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TRANSCRIPT
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EXTENDED FINITE STRIP METHOD FOR PRISMATIC «
PLATE AND SHELL STRUCTURES
by
GHULAM HUSAIN SIDDIQI, B.E., M.S. in C.E.
A DISSERTATION
IN
CIVIL ENGINEERING
Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for
the Degree of
DOCTOR OF PHILOSOPHY
Approved
May, 1971
7 3 \91I
ACKNOWLEDGMENTS
I am deeply indebted to Dr. C.V. Girija Vallabhan
for his guidance and counseling during this investi
gation. I am grateful to Dr. Kishor C. Mehta, Dr. James
R. Mcdonald, Professor Albert J. Sanger and Dr. Donald
J. Helmers for their advice and helpful criticisms. I
am also grateful to Mr. Gary A. Lance and Mr. Sherrill
Alexander for -their assistance in drafting work.
11
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
LIST OF TABLES
LIST OF FIGURES
GLOSSARY OF SYMBOLS
I. INTRODUCTION 1
Review of Literature 2
Finite Strip Method 4
The Extended Finite Strip Method . . . . 4
Comparison with Finite Eleiaent Method . . 7
Scope of this Study 7
II. EXTENDED FINITE STRIP METHOD 8
Introduction 8
Some Basic Theorems in Solid Mechanics . 9
Equilibrium Problem in a Continuous
Systems 14
Trial Solutions with Undetermined
Parameters 16
Ritz Method 19
Types of Structures 20
Geometry and Frames of Reference . . . . 22
Theory of Thin Plates 24
Rib Elements 2 8
Philosophy of the Extended Finite Strip
Method 30
iii
IV
Page
Polynomials 31
Base Functions 37
III. STIFFNESS MATRIX EQUATION 45
Introduction 45
Stiffness Matrix Equation of Membrane Action 48
Stiffness Matrix Equation of Bending Action 54
Combined Stiffness Matrix Equation . . . 61
Rib Stiffness Matrix 62
Transformation to Global Coordinates . . 6 4
Overall Stiffness Matrix 66
Nodal Line Forces 66
Solution by Gaussian Elimination . . . . 68
IV. ANALYSIS OF PROBLEMS WITH KNOWN SOLUTIONS AND CONVERGENCE TEST 69
Introduction 69
The Displacement Contributions of the
Base Functions 70
Plate Structures under Bending Action . . 71
Convergence Test 74
Structures under Combined Membrane
and Bending Action 77
Rib Attachment 7 8
V. APPLICATION OF THE EXTENDED FINITE STRIP METHOD 84 Rectangular Plate with Overhang 8 4
V
Page
A Folded Plate Structure 86
A Circular Cylindrical Shell with Canopy . 86
A Continuous Plate with Settling Support . 102
Nondimensional Coefficients 10 2
VI, CONCLUSIONS, OBSERVATIONS AND RECOMMENDATIONS 106
LIST OF REFERENCES 110
APPENDIX 113
A. Stiffness, Force and Transformation Matrices 114
B. Nondimensional Coefficients for Rectangular Plates 130
LIST OF TABLES
Page
1. Base Functions ip and fi for Symmetric Membrane Action 41
2. Base Functions ip and n for Antisymmetric Membrane Action 41
3. Base Functions for Bending Action 44
4. Base Functions for Hinged-Clamped Condition , 4 4
5. Base Functions for Clamped-Clamped
Condition 44
6. Values of a in Plate 1 73
7. Values of a , 6 and B at Points 1 and 2 X y
m Plate 2 73 8. Values of a at Points 1 and 2 in Plate 3 . . 75 9. Values of a at Points 1 and 2 in Plate 4 . . 75
10. Corner Supported Plate Carr'ying Uniformly Distributed Load. Convergence Test Data at Center Point 76
11. Plate Hinged Along Opposite Edges (Free Along Other Two) Carrying Uniformly Distributed Load. Converqen-ce Test Data at Center Point 77
VI
LIST OF FIGURES
Page
1. Pifismatic Plate and Shell Structures and
Their Geometry 21
2. Global and Elemental Frames of Reference . . 23
3. Ribs Along x-edges of Strips 2 3
4. (a) Symmetrically Displaced Shape of Membrane Action 39
(b) Antisymmetrically Displaced Shape of Membrane Action . . . . . . . . . . 39
5. (a) Symmetrically Displaced Shape of Bending Action 4 3
(b) Antisymmetrically Displaced Shape of Bending Action 4 3
6. Transformation of Coordinates 66
7. Plate Structures of Different Bomidary Conditions 72
8. Cylindrical Shell Structure 79
9. (a) Plots of Stress Functions of the Cylindrical Shell 80
(b) Plots of Stress Functions of the
Cylindrical Shell 81
10. Square Hinged Plate with Elastic Ribs . . . 82
11. Isotropic Rectangular Plate with Overhang . 85
12. Folded Plate Structure with Elastic End-Ribs and North Light Window Details 87
13. (a) Displaced Shapes of the Folded Plate Structure 8 8
(b) Transverse Stress Resultant T (lb/ft)
Along Mid Section of Plates 89
Vll
Vlll
Page
(c) Longitudinal Stress Resultant T Y
(lb/ft) Along Mid Section of Plates . . 90
(d) Membrane Shear T in Plates Along
End Section 91
(e) Transverse Moment M (Ib-ft/ft) Along
Mid Section of Plates 92
(f) Longitudinal Moment M (Ib-ft/ft)
Along Mid Section of Plates 9 3
(g) Twisting Moment M (Ib-ft/ft) in xy
Plates Along Support Section 9 4
14. (a) Axial Force P (lb) in Rib Members . . 9 5
(b) Shear Force V (lb) in Rib Members . . 96 z
(c) Moment M in Rib Members 9 7
15. Cylindrical Shell Structure with Canopy . . . 98
16. (a) Displaced Shape of Shell Along Mid
and End Sections 99 (b) Stress Resultants in Shell Along
Mid Section 100 (c) Moment Resultants in Shell Along
Mid Section 101 17. Continuous Plate Subjected to Support
Settlement 103
GLOSSARY OF SYMBOLS
A = Area of rib; amplitude of displacement.
a = Length of a structure and a strip
in y-direction.
a. = Flexibility constant at point-i.
a. . , [A] = Flexibility influence coefficient; matrix of these coefficients.
B = Overall width of a structure in
x-direction.
b = Width of a strip element.
B. = A linear differential operator at point-i on the boundary.
L.BJ = Matrix of transformation from generalized displacements to strains and curvatures.
c. = Undetermined parameters.
LC I = Matrix for transformation from generalized displacements to displacements .
D = Domain.
d. = Displacement of a point-i.
{d} = Column matrix of displacements.
°x' ^y' °xy' ^1' ' = Plate rigidities.
[D] = Matrix of plate rigidities.
E , E , E, = Young's Moduli of elasticity. X y X
E , E , E, = Elastic constants of an orthotropic "" y ^ body.
[EJ = Matrix of elastic constants.
ix
x
f = Force function in a domain.
{f} = Column vector of generalized forces referred to elemental coordinates.
{f} = Column vector of generalized forces
referred to global coordinates.
G = Shear modulus.
g. = Prescribed boundary conditions.
[G] = Matrix of force amplitudes.
{H} = Column vector of base functions.
I = Moment of inertia of rib about y-centroidal axis.
^i-; I [K] = Stiffness influence coefficients; matrix of these coefficients,
L = Distributed line loads applied transversely on a strip-
L = Amplitudes of L-loads in the function space.
L^_ = Linear differential operator of 2"" order upto 2m.
M , M , M = Moment resultants in x- and y-X' y' xy
P. 1
{P}
Qx '
q /
«y
q^ m
directions and twisting moment.
{M} = Column vector of moment resultants.
N, N , N = Nodal line force and its components
(N,)^ , (N ) = Amplitudes of N-forces in the X m ' z m £ j_ •
function space.
= Point load at point-i.
= Column vector of P-loads. = Shear resultants in x- and y-
directions.
= Distributed loads; amplitudes of these loads in the function space.
XI
R = Residual of the governing equation.
r = Centroidal distance between a strip and a rib.
T , T , T = Stress resultants in x- and y-X V XV
- ^ directions and in xy-plane,
t = Thickness of a strip element.
t' = Depth of a rib element,
U = Strain energy.
V = Potential energy of loads. u, V, w = Displacements of a point on the
middle surface in x- , y- and z-directions respectively.
u, V, w = Displacements of a point on the middle surface in x- , y- and z-directions respectively.
u', v", w' = Displacements of a point in the plate in x- , y- and z-directions.
X, Y; X^, Y^ = Body forces in x- and y-directions; amplitudes in the function space. m m
X, y, z = Coordinates of elemental frame of reference.
X, y, z" = Coordinates of global frame of reference.
a m Shape factors and nondimensional coefficients.
3 ; 3 , 3 , 3 = Shape factors; nondimensional y ^y coefficients.
e , e , Y = Strains in x- and y-directions and y ^y shear strain in xy-plane.
{e} = Column vector of strains.
a , a , T = Stresses in x- and y-directions ^ ^y and shear stress in xy-plane.
Xll
{a} = Column vector of stresses,
V # V = Poisson's ratios in x- and y-direc-X' y tions,
6 = Displacement function in a domain,
{6} = Colximn vector of generalized dis-
lacements.
6 = Rotation about y-axis.
^^, ^^ = Base functions of y, m m • -
(f) = Potential energy functional; angle between elemental and global frames of references.
(x) = Column vector of curvatures.
CHAPTER I
INTRODUCTION
Prismatic plate and shell structures are widely used
in Civil Engineering practice. The plates when formed into
ribbed slabs, or assembled into box girders have a wide
application to roofs, floors, canopies and bridges. Because
of their economy, folded plate and shell structures are used
extensively in industrial and commercial buildings.
The extended finite strip method developed in this
study presents an integrated approach for analysis of pris
matic plate and shell structures. This method has the capa
bility of analyzing the structures which: a) displace along
any or all the edges, b) are carried on integrally built-in
columns or supported on corners or along the edges, c) employ
elastic rib or stiffeners along the edges, d) carry any
type of loads-distributed, live or concentrated—at any
location, e) employ orthotropic material, f) employ details
such as changes of width and thickness from one strip to the
other, g) use special features such as north light windows
over the span lengths, and h) are subjected to differential
settlement at their support. There are several methods and
some graphical and tabular aids of analysis available, but
they have limitations in analyzing structures defined above.
1
Review of Literature
A review of the literature available on the subject,
in the form of methods and techniques, tables and charts
that can be directly applied to the analysis of a prismatic
plate or shell structure, is presented here. The scope and
limitations of these methods are also assessed.
On the subject of thin plates in bending Timoshenko
[1], Nadai [2], Margurre et al. [3] and ACI-Standard [4]
have tables and charts for nondimensional coefficients
which can be used directly for the analysis of a plate. The
information presented is for isotropic material properties
except in Ref. [1] where orthotropic properties are con
sidered. However, plates with complicated boundary condi
tions are not considered and the information provided is
limited.
Folded Plate Structure
Many methods for analysis of folded plates have been
developed; a review of this information is found in the ASCE
Task Committee report [5] where a modified version of
Gaafar's method is recommended for design purposes. This
method is difficult to program and is not applicable to
small span-width ratios [6]. The elasticity method origi
nated by Goldberg and Leve [7] was applied by DeFries-Skene
et al. [8] as a stiffness approach, and was presented in
Refs, [9], [10] and [11] as a finite difference technique.
The rectangular finite element technique used by Zienkiewicz
and Cheung [12] was applied to the study of the folded plate
behavior by Rockey and Evan [13]. Cheung introduced the
finite strip method [15] and later applied this method to
analysis of folded plates [6], All these methods assume the
end-diaphragms to be rigid in their plane and free to rotate
normal to their plane; Williamson [14] has modified the
Goldberg et al. method [11J to analyze the folded plates
supported by flexible end-diaphragms. The finite element
and finite strip methods, are the only ones capable of ana
lyzing folded plates with orthotropic properties.
Prismatic Shell Structure
In references [16], [17], [18] and [19] tables are
furnished that permit analysis of uniformly loaded, simply
supported, i.e., with rigid end stiffeners, single barrel
shells of uniform circular cross-section and made from iso
tropic, homogeneous materials. All these methods of analy
sis require a large amount of computational effort. In
Ref. [20] a computer method is presented for analyzing
cylindrical shells of various cross-sections by approximat
ing the section by a series of circular segments. In the
field of finite elements Clough et al. [21] and several
others have developed computer methods for analysis of these
structures while Mircea Scare [22] has expounded finite dif
ference techniques toward this end.
Finite Strip Method
The finite strip method was developed by Cheung [15].
This method divides the domain of a plate structure into a
number of rectangular strips. The displacements of these
strips are modeled by the product of two exclusive functions
of the coordinates. One of these functions is a polynomial
expressed in terms of undetermined parameters while the
other is a series of base functions which a priori satisfy
the boundary conditions in that direction. The stiffness
matrix equation of a strip is developed by the Ritz method
in which the potential energy functional of the strip is
minimized with respect to the undetermined parameters.
Because of the base functions used the finite strip
method [15] has limited application to the plate and shell
structures. Any plate or shell structure which is subject
to translational displacement along the transverse edges of
the strips cannot be analyzed by this method.
Extended Finite Strip Method
The method proposed in this study is an extension of
the finite strip method introduced by Cheung [15] and will
be referred to as the Extended Finite Strip (EFS) Method.
The above mentioned restrictions on the finite strip method
are removed. In addition rib elements are introduced along
the strip edges to generalize the method's application to
prismatic plate and shell structures.
The method, briefly speaking, divides the structure
into a finite number of strips along the length of the struc
ture; each strip element, therefore, is bounded by two nodal
lines. The membrane and bending actions of a strip are
plane stress actions. For thin plates of linear elastic and
orthotropic materials, subjected to small deflections, these
actions are independent and superposition is valid. A nodal
line has four degrees of freedom: the displacements along
longitudinal and transverse directions (y- and x-directions)
of a strip pertaining to membrane action, displacement
perpendicular to the plane (along z-direction) and rotation
about the longitudinal axis of the strip pertaining to
bending action. The governing differential equations of
membrane and bending action are written in terms of these
displacements which are functions of the x- and y-coordinates
The displacement functions in the governing equations are
replaced by the product of two exclusive functions of x and
y to obtain a "trial solution." The functions of y are a
set of linearly independent base functions spaning the
domain in y-direction and satisfying the essential boundary
conditions (including displacement along the edges parallel
to x-direction). The functions of x are polynomials
developed in terms of amplitudes of the base functions for
each degree of freedom permitted at the nodal line. These
amplitudes form the undetermined parameters of the trial
solution. The "best" solution is obtained by the Ritz
method in which the potential energy functional of the strip
is minimized with respect to the undetermined parameters.
The process of minimizing the functional yields a discrete
analogue of the governing equations of the equilibrium,
i.e., a set of simultaneous equations of equilibrium in
terms of undetermined parameters. This set of equations is
the stiffness matrix equation of the strip.
By the usual procedure of transformation, the stiff
ness matrix equations of the individual strips are trans
formed to global coordinates and assembled to obtain the
overall stiffness matrix equation. The solution of this
matrix equation is obtained using Gauss elimination, which
yields values of the undetermined parameters.
The displacements, stress resultants and the moment
resultants at a point in the strip are functions of the
undetermined parameters of its bounding nodal lines. The
displacements, stress and moment resultants are, therefore,
evaluated using these functions.
Comparison with Finite Element Method
The finite element method, which is one of the most
powerful tools of stress analysis available, may be used to
analyze the prismatic plate and shell structures considered
in this study, even when they have cut-outs and local varia
tions in thickness. The extended finite strip method is not
applicable to structures of this type. However, the method
has certain advantages over the finite element method:
smaller number of unknowns [15] , and the displacements in
X- and z-directions are the only ones to undergo transforma
tion so that their compatibility is not affected by this
transformation.
Scope of This Study
Only the prismatic plate and shell structures which
have complete freedom of rotation about the transverse edges
of a strip are considered in this study. However, the means
to develop stiffness matrix equations of structures, which
are clamped against these rotations at one or both edges,
is indicated.
Nondimensional coefficients for deflection, stress
and moment resultants of certain rectangular plate structures
are furnished in the form of charts and tables.
CHAPTER II
EXTENDED FINITE STRIP METHOD
Introduction
An explanation of the philosophy of the extended
finite strip method and the development of its basic prin
ciples is presented in this chapter. To insure completeness
of this discussion some basic principles and definitions and
some consequent deductions in solid mechanics [23, 24] are
repeated here. The solution of equilibrium problems by
trial solutions using undetermined parameters is discussed
briefly. One of these, the Ritz method, which is based on
the stationary functional approach, is adopted for this
study. The types of structures that can be analyzed by this
method, their geometry and the frames of reference adopted
are defined. A brief review of the theory of thin plates,
with rib elements attached along the transverse edges, is
made.
