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TRANSCRIPT
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1
There are three important steps in the computational modeling of any
physical process:
(i) Problem definition,
(ii) mathematical model, and
(ii) computer simulation.
The first natural step is to define an idealization of our problem.
The second step of the modeling process is to represent our idealization of the
physical reality by a mathematical model: the governing equations of the
problem.
After the selection of an appropriate mathematical model, together with
suitable boundary and initial conditions, we can proceed to its solution.
The three classical choices for the numerical solution of PDEs are:
(i)FDM
(ii) FEM
(iii) Finite volume method (FVM).
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Solution Methods
The solution methods can be classified in three categories:
Analytical Methods, Approximate Methods, and Numerical Methods.
Analytical Methods:
They provide closed form exact solutions to the mathematical model of engineering
problems. They can be used only if the geometry, loading and boundary conditions of the
problem are simple. Integration methods and other analytical solution methods of
differential equations are the examples of the analytical methods.
Approximate Methods:
They provide closed form approximate solutions to the mathematical model of engineering
problems. They can be used only if the geometry, loading and boundary conditions of the
problem are simple. Ritzs method, Galerkins Method, Collocation Methods, Least Square
Method, Moment Method, Kantrovichs Method, etc.
Numerical Methods:
They provide discrete form approximate solution to the mathematical model of engineering
problems. They can be used to solve the problems with relatively complex geometry,
loading and boundary conditions. In particular finite elements can represent structures of
arbitrarily complex geometry. Finite Difference Method, Finite Element Method,
Boundary Element Method, etc.
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A Finite Element method is a numerical technique to obtain
an approximate solution to a class of problems governed by
elliptic partial differential equations.
Such problems are called as boundary value problems.
The finite element method converts the elliptic partial
differential equation into a set of algebraic equations which
are easy to solve.
-
The initial value problems which consist of a parabolic or
hyperbolic differential equation and the initial conditions
(besides the boundary conditions) can not be completely solved
by the finite element method.
To solve an initial value problem, one needs both the finite
element method as well as the finite difference method where the
spatial derivatives are converted into algebraic expressions by
FEM and the temporal derivatives are converted into algebraic
equations by FDM.
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SPATIAL DISCRETIZATION ANALYTICAL
Polynomials Legendre Fourier series Bessel Chebyshev Etc.
NUMERICAL Dynamic relaxation Finite difference Mesh free Collocation DQM Finite element MWR Finite strip ETC
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6
SPATIAL DISCRETIZATION ANALYTICAL
Polynomial Legendre
Fourier Series Bessel
Chebischev Etc
NUMERICAL Dynamic Relaxation Finite Difference
Mesh Free Collocation
D Q M FINITE ELEMENT
M W R Finite strip
METHOD OF SOLUTION Equilibrium Method
Energy Method
Conservation of Energy
Rayleigh Method
Minimum Potential Energy
Virtual Displacement Method
Finite Difference Method
FINITE ELEMENT METHOD Lagrangian Multiplier Method
Ritz Technique
Levy’s Solution
Galerkin Method
Differential Quadrature Method
Boundary Characteristic Orthogonal Polynomials (BOCP)
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MECHANICAL COMPONENTS / ELEMENTS
BEAMS, PLATES, SHELLS
VERY THICK – 3D
THIN
MODERATELY THICK – 2D
THICK
VARIATION OF FUNCTION IS
INDEPENDENT OF Z CO-ORDINATE
ANALYSIS
ELASTICITY EQUATION
U(X,Y), V(X,Y), W(X,Y), ᶲ(X,Y), ᴪ(X,Y)
LAMINATED STRUCTURES SEMI- ANALYTICAL – 1D
(FINITE STRIP)
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TWO ASPECTS
FORMULATION SOLUTION
BALANCE LAWS
VARIATIONAL PRINCIPLES
ALL PROBLEMS CAN BE MODELLED
BY DIFFERENTIAL EQUATION
EQUATIONS
COUPLED NON LINEAR
DYNAMICAL SYSTEM
EVEN TODAY WELL FORMULATED
PROBLEMS STRIVE FOR QUICK AND
EFFICIENT SOLUTIONS
1. ANALYTICAL
2. NUMERICAL
VERY FEW NON-LINEAR BOUNDARY
VALUE PROBLEMS HAVE ANALYTICAL
CLOSED FORM SOLUTIONS IN SIMPLE
FUNCTIONS
-
PROBLEMS ANALYSIS
RESPONSE STABILITY
STATIC DYNAMIC STATIC
TIME DOMAIN FREQUENCY DOMAIN
(EIGEN VALUE ANALYSIS)
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NON LINEARITY
• MATERIAL
ELASTIC, ELASTO PLASTIC
VISCO PLASTIC, TEMPERATURE DEPENDENT
• GEOMETRICAL
VERY LARGE, MODERATELY LARGE
SMALL STRAINS, SMALL ROTATIONS
LARGE DEFLECTIONS
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NON LINEAR VIBRATION
MECHANICAL
LOADING
THERMAL
LOADING
NONAXISYMMETRIC
PROBLEMS
AXISYMMETRIC
PROBLEMS
(ζ = 0)
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TIME DOMAIN
EXPLICIT
EULER
RUNGE KUTTA
DIFFERENCE METHODS
ETC.
