresses are
'bx= 'bx 1 r' r- 1- ii^'n'= ± -TTjr: r- Z E b COS X
2 (1-v V ) m n mn aX y
rm2(-l)^''l+(m2 + ^^!-^) cos^y]3^ ^
(31)
33
by ^2^ 2 (l-\v^) m n mn ""^^a
'
r
Xb^J
(32)
andX X. D
bxy ^ . Yn ^ _ y x^ " .
"''^r^L!h_, '^-''x^y' 2np^B^
^n
E Z b run sm—r--y sm—^xm n mn b a
m = 1 , 3 , 5 . . . .
Xl ~~ -^Z ^ i -^ • • • m
The nondimensional total stresses are therefore, from (12)
a* = a* ± a^tx X bx
ty "^y -''by (33)
T* = T* X T*txy xy bxy
3.4 Effective Width
The effective width of a rectangular plate which has
buckled is defined as that width of a uniformly stressed
phantom plate of the same thickness stressed to the same
maximum stress and sustaining the same total force as the
real plate. The effective widths a and b , in the y and x
34
directions respectively are hence defined as
Nao = a -T^ C34a)e ye n
Nho = b -^ (34b)e xe h
where a is the edge membrane stress in the y-directionye ^ ^
(a at X = ± -tt) and a is the edge membrane stress in they 2' xe ^
x-direction (a at y = ± -5-) . Substitution of (25) evaluated
at X = ± p- into (34a) gives
a N*_e = YpM-i)Pc
N* + 3(l-v V )E E 22 cos^yy X y p q,it,^ 222, »»i+\ b"'
(P + 2Yp^p^q'' + p q )
(35a)
P/ q = 0, 1, 2..,.
or
!e^ ^2
(-l)P"^ P'(-l)%q
2qTrN* + 3(l-v V ) [Z
^ ^^ C +Z E E3 Icos^yy X y p p2 p,0 Pq=i(pV2Yp^p2q2+p'*qM ^
substitution of (24) evaluated at y = ± j into (34b) gives
b N*e _ Xb
2 q ^
N* + 3(l-v V )Z E ^ ^"-^^ ^ Sq ^2pTTX X y p q ^-^ cos-^^—
X
(p- + 2Yp'p'q' + p'qM^
(35b)
p , q = 0/ 1, 2,,..
35
or
b N*e _ X
N* + 3(l-v V )[Ei-ii-Go,q + E I EH_] cos^^x
"" X y q g2 p=i^ (p^* +2Yp2p2q2 +p^qM ^
Clearly, the effective widths a and b are not constant over
the y and x-directions respectively. Neglecting the small
amount of change due to the periodic terms in y or x, taking
only the average values into account [15] , the effective
widths take the form
a N*e _ ya
N* + 3(l-v V )E ^^^
Cy X y'p 2 P,o
(36a)
p = , 1, 2 . . . .
and
b N*e X
N* + 3(l-v V )Z ^^'
CX X y q „2 o,q
(36b)
q=0/ 1/ 2....
n
3.5 Principal Stresses
For the state of plane stress, the maximum and minimum
principal stresses, a, and o^r are given by
36
1,2
G + a
2 - N
- aX i)- + T
xy
Substituting the nondimensional total stresses, equations
(33) , the nondimensional principal total stresses are
1/2
a*' + a*'tx ty
r*' - rr*'r* - o'
2 txy (37)
where
1,21,2
[IL!iL.3
b^Jx
and
txtx
'* o2P'^B
r*' = a*ty ty
* I
txy= txy
. YP'3
37
4. Numerical Solution
4 . 1 Analytic Solution
Shultz [15] found that eight deflection terms were
sufficient to describe a simply supported plate up to an
aspect ratio of 8. In this solution eight deflection terms
in the y-direction and one in the x-direction were assumed.
From equation (15) with m = 1 and n = 1,2,3. ..8, the
deflection of the plate takes the form
w = h cos—X [b, - (1+cos-T—y) + b, „ (-l+cos-r—y)a 11 b -^ 12 b -^
+ b^3(l+cos^y) + b^^(-l+cos^y) + b^^^ (1+cos^y)
I i_ / T .1277 X , , ,,
,14tt » , , It, 16Tr .+ b^g (-1+cos-^y) + b^-, (1+cos-^—y) + b^^g (-1+cos-g—y)
Similarly, the right hand side of equation (16) is ex-
panded for m, i = 1; n,j = 1,2, 3... 8. The coefficients C
are determined by collecting terms on the right hand side of
equation (16) and matching the coefficients to the series
on the right hand side of equation (17) for p = 0, 1;
q = 0, 1, 2... 16. The coefficients X and Y are listedq q
in Table 1 where the coefficients C are given bypq ^
C = Y + (-l)Px (38)pq q ' ' q
P = 0, 1
q = 0, 1, 2. . . .16
38
Substitution of coefficients C into equation (22)
using equation (19) and with va, r = 1; n, s = 1,2,3. ...8
produces the following equation after considerable simpli-
fication
N*Z (2s)2N*bT + Z -^, - —- Z E b, {2(~1)^"*"^s ylfS s 4l,s uns l,n
+ ri+2p2Ti(2s)2 + pV2s)'*]6^^} + -^ Z Z b (-D'^'^'^N*Ilo If II o X / ii ^
+ 2(-)^-^ Z Z b, (-1)^"^^S* + (-)^-^ Z b, n(-4- + -^)S*^Tj' 2ns l,n IT 2 ,
l^n n+s n-s'p p n+s
+ Z Z b, {2(-l)^"^^n^[A, C, + 2A C ]
n s l,n l,n l,n o,n o,n
+ 2(-l)^'*"-^s^[A, C^ + 2A C ]^ l,s l,s o,s o,s
+ (s-n)MA^^^^^C^^^^^ + 2A^Jn-s|CoJn-s
+ (s+n)^[2A ^ C , + W«AtI
„|C,I
^i]}^ o,n+s o,n+s 1, n-s l,In-s|
- Z (-1)^"^^ ig* = (39)S TT
n,s = 1,2,3. ...8
where W=1.0ifn7^s
W = 2.0 if n = s
and4<1 - \\^
A
39
3
where
^^ (p** + 2Yp2p2q2 + q'»p'»)
<t>= A - ^
pq pq pqQJ,
Equation (39) gives a set of eight simultaneous, non-
linear equations for the eight coefficients b^^, ^12' ^13'
^14' ^15' ^16' ^17' ^"""^ ^18-
4 . 2 Computer Solution
Equation (39) is programmed on the IBM System 370 model
165 using Fortran IV level Gl. The subroutine ZEROIN from
the M.I.T. MATHLIB program library is used to solve the
system of simultaneous, non-linear equations. This sub-
program uses an iterative method of solution and convergence
is achieved when the difference between two successive values
-12of b IS less than 0.5x10 .nm
The foregoing analysis is for the completely general
case with all the loadings as indicated in figure 1, however,
the computer solutions were restricted to certain specific
cases.
40
Basically, two programs were written to calculate the
deflection coefficients. The first program produces curves
of deflection at the center of the plate, effective width
and bending moment in the y-direction at the center of the
fixed supports versus inplane edge compressive load N* with
Q* as a parameter, N* = and S* fixed. Curves of this
nature, for a range of orthotropic parameters y and ri,
result in a set of design charts. The other program permits
investigation of the behaviour for any set of loading con-
ditions with N* = 0. Deflection along the centerline x =
and total principal stresses along the centerlines x =
and y = are plotted.
41
5. Results
5.1 Design Charts
The design program was used to generate a set of de-
sign charts with N* = S* = 0. Virtual plate aspect ratios
of 1/1.5, 1/2, 1/4 and 1/6 are presented. For each aspect
ratio a complete set of curves for orthotropic material
parameters y = n = 1 • and y = 4.0, n = 0.5 are given.
Description of design charts:
5.1.1 Charts of Deflection
The nondimensional deflection at the center of the
plate, from equation (15) with x = y = 0, is plotted versus
the inplane compressive edge load N* with the lateral load
Q* as a parameter. Figures 2,3,8,9,14,15,20 and 21 give the
deflection.