Finally the basic principles of the extended finite
strip method are expounded and the development of function
space to represent the displacement and stress and moment
resultants in the domain of a finite strip is considered.
8
Some Basic Theorems in Solid Mechanics
The whole edifice of linear elasticity is built on
the concept of a linear elastic solid. The definition of
such a solid is based on the following three hypotheses
[23]:
Hypothesis I. The body is continuous and remains continuous
under the action of external forces. This means that within
a solid the neighboring points remain as neighbors under any
loading conditions. In other words no holes or cracks open
up in the interior of the body under the action of external
loads. Mathematically the hypothesis implies that displace
ment at a point, expressed as a function of the coordinates
and the first derivative of the function at a point, remain,
continuous before and after the application of the external
loads.
Hypothesis II. If a body in static equilibrium is acted
upon by a set of forces—P , P , . . . , P^^—the displace
ment, d, of an arbitrary point within the body in an arbi
trary direction is given by
n d = E a. P. i = 1, . . . , n (2.1)
1 1 1
where a , a , . , , , a are constants independent of magni-1 2 n
tudes of P , P , . , , , P (Hooke's Law). The constants, 1 2 n
however, depend upon the location of the point at which the
10
displacement is measured, and upon the location and direc
tion of application of an individual force.
Hypothesis III, There exists a unique unstressed state of
the body to which the body returns whenever all the external
loads are removed. A body that satifies the above three
hypotheses is said to be a linear elastic solid.
A number of principles can be deduced, from these
three hypotheses. Some important deductions are mentioned
here without establishing proof for them. Proofs are given
by Fung [23] ,
Principle of superposition. By a combination of Hypotheses
II and III it can be shown that Equation (2.1) is valid
irrespective of the order in which P.(i=l,n) are applied.
A constant a. depends upon load P. only and is independent
of the rest of the loads in the set. This is the principle
of superposition of load-deflection relationship.
Uniqueness of total work done by the forces. -The displace
ment at the point of application of a force measured along
its direction is defined as the corresponding displacement.
The total work done by a set of loads in going through
their corresponding displacements is unique irrespective of
the order of application of these loads.
11
Maxwell's reciprocal relation. The corresponding displace
ment at point i due to a unit load at point j, in a body, is
denoted as a^. and is called the flexibility influence co
efficient. The total displacement at point i, due to a set
of loads is
d^ = Za^j P. j = 1, . . . , n (2.2)
Maxwell's reciprocal relation states that the flexibility
influence coefficients for corresponding forces and dis
placements are symmetric. In symbolic form
a^. = aji (2.3)
Betti-Rayleigh reciprocal relation. This theorem which is a
corollary of Maxwell's reciprocal theorem, states that the
work done by a set of forces in going through corresponding
displacements produced by a second set of forces is equal to
the work done by the second set of forces in going through
the corresponding displacements produced by the first set of
forces.
Strain energy. For a body going through an isothermal and
adiabatic deformation process, the work done by the external
forces is equal to the change in internal energy. If the
internal energy is reckoned as zero in the unstressed state
then the change in internal energy is called the strain
12
energy stored. The strain energy U is given by the equation
U = 1/2 ZEa^j P^ Pj i,j = 1, . , , , n (2,4)
or in matrix notation
U = 1/2 {P}^ [a] {P} (2,4a)
where {P} denotes a column matrix of the P forces and [a]
denotes the square symmetric matrix formed by the influence
coefficients a.. , The superscript t to a matrix symbol
denotes its transpose. The strain energy in a linear elas
tic solid is independent of the order of application of the
loads.
Positive definiteness of strain energy and uniqueness of
solution. For a linear elastic solid defined above there
exists a strain energy function U which is expressible in
terms of displacements d. For a solid body to have a
stable, natural state, such as the unstressed state of a
linear elastic solid, the strain energy function must be
positive definite; i.e., it must be non-negative, and zero
only in the natural state. The positive definiteness of U
implies that the determinant of matrix [a] is always posi
tive. The inverse of the matrix [aj therefore exists, which
leads to the theorem of uniqueness of solution. This theo
rem states that for a linear elastic solid there exists a
one-to-one correspondence between the elastic deformations
13
and the forces acting on the body. A relation analgous to
Equation (2.2) may be written as
^i " ^ ij ^j j = 1, . . . , n (2.5)
where k.. are stiffness influence coefficients. Both [aJ
and [K] for the system are symmetric.
The strain energy U in terms of stiffness influence
coefficients is given by
U = 1/2 EE k..d.d. , i, j = 1, . . . , n (2.6)
or in matrix notation
U = I {d}^ [K] {d}, (2.6a)
where {d} is column matrix of displacements d. , and [K] is
a square symmetric matrix of the stiffness coefficients k..
Potential energy functional.--A potential energy functional,
^ , can be assigned to any geometrically compatible state of
the elastic system according to the formula:
$ = U-V (2.7)
where U is the strain energy of the system as already
defined and V is the potential energy of the prescribed
loads.
14
Minimum Potential Energy Theorem.—This theorem is stated
here without proof. Of all admissable sets of displacements
satisfying ;the boundary conditions, the one which also
satisfies the equations of equilibrium is distinguished by
the minimum value of the potential energy functional. The
equations for determining the minimum.
^^ = E kijdj-P^ = 0 i,j = 1, . . . , n (2.8) 9di j
or E k^.d. = P^ i,j = 1, . . . , n , (2.8a)
are identical with the equilibrium conditions. The Minimum
Potential Energy Theorem leads to several approximate methods
of solution of complicated problems in elasticity and struc
tural mechanics. The stiffness coefficients are the vehicles
through which this principle is established and Equation
(2.8a) automatically yields these coefficients.
Equilibrium Problem in a Continuous System
In the case of equilibrium problems in a continuous
system, the relations of Equation (2.8a), which are for a
discrete system, become a set of differential equations of
the type
L (6) = f in domain D . (2.9) 2m
15
In physical problems in solid mechanics 6 is the dis
placement function and f is the force function. A solution
to the equilibrium problem lies in determining a function
6 which satisfies Equation (2.9) and also meets the boundary
conditions,
B^ (6) = g^ i = 1, . . . , m (2.9a)
on the boundary of the domain D. The symbol L (6) operates
on function 6 and its ordinary and partial derivatives
up to order 2m. The symbol B. (6) operates on function
6 and its derivatives up to order 2m-l, which when evalu
ated at each point i on the boundary, satisfy the prescribed
values g. . ^1
The boundary conditions of an equilibrium problem
are divided into two categories: the conditions which
satisfy geometric compatibility at the boundary, and the
conditions which satisfy the force balance conditions at
the boundary. The conditions of the first category are
called essential boundary conditions while the others are
called natural boundary conditions.
An equilibrium problem is said to be linear when its
governing equations and the boundary conditions are linear.
16
Trial Solutions with Undetermined Parameters
There are several approximate procedures for the
solution of equilibrium problems in continuous systems. One
category of these methods utilizes trial solutions with
undetermined parameters [24]. In these methods trial solu
tion in terms of undetermined parameters is selected. The
trial solution represents a whole family of admissible
approximations which simulate 6 within the domain and
satisfy the boundary conditions. Each of the various pro
cedures has different criteria for picking out the "best"
approximation of the selected family of admissible functions.
For a linear equilibrium problem a trial solution
for 6 has the form
6 = Ec. y. i = 1, . . . , n (2.10)
where the ^. are known linearly independent functions in the
domain D satisfying the boundary conditions and the c^ are
the undetermined parameters. There are two basic criteria
for fixing c. . In one of these c are so chosen to make ^ 1 1
weighted averages of the equation residual vanish, and in
the other c. are so chosen as to give a stationary value to
the potential energy functional of the system. Application
of either criteria results in a set of n simultaneous
equations in c. . These methods, therefore, reduce an
17
equilibrium problem in a continuous system to an approxi
mately equivalent equilibrium problem in a discrete system
with n degrees of freedom.
Weighted Residual Methods
The trial solution of Equation (2.10) is selected to
satisfy both the essential and the natural boundary condi
tions. The equation residual R in terms of the trial solu
tions is
R = f-L (Ec.^.) i = 1, . . . , n (2.11) 2m 1 1
For the exact solution the residual is identically
zero. Within a trial family, however, a "good" approxima
tion is one which renders R small. Any one of the following
four methods may be used to make the weighted averages of
R vanish.
Collocation.- -The residual is set equal to zero at n arbi
trary points in the domain D . This technique results in n
simultaneous equations for determining c. .
Subdomain. The domain D is subdivided into n subdomains,
according to an assumed pattern. The integral of R over
each subdomain is then set equal to zero thus obtaining n
simultaneous equations for determining the c^ .
18
Galerkin.—This method is based on the mathematical concept
that m^ and R are orthogonal over the domain D . Expressly
this condition is written as
•'D ""i RdD = 0 i = l , . . . , n (2.12)
whereby n simultaneous equations are obtained for determin
ing the c. .
Least Squares.—According to this method the integral of the
square of R is minimized with respect to the undetermined
parameters c. to provide n simultaneous equations.
Be 1
: / . .2 R^dD = 0 i = 1, . . . , n . (2.13)
For a particular trial family which satisfies all
the boundary conditions these methods produce slightly dif
ferent approximations. The Galerkin method yields results
that are superior to those obtained from the other methods
of this category. These methods may yield meaningless
results if the trial family satisfies only the essential
boundary conditions. In this regard these methods are
restrictive in their application to the equilibrium prob
lems.
19
Ritz Method
In this method a trial family is so chosen as to
satisfy the essential boundary conditions only. This trial
solution need'*not satisfy the natural boundary conditions.
The method consists of expressing the potential energy, $ ,
of the system in terms of the assumed trial solution of
Equation (10) and extremizing ^ with respect to the undeter
mined parameters c. . Even if the assumed trial solution
violates the natural boundary conditions, the extremization
of the functional ^ ensures the "best" result out of the
assumed trial solution. This procedure provides n simul
taneous equations for determining the c. . In matrix nota
tion these equations are written as
[K] {C} - {f} = 0 (2.14)
or [K] {c} = {f} (2,14a)
where [K] is the stiffness matrix of the system, {c} is the
column matrix of undetermined parameters, and (f} is the
column matrix of "distributed forces," The undetermined
parameters determine the amplitude of displacements and the
degree of freedom of the system and, therefore, are recog
nized as the generalized displacements. The force vector
{f} is to be expressed in the same function space. Equation
20
(2,14a) is referred to as the stiffness matrix equation of
the system.
The Ritz method has the advantage over the weighted
residual methods in requiring satisfaction of only the essen
tial boundary conditions by the trial solution. This helps
to simplify the selection of trial solution.
The value of $ for a true solution is the minimum so
that the value of $ obtained from a trial solution always
yields an upper bound to $ , The relative "goodness" of one
trial solution over another can, therefore, be judged by
smallness of the $ value. The superiority of the Ritz
method in this regard over the other methods is demonstrated
in literature [24].
Types of Structures
All the prismatic plate and shell structures, in
which the boundary conditions along two parallel edges can
a priori be satisfied by suitable base functions, are
amenable to solution by the extended finite strip method.
Such prismatic plate and shell structures are represented
by an assembly of flat strip elements as shown in Figure 1.
Figure 1(a) shows the variation in strip widths and thick
nesses, and the types of loads that can be applied. The
structure of Figure 1(b) has stiffening ribs along edges
21
Modal Lliie
t ^
o o x strip Element
(a) Plate Structure with Varying Plate Thickness and Different Loads.
North Light
Elastic
Strips of zero thickness with ribs
(b) Folded Plate Structure Columns and North Light
with Elastic End Ribs, Built-in
(c) Cylindrical Shell Structure
A QhplL Structures and their Geometry. Figure 1. Prismatic Plate and Shell Structur
22
parallel to the x-axis and is supported by monolithically
built in columns.
Geometry and Reference Frames
Figure 2 shows a prismatic structure with two frames
of reference: the global and the elemental. The global
frame of reference is denoted by a bar over the letter.
The x-f y- and z- axes are oriented as shown in Figure 2.
Directions indicated are the positive directions of these
axes. The same triad is chosen for elemental frame of
reference also. The x- and y-axes in this frame are con
tained in the plane of the strip element while the z-axis
is normal to this plane. The x-axis is oriented along the
edge where boundary conditions are a priori satisfied by the
base functions. The directions parallel to the x-axis and
y-axis are referred to as transverse direction and the
longitudinal direction respectively for both the structure
and a strip element. The y-axis in global and elemental
frames of reference always remain parallel to each other.
The dimensions denoted by a, b, and t in Figure 1(a)
are length, width an^ thickness of a strip element (along
X-, y- and z-axes) respectively. Translational displacements
parallel to x-, y- and z-axes of a point on the middle plane
of a strip are denoted by u, v and w respectively. The
23
Figure 2 . Global and Elemental Frames of Reference
Nodal L ine- i
? (c) (d)
Figure 3 . Ribs along x-edges of StripJ
24
rotational displacement about the y-axis of a point is
denoted by 0 .
Theory of Thin Plates
Because a strip element is considered to be a thin
orthotropic plate an outline of the theory of such a plate
is presented here. The equations for membrane and bending
displacements, the resulting strain and stress relations
and the consequent equations of stress and moment resultants
are also given here without derivation [1, 23]. For small
deflection theory and linear elastic material analyses
considering both membrane and bending actions are based on
Kirchhoff's hypothesis. This hypothesis states that every
straight line in the plate that was originally perpendicular
to the middle surface remains straight and perpendicular to
the deflected middle surface after the strain.
Application of Kirchhoff's hypothesis leads to the
following expressions for the displacements u', v' and w'
of a point (x,y,z) in a plate in terms of the displacements
u, V and w of the corresponding point (x,y,o) in the middle
surface
w' = w
u' = u - 1 ^ Z (2.15) 9x
V • = V - ^ Z
9y
25
The strain components at a point are
3u' 8u a^w' „ X 8x
8v' ""y 3y
8x
8v 8y
8X-2
a ^ w ' 8y^ (2.16)
^ 8u' 8V' 3u 8V _ 3 w' 'xy 3y 3x ^ 3y 3x' 3x3y
To make this development general, the plate material
is considered to be orthotropic. Orthotropy is a paticular
case of anisotropy in which elastic properties of a material
in two mutually perpendicular directions are defined. | 1, 2 8]
The elastic properties of such a material in two dimensions
are completely defined by five constants out of which four
are independent. The elastic constants are E , E (Young's
moduli), V , V (Poisson's ratios), and G (shear modulus). X y
The constant v is dependent one and is given by
V E = V E = El X y y X ' (2.17)
The stress-strain relation of such a material is shown
to be
T.
xy
E. X 1 - v V
X y E
.1 1 - v V"
X y
0
1-v X
^ 1 - v
X
0
V y
V'
y
0
0
X
< e M2.18)
Y xy
26
or
E E X
1 0
El E 0 ^ y
f ^
< e^ ;> . (2.18a)
xy
It may be noted that in isotropic condition E = E = E ,
V = V = V and G = E/2(l+v). X y
The stress resultants (force per unit length),
denoted by T , T , and T are - X y xy
X
t/:
-t/:
^n ^x ,3u , 3v, "x ^^ = T^TT^ <33r + "y 3?'
X y -^ -^
.t/:
-t/:
a dZ y
E t y (^ + V — )
l-v._v.. 3y X 3x'
3v 3u,
X y
(2.19)
t/:
xy T dZ xy
/3u , avv <5t (37 + 33?)
-t/;
27
The moment r e s u l t a n t s (moments p e r u n i t l e n g t h ) ,
d e n o t e d by M , M a n d M a r e X y '^y
M, X
. t / 2 -E t a^ Z dZ = j , ^ ^ ^ v
X 1 2 ( l - v ^ v . ) . 2 - t / :
2 2 / 3 ^ . . 3 j v .
x ' y ^ 3x2 y 3y2
= - ( D „ 1 J 1 - + D i ^ ^ ) ^ 3y^ 3y '
M . t /
- t /
2 -E t a Z dZ = TTT^ y 1 2 ( l - v ^ v ^ ; 3 ^
2 2 . 3 w , 3 w . ( -r^ + V^. 7-) 1 2 ( l - v „ v , J \ , ^ X 3^2
X
= _(D i ! i i + D , i ^ ) y 3y2 S x ^
( 2 . 2 0 )
t/: M xy -
-t/;
3 2 2 n J3n o G t , 3 W . I T - . / 9 ^ \
^xy 2 <i2 = 2 3 ^ ( 3 ^ ) = 2 D^y ( 3 3 ^ ) •
From t h e moment e q u i l i b r i u m e q u a t i o n s
3M Q, J&
3M X
3y 3x ^ (D i ! w + (D^ + 2D ) ^ } 3x X , . . 2 ^ x y ' ^
3x^ 3y
^ (D ^ ^ + D ^^ ^ 3x^
2 I 3 W
2 9y
} , ( 2 . 2 1 )
3M 3M yx _ X 3y " 3x
3 _ (D l_w + D' i - ^ } 3y ^ . . .2 . . .2 9y 3x
28
Equations (2.19) represent the membrane forces.