IMPLICIT
NEWMARK – β
WILSON – θ
HOUBOLT
PARK STUFFY
ETC.
RESPONSE STABILITY
BIFURCATION
CHAOS
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Virtual Work Principle
The displacements are called virtual because they are imagined
to take place.
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Virtual Work Principle
When a mechanical system experiences variations in its configuration, it is said to undergo
virtual displacements. These displacements need not have any relationship to the actual
displacements that might occur due to change in the applied loads. The displacements are
called virtual because they are imagined to take place.
(1)
The external virtual work done due to virtual displacements δu in a solid body Ω subjected
to body forces f per unit volume and surface tractions t per unit area of the boundary is
given by
(2)
The total internal virtual work is obtained by integrating the over the entire volume of the
body and is given by
(3)
The principle of virtual displacements can be stated as:
If a continuous body is in equilibrium, the virtual work of all actual forces inn moving
through a virtual displacement is zero:
(4)
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Isoparametric Elements
Element is based on the transformation of the parent element in
local or natural coordinate system to an arbitrary shape in the
Cartesian coordinate system by use of shape function of the
rectilinear elements in their natural coordinate system and the
nodal values of the coordinates.
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16
Governing Equations 0 qKPKqM Ge
tPPtP tS cos
crS PP crt PP
and are static and dynamic load factors
,
Equation of motion
0cos qtKPKPKqM GcrGcre
Equation reduces to
042
1 2
qMKPKPK GcrGcre
(1)
(4)
(5)
(2)
(3)
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(i) Free vibration:
= 0, = 0 and 2/
02 qMKb (1)
(ii) Vibration with static axial load:
= 0 and 2/
02 qMKPK Grcb (2)
(iii) Static stability:
= 1, = 0 and = 0
0 qKPK Grcb (3)
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Buckling, Vibration Dynamic Stability
(i) Governing equations for free vibrations are
0 qKqM b (1)
(ii) Governing equations for vibrations with in-plane loads are
0 qKPKqM Gb (2)
(iii) Governing equations for Static stability or buckling
0 qKPK Gb (3)
(iv) Governing equations for Dynamic Stability
0cos qtKPKPKqM GcrGcrb (4)
where bK , GK , M are overall elastic stiffness, geometric stiffness, and mass matrices
respectively, q is the displacement vector. The elements of overall matrices in equations
(1), (2) and (3) can be generated through the assembly of corresponding element matrices.
The eigenvalues of the above equations give the natural frequencies and buckling loads for
different modes. The lowest values of frequency and buckling loads are termed as the
fundamental frequency and fundamental critical load of the structure.
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The in-plane load tP may be periodic and can be expressed in the form
tPPtP tS cos (5)
where SP is the static portion of P. tP is the amplitude of the dynamic portion of P and
is the frequency of excitation. The static elastic buckling load crP is the measure of the
magnitudes of SP and tP ,
crS PP , crt PP (6)
where and are termed as static and dynamic load factors respectively.