5.1.2 Charts of Effective Width
The nondimensional effective width given by equation
(36b) is plotted versus the inplane compressive edge load
N* with the lateral load Q* as a parameter. Figures 4,5,10,
11,16,17,22 and 23 give the effective width.
n5.1.3 Charts of Bending Moment
The nondimensional bending moment M* at the middle ofy
the supports y = ± y and x = 0, equation (27) , is plotted
versus the inplane compressive edge load N* with the lateral
load Q* as a parameter. Figures 6,7,12,13,18,19,24 and 25
give the bending moment.
42
5. 2 Behaviour Charts
The behaviour program was used to investigate a few
general sets of loading conditions with N* = 0. Virtual
plate aspect ratios of 1/1.5 and 1/1.25 are presented for
orthotropic material properties y = ri = 1.0.
Description of behaviour charts:
5.2.1 Charts of Deflection
The nondimensional deflection, from equation (15) , along
the centerline of the plate (x = 0) is plotted versus y/b
with either S* or N* as a parameter and Q* fixed. Figures
26,27,36,37,46, and 47 give the deflection.
5.2.2 Charts of Maximum and Minimum Total Principal Stresses
The nondimensional total principal stresses, equation
(37) , along the centerlines on the bottom surface of the
plate (z= —^) are plotted versus y/b (x = 0) and x/a (y = 0)
with either S* or N* as a parameter and Q* fixed. Figures
28-31, 38-41, and 48-51 give the stresses for x = and
Figures 32-35, 42-45, and 52-55 give the stresses for y = 0.
6. Discussion of Results
6.
1
Orthotropic Properties
Reference [10] contains a complete discussion of the
rigidity coefficients D , D and D and compliance coef-^ J- X Y xy ^
ficients J , J and J which are used to express the non-x' y xy ^
dimensional parameters and orthotropic material coeffi-
cients. Possible approximate formulas for their calculation
are given in that reference. The two extreme cases of
orthotropic plate coefficients are the isotropic case where
ri = Y = 1 • sri^ the grid (intersecting beams, no plate)
n = and Y = °°« Most plate-stiffener combinations fall
between ri = y = 1*0 ^rid n = 0.5/ y - 4.0; hence the choice
of parameters for the design charts.
Comparison of Figures 2 to 25 as to the nondimensional
load parameters N* and Q* indicates as p decreases the
assigned values of N* decrease and Q* increase. N* and Q*
are indirect functions of the aspect ratio since they are
nondimensionalized by the plate edges a and b respectively.
The ranges of values used in the design curves were predi-
cated on achieving a certain minimum value of effective
breadth in the calculations.
6.
2
Comparison with Existing Solutions
The first set of design curves to be produced were
those for the case p=Y=n=1.0, Figures 61,62 and 63,
for N* with Q* as a parameter and S* = N* = 0. This choicey ^ x
of parameters permitted comparison with the solution obtainef
44
by Mansour [10] . The number of coefficients retained in
this analysis (only one in the x-direction vice two by
Mansour) does not favor the solution for the square plate
case, however, the results of Figures 61, 62 and 63 agree
to within 3% of the figures 1, 2 and 3 of Mansour [10]
.
The deflection coefficients obtained in reference [10] show
that the one term assumption only neglects terms that are
less than 5 1/2% of the primary deflection term b, ^ which
accounts for the reasonable agreement in this case.
For the loading conditions of the design curves.
Figures 2-2 3, comparison was made with the solution of
Levy [5] . Levy calculated deflection at the center of the
plate and effective width for the isotropic plate of
3=1.5, Q*=S*=0 and the same boundary conditions as
this investigation. Comparison of Levy's results. Figure 3.
and Table 11 of reference [5] , with the curves for Q* =
from Figures 2 and 3 is graphically displayed in Figures 64
and 65. Correlation of the two results is very good,
6.3 Design Charts
6.3.1 Charts of Deflection
The deflection at the center of the plate is always the
maximum deflection since only the first deflection mode is
encountered in these cases. With reference to the curve
Q* = in the deflection charts, the plate remains unde-
flected until the critical compressive load N* is reached.
Increasing N* further buckles the plate and plate deflection
I
45
increases rapidly. Hence, intersection of the curve Q* =
and the N* axis gives the lowest buckling load. The non-
linearity of the curves is evident. In the small deflection
w 1range, r- < -x-, a doubling of the lateral load Q* will double
the deflection, however, in the large deflection range the
lateral load effect becomes nonlinear with decreasing incre-
ments of plate deflection for increasing additions of load.
As p goes from 1/1.5 to 1/6 the magnitude of the maxi-
mum deflection of the plate is considerably reduced. The
plate becomes more of a beam supported all around and hence
"stiffer" than the slightly rectangular plate. The effect
on the deflection of increasing Q* as p -^ 1/6 becomes less.
Comparison of buckling loads and the range of N* values
over the four aspect ratios shows that for p = 1/1.5 the
maximum value of N* considered is four times the bucklingX ^
load while it is less than twice the buckling load for
p = 1/6. The reason for this selection of N* values is^ X
that the range where the solution is reliable decreases as
p -> 1/6. Convergence to the correct solution becomes in-
creasingly more difficult. In addition, the 8 term solution
begins to become insufficient for p = 1/6 where the eight
term, t),«, has increased to a value almost 6% of the primary
term, b-.^, for the highest set of N* and Q* considered.
6.3.2 Charts of Effective Width
By definition, the plate is fully effective in carrying
the external compressive edge load until it buckles. With
46
reference to the effective width figures, b /b = 1 for Q'^ =
until the buckling load is reached. Extensions of the
curves for Q* 7^ at values of N* less than the buckling
load are meaningless and should be ignored. After buckling,
the effectiveness drops off as the plate deflects. Compeiri-
son of different aspect ratios shows that the effective
width decreases more rapidly and takes on significantly
lower values a p -> 1/6.
6.3.3 Charts of Bending Moment
The bending moment M* at the middle of the fixed sup-
ports y = ±-J
is calculated because it is expected to be
large. M* is equal to M* at these points since the curva-
ture along these supports in the x-direction is zero (M* =0)
,
^ohence equation (28) is plotted. With reference to the
bending moment figures, the bending moment is zero until
the plate deflects. Increasing Q* increases the bending
moment in all cases with the nonlinearity associated with
large deflections.
6.4 Behaviour Charts
Unfortunately, there are no existing solutions to which
the results of shear loading may be compared for this par-
ticular set of boundary conditions. Levy [7,8] and Payer
[11] considered the simply supported isotropic case. How-
ever, the results obtained are consistent with those antici-
pated in the magnitude and shape of the center line
47
deflection curve and values of principal stresses.
Results are given only for p = 1/1.5 and 1/1.25.
Investigation of the solution revealed that for virtual
aspect ratios greater than about 1/1.75, the plate will have
one symmetric buckle about the diagonal while for smallei*
values of p antisyiranetric buckling shapes are experienced.
The choice of one term in the x-direction limits the solution
to some degree in the former case*, however, it cannot begin
to approximate the true solution in the later case. Hence,
the narrow range of investigation for p.
Additionally, with reference to the nondimensionalizing
of the total stresses, sections 3.3 and 3.5, to present the
results in a concise and meaningful form, the solution is
restricted to the isotropic case where y = t\ = 1.0 . This is
a result of the isotropic aspect ratio 3 appearing expli-
citly in the nondimensional equations. Nondimensionalizing
the bending and membrane stresses to eliminate 3 will result
in a set of incompatable nondimensional stresses which must
be plotted and evaluated separately, thus increasing the
number of charts required for each specific case.
Initially, the total principal stresses at both the top
and bottom {z = ± -y- respectively) of the plate were investi-
gated. Figures 56-60 show the results for p = 0.667, y = n =
1.0, S* = 14 (above critical load) and N* = Q* = 0. Using
the Von Mises yield criteria for the biaxial case, a = 0,
*Note: The assumption of the one team solution in the x-direction will model a symmetric buckle about the platecenterline, not the diagonal.