Equations (2.20) the bending forces and Equations (2.21)
the transverse forces in terms of middle surface displace
ments .
Rib Element
Conventional folded plate and cylindrical shell
structures are designed and constructed with rigid end
diaphragms along the x-edges. To permit analysis of
structures which have flexible stiffeners along the edges
beam elements which are subjected to axial and bending
deformations in xz-plane are considered in the study. The
rib elements are attached to the soffit of the strip along
both x-edges. They serve two purposes: elastic end ribs
in lieu of rigid end diaphragm and integrally built-in
columns. To simulate the conditions of integrally built-in
columns the thickness of strip elements with elastic end
ribs in particular location is equated to zero. This is
shown in Figure 1 (b). Figure 3 (a) shows such a rib of
cross sectional area A, moment of inertia I about its
centroidal axis parallel to y-axis, and depth t'. The
distance from the middle surface of the strip to the cen
troidal axis of the rib is denoted by r and is
r = I (f+t) . (2.22)
Figure 3 (b) shows the displaced state of the rib in
terms of strip-displacements along x-edge. The displace
ments of middle surface of the rib in xz-plane at a nodal
29
line i are denoted by uf, w5, and ef, The displacement at
nodal line i and j in terms of plate displacements along
x-edges are
{6^} = <
r u. 1
r w. 1
1
U
r w .
3
> =
9" ^J
0 -r 0 0 0
0 1 0 0 0 0
0 1 0 0 0
0 0 1 0 -r
0 0 0 0 1 0
0 0 0 0 0 1
u • ^
-li
^-ii
-11
^-ij
U2.23)
w -ij
-Ij
It may be pointed out here that for r=o, the
centroidal axis of the rib coincides with the middle
surface of the strip as shown in Figure 3 (c) or the rib
becomes a separate body which is not integrally built
with the strip as shown in Figure 3 (d). In. the litera
ture on this subject the ribs are attached to the plate
structures in this fashion even though the structures are
not constructed this way. A rib attached to the soffit
has a relative stiffness of an effective L-Section which
is much higher than that of a rectangular section. The
deformation pattern of a rib established in Equation (2.2 3)
excludes its rotation about x-axis.
30
Philosophy of the Extended Finite Strip Method
The philosophy of the Extended Finite Strip method
is similar to that of the Kantrovich Method (25) for
approximate solutions of partial differential equations
of a function. In this study the governing partial diff
erential equation is in terms of the deflection function.
This equation is reduced to an ordinary differential
equation by expressing the displacement function as the
product: of two exclusive functions f(x) and ij; (yj so that
these functions satisfy the essential boundary conditions.
According to linear elastic theory of plates and
for shiall deflections, the membrane and bending action
in a plate or a strip element are independent actions.
The governing differential equation in membrane action
is expressed in terms of displacements u and v which are
functions of x and y, while in bending the action is
expressed in terms of the deflection w, which is also a
function of x and y. Each of the displacements u, v and w
are, therefore, expressed as products of a polynomial
and a set of linearly independendent base functions.
The polynomials [s] model the displaced shape of
a strip in the x-direction and satisfy boundary conditions
along the bounding nodal lines of the strip. The base
functions simulate the displaced shape in the y-direction
31
and satisfy the essential boundary conditions along edges
parallel to x-axis. The polynomials for membrane and
bending actions are developed sep4rately.
Polynomials,'
Membrane action: The u- and v-displacement of a point
in the middle surface of a strip element are expressed
as the sums of u_- and v -displacements respectively m m ^ jr J
c o n t r i b u t e d by i n d i v i d u a l b a s e f u n c t i o n s .
u = E u^ m ^
m = - 1 , 0 , 1 , . . . , 6 (2 .24)
V = Z V m ^
The displacements u and v are modeled as the product ' m m
of a first ordered polynomial and the appropriate base
function for the displaced shape.
U = (ai+ ap x) ij; (y)
v^ = (3i+ 32 X) ^^(y) m= -1,0,1,...,6 (2.25)
where a. and 3. are shape factors and 4 j (y) and ^ (y) are
the base function which are discussed in the next section
The first expression of the Equation (2.25) at a point
along the nodal lines i and j, where the base function is
unity, yields the amplitudes
32
mi ^
u . = ai + b ao m] ^ ^
(2 .26)
E q u a t i o n s (2 .26) e x p r e s s e d i n m a t r i x n o t a t i o n a r e
u . mi
u .
0
b <
a i
>
a2 ^
t .
(2.26a)
The vector [u . u .] is the vector of undetermined * mi mj
parameters and [ai 02] the vector of shape factors.
The shape factors are expressed in terms of undetermined
parameters by inverting the matrix in Equation (2.26a) as
' a i
<
a2
> =
0
-1/b 1/b
U . mi
> .
u . mj
(2.27)
By similar logic the 3-shape factors are expressed in
terms of the undetermined parameters v^. and v .. ^ mi mj
3 i
<
32
> =
0
-1/b 1/b
V . mi
V . mj
(2.28)
33
The displacements u . and v . for m=-l are the amplitudes
of the displacements along the x-edges. The displacements
u and V of Equation (2.25) expressed in terms of undeter-m in
mined parameters are
% = ^ (l-¥^ % i ^ E-^mj> m ^
V = { (1- ) v^. + -^ v^.} ^iy) m b mi b mj m -
(2.29)
L i m i t a t i o n s of L i n e a r P o l y n o m i a l : The s h e a r s t r a i n , Y„. ,^ i n xy
the s t r i p e l e m e n t i s
^ 3u ^ 3v ^xy 3y 3x
( 2 . 3 0 )
This expression for the assumed displaced shape is
+ -^{(31 + 32: ) ^^W ^
= (ai+a2x) (ip ) + 32^1^ y
(2.31)
where {ib ) is the first derivative of the function ^ m y
with respect to y. The shear strain in this modeling of
the displaced state is a linear function of x when in
actuality it is a second ordered function of x. This
in essence renders the strip more rigid in its rotation
about z-axis. Use of higher order polynomial will obviate
34
this situation, but requires introduction of one more
undetermined parameter and causes consequent increase of
the band width.
If the plate subject to membrane action is divided
into two or more strip elements the overall distribution
of shear strain, although linear over an individual strip,
approximates to a parabolic distribution over the entire
plate. The idea of a higher degree polynomial, therefore,
is not pursued further in this study.
Bending action: The displacement w at any point in the
middle surface of a strip element normal to plane of
the strip expressed as sum of w displacements contributed
by individual base function as
w = 2 w m= -1,0,1,...6 . (2.32a) m m
Similarly the slope in x-direction, 6,is expressed as
e = Z e m= -1,0,1,...6 . (2.32b) m m
The displacement w is modeled by the product of a third
degree polynomial in x and the appropriate base function
for the displaced shape as
2 3 w = (ai+aoX- +aciX +a^x ) ijj(y) • (2.33) m m
35
A third degree polynomial is uniquely defined in the
domain by four constants at the boundary. Since there are
four constants (the undetermined parameters) involved at
the two nodal lines of a strip the polynomial of Equation
(2.33) is chosen. The slope, e , in x-direction at any point,
the first derivative of w with respect to x, is m "^
3w m 9
m " ~J^ " (a2+2a3X+3a4X ) i|;(y) (2.34)
Two undetermined pa ramete r s w . and e . , t h e r e f o r e , ^ mi mi
are introduced at each nodal line i defining the amplitude
of nodal displacements and rotations. The amplitudes, at
a point on the nodal lines i and j where the value of base
function is unity, expressed in matrix notation are
w . mi
e . mi
w . mj
^mj
> ^=
1
0
1
0
0
1
b
1
0
0
2 b
2b
0
0
b^
3b2
"2
^ 3
a^ . ^
(2.35)
To express the shape factors in terms of undeter
mined parameters, the matrix in Equation (2.35) is inverted.
The shape factors, therefore, are
36
a i
a2
^3
a^
0
0
0
0
0
0
- V b 2 - V b V b 2 - V b
^ / b s V b 2 - 2 / b 3 V b 2
w . mi
m i , (2 .36)
w .
e . mj
S u b s t i t u t i n g i n t o E q u a t i o n ( 2 . 3 3 ) , t h e v a l u e s of a from
Equat ion (2 .36) and c o l l e c t i n g t h e t e r m s , t h e d i sp lacemen. t
w becomes m
r /n 3 2 ^ 2 3 . W = { ( 1 - — r - X + X ) W . +
m , 2 , 3 m i b b
2 2 1 3 ( x - — X + V x"*) e . +
1 , 2 m i
, 3 2 2 3, ( X - X ) w . + b^ b= ""
( - ^ - ' " ^ " ' ^ '"3' V W
( 2 . 3 7 )
The w and e f o r m= - 1 r e p r e s e n t t h e a m p l i t u d e m m
of t h e s e d i s p l a c e m e n t s a l o n g t h e x - e d g e s and c o n s t i t u t e
the r i g i d body d i s p l a c e m e n t s of a s t r i p .
37
Base Functions
The base functions, as was stated earlier, must
simulate the displacement in y-direction and satisfy the
boundary conditions along the edges parallel to x-axis.
The boundary conditions along x-edges may be such as to
permit: a) the displacement without bending deformation
of a nodal line or in other words the displacement of
nodal line itself, b) complete freedom of rotation about
x-edges and c) complete clamping of rotation about one or
both x-edges.
The first of these conditions is satisfied by the
first two base functions, Th (y) and ^ (y) , m=-l and 0. m m '
Physically speaking the later two conditions result in
hinged-hinged, hinged-clamped (against rotation) or
clamped-clamped nodal lines. Each one of these specific
cases is properly represented by a set of base functions
The upper limit on the value of m is fixed at 6
with the understanding that contributions from the higher
base functions are insignificant. The results confirm thin
assumption.
The base functions for the hinged-hinged condition
only are discussed here. Function space for the other
two conditions can be developed by replacing the functions
38
ip , (m=l, 6) by Vlasov functions [5J . These functions,
for the sake of completeness of this study, are listed in
Tables 4 and 5.
Base functions for membrane and bending action are
selected separately. Any asymmetric displaced shape, in
membrane or bending action, can be split into a combination
of symmetric and antisymmetric displaced shape. The
function shape for symmetric and antisymmetric cases are,
therefore, developed separately.
The displaced shape of a strip element is defined
by the deformation of its nodal lines. In Figure 4 the
nodal line-i can be thought of as displaced without bend
ing by u_^. in x-direction and v_,. in y-direction. In
general instead of talking about the displaced shape of a
strip element a reference is made to the displaced shape
of a nodal line.
Membrane Action
A displaced shape in membrane action is split into
two parts: the symmetric and antisymmetric displaced shapes
The function space for each part is developed seperately.
Symmetrically Displaced Shape: Figure 4 (a) depicts the
symmetrically displaced shape of a strip element subject
to membrane action. Each nodal line has two degrees of
39
Axis of Symmetry
NodalLine-i
Figure 4(a). Symmetrically Displaced Shape of Membrane Action.
oi X K i ' °»=2,4,6 H-H Y
1 • / 1 / 1 / • / 1 / 1 / 1 / 1 /
\J
/ •
/ /
/ •
/ /
/
Line-i
..J
u-— Axis of Antisymmetry
Figure 4(b). Antisymmetrically Displaced Shape of Membrane Action
40
freedom, u and v. This figure shows- the displacement
shape is comprised of two parts: 1) displacement without
bending, and 2) bending deformation.
The base functions for u- and v-displacements which
a priori satisfy boundary conditions along x-edges and
span the domain in only y-direction are respectively
denoted by ^_(y) and ^_(y) / and given in Table 1. ij; (y),
or in short, ]h define the u- displacement along a m '
nodal line while fi (y) , or fi , define the v-dia^lacement
along the same nodal line.
Antisymmetric Displaced Shape: Figure 4 (b) depicts
antisymmetric displaced state of a strip element. The
base functions which a priori satisfy boundary conditions
along x-edges and span the domain only in y-direction are
respectively denoted by \l) and n . These functions are •* "* m m
given in Table 2.
It may be pointed out here that these functions
have been developed to make this discussion complete.
The development of stiffness matrix equation for this
displaced shape in membrane action, however, is left out
of this study.
41 T a b l e 1 , Base F u n c t i o n s \l) and fi f o r Symmetric Membrane
A c t i o n .
m
- 1
i> m
Shape m
- 1
fi m
-2Z
Shape
§ i n I Z a
sin isX a
c o s l ^ a
cos 37TX
s i n l u x a cos
STT
Table 2 , Base F u n c t i o n s i| and Q f o r A n t i s y m m e t r i c Membrane A c t i o n ,
m ^
m Shape m Q
m Shape
- 2 Z a
s i n 2Try a •
cos 2 Try a
s i n ITTX a
s i n 6TTy a
cos 27Ty a
cos ^Try a
42
The displacements u and v are individually
approximated by a set of four base functions. This,
therefore, leads to eight degrees of freedom at a nodal
line in membrane action.
Bending Action
A displaced shape in membrane action is split
into two parts: the symmetric and antisymmetric displaced
shapes. The function space for each part is developed
separately. Figure 5 (a) depicts symmetrically displaced
shape of a strip element under bending action. Each nodal
line has two degrees of freedom viz. w and e. Figure 5 (b)
depicts the antisymmetrically displaced shape of a strip
element under bending action.
The displaced shapes of Figure 5 are comprised of
two parts: displacement without bending, and bending
displacement. The base functions which a priori satisfy
boundary conditions along x-edge and span the domain'in
y-direction are denoted by iJ; . These base functions for
- m
symmetric d isp laced shape and those for antisymmetric
displaced shape are given in Table 3.
43
Nodal Line-i
Axis of Symmetry
mi
ni=l,3,5
«-HJ
Figure 5(a). Symmetrically Displaced Shape of Bending Actioni
Axis of Antisymmetry
Figure 5(b). Antisymmetrically Displaced Shape of Bending Action
44
Table 3, Base Functions for Bending Action,
Syr
m
- 1
1
3
5
MTietric c a s e
m
1
Try s m —^
a
s i n i ^ a .
5 Try s m —*-
a
A n t i s y m m e t r i c c a s e
m
0
2
4
6
m
1 - 2 x -a
2TTV s m — ^
a
4Try s m j ^
a
s i n ^ ^ a
Table 4, Base Functions for Hinged-Clamped Condition,
m
- 1
0
1 t o 6
^m
1
1 -
s i n
4m+l V = J—
a
m a
u y . , m- • n s i n h
•" a s i n u^
^m s i n h \xj^
Table 5. Base Functions for Clamped-Clamped Condition.
m ^ m
-1
0 - 2X a
Ito 6 s m - ^ - smh - ^ - n^(cos - ^ cosh -^) ^m^
2m+l ^m - "2" TT
s m p^ - sinh y^
" n cbs vi - cosh y im m
CHAPTER III
STIFFNESS MATRIX EQUATION
Introduction
The formation of the stiffness matrix equation
of a strip is done in several steps. The stiffness matrix
equation referred to an elemental frame of reference for
membrane action of a symmetrically displaced strip,
is developed in the first step. The superscript 'ms'
refers to membrane action of a symmetrically displaced
shape. The symbol [K^^] is the stiffness matrix; {6 }
contains the generalized displacements of the strip and
ms
{f } generalized forces expressed in generalized co
ordinates. The stiffness matrix equation for membrane
action of an antisymmetrically displaced strip is
[K"^] {6™^} - {f"^} = 0 . (3.2)
The superscript 'ma' refers to membrane action of an
antisymmetrically displaced shape. Since no particular
need of antisymmetrical membrane action was foreseen
in the scope of this study. Equation (3.2) is not
developed in detail. However, it may be developed in
a manner similar to Equation (3.1) by using the base
45
46
f u n c t i o n s of T a b l e (2) i n s t e a d of t h o s e of T a b l e ( 1 ) .
The s t i f f n e s s m a t r i x e q u a t i o n r e f e r r e d t o an
e l e m e n t a l f rame of r e f e r e n c e f o r b e n d i n g a c t i o n of
s y m m e t r i c a l l y and a n t i s y m m e t r i c a l l y d i s p l a c e d s t r i p s .
,bs- .bs b s [ K " ^ J {6"^} - {f^^} = 0 ,
and ,ba .ba ba [ K " " ] {6""} - {f^^} = 0 ,
( 3 . 3 )
( 3 . 4 )
respectively are developed in the second step. The
superscripts 'bs' and 'ba' refer to symmetrical and
antisymmetrical bending action respectively.