Equation (4) represents a system of second order differential equations with periodic
coefficients of Mathieu-Hill type. The solution of this equation may be bounded or
unbounded, which depends on the combination of the values of dynamic load factor and
frequency of excitation. The development of regions of instability arises from Floquet’s
theory, which establishes the existence of periodic solutions. The boundaries of the dynamic
instability regions are formed by the periodic solutions of period T and 2T, where T =2
/ . The T periodicity is achieved with a solution in the form of a trigonometric series
6,4,22
cos2
sin2
1)(
0
k
tkkb
tkkabtq (7)
the coefficients of sin2
tand cos
2
t lead to a series of algebraic equations for the
determination of instability regions. Principal instability region, which is of practical
importance corresponds to k = 1 and for this case, the instability equation leads to
042
1 2
qMKPKPK GcrGcrb (8)
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The in-plane load tP may be periodic and can be expressed in the form
tPPtP tS cos (5)
crS PP , crt PP (6)
Equation (4) represents a system of second order differential equations with
periodic coefficients of Mathieu-Hill type. The solution of this equation may
be bounded or unbounded, which depends on the combination of the values of
dynamic load factor and frequency of excitation. The development of regions
of instability arises from Floquet’s theory, which establishes the existence of
periodic solutions. The boundaries of the dynamic instability regions are
formed by the periodic solutions of period T and 2T, where T =2 / . The T
periodicity is achieved with a solution in the form of a trigonometric series
6,4,22
cos2
sin2
1)(
0
k
tkkb
tkkabtq (7)
the coefficients of sin2
tand cos
2
t lead to a series of algebraic equations for
the determination of instability regions. Principal instability region, which is
of practical importance corresponds to k = 1 and for this case, the instability
equation leads to
042
1 2
qMKPKPK GcrGcrb (8)
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21
The generalized stress-strain relationship for a plate element is
ppp D (3.4.6)
where the stress resultant vector is
yx
My
Mx
Myx
Ny
Nx
NTp (3.4.7)
)1(4.20000000
0)1(4.2
000000
00)1(24
00000
000)1(12)1(12
000
000)1(12)1(12
000
00000)1(2
00
00000011
00000011
3
2
3
2
3
2
3
2
3
22
22
Et
Et
Et
EtEt
EtEt
Et
EtEt
EtEt
D p
Using the isoparametric coordinates, the element stiffness matrix is expressed as
1
1
1
1
ddJBDBK pppT
ppb (3.4.8)
and JJ p .
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22
Consistent Mass Matrix
dz
zz
zz
z
z
t
t
m p
2
2
000
000
0000
000
000
2/
2/
120000
012
000
0000
0000
0000
3
3
t
t
t
t
t
m p
dydxNmNM PT
eP
The element mass matrix can be expressed in iso parametric coordinate as
1
1
1
1
ddJNmNM ppT
ep
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23
Geometric Stiffness Matrix
22
2
1
2
1
x
w
x
u
x
uX
GpEp
0
0
2
1
2
1
2
1
2
1
2
1
2
1
222
222
y
V
x
V
y
U
x
U
y
W
x
W
y
V
y
U
y
W
x
V
x
U
x
W
y
Wx
W
x
V
y
U
y
Vx
U
9
1rrqrB PGP
dzdydxBBK PGp
T
PGpG
Geometric stiffness matrix expressed in isoparametric coordinates
1
1
1
1
ddJBBK ppGPT
pGPG
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24
120
12000
012
012
00
120
12000
012
012
00
0000
0000
33
33
33
33
tt
tt
tt
tt
tt
tt
yxy
xxy
xyy
xyx
yxy
xyx
P
Stiffener Element Formulation for Eccentrically Stiffened Plate
x
rN
rNx
rN
x
rN
x
rN
rBS
0000
000
0000
0000
dxBDBeK SST
SS
2.1/000
000
00
00
S
S
SS
SS
S
GA
GT
ESEF
EFEA
D
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LOAD VECTOR
Let )( tF = Equivalent nodal forces
If the distributed force is of uniform intensity Op ,
then the load vector is given by
ddJtpNF OT
e
Equivalent nodal forces if concentrated are
expressed as:
JNPP Tor
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26
MODE-1
MODE-4 MODE-3
MODE-2
MODE-5 MODE-6
Mode shapes of Longitudinal stiffened plate subjected to partial
edge loading
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Computer Program for Buckling Analysis
The geometric stiffness matrix is essentially a function of the
in-plane stress distribution
(a) First for the static analysis and second for the buckling
analysis.
(b) For non-uniform in-plane edge loading case, a three-point
integration scheme is adopted
(c) The overall elastic stiffness matrix, geometric stiffness
matrix and mass matrix are stored in a single array where
the variable bandwidth profile storage scheme is used.