48
(a*) 2 = (a*) 2 + (a*) 2 - a* a* (40)
a combined stress may be calculated to evaluate the relative
importance of the total stress field at the top or bottom
of the plate. With reference to Figures 57-60, the table
of (Oq)*^ is as follows:
y=x=0 y= ± ~r x=0 x= ± ^, y=0
Top 12.4 17.7 10.8
Bottom 12.2 19.5 10.8
The combined stresses are slightly greater at the top
than at the bottom for the center of the plate, however, the
magnitude and difference is considerably greater at the
fixed supports with the bottom stresses being of higher
value. By the nature of the upwards (+z) buckling mode,
this is the expected result. Due to the higher combined
stress at y = ± ^, the behaviour charts were plotted only
for the principal stresses at the bottom of the plate
(z = - 2^ •
Figures 26-45 show the effect of increasing shear loads
above the critical load for p = 1/1.5 and 1/1.25. Inplane
edge loads N* = N* = and cases are presented for both Q* =
and a finite value. Increasing S* and/or Q* increases the
deflection and total principal stresses throughout the plate.
Figures 46-55 show the effects of completely general
sets of loading conditions with increasing inplane edge
compressive loads N* above the critical load. Comparison
of the charts for the sub-critical (S* = 6) and super-critical
49
(S* = 14) shear loads clearly indicate the influence of
shear on the shape of the deflection curve and values of
principal stress. Increased shear increases deflection and
principal stresses. In addition,, the highest maximujn
principal stresses are observed to occur at the fixed
support, y = ±-J,
6. 5 Examples Demonstrating Use of the Charts
The following examples are given to demonstrate the use
of the design and behaviour charts. The approximate formu-
las for calculation of the rigidity and compliance coeffi-
cients are discussed in detail in reference [10] . The
formulas are listed in Table 2.
6.5.1 Design Example
Consider the following orthotropic characteristics and
nondimensional loads for the stiffened plate:
p = 0.5, Y = 4.0, n = 0.5, N* = 4.0, Q* = S* =
From Figure 9
TT^D
The critical load N = 1.8X 2c a""
The center of the plate deflection
w = 1.64h
where h is the average stiffener's depth plus the
plate thickness.
From Figure 11
The effective width b = 0.505be
50
Edge membrane stress (equatilons 24,34b and 35b) is
X b^' T^/T 2. 0.505 10.9e e 12(l-v )
^ . 0.73 :!I-^
^e a^Jy
From Figure 13
The bending moment at the middle of the fixed support
M* = -102y
The maximum bending stress at this point (equation 29)
is6M* hD D
a, = ± -^ . -^ =q: 612 -^
^y h^ b^ hb^
6.5.2 Behaviour Example
Consider the following orthotropic characteristics and
nondimensional loads for an isotropic plate:
p = 0.667, Y = n = 1.0, N* = 3.0, Q* = 8.0, S* = 6.0
From Figure 4 6
The maximum deflection at the center of the plate w =
1.74 h
From Figures 48 and 50
The maximum principal stresses occur at the fixed sup-
ports Y = ± J, X = and are
a, = 6.501 ^2b^J
X
1 oifi'^ha^ =-1.542
b'^Jx
51
and the combined stress
^0 = ^1^ + ^9 - ^1^9 = 54.6 [^^-^]2 ^ ^2 ^ ^2 _ ^ ^ _ rx ^ ^^ iL_] 21^2
X
52
7 . Conclusions and Recommendations
The problem of a large aspect ratio orthotropic plate
with the short edges fixed and long edges simply supported
has been modeled for the case of inplane edge compressive
and lateral loads and subcritical edge shear loads. The
design charts provide valuable information of use in estab-
lishing design criteria for the plate-stiffener combination.
From the behaviour Figures 46 to 55, it is evident that
the highest principal stresses occur at the fixed supports
y = ± 2-. The design Figures 2 to 25 do not present informa-
tion to permit the calculation of the membrane stresses at
the fixed supports. Design charts of a / edge membrane^e
stress in the y-direction (denominator of equation (36a) )
,
should be produced.
As noted in the results, the one term assumption in the
x-direction will not model the deflection of a plate with
edge shear as the predominate loading. As the major point
of this investigation was to model a large aspect ratio
plate, the solution chosen was an inevitable consequence
when the large range of p is considered. It is recommended
that to model a large aspect ratio plate subjected to shear
loading, close attention must be paid to the important
deflection terms that should be included in the solution.
Levy [8] confined himself to a B = 1 . 5 and used 14 selected
terms in the deflection equation.
53
Additionally, the eight term solution is just suffi-
cient to model the case for the orthotropic plate aspect
ratio of 1/6.
54
8. References
1. Bleich, F. and L.B. Ramsey, Buckling Strength of MetalStructures , McGraw-Hill, New York, 1952.
2. Bleich, F. and L.B. Ramsey, A Design Manual on theBuckling Strength of Metal Structures , Technicaland Research Bulletin No. 2-2, SNAME, May 197 0.
3. Coan, J.M. , "Large-Deflection Theory for Plates WithSmall Initial Curvature Loaded in Edge Com-pression," Journal of Applied Mechanics, June1951, p. 143.
4. Levy, Samuel, "Bending of Rectangular Plates with LargeDeflections," NACA Report Number 737, 1942.
5. Levy, S. and P. Krupen, "Large-Deflection Theory forEnd Compression of Long Rectangular PlatesRigidly Clamped Along Two Edges," NACA TechnicalNote No. 884, January 1943.
6. Levy, S., D. Goldenberg and G. Zibritosky, "SimplySupported Long Rectangular Plate Under CombinedAxial Load and Normal Pressure," NACA TechnicalNote No. 949, October 1944.
7. Levy, S., K. Fienup and R. Woolley, "Analysis of SquareShear Web Above Buckling Load," NACA TechnicalNote No. 962, February 1945.
8. Levy, S., R. Woolley and J. Cornick, "Analysis of DeepRectangular Shear Web Above Buckling Load,"NACA Technical Note No. 1009, March 1946.
9. Mansour, Alaa, "On the Nonlinear Theory of OrthotropicPlates," Journal of Ship Research, December1971, p. 266.
10. Mansour, Alaa, "Post-buckling Behavior of StiffenedPlates with Small Initial Curvature Under Com-bined Loads," International Shipbuilding Progress,June 1971 and Dept. of NAME, MIT, Report No.70-18, Oct. 1970.
11. Payer, Hans Georg, "Notes on the Buckling and Post-buckling Behavior of Deep Web Frames," MarineTechnology, July 1972, p. 302.
12. Schade, Henry A., "Bending Theory of Ship BottomStructures," Trans. SNAME, vol. 46, 1938, p. 176.
55
13. Shade, Henry A., "The Orthogonally Stiffened PlateUnder Uniform Lateral Load," Journal of AppliedMechanics, Dec. 1940, p. A-143.
14. Schade, Henry A., "The Effective Breadth of StiffenedPlating Under Bending Loads," SNAME, vol. 59,1951, p. 403.
15. Schultz, H.-G., "Post-buckling Behavior of Wide ShipPlating," Report No. NA-64-3, Department ofNaval Architecture, University of Californiaat Berkeley, February 1964.
16. Timoshenko, S. and S. Woinowsky-Krieger , Theory ofPlates and Shells , McGraw-Hill, New York, 1959.
17. Yamaki, Naboru, "Postbuckling Behavior of RectangularPlates with Small Initial Curvature Loaded inEdge Compression," Journal of Applied Mechanics,September 1959, p. 407.