The membrane and bending stiffness matrix equations
of symmetric displaced shape are superposed in the third
step tp obtain the combined stiffness matrix equation of
the strip as
ms
K
0
bs <
f ^
^ms
\ - <
,^^ \
r ->
^ms
^bs J
V = 0 • (3.5)
Equat ion (3 .5 ) i s r e a r r a n g e d t o g roup t h e e l e m e n t s of
{6 } and {6 } p e r t a i n i n g t o n o d a l l i n e s i and 3
t o g e t h e r as
[ K ^ ] {6^} - {f^} = 0 (3 .6)
or [ K ^ ] {6^} = {f^} (3 .6a )
47
The superscript 's' refers to symmetrically displaced
shape. The Equation (3.6) is referred to an elemental
frame of reference and is called the combined elemental
stiffness matrix equation of the strip.
The ribs or stiffners employed along the x-edges
of a structure are considered in fourth step. The ribs
are considered to displace in xz-plane alone. The
stiffness matrix elements of such a member in terms of
its planar displacements are established in the litera
ture [26] . These stiffness elements are evaluated in
terms of the corresponding plate displacements according
to Equation (2.23) in this step.
The transformation of the elemental stiffness
matrix equation to a global frame of reference is
effected in fifth step. The transformed equation,
[ic J {6^} - {f^} = 0 (3.7)
or [ic ] {6^} = {f^} , (3.7a)
is referred to as the generalized stiffness matrix
equation.
In the sixth step the generalized stiffness
matrix equations of individual strips are assembled to
form an overall generalized stiffness matrix equation of
the entire structure. This matrix is a banded matrix of
48
half-band width equal to thirty two.
The nod al line forces N are considered in the
seventh step- The generalized forces referred to a global
frame of reference due to these applied nodal forces are
developed and added to {f^} in Equation (3.7)
The solution of Equation (3.4) and (3.7) is
obtained on the computer by employing a Gaussian elimina
tion process using only the half-band width of the stiff
ness matrix.
Stiffness Matrix Equations of Membrane Action
Symmetrically Displaced Shape
Base Functions: The function space of the u and v
displacements is spanned in y direction by \l) - and
fi - base functions respectively. These base functions
for a symmetrically displaced shape are given in Table 1
Displacements: The displacements u and v of Equation
(2.29) when expressed in matrix notation are
r "1 u m
> =
V
m
(1-x/b)^
0
m .0
(1-x/b)^
{x/h)P m
m
0
(x/b)^ m
u . mi
V . mi
u mj V .
L "^3 J (3.8)
49
or ,m .ms = [c-j (C • (3.8a)
The vector of total displacement, therefore, is
U
V > ,m
'-1 ,m .m
ms [C"^] {d"" }
.m ,ms -1
.ms 'i
,ms 3
.ms
V
(3.9)
Stra in : The s t r a i n vec tor {e } corresponding to the m IT 3
displacement vector Tu v 1 obtained from ^ m m-i
Equation (2.16) by substituting therein a value of z
equal to zero is
"b V
0
(1 - ^ ) ^ b ^my
{e } = m
xm
'ym
3u m
3x
3v m
Y xym J
3y
3u
3y
0
X, (1 - Bfi b my
D m
b ^m
0
^b - ^
, 3V m I m 3x
my
0
b my
b m
u mi
mi
u mj
V . m:^
(3.10)
50
or ,ms (3 .10a)
The s t r a i n v e c t o r {e} c o r r e s p o n d i n g t o [u v j ^ ,
t h e r e f o r e , i s
{ £ } = B [• ' - 1 B m B m B m ] ,ms
.ms
' i .ms ^3
.ms
= [B"^J {6"^^}
(3 .11)
Stress: The stress vector {a } corresponding to {e } m m
is given by Equation (2.18a)
m
xm
< a > = E ' ym '
xym
m (3.12)
where [EJ is the square matrix of elastic constants in
Equation (2.18a). After substituting for {e^} from
Equation (3.10) the Equation (3.12) becomes
,ms (oj = [E] LB"] {6""-} (3.13)
51
The stress vector {a} corresponding to {e} is, therefore,
written as
io) = [ E ] [B:::^ B m B m B™]
, 6
.ms ' -1
ms 1
.ms >3
.ms
.ms = [EJ [B^] {6" } (3.14)
Forces: The membrane forces, i.e. forces acting in the
xy-plane of a strip, are the body forces distributed over
the entire plane. The components of these forces in
X- and y-directions are denoted by X and Y respectively.
These components expressed in the chosen function space
are
X = X_^ i|;_ + X^ ij; + X3 rp^ + X^ i^
and Y = Y , f2 T + Y. Q. + Y. J2-, + Y^ fit--1 -1 1 1 3 3 D O
(3.15)
where X and Y are amplitudes of these forces in the m m
function space. It may be pointed out here that for
uniformly distributed forces X and Y^ , m=l,3 and 5 • m m
are zero and X_, and Y , are respectively equal to the
52
magnitudes of X and Y. The body forces in matrix
notation are
X
Y
X_^ 0 Xj o X3 0 X^ 0
0 Y^^ ° ^1 ° ^3 ° ^5
^
^
-1
-1
^
n.
ip.
^,
^,
$7,
=- [G] {H} ' (3.15a)
Strain Energy: The strain energy U of a strip of
thickness t for the plane stress condition is
U" = i^tf^ f^ {e}^ {0} dx dy. •'o -/o
(3.16)
Substituting for {e} and {0} from Equations (3.11) and
(3,14), the stxain energy becomes
U-" = )5tpp{6"'^}^ [B^J^ [E] [B™] {6™=} dx dy V/Q •'O
(3.16a)
53
Potential Energy: The potential energy v" of the body
forces going through displacements [u vj is
V = / f C} ih dx dy. (3.17)
Substituting for \u v] and [x Y J from Equation
(3.9) and (3.15a), the potential energy becomes
:r 'o •'o
-a ID
V^ = J J {6"" } (c"" } {G} {H} dx dy. (3.17a)
Extremum of Potential Energy Functional: The potential
energy functional of the strip for the membrane action
is
$ = u^ - V^ . (3.18)
Differentiating $ with respect to the undetermined
parameters {6^^} and equating the derivatives to
zero, yields the equation
3(6^^}
,ms n r .ms , r i s or [K^^^J {6" } = {f '"} - (3.19a)
The Equation (3.19) is the same as Equation (3.1).
The stiffness matrix [K^^] and the generalized force
vector {f" }; evaluated from Equation (3.19a) are given
in Appendix A.
54
Antisymmetric Displaced Shape;
The function space of the u and v displacements
is spanned in y-direction by base functions ij; and n ^m m
(m=0,2,4 and 6) respectively. These functions for an
antisymmetric displaced shape, are given in Table 2.
The Equation (3.2) can be developed by using above
base functions in the derivation of Equation (3.19a).
As explained earlier this derivation is, however, left out of this study.
Stiffness Matrix Equations of Bending Action
Symmetrically Displaced Shape
Base Functions: The function space of the w displacement
is spanned in the y-direction by base functions \p ,
These functions for a symmetrically displaced shape are
given in Table 3.
Displacements: The displacement w of Equation (2.37)
when expressed in matrix notation is
m ^ i < ^ > % f^Cx)*^ f3(x)*^ f4(x)*^ w .
mi
mi > •
w m3
e L mj
(3.20)
55
where f (x) = (l-3_ x^ + 2. 3) ,f (x) = (x- x^ + . x ^ , K2 u3 b b '
f (X) = (^ x^ - ^. x ^ , and f^(x) = (- ^ x^ + i^ x^) .
b-
Equation (3.20) written in compact form is
w = [C^l {6^^} m ^ m -• m (3.20a)
The t o t a l displacement w, t h e r e f o r e , i s
w = [c' ,b ^b 5
.bs 0
.bs <5i
.bs •53
>bs
bsi r.bs = [C^^J { 6 ^ ^ } (3 .21)
Curvatures: The curvatures of a strip element in terms
of w are
(x} = < -
-(c^ ) _1 XX
-(c^i)
2(c''j)
yy
xy
2 3 W
2 3X
2
^ 2
3y 2
3 W 3x3y
1 XX 3 XX
-(c^) yy -(cb
yy
2(C^)xy 2(C^)xy
s'xx
-(c^) yy
2(C^) 5' xy
^^bs^ <5_i
.bs <5 1
6 .
jcbs ^5
(3.22)
56
or = [B^] (6^^ , (3.22)
where the subscripts x and y denote partial derivatives
Moments: The moment curvature relation is obtained from
Equation (2.20) as
M X
{M} = < M > = y
M xy
X
°y °
0 D x^
V
2 3 W
2 3x 2
3 W 2
3y
3 w 3x3y
bs = [Dj {x} = [DJ [B^] {6 } (3.23)
Loads: The transverse loads (parallel to z-axis)
carried on the surface of a strip may be either distri
buted, line and/or point loads. These loads are denoted
by symbols q,L, and P respectively. The load L is
/ located at distance x* from the origin of the elemental
frame of reference while the load P is located at
distances x and y . All loads considered are symmetrical o - o
The distributed loads are expressed in the chosen function
space as
57
<J = ^-i*_i+ ^i *i + q, "(-J + q *
= [q_i qi - s]
5 " 5
1| -1 1= [G^] (H) ,
^
^
4> (3.24)
where q_ ,q ,q and q are the amplitudes of q in 1 1 3 5
the function space. The line loads in the function
spaca are
L = [L L "1 1 S ^.
^
i>
L 5
= [G^J {H}
(3.25)
where L , L , L , and L are the ampli tildes of L " 1 1 3 5
in the function space. The concentrated load is
P = P ^ = p ~i ~i ~i
(3.26)
where P_ is the amplitude of the load in the function
space. For uniformly distributed and uniform line
loads, the amplitudes q and L , m=l,3 and 5 are equal ^ ^m m ^
to zero, and q_ and L_ are equal to magnitudes of q ai d
L respectively.
58
Strain Energy: The strain energy due to bending U^^, of
a strip is
bs r^ r^ t U = h j J {M}^ (x) dx dy. (3.27)
o •'o
Substituting for (x) and {M} from Equation (3.22) and
(3.23), the strain energy becomes
U^^ = hf^ [^ {6^^}^ [B^]^' D B^ {6^% dx dy. (3.27a) •'o *o
Potentential Energy: The potential energy v of the
transverse loads going through w displacements is
a ^b
o -o V^^ =f f wq dx dy + r w. L dy + P (W ) (3.28)
where w is the value of w at y= a/2 and x = x and
w is the value of w at x=x and y=y . Equation (3.2 8) 0 0 0 ^
in terms of expressed values of w, q, and L from Equations
(3,24), (3.25) and (3,26) yields
o o V = =/7'^. {6'==} [C^^J^ [GS]{H) dx dy
f / U"^^)^ rcf]^[G^]{H] dy +{6'=^^[cfjt P. o
(3 .28a)
59
Extremum of Potential Energy Functional: The potential
energy functional of the strip for symmetrically dis
placed bending action is
. ,bs -±)s $ = U - \r . (3,29)
Differentiating the functional$ with respect to the
bs undetermined parameters {6 } and setting the derivatives
equal to zero yields the equation
or [K^S] 16^^)= {f^=} . (3.30a)
Equation (3.30) is the same as Equation (3.3). The
Stiffness matrix [K J and the generalized force vector
{f } evaluated from Equation (3.30i) are given in
Appendix A. The force vector is listed there in three
parts for uniformly distributed q and L loads and point
load P respectively. It may be pointed out that for a
partial distributed load applied ov^r an area defined by
X and X and y and y , the double integral in Equation
(3,28a) is evaluated over these limits. Similarly for
a partial-line load applied over length y to y the
single integral of Equation (3.28a) is evaluated over
these limits. The generalized force for a partial-
60
distributed load is, therefore, obtained by substituting
for variable 'a' and 'b' the values of (y - y ) and (x2- x^)
respectively. In the same way for the partial-line
load the variable 'a' is replaced by the value (y - y )
to obtain the corresponding generalized force vector.
Antisymmetric Displaced Shape
The function space of the w displacement is
spanned in the y-direction by base functions ip . These
functions for the antisymmetric displaced shape are
given in Table 3,
The Equation (3.20) through (3.2 8) are developed
using these base functions instead of the base functions
of the symmetrically displaced shape. Since the prinr
ciples and the reasoning involved in the derivation are
the same, these steps are not repeated here. The counter
part of Equation (3.3 0) in this case is
i l _ _ = [K^^] (6^^} " {f^^} = 0 (3.31) 3{6^^}
or [K^^] {6^^} = {f^^} (3.31a)
where {6^^} = [6^^ 6 ^ 6 ^ 6^^f. The stiffness matrix 0 2 k 6
[K^^] and the generalized force vector f ^ are also
listed in Appendix A. The generalized force vector in
this case also is given in three parts for the uniformly
61
varying distributed and line loads, and the point loads.
The same comments for partial loads hold for this force *
vector as for the one of the previous case. It may be
noted that q and L are the amplitues of uniformly vary
ing loads in the base function ip , and that the anti-0
symmetrical point loads occur in pairs.
Combined Stiffness Matrix Equation
The membrane and bending stiffness matrix equations
of the symmetrically displaced shape are superposed,
according to Equation (3.5), to obtain the combined stiff
ness matrix equation of the strip. The generalized
r ms bs 1 displacement vector of Equation (3.5) , L<S <S J is
rearranged in the form
{6S} = [u_^^ v_^^ w_^^ e_^^ u^^ v^^ ^5i %i-
"-ij ^-Ij ^-IJ '-Ij ^^3 "5j -5j ^j]
(3.32)
The Equation (3.5) is rewritten in the function space
defined by {6^} to obtain the Equation (3.6).
The antisymmetric stiffness matrix equation for
bending action does not have its counterpart for the
membrane action and, therefore, this operation of
combining is not dotie in this case. However, the {6 }
t
62
is rearranged to group together the generalized displace
ments pertaining to i and j nodal lines. The rearranged
vector in this case, is
{6 ^} = [w_ . e_ . ... w . 0 . w - . e ,....- w . e .i^ -11 -il 51 51 -Ij -Ij sj SD-"'
(3.33)
The Equation (3.31) is written in function space defined
by {6^^} as
[K^^J{6^^} - {f^^} = 0 (3.34)
or [K^^]{6^^} = {f^^} . (3.34a)
This equation is referred to both global and elemental
frames of reference since both happen to coincide in this
case.
Rib Stiffness Matrix
The rib members are assumed to displace in the
xz-plane alone. The displacements outside of this plane
are neglected in this analysis.
Displacements
The transformation matrix of Equation (2.23) is
r s modified to express {6 7 in generalized coordinates {6 } This modified transformation matrix is denoted by [T^J .
63
The vector {6 } is written as
{6^} = [T ] {6^} 1
(3.35)
The transformation matrix [T ] is given in Appendix A.
Stiffness Matrix
The stiffness matrix [K] of the rib-element [26]
in {6 } coordinates is
[K] =
E A X 0
12E I 6E I X y X Y
4E I X y
Symmetric
-E A X 0
0 -12E I 6E I X y X y
. -6E I 2E^I 0 X y X y
E A X
0 0
12E I -6E I X y X y
4E I
J (3.36)
64
The stiffness matrix [K ] in generalized coordinates is
[K^J = [ T ^ ] ^ [K] [T^] (3.37)
and is given in the Appendix A. The stiffness matrix
[K ] is now superposed on the [K ] matrix to yield a new
matrix
[ K " ] = [K=] + [K^] (3.38)
which models the membrane, bending and rib stiffness for
a symmetrically displaced shape of the strip element.
Equation (3.6a) is now written as
[K^^] : {6^} = {f^} (3.39)
Transformation to Global Coordinates
The y- and y-axes in the global and elemental
Coordinates remain parallel to each other so that no
transformation of v- and 6- displacements from elemental
to global coordinates or vice versa is required. The
transformation of coordinates takes place in the xz-
plane alone. Figure 6 shows the elemental frame of refer
ence rotated in a positive direction through an angle 4)
with respect to the global frame. The u and w components
of a displacement in the xz-plane are transformed to u and
w in xz-plane. The relationship between u and w, and
65
u and w is
H _ < Iwj
COS<J)
-sincf)
sin(|)
COS(j)
/- _ ^ h <_ }
w —
( 3 . 4 0 )
The Equation (3.40) i s expanded to e f fec t t h i s t r ans fo r
mation between the genera l ized displacements {6^}
referred to elemental frame and the corresponding general
ized displacements r e fe r r ed to global coordinates {6 } .
(6^) = [T^] a^} (3.41)
where [ T ] is the transformation matrix. This matrix is 2
also given in Appendix A. The combined s t i f f n e s s matrix
equation of a s t r i p element re fe r red to global coord ina tes ,
there fore , i s
[T ] [K^'^l [T J^ {6^} - [T ]{ f^}= [ K ^ ] {6^} - {f^} = 0 2 -• 2 2
( 3 . 4 2 )
or [ K ^ J {6^} = {f^} (3 .42a)
It may be pointed out that this transformation is effected
in the computer. The Equation (3.42a) is the same as
Equation (3.7a).