-
Solution Technique
• The static equations of equilibrium in the form of = is solved by Cholesky
decomposition procedure
• The algorithm contains three subroutines, REDUCE, FORSUB, BACKSUB.
Subroutine Subroutine REDUCE decomposes a symmetric matrix in the
variable bandwidth store L (NK) with address sequence LD (N).
• On exit the Cholesky lower triangular matrix appears in L(NK) except in the
case of a reduction failure.
• Subroutine FORSUB solves by forward substitution LV = U, where L is a lower
triangular matrix in the variable bandwidth store L (NK).
• Subroutine BACKSUB solves by backward substitution L V = U.
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Crouts Triangular method
A X = B
LUX = B
UX = V
LV = B
L and U is computed as:
1. First row of U
2. First column of L
3. Second row of U
4. Second column of L
5. Third row of U
6. First row of U
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30
COMPUTER PROGRAM
The governing equation of the structural behaviour subjected to in-plane stresses are
obtained by adopting Mindlin’s plate theory
Nine noded isoparametric quadratic element with five DOF(u, v, w, x and y) per node is
employed in the analysis.
The structure is divided into a two dimensional array of rectangular elements. The element
matrices of the stiffened plate element consist of the contribution of the plate and that of the
stiffener. Element elastic matrices and mass matrices are obtained with 2 x 2 sampling point
related to shear strain components to avoid possible shear locking.
The geometric stiffness matrix is essentially a function of the in-plane stress distribution in
the element due to applied edge loading.
Element matrices are assembled into global matrices using skyline technique.
Eigenvalues are obtained by simultaneous iteration due to Corr & Jenning.
-
Solution Technique for
static, bending, buckling, free vibration,
vibration with load and dynamic stability
(a) Solution procedure for linear static analysis
The stiffness matrix is stored as one-dimensional array through skyline storage scheme.
This scheme eliminates zeroes within the band after the last non-zero value and reduces the
storage requirement. The static equations of equilibrium in the form of A X = B is
solved by Cholesky decomposition procedure according to the algorithm presented by Corr
and Jennings and Subspace Iteration Technique. .
The algorithm contains three subroutines, REDUCE, FORSUB, BACKSUB. Subroutine
Subroutine REDUCE decomposes a symmetric matrix in the variable bandwidth store L
(NK) with address sequence LD (N).
On exit the Cholesky lower triangular matrix appears in L(NK) except in the case of a
reduction failure.
Subroutine FORSUB solves by forward substitution LV = U, where L is a lower triangular
matrix in the variable bandwidth store L(NK).
Subroutine BACKSUB solves by backward substitution L T V = U.
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xxA
Where 2
1 1,
andLXLMLA
TT
This represents a standard eigen values problem and simultaneous
iteration technique has been used to extract the eigenvalues and
eigenvectors.
The methodology is explained as follows:
1. Set a trial vector U and orthonormalize
2. Backward substitute UXL
3. Multiply XMY
4. Forward substitute YVL T
5. Form VUB T
6. Construct T so that ijt = 1 and 22
jiiijiji
ji
ijbbsbb
bt
7. Multiply TVW
8. Perform Schmidt orthonormalisation to derive U 9. Check tolerance U -- U 10. If not satisfactory, go to step 2
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Subroutine LSMP
.
The element rigidity matrix for the plate element is evaluated in this subroutine
based on the mechanical properties of the plate.
The shapes functions at the nodes of the plate element are computed are
computed in subroutine BBP.
The subroutine generates the strain displacement connectivity matrix, i.e. the [B]
matrix of the plate element using the shape function which is called from the
subroutine BBP, and their derivatives obtained from Subroutine SHAPE.
This subroutine generates the elastic stiffness matrices based on the nodal co-
ordinates, shape functions and the rigidity matrix of the plate element.
The global co-ordinates and the nodal co-ordinates facilitate the computation of
these matrices. The subroutine GEN that generates the co-ordinates of the nodes
and nodal connectivity, SHAPE (shape function of the plate element), are used in
this subroutine for the evaluation of element stiffness matrix.