18. Kantorovich, L.V. and V.I. Krylov, Approximate Methodsof Higher Analysis , P. Noordhoff LTD, Holland,1964.
56
TABLE 1 - COEFFICIENTS Cpq
C = Y + (-D^Xpq q q
p = 0,1
q = 0,1,2. . .16
where X and Y are given byq q ^ -^
X^ = b?T +4b?^ + 9b2 +16b2 + 25b^2p + 3 6b,2^ + 49b2 + 64b2o 11 12 13 14 15 16 17 18
Y = -Xo o
X, = 4bTTbT^ + 12b, ^b,, + 24b,^b, .+ 40bT.b, £.+ 60b, ^.b^^1 11 12 12 13 13 14 14 15 15 16
-"bi2bi3 - 25b^3b^4 - 41b^^b^5 - eib^^b^g - 85b^gb^, - llSb^^b^^g
^2 =-"^il
"^'^ll'^lS
-^ "''12''14 + 30bj_3b^5 + ^Sb^^bj^g
+ 70b^5b^,+ 96b^gb^g
Y^ = Sb^^'-l'll^ "^12 - "=13 - ^14 - ^15 + "^le - \l + \b^
-^il -"''ll'=13
- 34bi3b^5 - 52b^4b^5- 74b^5b^,
- "°'>16''l8
^3 = -*^l''l2" ^\l\4 -^ 20b^2''l5 * 36b^3b^g + 56bj^^b^^
+ BOb^jbj^g
57
Y3 = l«b^3'-hl ^ "^la- ^3 +^4 -^5 - ?16 -"=17 ^^s'
+ 64b^^b^g
Y, = ^''•^14^-^11 "- ^12 -^13 •" ^14 - ^15 -^ ^16 - ^17 " 2^18^
-lOb^TbT- - 26bTTbTt. - Ab^ ^ - 40bT^b, ^ - SSb^-bT-11 13 11 15 12 12 16 13 17
Y5 = 50b^5(-b3^^ + b^2 - t-is + 1^14 - "15 ^''le
- '^17 * \s'>
X, = -10b, ,b,, + 14b, ,b,, - 16b, ^b,. + 32b,,b,„ - 9b11 15 11 17 12 14 12 18 '13
Yg = 72b^g(-b^^ + b^2 - "=13 ^ \4 - hs ^ ^6 - '^17 " "^is'
h = -l^b^^b^g H- 16b^^b^g - 20bj^2'=^5 - 24b^3b^^
^7 = SS''17(-^1 -^ h2 - ^3 ^ h4 - ^5 ^ ^6 " ''n "^'='lB>
-37b^jLb^g - 65b^^b^3 - 29b^2''l5 " 25bi3b^4
Xg = -14b^^b^, - 24b^2b^g - 30b^3b^5 - ISb^^
58
Yg = 128b^g(-b^^ + b^2- - ^3 " ^4 - ^5 ^ ^6 " ^7 "^ ^8^
-50b, ,b,_ - 40b, ^bT^ - 34bT.b,c.- 16b,^.11 17 12 16 13 15 14
Xj =-16b,^b^g - 28b^2b^^ - 36b,3b^g - 40b^,b^5
Yg =-65bj^^b^g - 53b^2'=^^ - 45b3_3b^g - 41b3^^b^5
^10 =-««h2^8 - ^^\3\l - 52bi4'=16 - "bJs
^11 =-^2'^i3''i8 - ^^^u^n - ^o'^is^ie
^11 =-"'"l3''l8 - "'"I4''l7 - "'^is'^ie
X^2 =-64bi4h8 - ^°h5^7 " ^Sb^^
Y^2 =-80b^4b,g - 74b,5b,, - 36bjg
X^3 =-B0b,5b^g - 84b^gb^^
"13 =-8«'=i5'=i8 - ^^he^n
^14 -^^^e'^is - ""^It
"l4 =-"°^6'^18 - "l^iv
^^15 ° ^^^''l7''l8
"l5 = -113^7^8
59
^16 = '^'Ks
^16 " ^16
60
TABLE 2 - RIGIDITY AND COMPLIANCE COEFFICIENTS
EIX
XX
EID
y
p=^
ab
1» I s '
_Z XI s
\ X y
n = xy
JD D . X yX y ^ ^
1 1px pyI I
X Eh
J -y Eh
xy1 , 1+vE ^ h
V.- -]
P h
Y = xy(1 + V) ^
h hX y - V-
h hX y
J JX y
h =2h h
X yh + hX y
61
TABLE 3 - INTEGRALS
b/2 a/2
/ / COS—X COS—X axdy = -rr m = r^ a ^ 8
= m 7^ r
b/2 a/2 -
/ ; COS—^x COS—^x cos-^r—-y dxdy =^ ^ b ^ ^
b/2 a/2r r IHTT rTT 2n7r - , ./ / COS—^x cos—^x cos , y dxdy =
^ ^ b ^^
b/2 a/2 ^ _
/ ; cos—^x cos—X cos-T—-y cos-^r—-y dxdy =v-7rj m=r, n=sQ Q
a a b-^ b-^-'ie '
= mf^r or Ht^s
b/2 a/2 ^ ^r f n\7T 2piT rTT 2q7T _, , -.
/ / cos—X COS—^-—
X
cos—X COS—c^y dxdy =^/^ a a a b
b/2 a/2 ^ ^ ^/ / COS—^x cos—7—y dxdy =
^ ^
b/2 a/2^^ ^j^ ^
/ / COS—X dxdy = ;^---(-l) r odd^ ^ 2r7T
= otherwise
/ / cos—^x COS—£--x COS—^x COS—g—y cos-r-—y dxdy^ ^ ^ b ^ b ^
^
ab , ^= -^i m+r = 2p, q = s
= Yo'l^^-^i ~ 2pf q=s, m > r
ab 1I ^= •jjM^"!^! = 2p, q=s, r > m
= otherwise
62
b/2 a/2 ^^ 2p7T . rir 2nTr 2qTr , ,
/ / cos—X cos X cos—X cos-4r-Y cos—5—y dxdy
ab, ^= j2 ' in + r = 2p,q = n
= jy; [m - r| = 2p, q = n, m > r
abI I ^= jy^ [r - m] = 2p, q = n, r > m
= otherwise
b/2 a/2 - o o/ / sm—^x sm-^-—X cos—^x sm-^r-^y sm-^y dxdy
^ ^ ^ ^ ^
ab, ^= -s-T ' m + r=2p,n = q
abI I o= r^ > |m - r[ = 2p, n=q, m > r
= -T?' I
•'^ - m] = 2p, n=q, r > m
= otherwise
, , ITITT 2p7T riT 2n7T 2q7T 2stt -3 -,
/ / cos—^x COS X COS—X COS—r-'-y COS—c-^y cos ,-y dxdy
r. r. a a a b. b-' b-' ^
ab64
; m + r = 2p, n + s = q
= -^
I
m + r = 2p,|n - s|= q, n > s
= -^
)
m + r = 2p, I
s
- n|= q, s > n
= -^) |m - r|= 2p, n + s = q, in>r
= -^J |m - r|= 2p,|n - s|= q, m>r, n>s
= -^t |m - r|= 2p,|s - n|= q, m>r, s>n
= -^1 |r - in|= 2p, n + s = q, r>m
63
ab•64 ) [r - in[= 2p,|n - s = q, r>m, n>s
~i |r - m|= 2p, |s - n = q, r>m, s>n
= otherwise
b/2 a/2
/ /. mTT . 2p7T riT . 2n7T . 2qTr 2stt j ,sm—^x sin--E—-X cos—^x s.in—?—-Y sm—^y cos-t-—y dxdy
a a a b-^ b-' b-'-'
ab,
«= -^ ) m + r = 2p, n + s = q
64' m + r = 2p, n - s = q/ n > s
ab-64' m + r = 2p, s - n = qf s > n
ab~ 64' m - r = 2p, n + s = q/ m > r
ab" 64' m - r = 2p, n - s = q. in>r, n>s
ab="64 ' m - r = 2p, s - n = q/ m>r, s>n
ab-64' r - m = 2p, n + s = q/ r > m
ab=~64' r - m = 2p, n - s = q/ r>m, n>s
ab" 64' r - m = 2p, s - n = q/ r>m, s>n
= otherwi.se
b/2 a/2
/ / sin—^x cos—X sm—7—y dxdy
niT
m+r
m-r
niT
=
2 m-r
odd , n odd
odd, n odd
otherwise
64
, ^ ^' '. infT rir .. 2nTT 2s7t _ ,
/ / sm—X cos—X sm—r—-y cos-r—y dxdya a D D
ab ^ 1 , , 1 . 1 . m+r ,, . ,j= (-T—) (—r— + ) » -TT- odd, n+s odd
o 2 m+r n+s n-s '' 2 '
27T
ab , 1 , , 1 , 1 V m-r ,, , ,
,
=( ) (—;— + ) . —TT— odd, n+s odd
o„2 m-r n+s n-s' '' 2 '
27r
= otherwise
65
Table 4' - Computer Symbols
Fortran Symbol Variable
B(N) b^ ^l,n
RHO p
GAMMA Y
ETA n
QSTAR Q *
NXSTAR N*
NYSTAR N*
SSTAR S*
W/H w/h
AE/A a^/a
BE/B b^/b
MYSTAR M*
SIGMA STAR a* or a*
CPQ Cpq
APQ A^^pq
RANG Max. value of N* or S*
RANGQ Max. value of Q*
66
A^
-9kk *-
i r t t i t I *
NNyx
-^ X
Figure 1
67
wh
J . ^ —
^^2.4 - J^^
/^^°*1.6 - yY//
y^X / / P = 0.667
y^y / / Y = 1.0
0.8 - y^ / / n = 1.0
y^ 1 Q* = 0,4,8,12
/\J m \J ""
1 1 1 1
0.0 3.0 6.0 9.0 12.0
wh
0.0 3.0 6.0
N*X
9.0 12.0
Figures 2 & 3
68
1.00
b
b
0.85 -
0.70 -
0.55 -
0.40
2.0
1.00
0.85
0.70 -
0.55 -
0.40
2.0
4.5
4.5
7.0 .5
7.0
N*X
12.0
9.5 12.0
Figures 4 & 5
6920 -,
-40 -
M*y
-100 -
-160 -
-220
-50 -
M*y
-110 -
-170 -
-230
0.0 3.0
p = 0..667
y = 1.