66
Over all Stiffness Matrix
The overall stiffness matrix of a structure is
assembled from Equation (3.42a) of each strip by super
posing them at their common nodal lines. Such a matrix
is a banded matrix
Figure 6. Transformation of Coordinates.
of one-half band width of thirty-two elements. This
matrix is, therefore, stored in a rectangular array
thirty-two locations wide.
Nodal Line Forces
The line forces N along nodal lines of uniform
intensity are applied in the xz-plane along the nodal
lines. These forces have two components N^ and N^
referred to the global coordinates. These components
expressed in the corresponding function space are
^X = N^ _/-! ^ \ l n +N^3 n^<^X5 ^5
^Z = ^Z _^^-l + N^i n + N^^ ^3+N^^ ^3
(3.43)
where N , and N ^ are amplitudes of N and N in the xm zm ^ X z
function space. N ^ and N , m=l,3,5 are equal to zero xm zm ^
when N and N have a uniform intensity; and since \IJ_=1 X 1
67
(N ) = N and (N ) = N . These forces, applied at the
i-th nodal line, are written in the global generalized
displacements coordinates of this nodal lines as
N X
N
/^ ~\
(N ) 0 X -1
0
^NJ.i 0
(N^)^ 0 (N )3 0 (N ) .
0
0
.N
0 0
0 0
= LG^'J{H} »
0
0 0
(N^)^ 0 (N )3 0 (N^)^
0
0
0
4
0
Q
0
0
-1
>
-1
0
0
0
5 >
3
(3.44)
68
The potential energy V^ , of these loads is
V^ = / {6^}^ {H} [G^J {H} dy (3.45)
—s where {6^} denotes the global generalized displacements
of the i-th nodal line. Differentiating V^ with respect —s
to {6^} , the vector of the nodal generalized line force
is obtained as
3{6?} ^n = (f"} (3.46)
The force {f } for uniformly distributed line loads
along the nodal lines is also,given in the Appendix A,
{f } is superposed on the overall force vector at proper
location before running a solution of the overall
stiffness matrix equation.
Solution by Gaussian Elimination
The overall stiffness matrix equation is solved
for the global generalized displacement by Gaussian
ep-imination procedure using half band width storage.
69
CHAPTER IV
ANALYSIS OF PROBLEMS WITH KNOWN SOLUTIONS AND CONVERGENCE
T^ST
Introduction
The extended finite strip method is applied to
several problems whose solutions are reported in liter
ature. The purpose of this application is to obtain a
check on the validity of the stiffness matrix equation,
compare the results with those obtained from other
methods (which are also approximate) and to study the
convergence characteristics of the displacements and the
stress and moment resultants. A study of contributions
to the displacements from the individual base functions
is also made.
In the case of homogeneous isotropic rectangular
plates of overall dimensions a and B, the displacement
w normal to the plane of plate and the moment resultants
M , M and M at a point n on the middle surface of the X ' y xy ^
plate are expressed as [l]
w_. = a q a^/D n n
f ^ M xn
' \n
L xy>nj
* = ,
xn
^yn
. xyn -
> q a
(4.1)
70
where q is the uniformly distributed load carried by the
plate, D is the flexural rigidity of the plate and
°n' ^xn' ^yn' ^^ xyn ^ ® ^^® nondimensional coefficients
When the plate carries a concentrated load P the displace
ment and the moment resultants at a point n are given by
w = a P a^ n n
M xn
< M
& xn
yn > = V3
M xyn
yn
B. xyn
(4.2)
The nondimensional coefficient for deflection
and moment resultants at critical points on the plate
structures are evaluated and compared with the available
information.
The Displacement Contributions of the Base Functions
The a-coefficient for mid point displacement of
a square corner supported plate carrying uniformly is
tributed load is 0.02537. This value is the sum of
individual contributions from the four base functions.
These contributions are: 0.017575, 0.007642, .000201
and -.00004 8 respectively. It may be noted that the
contribution from the fourth base function is
71
significantly small. The omission of higher base
functions, therefore, is justified.
Plate Structures Under Bending Action
Figure 7 shows four square plates of uniform
thickness. Each has different boundary and loading
conditions. The graphical convention used is also
explained on this figure. The description of an individual
plate and results of its analysis follows.
Platd 1
This plate is a homogeneous, isotropic square
corner supported plate carrying a uniformly, distributed
load. The results of the present analysis along with
other results reported in the literature are compared
in Table 6. The results of Marcus [27] and Lee and
Balletros [27] are experimental. A value of o.3 is used
for Poisson's ratio v-.
Plate 2' — — «
This plate is a homogeneous, isotropic square
plate with opposite edges hinged, carrying uniformly
distributed load. The results of present analysis along
with available information are tabulated in Table 7.
A value of v equal to 0.3 is used.
'>>]/ ^\ys\y\Lf\ly\i^slys\yQ
^
^
sC ^ V V "W^ q
I r
72
Plate 1 Plate 2
\^j^.^LJL:^ L^L^
^
Plate 3 Plate k
Graphical Notation:
Support at corners only-
Point
Nodal line, edge or part length of edge
Fixed
L ©
N N N N N
Hinged
Free
Figure 7. Plate Structures of Different Boundary Conditions
Table 6. Values of a in Plate 1.
73
Extended Finite Strip Method
Nodal^ Lines
2
3
4
11
a at Point 1
0.01757
0.01757
0.01757
0.01758
a at Point 2
0.02533
0.02537
0.02537
0.02537
Description
Finite
Element [27]
2 x 2
4 x 4
6 x 6
Marcus [27]
Lee and
Balleteros [27]
Other Methods
a at Point 1
0.0126
0.0165
0.0173
0.0180
0.0170
a at Point 2
0.0176
0.0232
0.0244
0.0281
0.0265
Table 7. Values of a, 6 and 6 at Points 1 and 2 in Plate 2. ^ ^
Extended Finite Strip Method
Nodal^ Lines
2
3
4
11
a at Point 1
0.01487
0.01487
0.01487
0.01487
6^ at
Point 1
0.1440
a at Point 2
0.01299
0.01300
0.01300
0.01300
6 at X
Point 2
0.1218
Point 2
0.0233
Other Methods
Description
Timoshenko
[1]
Description
Timoshenko
[1]
a at Point 1
0.01509
6 at X
Point 1
0.1318
a at Point 2
0.01309
1
b at X
Point 2
0.1225
1 B at
y Point 2
0.0271
Nodal line in half plate
74
Plate 3
This plate is a homogeneous, isotropic square
cantilever plate carrying uniformly distributed load.
The results of analysisl along with other results reported
in literature are tabulated in Table 8. A value of v
equal to 0.3 is used.
Plate 4, w
This plate is a homogeneous, istropic square
cantilever plate carrying a set of antisymmetric point
loads at the outer corners. The results are presented
in Table 9 and the comparison is made with results from
another solution. A value of 0.3 is used for v .
Convergence Test
The convergence characteristics of displacement
and moment resultant values from analysis of rectangular
corner supported plates and plates hinged along opposite
edges were studied. Three overall width to length ratios
of these plates were chosen viz. 0.3, 0.6, and 1.0. The
results from analysis of these plates with increasingly
finer division of the strips, are given in Tables 10 and
11 respectively. Some generalized conclusions regarding
convergence may be drawn from these tables.
75
Table 8. Values of a at Points 1 and 2 in Plate 3
Extended Finite Strip Method
Nodal Lines
2
3
4
5
11
a at Point 1
0.12550
0.12594
0.12619
0.12629
0.12640
a at Point 2
0.12677
0.12771
0.12803
0.12815
0.12828
Other Methods
Description
Finite Element [27] 3 x 3
Finite Difference [27]
Livesley and Birchall 5 x 5
Experimental [27]
Leissa and Niendenfuhr
a at Point 2
0.1250
0.1250
0.1250
Table 9. Values of a at Points 1 and 2 in Plate 4
Extended Finite Strip Method Other Methods
Nodal Lines
2
3
4
11
a at Point 1
0.26441
0.26947
0.27008
0.27030
Assumed Displaced Shape;
w = A (1-cos 1 ) (1- ^)
Solution: A = 192 Pa' TT'* + 96T:2(I _^) Q
and a = 0.25242
76
Table 10. Corner Supported Plate Carrying Uniformly Distributed Load. Convergence Test Data at Center Point.
B / a
0 . 3
0 . 6
1.0
Mult
D i v i s i o n N . L . i
3
4
5
6
2
3
4
5
6
7
8
9
10
3
4
5
6
7
8
9
10
11
l i p l i e
b / a
0 . 0 7 5 0
0 . 0 5 0 0
0 ^ 0 3 7 5
0 . 0 3 0 0
0 . 3 0 0 0
0 . 1 5 0 0
0 . 1 0 0 0
0 . 0 7 5 0
0 . 0 6 0 0
0^0500
0 . 0 4 2 9
0 . 0 3 7 5
0 . 0 3 3 3
0 . 2 5 0 0
0 . 1 6 6 7
0 . 1 2 5 0
0 . 1 0 0 0
0 . 0 8 3 3
0 . 0 7 1 4
0 . 0 6 2 5
0 . 0 5 5 6
0 . 0 5 0 0
r
w max
0 . 0 1 4 1 6
0 . 0 1 4 1 2
0_.01408
0 . 0 1 3 8 4
0 . 0 1 5 0 3
0 . 0 1 5 0 3
0 . 0 1 5 0 3
0 . 0 1 5 0 4
0 . 0 1 5 0 4
0 ^ 0 1 5 0 1
0 . 0 1 5 0 1
0 . 0 1 4 9 9
0 . 0 1 4 9 8
0 . 0 2 5 3 7
0 . 0 2 5 3 7
0 . 0 2 5 3 7
0 . 0 2 5 3 7
0 . 0 2 5 3 6
0 . 0 2 5 3 6
0 . 0 2 5 3 6
0 . 0 2 5 3 5
0 . 0 2 5 3 5
q a V D
(M ) X max
0 . 0 0 5 6 9 9
0 . 0 0 5 5 3 5
q_.og5509
0 . 0 0 5 5 8 8
0 . 0 3 5 7 6
0 . 0 3 4 4 4
0 . 0 3 4 4 5
0 . 0 3 3 0 9
0 . 0 3 2 9 4
0^03287
0 . 0 3 2 8 8
0 . 0 3 2 8 5
0 . 0 3 2 5 5
0 . 1 1 5 3
0 . 1 1 2 0
0 . 1 1 0 8
0 . 1 1 0 3
0 . 1 1 0 0
0 . 1 0 9 8
0 . 1 0 9 7
0 . 1 0 9 6
0 . 1 0 9 6
(M ) y max
0 . 1 2 3 6
0 . 1 2 3 1
0^1228
0 . 1 2 0 6
0 . 1 1 9 5
0 . 1 1 9 1
0 . 1 1 8 8
0 . 1 1 8 7
0 . 1 1 8 7
0^1184
0 . 1 1 8 4
0 . 1 1 8 3
0 . 1 1 8 2
0 . 1 0 8 6
0 . 1 0 7 6
0 . 1 0 7 3
0 . 1 0 7 1
0 . 1 0 7 0
0 . 1 0 6 9
0 . 1 0 6 9
0 . 1 0 6 9
0 . 1 0 6 8
(M ) xy max
0 . 0 2 8 1 6
0 . 0 2 8 0 9
0^02806
0 . 0 2 7 9 0
0 . 0 6 1 9 8
0 . 0 6 2 0 3
0 . 0 6 1 9 1
0 . 0 6 1 9 1
0 . 0 6 1 9 2
0_^g6183
0 . 0 6 1 8 3
0 . 0 6 1 8 4
0 . 0 6 1 8 1
0 . 1 0 7 3
0 . 1 0 7 3
0 . 1 0 7 2
0 . 1 0 7 1
0 . 1 0 7 0
0 . 1 0 7 0
0 . 1 0 7 0
0 . 1 0 6 9
0 . 1 0 6 9
qa^
Node lines on half plate.
77
Table 11. Plate Hinged Along Opposite Edges (Free Along Other Two) Carrying Uniformly Distributed Load Convergence Test Data at Center Point.
B/a
0.3
0.6
1-0
Mult
Division N.L\
3
4
5
6
2
3
4
5
6
7
8
9
10
11
2
3
4
5
6
7
8
9
10
11
iplie.
b/a
0.0750
0.0500
0.0375
0.0300
0.3000
0.1500
0.1000
0.0750
0.0600
0.0500
0.0429
0.0375
0.0333
0.0300
0.5000
0.2500
0.1667
0.1780
0.1000
0.0833
0.0714
0.0625
0.0556
0.0500
r
w
0.0001043
0.0001043
0.0001043
0.0001043
0.001658
0.001658
0.001658
0.001658
0.001658
0.001658
0.001658
0.001658
0.001658
0.001658
0.01299
0.01299
0.01299
0.01299
0.01299
0.01299
0.01299
0.01299
0.01299
0.01299
qaVD
(M ) X
0.01151
0.01126
0.01117
0.01125
0.02512
0.04542
0.04440
0.04404
0.04388
0.04379
0.04374
0.04369
0.04367
0.04367
0.06833
0.1256
0.1228
0.1228
0.1213
0.1210
0.1209
0.1208
0.1207
0.1206
(My)
0.003227
0.003149
0.003122
0.003110
0.005623
0.01173
0.01142
0.01132
0.01127
0.01124
0.01123
0.01122
0.01121
0-01122
0-007253
0.02452
0.02365
0.02335
0.02321
0.02314
0.02309
0.02307
0.02307
0.02307
qa2
(M ) xy max
.001091
.001091
.001091
-001092
0.006245
0.006337
0.006344
0.006344
0.006344
0.006339
0.006333
0.006328
0.006326
0.006303
0.01917
0.01961
0.01967
0.01968
0.01968
0.01968
0.01967
0.01966
0.01964
0.01962
•
Node lines on half plane
78
1) The values of displacements stablize with the
division of (half plate) into 3 or 4 strips only.
2) The values of moment resultants stablize with
division (of half plate) into 6 or 7 strips.
3) For the ratio of strip width to length (b/a)
less than 0.05, the end results from 15 digit arithmatic
become inaccurate due to round off errors on the machine.
Structures Under Combined Membrane and Bending Action
A single span, single barrel, cylindrical shell
[l6,20] shown in Figure 8 was analyzed with three different
meshes viz. division into 4,8, and 12 strips. The results
of u and w displacements, referred to global coordinates
and the stress resultant T from three mesh sizes are
plotted on Figure 8. The results from the 12-mesh size •
analysis are plotted in Figures 9 (a) and (b) and compared
with the References [l6] and [20] .
The Rib Attachment
Figure 10 shows a square plate hinged along two
opposite edges and supported by elastic beams along the
other two. Figure 10 (b) shows the rib attached to the
plate with its centroidal axis conincident with the
middle surface of the plate. The value of parameter r
is zero in this case. Figure 10 (c) shows the rib attached
79
I J. i 1 A 1 J. ^
R=31.0'
L * * X A A A X
I i I 1 - ^ - -25 psf UT psf
a=62.0'
jLRigid End-St i f fener
o H
iDlH
6.0
U.O
2 .0
0 .0
12.0 1 o n - ^ jLa • ' - ' i
ft n. O . Lr "
o H
X U.O '
n n . U. U'
zq
8.0
J-o
X
k.O-
0-
-2 .0J
1
1 I I I
LwraXLL
Legand:
12 mesh-size
• 8 mesh-size
O k mesh-size
u and w are r e f e r r e to glor-dl fraTie of reference
Figure 8. C y l i n d r i c a l She l l S t ruc tu re ,
80
8.0
J-o
X
6.0
k.o
2.0
0.0
-2.0 ^ • 1 . •
0 Ref. [16] X Ref. [20]
CO
O H
0.0
-1.0
X -2.0
-3.0
1 * 1 "^^L, 1 1 1 • I 1 j 1 1
Figure 9(a). Plots of Stress Functions of the "ylindri-al Thell.
81
00 O
X
X
0.0(
- 2 . 0 .
• ) | . 0
^ n -- D . O
-8 0-
-
1 • /
^ ' 1
. EFS
© Ref. [16] X Ref. [20]
1
0 . 0
CO
o
X
X - 2 . 0
- l i . O
• v j
I I ^ ^ : ; ; ^ ~ • - • » I • I
I I I I I I \ * \ \ *
Figure 9 ( b ) . P l o t s of S t r e s s Func t ions of t h e C y l i n d r i c a : 5:h"l l .
82
Middle Plane
K . Centroidal Axis
1.0'
(c) Section at A
f ^ sSJ \ . t=0.5' A J -^ 4^r=0.0
1.0'
(b) Section at A
P r o p e r t i e s
r
A
E
V
D
I y
A=EI /aD y
Rib of F i g . 10(b)
0 .0
1.76
87 .36
0 . 3
1.00
0.U57
l+.OO
Rib of F ig . 10 (c )
0 .88
1.76
87-36
0 . 3
1.00
0.i457
U.OO
r
0 . 0
0.88
Method
EFS Method Ref. [ l ]
EFS Method
«1
.001+712
.OOU72O
.OOI151I+
^xl
.05350
.05280
.05193
^ 1
.OU63O
.OUU7O
.OU69O
Figure 10. Square Hinged Plate with Elastic Ribs
83
to the soffit of the slab when r = 0.88'. The compari
son of the displacements and moment resultants obtained
from analysis by EFS method with the available informa
tion [1] is done on Figure 10. No published data are
available to compare results obtained from analysis by
this method of a plate supported by elastic ribs attached
to the soffit.
84
CHAPTER V
APPLICATION OF THE EXTENDED FINITE STRIP METHOD
The Extended Finite Strip Method is applied to
several structures of different types to demonstrate the
verstality and potential of the method. The method is
also applied to several rectangular plate structures
having various boundary conditions to develop non-
dimensional coefficients for displacement, stress and
moment resultants at critical points on these plates.
The structures analyzed for the demonstration are
a) a rectangular plate with overhang, b) a series of folded
plates, c) a circular cylindrical shell with a canopy,
and d) a continuous plate subjected to support settlement.
Detailed discussion of these structures follows.
Rectangular Plate with Overhang
An isotropic rectangular plate of constant thick
ness, hinged part length along two parallel edges and
over hanging the remaining length, and fixed along the
fourth edge, is shown in Figure 11. The deflected shape
of this plate obtained from analysis by EFS, along
various lines shown is plotted on the same figure. A
value of V equal to 0.3 is used.
85
\ L > N U N l / \ l / N b \ l / N b \ U \ b s b N l ^ x I X x l / N l x s l x v l / N l / s b x l / q
w = a - f for
- - < T >
= 1.5
Deflection Along Line 1
0
.01
.02
.03 -j-
a
•^—^ 1—t-1.0a 1.5a
Deflection 'i 1 ong Line 2
Deflection
Along
Line 3
Figure 11. Isotropic Rectangular Plate with Overhang.
86
A oljded Plate Structure
A single span folded plate structure supported on
integrally built-in columns and employing i) the conven
tional rigid end diaphragm, ii) the unconventional
elastic end ribs of various sizes, and iii) a north
light window feature, is shown in the Figure 12. The
deflected shapes of these structure in the xz-plane, along
the middle and end sections are plotted in Figure 13 (a).
The maximum values of the stress and moments resultants
along the midline of these structures are plotted in
Figures 13 (b) , (c) , (d) , (e) , (f) and (g) . Values of
these resultants in the rib members are produced in
Figures 14 (a) , (b) and (c) .
A Circular Cylindrical Shell with Canopy
A circular cylindrical shell conventionally
supported by rigid stiffeners in the transverse direction
and employing a circular cylindrical canopy on both ends
is shown in Figure 15. The deflected shape of the
structure in xz-plane is shown on the Figure 16 (a).
The maximum values of stress resultants are plotted on
Figure 16 (b) and those of moment resultants in Figure
16 (c).
87
3.0' 1.0' 2.0' 1.0
0- psf
50.0'
Elevation
2'-3'
I n" 5'-0
3'-0'
9'-0'
Description
Diaphragm Rib No. 1 Rib No. 2 Rib No. 3
Figure 12. Folded Plate Structure vith Elastic End- ibs a-North Light Window Details. '
88
EJ Rigid End Diaphragm
A Rib No. 1
V Rib No. 2
O Rib No. 3
X North Light Window with Rib No. 3
(i) Displaced Shape Along Mid-Section.
(ii)
Figure 13(a).
Displaced Shape Along End-Section.
Displaced Shapes of the Folded Plate Ttrujlur-
-2.0
cn O
X
X
-k.o
-6.0
-10.0
Figure 13(b). Transverse Stress Resultant T (H^/ft) Alcr.c "'i Section of Plates.
90 12.0
10.0
-2.0
Figure 13(c). Longitudinal Stress Resultant T (lb/ft) Alon? Section of Plates.
k.o 91
2.0
0.0
-2.0
CO C H X
° -U.O
-6.0
-8.0
-10.0
-12. d
\ M ^
^Vee Kdge
Q R i g i d End Diaphragm A Rib No. 1
V Rib No. 2
O Rib No. ']
• Nor th L igh t Window w i t h Rib No. 3
3 Nor th L igh t Edges
>o^c
1 1 / \ 1 / 14-. \ ^ / \L/\\ * / jw \ \ \ / / fv \ \ \ ' / i III G \ \ ' / Mil
1 CK T
Fold 1 Line I
drff ^*^ 's<r^ \
/ / _ /
7/ i* /
Fold / Line /
Center Fold
Figure 13(d). Membrane Shear T (lb/ft) in Plates A.rr.c ti.o ?.rKi Section.
9 2
-8.0
CNI
o H -12 .0 X
X
- ] 6 . 0
-20.0
-2U.0
-28.0
Figure 13 (e ) . Transverse Moments M ( i b - f t / f ' t ) Alor.f, " i i of the P la te s .
i-Tect icn:
93
-2.0
Figure 13(f) Longitudinal Moment M (Ib-ft/ft) Along Mid-f-ecti, n of Plates.
9 4
10.0
CO
O H
2 .0
0 .0
- 2 . 0
m Rigid End Diaphragm
A R i b No. 1
V Rib No. 2
Q Rib No. 3
O North Light Window with Rib No. 3
J North Light Window Edge
Fold Line
Center Fold
Figure 13(g). Twisting Moment M (ib-ft/ft) in Plates Alcrp L urrc: Section.
95
^ ^ ^
1.0
0.0
in O
X
(^ X
-1.0
-2.0
Columns Fold Line
Fold Line
Center Fold
A Rib No. 1
V Rib No. 2
Rib No. 3
O North Light Window with Rib No. 3
Figure lU(a). Axial Force P (lb) in the Rib l':er^.\ ers.
96
J-o
X
t^
-3.0
Figure lU(b). Shear Force V (lb) in the Rib Elements
97
0 North Light Window with Rib No. 3
Fold Line
Center Fold
Figure ll+(c). The Moment M (lb-ft) in the Rib Members.
98
I 1 1 11 M i i 1 1 i M 1 m X'° ^''
2 1 /2" ^
Rigid — End
Stiffeners 31.2 ps^
R=8.0'
12.0'
V Uo.o'
figure 15. cylindrical Shell Structure with Canopy, Fig
99
(i) Along Mid Section
\
(ii) Along End Section
Figure 16(a). Displaced Shape of Shell Along Mid and End fectlcr
100 1 0 . 0
CO
-p
§ H :3 CO <u K
<0 CO
-p CO
- 1 2 . 0
- 2 . 0
- 1 0 . 0
F igure l 6 ( b ) . S t r e s s R e s u l t a n t s in S h e l l Alont^ " i d - r e c t i c r
101
CO 4J
0 CO (U
K
+J C <u e o
6.0
U.O-
2.0!
0.0
§ - . 2 . 0
-k.o
-6 .0
- 8 . 0
-10.0
-12.0
Crown Springing
M -
^-O""^^
M iO xy /
0 M X 102 ^ X
Q M X 10^ y
A M X 102 xy
Free Edge
Figure l 6 ( c ) . Moment Resultants in Shell Along Mii-fection
102
A Continuous Plate with Settling Support
A rectangular plate continuous over two spans is
analyzed to study the distribution of displacements and
the moments due to settlement of the central support.
The plate structures and plots of w-displacements and
the moments M are shown in Figure 17.
Nondimensional Coefficients
A series of isotropic rectangular plates of
uniform thickness supported in different manners at the
boundaries, carrying different types of loads and bearing
overall width to length ratios of 0.2 to 2 are analyzed^
to obtain the nondimensional coefficients for the
deflections and moment resultants at critical points on
the plates. The results obtained are produced in
Appendix B in the form of tables 4 d in some cases in
plots. The description of these plates and the loads
carried by them is:
Case I: Corner Supported Plates:
a) Carrying uniformly distributed load .
b) Carrying a central point load.
c) Carrying uniformly varying load.
103
10.0'
H—nr lOTO'
0.1
Hinge
(b) Vertical Displacement w Along Mid-Section.'
Hinge
(c) M Along Mid-Section.
Figure IT. Continuous Plate Subjected tc ?upp;
Settler.ent.
104
Case II: Plates hinged along two parallel edges?
a) Carrying uniformly distributed load.
b) Carrying a central point load.
c) Carrying uniformly varying load.
Case III: Plates fixed along two parallel edges:
a) Carrying uniformly distributed load.
b) Carrying a central point load.
c) Carrying uniformly varying load.
Case IV: Plates fixed along one edge and corner
supported at the opposite edge:
a) Carrying uniformly distributed load.
b) Carrying uniformly varying load.
Case V: Plates fixed along one edge and hinged along
the opposite edge?
a) Carrying uniformly distributed load.
b) Carrying uniformly varying load.
Case VI: Plates hinged along one edge and corner
supported at the opposite edge:
a) Carrying uniformly distributed load.
b) Carrying uniformly varying load.
Case VII: Cantilever plates:
a) Carrying uniformly distributed load.
b) Carrying concentrated load at mid point
of free edge.
c) Carrying symmetrically applied
concentrated loads at outer corners.
105
d) Carrying uniformly varying load,
c) Carrying antisymmetrically applied
concentrated loads at outer corners.
CHAPTER VI
CONCLUSIONS, OBSERVATIONS AND RECOMMENDATIONS
The following conclusions may be drawn from this
study of the extended finite strip method.
1) Prismatic plate and shell structures which
are free to rotate about their transverse edges can be
analyzed by the method.
2) The inclusion of rib elements along the trans
verse edges makes the method versatile for analyzing
structures; a) supported on integrally built-in columns;
b) carried on elastic ribs instead of conventional rigid
end diaphragms and employing features such as north light
windows.
3) The number of unknowns involved in the method
is small for equally accurate results when compared to
the finite element method. The fine mesh analysis of the
folded plate structure, 50 ft x 41 ft overall projected
area, required only 240 unknowns.
4) The contribution from the fifth and sixth base
function are significantly small and therefore exclusion
of higher base functions is justified.
5) The convergence tests show that the values of
displacements stablize with the division of half structure
into 3 or 4 strips while the values of stress and moment
106
107
resultants stablize with this division into 6 or 8 strips.
6) In the case of plate structures in bending,
for a ratio of strip width to length (b/a) of 0.05, the
end results from 15 digit arithmetic may become inaccurate
due to round off errors on computer. However, the number
of strips required also depends upon the boundary con
ditions along the transverse edges.
7) The method also permits the study of redis
tribution of stresses due to support settlement.
8) Representation of loads, especially the
uniformly distributed, constant and varying loads is
simple because of the first and the second base functions
used.
Some observations about the method developed
indicating its capabilities and limitations are:
1) The method is set up for analyzing these
structures made from orthotropic material although an
analysis of such structure is not included in the report.
2) The function space used is nonorthogonal.
The first two base functions are nonorthogonal with the
remaining six while these six are orthogonal between
themselves. The terms in stiffness matrix equation
pertaining to the first two base functions, therefore,
get coupled with those pertaining to the remaining six
base functions. This results in increased band width of
108
32 elements of the stiffness matrix equation for each
case of symmetric and antisymmetric displaced shapes.
The Gram Schmidt process was employed to orthogonalize
the function space but due to nonorthogonality of the
first and second derivatives of these functions, the
matter was not pursued.
3) The method has specific application to the
folded plate roof structures of cold-formed steel which
are made from orthotropic plate members and supported by
elastic end stiffeners.
4) The method as developed here considers column-
members to be integrally built- in the xz-plane alone.
The following recommendations for future research
are made:
1) The stiffness matrix equation for anti
symmetrically displaced shape under membrane action may
be developed.
2) Function space of Tables 4 and 5 may be
employed to develop stiffness matrix equation of bending
action for hinged-clamped and clamped-clamped conditions
along the transverse edges.
3) The soil-structure interaction may be
developed to analyze pipe culverts and any other
pipe structures with end stiffeners that are buried under
the ground.
109
4) The method may be adopted to an advantage for
the study of stability and vibrational analysis of
prismatic plate and shell structures that have end
stiffeners.
LIST OF REFERENCES
1. Timoshenko,S.,and Woinowsky-Krieger,S., Theory of Plates and Shells, Second Edition, McGraw-Hill, New York,1959.
2. Nadai,A., Elastische Platten, Springer, Berlin,1925.
3. Margurre,K.,Woernle,H., Elastic Plates, Blaisdell Publishing Co. ,Massachusettes ,1969'.
4. American Concrete Institute,"Building Code Requirements for Reinforced Concrete," ACI Standard 318-6 3, Detroit,Michigan,196 3.
5. Report of the Task Committee on Folded Plate Construction," Journal of the StrUctural^Division, ASCE,December,196 3. '
6. Cheung,Y.K.,"Folded Plate Structures by Finite Strip Method," Journal of the Structural Division,ASCE,Vol. 95, No. ST12. Proc. Paper 6985',December, 1969.
7. Goldberg,J.E.,and Leve,H.L.,"Theory of Folded Plate Structures," International Association of Bridge and Structural Engineering,Publications,Vol.17,1957.
8. DeFries-Skene,A.,and Scordelis,A.C.,"Direct Stiffness Solution for Folded Plates," Journal of the Stri ictural Division,ASCE ,August,1964 .
9. Goldberg,J.,Glauz,W.,and Setlur,A.,"Computer Analysis of Folded Plate Structures," International Association of Bridge and Structural Engineering,Publications ,Seventh Congress,196 4.
10. Mast.P.,"New Method of Exact Analysis of Folded Plates," Journal of the Structural Division,ASCE, Vol.93,ST2,April,1967.
11. Goldberg,J.,Gutzwiller,M.,and Lee,R.,"Analytical and Model Studies of Continuous Folded Plates," Journal of the Engineering Mechanics Division,ASCE,Vol.9 4, EMS,October,1968.
12. Zienkiewicz,O.C.,and Cheung,Y.K.,"Finite Element Method of Analysis for Arch Dam Shells and Comparison with Finite Difference Procedures," Proceedings, Symposium of Theory of Arch Dams,Southampton University,1964,Pergamon Press,1965.
110
Ill
13. Rockey,K.C.,and Evans,H.R.,"A Finite Element Solution for Folded Plate Structures," Proceedings, International Conference on Space Structures, University of Surrey,1966.
14. Williamson,F.A.,"Stress Analysis of Folded Plate Structures with Flexible End-Diaphragms," Doctoral Dissertation, Department of Civil Engineering,Texas Tech University,May197o.
15. Cheung,Y.K.,"Finite Strip Method Analysis of Elastic Slabs," Journal of the Engineering Mechanics Division,ASCE,Vol. 94.No. EM6 .Df^nf^mh^r , l fi
16. American Society of Civil Engineers,"Design of Cylindrical Concrete Shell Roofs," Manual No.31, ASCE,New York,1952. " ~
17. Rudiger,D.,and Urban,D., Circular Cylindrical Shells, Teubner,Liepzig,19 55. ~
18. Holland,!., Design of Circular Cylindrical Shells, Oslo University Press,Oslo,1957.
19. Portland Cement Association,"Design Constants for Interior Cylindrical Concrete Shells," Advanced Engineering Bulletin No.1,PCI,Chicago,1960.
20. Scordelis,A.C.,and LO,K.S.,"Computer Analysis of Cylindrical Shells," Journal of the American Concrete Institute, Proc. V.61,May,1964.
21. Clough,R.W.,and Johnson,C.P.,"A Finite Element Approximation for Analysis of Thin Shells," International Journal of Solids and Structures,Vol.4,196 8.
22. Mercea Scare, Application of Finite Difference Equations to Shell Analysis, First English Edition, Pergamon Press,New York,1967.
23. Fung,Y.C., Foundations of Solid Mechanics, Prentice-Hall,Inc.,Englewood Cliffs,New Jersey,1965.
24. Crandall,S.H. , Engineering Analysis, McGraw-Hill, New York,1956.
25. Kantrovich,L.V.,and Krylov,V.I., Approximate Methods of Higher Analysis, Interscience Publishers, New York,19 56.
112
26. Willems,N.,Lucas,W.M., Matrix Analysis for Structural Engineers,Prentice-Hall,Inc.,Enqlewood Cliffs, New Jersey,1968.
27. Zienkiewicz,O.C.,and Cheung,Y.K.,"The Finite Element Method for Analysis of Elastic Isotropic and Orthotropic Slabs," Proc. Inst. Civ. Eng.,28,1964.
28. Troitsky,M.S.,Orthotropic Bridges Theory and Design, The James F. Lincln Arc Welding Foundation,Cleveland, Ohio,1967.