n = 1..0
Q* = 0, 4,8, 12
12.0
6.0
N*X
9.0 12.0
Figures 6 & 7
70
wh
J . z -
•
2.4 - ^y^/^^^Q*
1.6 - y^A/yy// P " °-^°
yyyy y = i-o
y//// n = 1.0
0.8 - yy^/ / Q* = 0,5,10,15
(^ — /tillJ » u —
2.0
wh
0.0 2.0
4.0 6.0 8.0
i,A -
^
2.4 -
/^^^Q*
^
1.6 -
y/y/ y = 4.0
y\/// n = 1.0
y^X// Q* = 0,5,10,150.8 -
0.0 ~1 1 1 1
4.0
N*X
6.0 8.0
Figures 8 & 9
71
beb
beb
1.02 -I
0.34 -
0.66 -
0.48 -
0.66 -
0.48 -
0.30
1.0 2.8
8.2
4.6
N*X
6.4 8.2
Figures 10 & 11
72
M*y
^\J —
P = n. RO
^^^"^^^\ \ Y = 1.0
^^^^i;;\^\\. n = 1.0
-60 - ^\!;XN\v Q* = 0,5,10,15
-140 -
\^X\/Q*
-220 _ ^\\
-300 - ^1 1 1 1
3.0
-70 _
M*y
-230
-150 _
-310 -|
8.0
N* Figures 12 & 13
73
1.60 -1
h
wh
1.20 -
0.80 -
0.40 -
0.45 -
0.00
0.0 0.6 1.2 1.8
N*X Figures 14 & 15
74
eb
beb
1.02 n
0.84 -
0.66 -
0.48 -
0.30
1.00
0.80 -
0.60 -
0.40 -
0.20 -t
0.9 1.3
2.5
T 1
1.7
N*
2.1 2.5
Figures 16 & 17
i
20 -I
-70 -
M*y
-160 -
-250 -
-340
80
75
0.0 0.6 1.2 1.8 2.4
-40 -
M*y
-160 -
-280 -
-400
0.0 0.6 1.2
N*X
1.8 2.4
Figures 18 & 19
76
1.60 -1
wh
wh
1.20 -
0.30
0.45
0.00
0.0 0.5 1.0
N*X
1.5 2.0
Figures 20 & 21
1
77
1.00
eb
0.80 -
0.60
0.40 -
0.20
1.00
1.00
0.80 -
0.60 -
0.40
0.20
1.00
1.25
1.25
0.167
1.0
1.0
0,50,100,150
1.50 1.75
0.167
4.0
0.5
0,50,100, 150
2.00
1.50
N*X
1.75 2.00
Figures 22 & 23
78
y
M*y
J u w —
50 -
200 -
p = 0.167
\\\/^*
450 - Y = 4.0
n = 1.0
Q* = 0,50,100,150 .
7nn — ^1 \j\) ""
1 1 1 I
0.0 0.5 1.0
N*X
1.5 2.0
Figures 24 & 25
79
p = 0.667 Y = n = 1.0 N * =
Y/B
Figures 26 & 27
QI
p = 0.667 y = n = 1.0 N* =
80
Q* =
S* = 13,14,16,20
Figures 28 & 29
81
p = 0.667 Y = n = 1.0 N* =
Y/B
Figures 30 & 31
82
p = 0.667 Y = n = 1.0 N * -
O —r-
if)
CE
cn
50
1
or.
CEt—
cr
I—
(
en
rv
Q* = 8
S* = 13,14,16,20
I-
Figures 32 & 33
83
p = 0.667 Y = n = 1.0 N* =
OJ
if)
crIT)
CO
en _.
Q* =
S* = 13,14,16,20
oI
rvj
OJ
01cr(—in
O".
z:
<n
LO
-.33 ,15 .18
X/R.35
:j
Q* = 8
S* = 13,14,16,20
-.bU1^.33
"T-
X/R.52
Figures 34 & 35
84
p = 0.80 Y = n = 1.0 N* =
C3
ino
o
in
to
OJ
3
-.bO
Figures 36 & 37
p = 0.80 Y = n = 1.0 N* =
85
Q* =
S* = 12,12.5,14,16
r/B
Figures 38 & 39
86
p = 0.80 Y = n = 1.0 N* =
Figures 40 & 41
87
p = 0.80 Y = n = 1.0 N* =
o -,
Q* =
S* = 12,12.5,14,16
7- .50
Figures 42 & 43
88
p = 0.80 Y = n = 1.0 N* =
o
U">
ccacI—en
cr
sI i
CD
Q*
S*
=
= 12,12.5,14,16
o
p
u")
.bD
CCCE}~in
crz:
I—
<
CD
33 -.151^
.01 .1! .52
X/R
Q* = 4
S* = 12,12.5,14,16
.0!
X/R.18 .35 .52
Figures 44 & 45
89
p = 0.667 Y = n 1.0 N* =y
(M
u->
'-.bO
DJ
CO
r(j
'-.50
Y/B
Figures 46 & 47
I
i
90
p = 0.667 Y n = 1.0 N* =y
Figures 48 & 49
91
p = 0.667 Y = ri = 1.0 N* =y
I -.50
Y/B
Figures 50 & 51
92
p = 0.667 Y = n = 1.0 N*y
X/fi
Figures 52 & 53
1
93
0.667
fM
x/n
CO
COcr.
if)
crz:
1—
I
in
nj
s*
Q*
N*X
14
8
1,2,3,4
qI 1.50 -.33 -.16 .01
X/R.35
Figures 54 & 55
1
i
1
94
p = 0.667 Y = n = 1.0 Q* = N* =
S* = 14
Y/B
Figure 56
95
p = 0.667 Y=n=1.0 Q*=N*=0
r/B
Figures 57 & 51
96
p = 0.667 Y = n = 1.0 Q* = N* =
Figures 59 & 60
97
wh
M*y
J . u
2.7 - ^^1.8 - y;^0.9
1
-^^.^ y^ / Q* = 0,2,4,6
0.0 /1 1 1 1
-80
0.0
-20 -
-40 -
-60
5.0 10.0 15.0 20.0
Y = n = 1.0
0,2,4,6
10.0
N*y
15.0 20.0
Figures 61 & 62
I
98
1.02 -I
ea
0.89 -
0.76 -
0.63 -
0.50
4.0 8.0
p==Y = n = i.o
Q* == 0,2,4,6
12.0 16.0 20.0
N*y
Figure 63
I
99
Q)
U
•HPL4
^1X5
100
•5C X U
Cn•HP4
0)
J3
101
9 . Appendices
102
APPENDIX A
Details of Derivation of Basic Equations of
Large Deflections
i
103
Definition of External Forces and Moments
Shearing force per unit length parallel to x axis
h/2
Q = / T dz^ -h/2
^^
Shearing force per unit length parallel to y axis
h/2
Q = / T dzy -h/2 i'^
Bending moments per unit length of sections of a plate per-
pendicular to X and y axis respectively
h/2M = / zcr dz^
-h/2 ^
h/2M = / za dzy -h/2 y
Twisting moment per unit length of section of a plate per-
pendicular to X and y axis respectively
h/2M = -/ ZT dz
h/2M = -/ ZT dzyx
.h/2 y^
104
Strain - Displacement Relations
Consideration of strain in the middle of the plate
dx-
u
u + —-dxdX
For large deflections, the effect of deflection on the
strains has to be included in the strain-displacement re-
lations. The accompanying figure shows a typical elongation
of a linear element of length dx due to displacement in the
X and z directions from bending.
Elongation = dl - 7r-<3x + -x-Cr— ) ^dx^ 8x 2 9x
The strain in the x direction is therefore
^x ~ 9x 2^dx'(a)
The strain in the y direction is likewise
^y ~8y 2^9y^ (b)
The shearing strain due to displacements u and v is
8u^
dvdy 9x
It can be shown that the shearing strain due to displacement
w is
1
i
105
9w ^ 9v£
The total shear strain in the middle plane is
"Yxy ay 8x 8x * 3y
For small angles the displacements u and v can be represented
by the change in displacement w as
u ~ - z -—9x
V ~ - z ^^—ay
Substitution into (a) , (b) and (c) gives
e3^2 2 ax
+ ^(ffr)' (2r
E„ =^ - 2
9y
9^w_^ i(9W) 2
y .„2 2 ay'
-9^ 9^^ + 9^ 9W'xy axay ax ay
*Numbered equations correspond to those used in the maintext.
Equilibrium of Forces and Moments
106
1. Lateral load q only
yx 9y -^
-K
*. M +8Mxy
xy 9xdx
M +—idyy ay
^
X
^Q + -^xX 9x
Q + —^<ayy ay
^
The above diagrams are superimposed to make one loading
condition. For equilibrium in the z-direction:
107
IF =z
8Q 3QX V ""
-dxdy + -5-^ydx + q dxdy =3x ^ 3y
9Q 8Q+ -^ = - q (a)8x 9y
Moment equilibrium about the x-axis
EM =0XX
8M 8M 3Q-^^^xdy 9^y^^ + Q dxdy + -^^^ydxdy + qdxdyO (dy) =
Neglecting the last tv7o terms since they are of higher order
gives
8M 3M_^EZ _ .^ + Q =0 (b)3x 9y y
Moment equilibrium about the y-axis and similarly neglecting
small terms
EM =0yy
9M 9My^ ^ _^ - Q =9y 9x ^x
2. Forces in the middle plane of the plate (Membrane Stresses)
The forces acting in the middle of the plate may have a
considerable effect on the bending of the plate and must be
considered.
108
I*—dx—
^
9N*^ N + -^x
X dX
X
N —X
Nxy
NN
^"xy '~TxN +-^r^^X
8NX* N + -T-^x
X 8x
8N-N ._. + -T^^yyx 8y
t 8N
y dy ^
Assuming no body forces or tangential forces
ZF =X
^Asxdy + -g^ydx =
ZF =y
8N 9N-r-^ydx + ^^^^dxdy =dy dx
and from symmetry noting N = N"^ ^ xy yx
109
8N 8nX
+ -^ =ax 9y
_Ji + -ii:^ =8y 8x
Equations (3) are independent of equations (a) , (b) and (c)
and can be treated separately.
Projecting normal forces N onto the z axis and taking
into account the bending of the plate and resulting small
angles
-N dy|^ + (N + -2idx) (|^ + A!w^x)dyX -^ 8x x 8x 8x ^2
dX
Similarly projecting normal forces N onto the z axis
-N dx|^ + (N + ^y) i^ + 9i^y)dxy ay y ay ^ ay ^^2
Neglecting higher than second order terms gives
Si 2 aN .iv-T
a w, , , X aw, , ,,.
^xTl^^^^ -^ -aF • 33?^^^^ ^^^oX
^2 aN ^XT a w , J , y aw, J , .N, dydx + -^ • TT—dydx (e)y9y2 ^ ay ay ^
Treating the projections of the shearing forces onto the
z axis the same way
(1) for N with slope of deflection surface in the yxy ^
direction on the two opposite sides of the element as
aw -1 aw, a^w ,
ir— and "5— + Tc—;^— dxay ay axay
i
110
xy9x8y -^ dx dy -^
(2) for N with slope of deflection surface in the xyx ^
direction on the two opposite sides of the element as
9w J 9w , S^w ,
ri 9^ ^ ^ j^ yx 3W, ,
Va^s^^^"" " "ly • ^^^"^
combining (1) and (2) gives
^2 9N . 9N -
01.T 9 w n -, . xy 9w, J , xy 9w, , ,jr»2N -r——-dxdy + —r—^ -r—dxdy + —r—^ tt—dxdy (f)xy8x9y -^ 9x 8y -* 3y 9x -^
Differentiation of (b) with respect to y and of (c) with
respect to x and substitution into (a) eliminates shearing
forces Q and QX y
a^M a^M a^M 32mX
_^yx ^ y xy = _ -
3^2 9x8y 3^a 9x9y
Noting that M = - M , adding expressions (d) , (e) and (f
)
to the load qdxdy originally defined and making use of (3)
S^M g^M a^MX _ 2
xy ^ ^ y^
9x^ ^^^^ 3y2
= -(5 + N ^ + 2N 1^ + N li^) (4)
^3x^xyaxsy y^^^
iI
Ill
Moment - Curvature Relations
Substitution of equations (2) for e , e and y in equations
CD gives*
p
^ ^"^xV 3x^ 2 3x Ygyz 2 y 3y
y r 9^W,1,^W^2 9^W
,1 ,9V7. 2t / v
^rr= "TilTrTr-I'Z + o-C-^;) - zv + o-v (^^] (a)
Y 1-V V r. 2 ^ oV X^ 2 ^ X dX^ X y 9y ^ dx
^ = -2zg|-4- + G |^.|^xy 9x3y 9x 9y
Taking the moments as defined before and integrating
h/2M = / ZQ dz^ -h/2 ^
h/2M = / za dzy -h/2 y
h/2M, ,^ = -/ ZT dz^y
-h/2 ^y
Even functions will drop out over the integration interval
leaving the same expressions for moments M , M and M as^ ^ X y xy
for a plate undergoing pure bending and only small deflec-
tions. The result is
*Numbers in parenthesis correspond to equations in the maintext.
ii
I
112
M = -D (^ + V ^) (6)
M = 2cr^xy 8x3y
where the rigicity coefficients D and D are defined as^ -^ X y
E h3D EX - 12(l-v V )X y
E h3D - ^-~
and
y - 12(l-v V )jt""x y
CZ ^^'12
"Equilibrium" Equation
Substitution of equations (6) into (4) gives
3""+ 2D 3"" + D 3'"
^ax* ''^ix^n^ ^ay"
= 5 + N 2i^ + 2N i^ + N 2iwXj^, xyaxay y^ya
where
2D =Dv +Dv +4Cxy X y y X
113
Substitution of the stress function defined by equations (5)
D i!^ + 2D -i-^ + D ^
-, d^F d^\7 ^d^F d^\7 ^ d^F a^w ,_,n + • — 2 • + • i7 J
"Compatibility" Equation
The forces N , N , and N in the middle plane of theX y xy ^
plate depend on the strain due to bending as well as the
external forces applied in the xy plane.