APPENDIX
A. S t i f f n e s s , Force and Transformation Matrices
B. Nondimensional Coeff ic ients for Rectangular P la t e s
113
114
APPENDIX A: STIFFNESS, FORCE AND TRANSFORMATION MATRICES
The stiffness matrix K"^^ and force vector {f" }
are subdivided into matrices of size eight as
,ms [K"^^] =
K ms 11
(K-)t
K ms 12
K ms 22
and .ms {f--} = <
(f
f
ms 1 ms
,bs The stifness matrix [K ] is subdivided into matrices of
size eight as
,bs [K^"J =
K bs 11 K
(K^^)^ K
bs" 12
bs 22_
,ba The stiffness matrix [K" ] is subdivided in the same way
,bs as [K^^] also. The superscript 'bs' is replaced by 'ba'
in this case.
The lower quadrant of [K^^] and [K-^^] are de
noted by [K^^] and [K^^] respectively. The elements
^m ^m
in these quadrants are functions of a parameter y^ and
they repeat themselves in the upper left and lower right
quadrants of [K^^ ] and [K^^] with a changed value of y^.
All these submatrices and the force vectors (f } and
{f^^} are produced on the following pages. The transfor
mation matrices [TJ and [T2] , the stiffness matrix [K ] ,
and the nodal force vector {f^} are given thereafter.
115
m-< 1.- ^
t—1
IH
IH 3 \ fd
CNJ
1
IH
IH Xi
\ fd
CN
r-i
IH
IH Xi \
fd
1
r - l •
IH
IH 3 fd
o IH Xi . — » CM
fd t= ro \ \ fd Xi "^ C>J ^-^ • ^ 1
+ IH
\
CN
1 O
IH Xi ^ ^ CM
fd t= ro \ \ fd Xi -^
^ +
o 1
- • IH
CN
IH O 'fd 3 ro ro \ \ Xi fd CN - - -•^ 1
r - l
IH 1
IH O 'fd 3 ro ro \ \ Xi fd
^ +
IH 1
IH 3 \ fd
CM
»—1
IH 1
IH 3 \ fd
r>j
1
IH 1
IH
3 fd
IH xn - - ^ CM
fd t= ro \ \ fd Xi ^ y ^ - ^
^ +
1 1—i
IH
\ CN
1
o Iw XI *—X CM
fd t= ro \ \ fd Xi "^ CM v - -
^ 1
+ IH
\ CN
IH O 'fd 3 ro ro
Xi fd 1 ^ - ^ + \
IH ^ O " ^ ^ ^
^ +
+ O . X^ I H fd —^ CN Xi .H CN \ \ Xi ( d CM
1 -—
IH ^ ^ ^ \ ^
;> t=
^ 1
o . x ' fd IH ^ ^ \ JQ Xi CNJ CM
fd —
^ +
IH 'fd o CN ^-^ H XI \ CM Xi \
CM ( d t= ^ ^ 1
IH
\ - -
1 1
IH — O fd ^
VD Xi \ CN
CM ( d
^ +
,
IH
:> \
1 +
+ o I H fd — KD Xi \ CN Xi \ CM fd t=
1
I
u - H
IH fd ^
\ CN Xi \
CM f d
^ +
1
1
1
116
r*-. (0 g -
» - ^ ^
^•—. 'j*
X **-* !
•SU ! L
I
i 1 1
} j .-4
1 lu
X, lu
3 t : i n \
\ n) 1 Ci
1
1 1
I 1
^ IW
X IW
3 t= i n \ (4
CM -
^ IW
X
(= ro \ n)
tN
! 1
i
1
^ IW
X lu
s ,—, t= [/) <>i * ^
Srs" 10 (N
^ "7
X "•.
I t j jvO
O ^—tH
> i j a IW « ^ t= (0 i n
ro 1 CN| 1 \ \ 1 Si n) \ CM ' T !
1
a 1 + 1
Iw ^ t=
i n \ CN ^ 1
O .. > i j a
|w « ^ t: (d i n
ro + CM
\ \ A «J «* "»•
^ O 1
^ IW
^ 1=
i n \ CM
*"
o > 1 ' ^
IW Si ^•^ OI (0 1=
ro 1 cn \ \ Si <o CM "a*
- *—
o + r .4
. t=
ro \
1 CM
! T 1
1
1 " 1 >i - -! IW X!
*~^ c^ Id t=
i CO + c n
i \ \ 1 Si <tJ
^ I -^^ —•
o
IW"
>-* f:
ro
CM *—
1
*—«» - 1
X ! ^• • r
j
rolcg
1 1 1 1 1
^ i
IW
1 1
. X IW .. ^ Si t :
i n \ (0
CM
r^
IW 1
, X IW <^ Si
t= i n
\ 10
CM • * ^
1
^ IW 1
IW^
3 t= CO \
1 (0 CM
1 1 1 1 I j
' ^ '• IW
X IW
"a p
ro
\ (0 CN
*—* 1
I ' 1 ' ^ ' ^ -^ ^ ^ - 4 . - ^ ^
" • ' • > < " >« X X — X ^ «H>r I O H ^ ^ H '
j a x>| : ja l <o l O K o 1 ( O l ^ CM
j a ~ j a l fl
t= I O I P CN
o*
i :
c ! ' 1 ' ^-x '
>, Si : j Iw ( ! ^ t= ; (0 i n 1
ro + (N 1 \ \ Si <a •V "a*
1 j ; i 1
i
^ ^ 1 ' i - ^ i '
-
O 1 ' i ' , ! •
- 1 Iw ' ! ' • - ' \ ^^ 1 1 1 ^ ' i i n ! 1 '
\ , 1 CM j ""^ i
1 •
" j . >l 3
1
1
IW « 1 , ^ 1 = (0 in 1 i 1
ro 1 CN 1 1 1 \ \ 1 ! ! j a 10 ! ! CM ^
^
_ 1 o + .-< Iw - -* ^
i n \ CM
O > i - «
|W Si .-^ c
i !
1
1 1
1 i 1 1
(0 1 +J 0 1 ' C H 1 (u (u 1 : 6 N 0)
•H 0
1
1 U +J : i 1
1 C nJ •H 3
1 C CP : -H W ! 1 « 1
fO t= B 0) 1 CO + cn 1 (U >-i 1 1 • \ • > 1 « <o : ^ "0 ' ^ I" ! 1 1 w - ^
^ 1 " i '
•t 1
— _ — \ .— - -
1 i ^ 1
1 I W 1
' ^ 1 j i t= ! 1
C? 1 1 I \ ' ' \ ^ j '
J . i ..
i o 1 ! .>< i:;- '
IW ^ 1
— t . i (0 t=
ro 1 <n i ' > > Si "O
CM ^
^ _ ^ 1
O +
Iw" '
^^ ^ ro
CM • ^ 1
117
an
X
(Ojfo
j a N (0 t :
CN cn flJiro
•0\ 10
CM I (Ojin in CN
ja h= lOJin
10 hn CMJCN
. > 1
Iw •* \ X ? t= ro
O *- I *
\ t: ro ^—^
+ --. X . <o Iw t l
Si \ CN Si
en
^ O
\ -a-
P^ > t: ro
ro >— '-^ I
0) M lO
Ul 4J a
- » -g (U
rH
w
c •rH C
• f ^ 10
e (U
o u N
o
(0 3 cr w
. > • Iw <•*•*«
lO (N r-t
\ X)
M t :
a>
O . XI
CN
\ lO
^-^
Iw
X 7> t= ro ro
'5 '(0
cn
CN \ (0
Iw
in
O
i n
Iw "(0 CN
l ^ '
Iw IW
Io vo
-« : \ + O
i n CN
X) CN
10" j in
— XI
o <0
CN i IW
I 3 CN
i >
+ ja CM
m CN
Iw'
7>
i n in
- ' I I
Iw
i > I >'
m
X I
Iw
ro CO
l > ' Iw ^rm^
10 vo \ X) M
1=
<n -
O J ' - . .
X) CN V 10
• ^ - '
+
> i n
O
. X Iw ** A CN
\ 10
+
10 vo \ Si
r4
t: in CN
Iw Ifl VO
in CM
CN
(0
+
in
- I 10 !
vo
in <N
O
ja CN
<0
+
IW \ ^ XJ XI CN
i n
"io . X vc Iw \ ^ XI
CN t= \ <n (0 - ^ — +
^ +
u
•p
w
;^ I iw 0) M
a Io vo
^ XI
CN • :
V, cn
— +
[ K^^] : *• 1 1 •*
118
( 1 2 a / b M D ^ ( 6 a / b 2 ) D ^
( 4 a / b ) D ^
- ( 1 2 a / b M D ^
- ( G a / b 2 ) D ^
( 1 2 a / b M D ^
y = IT 1
Symmet r ic
( 6 a / b 2 ) D ^ X
( 2 a / b ) D ' X
- ( 6 a / b 2 ) D
( 4 a / b ) D X
(24a/7rbMD + X
(12TT/5ab)D 1
(12a/7Tb2)D + X
( l l T T / 5 a ) D 1
-(24a/TTbMD X
- ( 1 2 T i / 5 a b ) D 1
(12a/TTb^)D X
+ ( T T / 5 a ) D 1
(6a /b^ )D^+(6y V 5 a b ) D X I 1
+ (13y "^b/TOaMD +
(12y V 5 a b ) D 1 xy
(12a/7rb2)D + X
( iT/5a)D 1
(8a/Trb)D + X
(4bTT/15a)D 1
- ( 1 2 a / T r b 2 ) D X
- (Tr/5a)D 1
(4a /Trb)D -
(b7T/15a)D 1
(3a /b^)D^+(3y V 5 a ) D
+ ( l l y '*bV420a^)D 1 y
+ (y V 5 a ) D 1 xy
(2a /b )D +(2y 2b/15a)D X 1 1
+ (y '*bV210aMD 1 y
+ (4p 2 b / 1 5 a ) D 1 x y
/^bsY
- ( 2 4 a / T T b ^ ) D X
- (127r /5ab)D 1
- ( 1 2 a / 7 r b 2 ) D X
-{TT/5a)D 1
(24a/TTbMD X
+ (12Tr/5ab)D 1
-(12a/Trb^)D X
- ( l l T T / 5 a ) D 1
- ( 6 a / b ^ ) D - ( 6 y V 5 a b ) D X 1 1
+ (9y '*b/140a^)D 1 y
- ( 1 2 y ^ V 5 a b ) D j j y
- ( 3 a / b 2 ) D - ( y 2 / 1 0 a ) D X 1 1
+ (13y '*bV840a^)D 1 y
- ( y V 5 a ) D 1 x y
( 6 a / b ' ) D ^ + ( 6 y V 5 a b ) D X 1 1
+ (13y ' *b /70a ' )D 1 y
+{12y ^ / 5 a b ) D I xy
I
(12a / i rb2 )D
+ (iT/5a)D 1
(4a/TTb)D
-(bTT/15a)D 1
- (12a /Trb2)D X
-(TT/5a)D 1
(8a/TTb)D^
+ (4Trb/15a)D 1
(3a/b^)D +(y V l O a ) D X 1 1
- ( 1 3 y '*bV840a^)D 1 y
- ( y V5a)D 1 ' xy
+ (a /b)D - (y ^b /30a)D
- ( y '*bV280aMD
- (y^2b/15a)D^y
- ( 3 a / b 2 ) D - - ( 3 y V 5 a ) D X 1 1
- d l y '•b%420aMD 1 y
- ( y / / 5 a ) D ^ y
( 2a /b)D +(2y 2b/15a)D X I 1
+ (y '*bV210aMD,,
+ (4y 2b /15a)D 1 xy
119
X5 ^ Q
i n . -« \ <o <o \
CN t : nH ~-'
~- +
XI
i n
Id
10 ro \
1= X)
Si
i n \ 10
CN
Q
Io
X Q
Si i n \ lO
00 —
^ +
10 ro \ X)
Si ->6
XI t=
i n \ (0
r>4
XJ Id
CM
XI i n \ Id
CM
— a
Id
XI i n \ Id
CN '-^
•^ +
XI Id
(= CM
XI
i n
Id
Q
Io
XI t=
i n \ 10
CN
Q
Io \ +
^ Q
X) 1= i n \ Id
CN — '
^ +
XI Id
CM
X o Q ^ ^ Id X3 ro (= \
i n t= \ Si Id ^
00 - ^ ^ +
X) t :
i n \ Id
CM
Q
"id \
^ +
Id i n X)
10
— + ro
^ j a 10
i n Si t= \ 10
00 vo ro
+ X
Q ^ Q
XJ
10
10 i n
1= ro
XI
i n
lO CM
Id
i
m Q
j a ' -t= Si
i n Id \ \ Id t :
^ CN CM iH
I ^ —' I
X
XI lO t : i n
ro \ \ t= Id X) ^ —'
^ I
^ a Si
Id
Id m
ro
X} Id i n
ro
X -^
Si
i n
Id ro
s I
XJ 1= Q
i n ^ \ <0 10 \
CN f :
•^ +
X Q Q — ^ 10
t : i n XJ \ ro ja \ «= Id ^
00 —' ^ +
+ X a
X Q Q ' -^ Id XI m t= \
ro (= \ Si Id 1 "
00 »- '
X Q
Si t= \ Id
00
XI Id
i n \ t=
vo ro
^ Q
X ^ Q Q
CI Id XJ m t= \ \ (=
Id ro *!• ro
I I
X Q - » ' Q —
XI
Id
Id i n
ro
I ro
Id
Id i n
t= Si
n •u o c u 0) (U
(U rH O W -tJ
tyi rH C lO
•rl 3 c cr •H w Id g 0) 0) M OS Id
X Q X Q
Q — X -^ Q ^ —
^.^ tn
Si (= \ 10
00
XI 10
i n
\ vo ro
,^ OJ
XI p
\ (0 ^
lO i n
\ t= ro CO
--' +
,^ «n
ia (=
\ 10
00 ^
1 ' 1
X) 10
i n \ (:
VO ro «^
1
1 Q * - v
< N
' XI sr
\ (0
"S-
i
Q
-^ 10 i n
\ t=
ro '- +
120
• • > t ' ~1
0 ) CM
X i CM «
^
S
CO •P o
0) <u g tq 0)
•H O H 4J
— & V T -
C f - 1 -d
•H :3
•H H fd g <u « fd
0) CO
Xi p.
fc<
t ' ro
II
CO
p-
, CO >o Xi :x
« J
1
1
in
II
in p.
u -H u -p 0)
w 1
t 1
1
I
1 1
1
1
121
{ f ' ^=} :
q - L o a d
( a b / 2 ) q - 1
( a b V l 2 ) q - 1
( a b / 2 ) q - 1
- ( a b V l 2 ) q — 1
L-Load
[ f
[ f
[ f
[ f
x
X
aL - 1
a L - 1
aL - 1
aL —1
P-Load
[f
[f
[f
[f
(ab/iT) q - 1
[f ^ ) L TT
[f (Sin^^J^)P a
(abV67T) q - 1
[ f X TT
[f X ( S i n l l ^ ) P a
(ab / i r ) q - 1
[ f X TT
[f ( S i n ^ ^ ^ ) P a
- (abV6TT) q - 1
[ f X 2a TT
) L [f ( S i n ^ ^ ^ ) P a
(ab/3TT) q - 1
[ f X 3TT
[f 3^yo (Sin^i-^lJ^)P a
(abVlSTT) q — 1
[ f X JTT
[f ,2iZo. (Sin:^^:i^)P a
(ab/3TT) q - 1
- ( a b V l S T T ) q - 1
(ab/5TT) q — 1
(abV30TT) q - 1
(ab/5TT) q - 1
- ( abV30TT) q - 1
[f X 2 a
I ) L
3TT ' _
[ f X 3TT
[f X 5TT ' .
[ f DTT
[ f X DTT
[f X 2 a .
[ f 3TTyo
(Sin^^^^J-ii-)P a
[f (S i n l l I ^ ) P a
[f (SinllIXo.)p
[f 5 T T ^
(Sin^^-^^J^)? a
[f ( S i n ^ ^ I ^ ) P
[f U 0
( S i n 5 ^ I ^ ) P a
122
11
( 4 a / b 3 ) D ^ +
( 9 6 / 5 a b ) D ^ y
( 2 a / b 2 ) D ^ +
( 8 / 5 a ) D ^ y
( 4 a / 3 b ) D ^ +
( 3 2 b / 1 5 a ) D ^ y
Symmet
- ( 4 a / b 3 ) D ^ -
( 9 6 / 5 a b ) D ^ y
- ( 2 a / b 2 ) D ^ -
( 8 / 5 a ) D ^ y
( 4 a / b 3 ) D ^ +
( 9 6 / 5 a b ) D ' xy
y2" = 2-n
r i c
( 2 a / b 2 ) D ^ +
( 8 / 5 a ) D ^ y
( 2 a / 3 b ) D -X
( 8 b / 1 5 a ) D ^ y
- ( 2 a / b 2 ) D ^
- ( 8 / 5 a ) D xy
( 4 a / 3 b ) D ^ +
( 3 2 b / 1 5 a ) D xy
(12a/Trb3)D + ' X
(24Tr/5ab)Dj
(6a/Trb2)D + X
(22Tr/5a)Di
- ( 1 2 a / T r b 3 ) D ^
- ( 2 4 T r / 5 a b ) D i
(6a/Tvb2)D^+
(2TT/5a)Di
( 6 a / b 3 ) D ^ + C 6 u 2 / 5 a b ) D i
+ (13 i i5b /70a3 )Dy
+ a 2 y 2 / 5 a b ) D ^ y
C6a/Trb2)D + ' X
(2Tr/5a)Di
(4a/TTb)D + ' X
t8Trb/15a)Di
-C6a/Trb2)D ' X
-(2TT/5a)Di
C2a/Trb)D X
- t 2 T r b / 1 5 a ) D i
( 3 a / b 2 ) D ^ + ( 3 u 2 / 5 a ) D i
+ ( l l u 2 b ^ / 2 4 0 a 3 ) D
2' xy
( 2 a / b ) D + C 2 p 2 b / 1 5 a ) D i
+ Cy5b3/210a3)D
+ (4M2b /15a )D xy ^ .