Again assuming no body forces and requiring load q is
perpendicular, equations (3) apply for equilibrium in the
middle xy plane
9N 8N
ax dy
(3)
-y-+ _^=Sy 8x
The corresponding strain components are those of equations (2)
X 3x 2^9x^
_ 8v . 1 ,8w.
2
/o^^y
-87
" 2^87^ ^2)
^xy By 8x 8x * 8y
114
Differentiating equations (2) and combining to eliminate
u and V results in
Using Hooke's Law to relate strain and forces N
N Nx_ _ y_
^x hE ^y hEX -^ y
N N
S hE X hE ^^'•^ y X
NY = -^^'xy hG
Differentiation of equations (a) and substitution into
equation (8) with use of equation (3)
3^N 9^N a^N- ^ + 2J . ^y + J ^X3^2 xy 8x3y y ^^2
29 w X 2 9 w 9^w
where
^^^y 9x^ 9y^
T - 1 T - 1
and
x E,,h ' y E h
xy Gh X y y x
Substitution of the stress function defined by equations (5)
115
APPENDIX B
EFFECTIVE WIDTH
f
Effective Width
116
t^
-•
: r
^
1
f? < -
-c yj
max
Figure a Figure b
Consider the simple case of a rectangular plate simply
supported on all edges where the loaded edge remains straight
in the plane of the plate at all times. If the load is below
the buckling load, the stresses will be distributed evenly
as shown in figure a. For post-buckling loads the center
of the plate will exhibit less compressive strain than the
edges because of large deflections of the center of the
plate. The stress distribution is shown in figure b. The
effective width relates the maximum stress o uniformlymax
distributed along a phantom plate sustaining the same total
load as the real plate, thicknesses being the same.
Therefore
a a = aae max
Where o is the average edge stress. For this particular case
117
h- ^ and
aj^ax - Oy^
Where a is the edge membrane stress in the y-directionye ^ -^
(a at X = ± -^r) . Substitution into the above relation givesy 2 ^
the result
Na a = a -T^ (34a)e ye h
Similarly, for loads in the x-direction
Nho = h — (34b)e xe h
ti
118
APPENDIX C
Details of Solution
1
119
1. Determination of Coefficients Cpq
Substitute deflection expression (37) into the right
hand side of equation (16) . The right hand side of equation
(16) becomes*
Term #1
= hljLL{^sin'^x[b,sin^Y + 2b^sin4?y + 3b,sin^y2V.2 al b-^ 2 b-^ 3 b-^
a'^b
+ 4b.sin-T-y + Sbr-sm-r—-Y + 6b-.sin-v—
y
4 b-' 5 b-^ 6 b"^
. -ji_ • 1417 , „, . 16Tr ^^
}+ 7b^sin^-y + 8bgSin-^y]
Term #2
- 14cos —xlb, + b,cos-r—y - b^ + b^cos-r—y + b-2, 2 al 1 b-' 2 2 b'' 3
a b
+ b-cos-T—y - b- + b.cos-r—y + b,. + bcCos—r—y - b^3 b-^ 4 4 b-' 5 5 b"* 6
. ,_ 12iT , , , 1471 , . , 1611 , r, 217+ bgcos-^y + b_ + b-cos-^y - bg + bgCos-^yj • [b^cos-^y
, A-L. 47T , rt, 67r , , ^, Sir , ^_, IOtt+4b^cos-T-y + 9b-cos-T—y + 16b.cos-^y + 25b_cos-^—
y
+36bgCos^y + 49b^cos^y + 64bgCos^y] } (a)
*Note: since m = 1 for all cases, for convenience b, isrepresented by b .
'
120
Each of the two terms of equation (a) is multiplied out and
like terms are collected in the form cos ^ y for
q = 0,1, 2... 16 using the function product relations
4sinasin3 = 2 cos(a-3) - 2cosCa+B)
4cosacos3 = 2 cos(a-3) + 2cos(a+3)
The first and second terms produce coefficients 2X and 2y^q q
respectively. Using the function relationships
. 2 71" 1 1 27Tsm —X = -^ - -sK^os—
X
a 2 2a9 7T 1,1 271"COS^—X = 77 + oCOS ^Xa 2 2a
there results
First terra:
„ 2qiT „ 2qiT 27TX cos-^y - X cos-^-y cos—^x
q b "^
q b "^ a
Second term:
Y cos—^y + Y cos-^y cos—^-x
q b -^ q b -^ a
Comparison with the right hand side of equation (17) , the
coefficients C arepq
C = Y + X0,q q q
C, = Y - Xifq q q
«
121
or
C = Y + (-l)Pxpq q q
p = 0/1
q = 0,1,2 16
2 . Application of the Method of' B.' G. Galerkin to Equation
(7)
Method of B.G, Galerkin (from reference [18] )
It is required to determine the solution of the equation
L(w) =
where L is a differential operator in two variables, which
solution satisfies homogeneous boundary conditions.
The approximate solution sought is in the formi
w(x,y) = Z h^ F^(x,y)n=l
where F (x,y) (n = l,2...i) is a system of functions chosen
to satisfy the boundary conditions and b are undetermined
coefficients. Consider the functions F (x,y) to be linearly
independent over a given region (no one of the functions can
be expressed as a linear combination of the others) . For
w(x,y) to be the exact solution of the given equation, it is
necessary that
L(w) =0
i
A
I
122
If L(w) is continuous, this requires that the expression
L(w) is orthogonal to all functions of the system F (x,y)
(n = l,2.,.i). With i constants, b. , bp...b., i conditions
of orthogonality can be satisfied.
Requiring that the two functions are orthogonal over a
region produces the following system of equations
//L(w)F (x,y)dxdyR
= f/Li I b^F^(x,y))F^(x,y)dxdy =
R n
(n = 1,2. . .i)
from which the coefficients b„ can be determined. Thesen
coefficients will define the solution w(x,y)
.
The method of B.G. Galerkin can also be obtained from
the principle of virtual work.
Application of the method of Galerkin to equation (7)
requires that the following equation be satisfied by all
functions f (x)g (y)
a/2 b/2 r,k^, •y'*^^cjit„
/ / [D ^^ + 2D _LJ^_ 4- D ^ ""
""dx' ""^dK^dy' ^dy If
8^F . a_^^ 2 S^^ 3^w _ 8_^ 8^w ,
~ ^ ~^ 2 \ 2 9x8y 8x8y ~
^ 2 .> 29y dx ^ ^ 8x^ ^y"-
X f^(x)g^(y)dxdy = (21)
123
Substitution of w and F, expressions (15) and (20)
respectively
a/2 b/2/ / Z 5^"^^ ^ S^"^^ ^TTiT.^^"^^ + cos-v—y] cos—-Xn n rs xinna mn d a
+ 2 D , h ZZ (-1-—) (— ) b cos—r—y cos—^xxy mn b a mn b -^ a
+ h D Z Z (-T-— ) b cos—J-—y cos—^x - qymn bmn b-^ a ^
u 3 V V V V /'l^^\ 2 /2qTT, 2 . w r/ n \ ^1+1 ,2n7T ,- h Z Z Z Z (—) (—?— ) (fc b I (-1) + cos—?—-yjm n p q a b ^pq mn b -^
2q7T miT 2piT• cos- V y cos—X cos-^E-—
X
b -^ a a
r; v. V V /2n7T. 2i. 2n'n" mTr- N h Z Z (-T—) ^b^^cos-^r—y cos—-xymn b mn b a
3 ,2nTT» 2 /2p7r. 2 J. i_ 2nTr 2q7T mTr 2p'n"- h Z Z {-^r—){-^^-)(b b_ cos-v--y cos—^y cos—^x cos—^-—
x
m n b a ^pq mn b -^ b -^ a a
— , „ „ /iHTT. 2i r / T \ n+1
,
2nTr , mTT- N h Z Z (—) "^b [(-1) + cos^r—'ylcos—^xxmna mn b-' a
r^T^-L. r^ r- /2nTr. ,m7T», . 2n7T . mTT- 2Sh Z Z (-y— (—-)b^^sin^r—y sm—-xm n b a mn id a
, -, 3 „ „ „ „ ,2mT. /mTTv /2pTT» ,2qTT. . , . 2nTT . 2qTT+ 2h'^ Z Z Z Z (—r—) (— )
(—^^-) (-^) li) b_ sin—r—y sm-^ymnpq baab ^pq mn b -^ b -^
. mTT . 2pTT T Tf T \ S+ 1 .2STT , ^"^ j j rt
• sm—X Sin—'=—x} •[ (-1) + cos—r—-yjcos—x dxdy =
a a Dam,r = 1,3,5....
124
The orthogonality properties of the trigonometric
functions must be used to evaluate the integral of equation
(21) . By employing the function relations
cosA + cosB = 2coS2-(A + B)coSy(A - B)
sinA + sinB = 2sin2-(A + B)coSy(A - B)
The trigonometric products may be expressed in typical forms
a/2r miT r-rr J/ cos—x cos—X dxQ a a
1 a/2= -r/ I2cos— (m+r)x + cos— (m-r)x + cos— (r-m) x] dx4q a a a
anda/
2
/ sm—X cos—X dx« a a
Ta/2
= x/ [2sin-(m+r)x + sin-(m-r)x - sin-(r-m) x] dx4 ^ a a a
where integration is carried out for all m and r. By
combining terms that integrate to similar values, these
integrals are equivalent to
Ta/2
=r^ [2cos-|m+rlx + 2cos-|m-r|x + 2cos-I r-mlx] dx4|^ a' ' a' I a' '
and
1 a/2x/ I2sin-|m+r|x + 2sin-lm-rlx - 2sin— I r-mlx] dx4q a' ' a' I a' i
||where the second two terms in each expression integrate to
I
125
zero if r > m or m > r respectively. Manipulation of the
trigonometric products on the right hand side of equation
(21) into these typical relations permits the reduction of
the equation into the convenient form of equation (22)
.
Table 3 lists all the specific integrals from equation (21)
with their respective values.
(
I
126
APPENDIX D
COMPUTER PROGRAMS
127
The computer program to solve the simultaneous equations
(39) is written in Fortran IV Level Gl for the IBM system
37 0. The program consists of the main and three subprograms
—
MATHLIB subroutine ZEROIN, and subroutines EVAL and CPQ.
Table 4 lists the important computer symbols used.
ZEROIN is a Fortran IV subprogram, from the M.I.T.
mathematical library, which computes the solution vector
b of a set of N simultaneous, nonlinear equations
NZF (b) = using double-precision arithmetic, b is obtained
by an iterative method beginning with estimated values of
the solution vector and iteration is performed as
k k k kwith the vector D = J 'FCb ) where F (b ) represents the
k kvector of function values at the point b and J represents
kan approximation to the Jacobian at b . Convergence is
tested with the expression
|b^+l - b^|2 ^ 0.5 X 10-i2|b^+l|2
Subroutine EVAL generates the eight simultaneous
equations and calculates the vector of function values for
a given solution vector b. EVAL is called repeatedly by
subroutine ZEROIN.
128
Subroutine CPQ is in turn called by subroutine EVAL,
with a solution vector b, to calculate the coefficients Cpq
that are used in the simultaneous equations.
For the starting values of the solution vector b, the
main program uses a well guessed set of values set in the
program. If the lateral load is not zero (Q* 7^ 0) (see
note 1) or the present loading conditions did not change
significantly from the last set, the previously computed
solution vector b is used for the starting values.
If ZEROIN cannot achieve convergence in 50 iterations,
the main program alters the initial set of starting values
and returns them to ZEROIN. Normally convergence is obtained
within the first 50 iterations, however, never more that a
second set of values is required.
Part 1 - Design Charts
The main program varies the nondimensional loads Q*
and N* or S* for a particular set of plate parameters.
Tables of nondimensional deflection coefficients, nondimen-
sional deflection at the center of the plate, nondimensional
effective width and nondimensional bending moment (y-
direction) at the middle of the supports (y = ± b/2) are
outputs. Additionally, CALCOMP plots of the last three
tables are produced.
129
Part 2 - Behaviour Programs
The main program reads the input parameters for 4 sets
of loading conditions on a particular plate and outputs
CALCOMP plots of nondimensional plate deflection along the
centerline (x = 0) and nondimensional total principal
stresses along the centerlines (x = and y = 0) . These
programs are completely general and can be used with any
set of loading conditions.
The design programs must be used with care. With
reference to equations (36a) and (36b) , the effective width
is only defined for N* 7^ or N* 7^ respectively. Hence/y X
variation of S* with N* = will give a value of zero for
the effective width. The design programs are written so
that no computation or plot of effective width will be
made for any variation of S* with N* fixed. Additionally,
deflection at the center of the plate and the y-direction
bending moment may not be significant parameters in this
case. It is recommended that the "behaviour" program be
used if variation of the shear load, S*, is desired.
A
130
Program Descriptions
Main Program 1 - Design (N* = 0)
Input one card per set of complete design curves set up as
follows:
Column Parameter
1-10 RHO
11-20 GAiyiMA
21-30 ETA
31-40 *NXSTAR (Enter Negative Valueto vary)
41-50 *SSTAR (Enter negative valueto vary)
51-60 RANG
61-70 RANGQ
*Note: Either N* or S* may be varied, but not both in the
same run.
Output
1. Tables
a. bl,n
b. r- at y = X
c. b (only if N* is varied)e -^ X
d. M* at y = ± fe, X =y ^
131
2. CALCOMP Plots
Tables b, c and d above,
3
.
Values
N* or S* = to RANG in 10 increments of RANG/20 and
Q* = 0, RANGQ/6, RANGQ/3 , RANGQ
Termination
A blank card as the last data card is required,
I
132
Main Program 2 - Design (N* = 0)
Input one card per set of complete design curves set up as
follows:
Column Parameter
1-10
11-20
21-30
31-40
41-50
51-60
61-70
RHO
GAMMA
ETA
*NYSTAR (Enter negative valueto vary)
*SSTAR (Enter negative valueto vary)
RANG
RANGQ
*Note: Either N* or S* may be varied, but not both in the
same run.
Output
1. Tables
a. b,1/n
2. CALCOMP Plots
wa. r-aty = x =
b. a (only if N* is varied)e -^
y
c. Mj«^ at y = ±J,
X =
I
133
3. Values
N* or S* = to RANG in 10 increments of RANG/20 and
Q* = 0, RANGQ/6, RANGQ/3 , RANGQ
Termination
A blank card as the last data card is required.
134
Main Program 3 - Behaviour
Input four car<
Co]Lumn
1-10
11-•20
21-30
31--40
41--50
51--60
61--70
Parameter
RHO \
GAMMA smust be same all 4 cards
ETA /
NSTAR (see note 2)
SSTAR
QSTAR
*TB
*Note: If TB other than zero, stresses at top of plate will
be computed. TB = zero defaults to bottom of plate.
Output
1. Tables
a. b,1/n
b. T-aty = x =
c. a? and ai at y = x =
2. CALCOMP Plots
a. g at X = for -I
< ^ < -f I
b. a* and a* at x = for - -^ < ^ < -f i12 2 a 2
I
135
c. a* and a* at y = for - -^ < - < + ^12 2 a 2
Termination
A blank card as the fifth card is required.
136
Subroutine EVAL
Two subroutine programs EVAL - one for N* = , the
other for N* = 0. The appropriate EVAL program must be
used with the corresponding main programs.
Subroutine ZEROIN & CPQ
Completely general subprograms. Use with all combina-
tions of main programs and EVAL.
NOTE 1
If the lateral load is equal to zero (Q* = 0) a possible
but trivial solution is that all the coefficients b =0.mn
Care must be exercised to avoid using b =0 for starting^ mn ^
values in this case. Additionally, under certain circum-
stances the solution readily converges to zero rather than
the appropriate non-trivial solution. The programm.er should
be aware of this possibility.
NOTE 2
The behaviour program may be used with either N* or N*
as a loading, but not both. The main program must be used
with the appropriate subroutine EVAL.
137
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138
MAIN PROGRAM 1 - DESIGN
(N* = 0)y
139
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A4298.145570
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3 2768 001 89863 8DUDLEY KNOX LIBRARY