€/
- (12a /T rb3 )D ' x
-C24Tr/5ab)Di
- (6a/Trb2)Dj^
- ( 2 T r / 5 a ) D i
(12a / i rb3 )D +
(24Tr/5ab)Di
- ( 6 a / T r b 2 ) D X
-C22Tr/5a)Di
- ( 6 a / b 3 ) D ^ - ( 6 p ^ / 5 a b ) D i
+ ( 9 p J b / 1 4 0 a 3 ) D
- ( 1 2 y 2 / 5 a b ) D ^ y
- ( 3 a / b 2 ) D ^ - ( u 2 / 1 0 a ) D i
+ ( 1 3 y 2 b ^ / 8 4 0 a ^ ) D
- ( y 2 / 5 a ) D ^ y
( 6 a / b 3 ) D ^ + ( 6 y 2 / 5 a b ) D
+ C13v2b /70a^)D
+ a 2 p ^ / 5 a b ) D ^ y
(6a/Trb2)D + X
(2Tr/5a)Di
(2a/ , rb)D^
- (2Trb/15a)Di
- ( 6 a / ^ b 2 ) D
-(2Tr/5a)Di
(4a/Trb)D + x
(8Trb/15a)Di
( 3 a / b 2 ) D ^ + ( p 2 / 1 0 a ) D i
- ( 1 3 p 5 b V 8 4 0 a )D
+ (P2 /5a )D^^
+ ( a / b ) D ^ - ( y 2 b / 3 0 a ) D i "+2 3
- ( U 2 b / 2 8 0 a )D y
- ( p 5 b / 1 5 a ) D ^ y
- ( 3 a / b 2 ) D ^ - ( 3 p 2 / 5 a ) D i
- d l u j b / 4 2 0 a ^ ) D
- ( y 2 / 5 a ) D ^ y
( 2 a / b ) D ^ + ( 2 u 2 b / 1 5 a ) D i
+ (P2bV210a^ )D
+ (4w2b/15a)D^y 1
123
1°
II I CM
! X I Q
XJ
10
XI
10 CN
I -^ X
Q ^ X Q " I
•0 in
VO
^ +
XJ ro \ 10
CM
10 i n \ Si r>i
XJ
i 10 I CM
10 i n
VO CM OJ I
Si
ro
lO
10 i n "V Si 0 0
XJ Id in \ CN
^ Q
XJ
\ Id
r<i
Id i n
vo
Q X '-.
Q XJ — Id <" i n XJ \ t= t=
\ CN Id r»
^ +
+ X
Q ^ Q
Id i n
vo
X Q Q — ^ Id XJ in t= \ ro JQ Id 0 0
^ ^ I w +
^x Q" Q ^ - ^ ja
tn Id XJ in 1= \ \ t= Id rM
XI
Id
CN
Id i n \ t=
vo vo
X Q ' - . Q
XI
\ Id
CN
Id i n
1= vo
- - X I
XJ m t= \ \ ^ Id CN
9 r»
I I
XJ
Id CN
Id i n \ VO VO
XJ
ro
Id
04
Id i n \ XJ
CM
X a Q
C4 Id
Si in
Id vo CM ^^
^ + m •tJ o c u 9) (U
+ I X
Q ^ Q
XJ
Id ro
XJ t= \ Id
vo
XJ 1= \ Id ro
Id i n
«X "O Q i n ^ f-H ja \ •: XJ Id ' 1 '
- ^ Q
XJ
\ Id ro
Id i n
1=
Q
+ X « Q i n
XJ
Id r j
XJ VO
X Q Q ^
XJ Id
i n \ 0 0
^ Q X Q
XJ
Id ro
Id i n XI
1=
Id vo
Si Id
i n \ 0 0
d Id XJ m
\ «= Id ^ ro ^
I
^ a Q i n ^ Q
Id i n
XJ
Id rM
X>
vo
XJ t= \ Id
ro
Id i n
XJ
I
Id in
Si
r H 0)
I
I c I - H
i.S \%
IS I
Id OI ja -
tc
X Q
j a
Id vo
^ Q XJ — Id
i n
0 0
XJ 10
i n
X Q Q —
a — XJ
Id vo
Si Id
i n \ 00
XI Id
i n
p ^
124
• • 1 — 1
fd CN
Xi <N « '""'
fd ^ jQ p.
Ui
(0 +J 0 c u Q) 0) e N 0)
rH O H -P
tnrH c: fd
•H ^ c \j*
•rH W fd S 0) 0) ^ Pi fd
t= • ^
II
J -
^
fd «fi Xi p.
«
!
^ vo
II
(O
P-
:
1 U
•H V-l -P
c r^
' 1 i \ ! I
; 1 1 i 1 '.
1 i i
1 1 ; i
1 2 5
{ f ' ' ^ } :
q - L o a d
( a b / 6 ) q
( a b V 3 6 ) q
( a b / 6 ) q
- ( a b V 3 6 ) q
(ab/2TT) q
[f
[f
[ f
[f
[f
X I
X
X
L - L o a d
(f)L 3 0
^4)L 3 0
f)L J 0
f)L 3 0
TT 0
P - L o a d
2 [ f
2 i f
2 [ f
2 [ f
2 [ f
X
X
X
( 1 - 2 X 1 ) p a
( 1 - ^ P
1 - 2 X 1 ( l - r i ! L ) p a
( i - ^ ) p a
(Sin^m
( a b V l 2 T T ) q
(ab /2TT)q
[ f X ^ ) L 7 T ' 0
[ f X TT 0
2 [ f X ,„ . 2TTyo ( S i n J
2 [ f X (S . 2TTy 0 m—*—
- ( a b ^ / 1 2 T T ) q [f X T T ' 0
2 [ f X (S i . 2TTyo i n •*—
( ab /4TT)q [f X TT 0
2 [ f X ( S i . 4TTyo i n — * - ^
( a b V 2 4 T T ) q [f X ^ ) L -^ 0
2 [ f X (S • 4TTyo m—^
( ab /4TT)q [f X
t TT 0 2 [ f
4TT' ( S i n H L o
- ( a b V 2 4 T T ) q
(ab /6TT)q
( a b V 3 6 T T ) q
(ab/6TT) q
- ( a b V 3 6 T T ) q
[f TT 0
[f 1 ^ ) L
[f X ^ ) L TT' 0
[f X
I ^ ) L
TT' 0
[ f ^ ) L TT 0
2 [ f X . 4TT' ( S i n H Z o
2 [ f X S i n i ^ ^
2 [ f (S 6TTy
i n — * -
2 [ f (S 6TTyo i n — * - ^
2 [ f X h 0
, _ . 6TTyo ( S m — i - i i
126
CD C7> O <X) r~^
W. *o
O -P
t o
O U
m
o -H -P fd o
M-I Ul
fd }^
EH
•H
-P fd
CM
CO
CM
irj eg
CM
(U ^ -p
M-I 0
+J CO 0 CcJ
0 M (U N
(!) V4 fd
(Q +J c (U g (1)
(U U3
I
cn
CO
J- in u) cvj .eg icM
H
ej ro
12
I
U) 0)
+J
c: •H TJ U o o u
O iH
o
<0
c: 0) 6 <u w
6 o u
c: o
-rH
•0
e o m (0
c U hi
(N
X •H U
S
-e-
sin
II
(0
-©•
cos
II
u
I
I u
U)
u 1/1
l / )
*«
u Irt
l/> I
I / )
I
I
u
u
to
m
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0) M to
U] • 1
d 0) 6 0)
i H
0)
tr> C
-H c
•H itJ e 0) p^
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0 -p
.H fO D rr 0)
«/>
128
«
CN
CO
IX)
CN
t n
CN
J -
CM
r—1
t o i-H
l—t
cn
CD
-
< X
w u
Si \ <
X
Si \ <
X
u
Si
< X u
O l
M
cn
\
H X
CM .H 1
CM
Si \
> 1
H X
w
<n
> i M
X w
o
X H W X
CM [ r ]
>-l OJ
1 + CM
Si \
> 1 M
X w VD
1
i3 \ <
X w
X X CM ^
u +
_—
XI \ <
X
1
Si \ <
w
J -
CN
w -P o C VH 0) (U B N (U
• - - I o <U -P
•H ::$
t n
CN
CM
Si \
>. M
X w 1
m
X) \ M
X
<N
. )
^ ^ _ J
vo CM
X X (M ^
t
CM
t
1
1 1
1
129
••5-n {f"}^:
aN
aN
(2a/TT)N
(2a/TT)N,
0
(2a/3TT)N X
(2a/3TT)N,
(2a/5TT)N X
0
(2a/5TT)N,
0
APPENDIX B: NONDIMENSIONAL COEFFICIENTS FOR RECTANGULAR
PLATES
Nondimensional coefficients for displacements
and moment resultants at critical points on rectangular
plates of homogeneous isotropic material and uniform
thi ckness are given, in tabular and graphical form on the
following pages. The information given is for plates of
overall width to length ratios of 0.2 to 2.0 supported in
different ways at their boundaries and carrying different
types of loads. These plates are listed in Chapter V.
130
131
>1 ,—1
0 M-I •H
D tn a •H >1 ,
u u fd
u (0
0 • fd rH 04
TJ O -P )-l
o D* CU :3 cn 1 ' 0 c u 0 1 U J
u fd iH 3 D> •
fd fd -P o O tS\ 0 « TJ
0 O 4J •H 13
Iso
tro
D
istr
i (a
)
M
1
w
CA
S
CM
>l
X oa
•-H
>1
oa
-1
X ca
(-1
3
PQ fd
in (X)
vo iH
o o
o ro r>-r-i rH
O
ro cr» rH O O
O
VO ^ fO iH O
O
(N
O
—»
-•
VO
o 00 (N O
O
O 00 CN CM rH
O
iH in in o o
o
00 o ^ fH o o
ro
o
I, r
^f
^
vo CM <y» ro o o
o o rsj rva rH
O
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vo iH iH O
O
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1
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ro r o o
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o
00 f-i vo r-t
o o
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o
^
ro in tr 00 o
o
in DO in o o o
o o vo in ro o rH <-\ r-i r-t
O O
iH 00 tr CM vo r» vo 00
o o o o
CN 00 iH O 00 t-i n-i CM O O
O O
00 <y>
o o
o iH t O rH
O
o ro r-o r-l
o
o 00
o r-i
H
O
r ro in CN
o o
o
rH
ro •
o II
:>
o o tr r 00 a>k f-i CM r-i r-i
O O
o o 00 ro ro o o o r-l rH
o o
o o rH 00 r- in ro vo rH rH
o o
rH VO ro CM r-i <T\ ro ro o O
o o
r-i rsj
rH t-i
a J-
fd cr
II
5
o <J\ o tr r-i
O
00 in vo cr> o
o
o CM
r« CTi r-i
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rH VO cri tr
o o
ro
r-i
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ro 00 CM cri o
o
o r-i r^ ro CM
o
(S\
r-CM vo o o
tr
r-i
C X
v^
r
^
c X 4
o tr ro vo r-i
o
ro o <Ti 00 o
o
o in r vo CN
o
vo CM <S\
r o o
in
r-i
CN
(d cr
c: >i
oa v/"
I I .
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• > • —
o o ro O vo 0> O CM tr IT r^ oc rH r-
o c
CM r-i CN tr in r-i 00 00
o o o o
o o tr 00 vo r O t ro ro
o o
rsi in iH cr» tr (T> CM O r^
o o
vo r
r-i r-i
>1
X oa
d" >i
X
» r^ 00 1 <s\ o 1 rH CM
» o o
r-\ tr vo 00 r ro r- r* o o
o o
o o r- r-i r^ 00
a\ ro ro tr
o o
vo in ro 00 in 00 r-i r-i
o o
00 CTs
r-i r^
1 •ri
c 3 •
CO MH W 0 0
c 0 ^ M U (d -ri
sz W +J
0
o ro OS rH CM
o
r-i r-i O r o
o
o CJ vo 00 tr
o
in cri CM CM
o
o
rsi
n U 0 C M-l 0 0 u C C © •
4-> 0 a-M -M •H c (d >-l H rH u 0 a cn Oi
•• fd >-i XI 0 to rH 0 :3 0 x: 0 0 M-l 4J
§ -
CO 4J 4J
CM
132
O X
CO
c
O O
c o
0.0 0.5 1.0 1.5 2.0
Ratio B/a
CASE - 1 ( a )
133
fd u -P c: 0 u
C •ri > 1 U u fd u cn 0 +J fd
r-i
(U
TJ 0 -p U o cu a, :3 in u 0 c: u o
u fd
rH :3 . CPTJ c »d fd o -P iJ o 0 TJ
cn, 0 o fd
• H >H 0 4 - P
o c
CM I
rg > i X
oa
o o v o r - o o ^ o r o v o o o o r H r H o o N v o c M o o r o r -r ^ r H ^ ^ f * ^ v o o o o ^ o ^ H r o t r l n v o v o ^ ~ » o o o o o ^ o ^ r ^ < y > C y » O O O O r H r H H r H r H r H r H r H r H r H r H r H
r H r H r H C M r M C N r g r s i c M r s j c N C M r M C N r N j r M r s i c M r M
o o o o o o o o o o o o o o o o o o o
<N I
o r-i X
r
f
oa
00 CM r-CM
ro rH ro vo
ro vo r CM
in 00 vo o
00 tr ro <T\
00 o tr 00
CM rsj r> r
00 <j\ r-i
r
vo 00 r vo
tr in tr vo
r-i 00 r-i VO
ro in <Tt in
00 in r* in
o c in in
ro • "^ in
ro r-i
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r-(Ti r-i in
r-i <T\ O in
in <y> cn tr
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CM I
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cn
ro
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in vo r-i
r«
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cri c o 00
<T> CM in 00
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CM • ^
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tr vo o CM
o o
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CM t-i
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in
vo o
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t r o o
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00 in vo
r^ CT> (T\ 00
CN CM O ro
in ro
m fd CM ro
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rM
ro o
II
CM
fd c
C C >i X >i X
cn ca ca
c X s
m
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c o
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u (d
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a o
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134
(M I
o X
w
9 O
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3.0
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1 0
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0.0 1
1 \
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r \
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1 1 I I 1 1
r k
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u 1 a
B
\
X
V = 0.3 W = aPa^/D
iWy 1= / / y i p
S y at Point 2
xy
^xh s , , _
\ ^
Qp
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Ratio B/a
2.0
CASE - K b )
135
CM I
o MH •H j::
D cn C •H >i u u fd u (fi 0 4-) fd
r-i
TJ 0 -p u o
02
>H 0 C >H O u U fd rH
cn c fd +J u . 0 TJ (Xi fd o
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p >1 O 5H cn fd H >
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X
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tr
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CN
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c rvj o •
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ro I
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r-i LO
CN CM ro ro L O l O L O l O L O L O l O L O
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CN ro "^ vo Ci ro LO CO o ro OJ rM vo
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CP c
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cn 0 tT'
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0 +J -H cn 0 Ou 04 O
cn
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r-i AJ 3 3 CnX> C-H
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ly Distr
trop
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o -H cn c M D
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fd
M M
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1
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> 1
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1
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CN
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oa
p - 4
1
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CM
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X oa
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CN
3
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O r-i X
«
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00 o o 00 rH i n CM <y> t r ro o ro O rH CM
o o o
o ^ o H CM 00 CN CM r o ^ rH t r rH ro in
o o o
t r o o r- r* vo (7^ rH r-t r rH <T> O r-i r-i
o o o
rH O O o o o KD <T\ r-i i n CM ro
O rH CN
vo ro vo o t r 00 CM O OJ o rH ro o o o
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0 ^ 0 4 0J6 o l LO 12. r 4 16 18 2 0
Ratio B/a
CASE - 1 1 ( a )
138
I
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CN ro ro ro tr
X
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CSS in
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