The element matrix is generated using 3 x 3 point Gauss quadrature scheme for
integration. Similarly it generates the consistent mass matrix of the plate element
based on the density ( ) value supplied in the input. This subroutine also
computes the geometric stiffness matrix of the plate element using the geometric
strain displacement matrix of the element
-
This subroutine computes the geometric stiffness matrix of the plate element using
subroutine BBG (geometric strain displacement matrix of the element), subroutine
STRESSP (stress matrix for plate element), subroutine ASSEM (assembles the
element matrices to generate the overall matrices) in a single array.
This subroutine generates the load matrix using the data given in input or
subroutine GEN. Subroutine SHAPE (shape function of plate element) is called here
to generate the data. The subroutine calculates the loading matrix due to a
transversely applied load, uniformly distributed load, point load, partial edge load
on the structure.
-
This subroutine computes the geometric stiffness matrix of the plate element using
subroutine BBG (geometric strain displacement matrix of the element), subroutine
STRESSP (stress matrix for plate element), subroutine ASSEM (assembles the
element matrices to generate the overall matrices) in a single array.
This subroutine generates the load matrix using the data given in input or
subroutine GEN. Subroutine SHAPE (shape function of plate element) is called here
to generate the data. The subroutine calculates the loading matrix due to a
transversely applied load, uniformly distributed load, point load, partial edge load
on the structure.
Subroutine LSMX, LSMY
Subroutine BBP
Subroutine BBX
Subroutine BBY
.
Subroutine STRESSP
Subroutine BDRY
Subroutine ASSEM
Subroutine R8USIV
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36
Problem As: Bending, Buckling,
Free Vibration,
Forced Vibration,
Vibration with Load,
Transient Vibration, Dynamic Stability,
Parametric Excitation
Material Isotropic, Laminated Composite, Stiffened,
Composite Stiffened Bare Plate/beam/Shell
Loading UDL, Varying, Concentrated,
Partial, Triangular,
Parabolic for In-plane loading,
out of plane loading, Transverse
Boundary Conditions SSSS, SSCC, CCSS, CCCC, SCSC, CCSS
Cantilever, Propped, Free, Restrained,
Unstrained
Cases 1D, 2D, 3D
Linear
Non Linear
Uniaxial, Biaxial, Combined
Coordinates Cartesian, Polar, Cylindrical, Local, Natural
coordinates
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37
Beam LINEAR, NON LINEAR, CURVED
Plate THIN PLATE, THICK PLATE,
MODERATELY THICK PLATE,
PLATE WITH SMALL DEFLECTION,
PLATE WITH LARGE DEFLECTION
SHELL CYLINDRICAL, PARABOLOID,
HYPERBOLOID,
CURVED SHELL,
SYNCLASTIC, ANTISYNCLAUSTIC,
SYMMETRIC, ANTISYMMETRIC
COMPOSITE PLATE
LAMINATED, SANDWITCH,
FUNCTIONALLY GRADED,
SYMMETRIC, ANTISYMMETRIC,
ANSYMMETRIC LAMINATED,
ANGLE PLY LAMINATED,
CROSS PLY LAMINATE
STIFFENED CONCENTRIC
ECCENTRIC (HAT, I, L, PLATE GIRDER)
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38
INDEXING
BBVD, NSTX,
NSTY
1-8 FOR PROBLEMS
NSTCODE CODE FOR STIFFENER/COMPOSITE,
0 FOR CONCENTRIC, 1 FOR ECCENTRIC
NBC Boundary condition code
0 for free edge, 1 for simply supported 2 for clamped
3 for free
4 for restrained, 5 for unrestrained movable (DOF)
NPROB Type of problem (with/without cutout )
0 For stiffened plate without cutouts
1 For stiffened plate with cutout., 2 for Composite plate
without cutout
3 Composite plate with cutout
4 Shell without cutout, 5 Shell with cutout
NPROBLEM For UDL type loading
1 For free vibration , 2 For buckling analysis
3 For static analysis, 4 For vibration due to axial load
5 For Dynamic stability analysis
For partial loading
6 For vibration and buckling analysis
7 For dynamic stability analysis
NTYPE Type of applied load on the structure
1 For uni-axial compression, 2 For bi-axial compression
3 For shear, 4 For combined uni-axial compression and
5 For combined bi-axial compression and shear
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39
INPUT:
GENERAL NODMN, NOMESH, NOBC, NRQD, E, ANU, T, RO,
NSTX, NSTY, X,Y, XG, YG, NOD,NELM
NP, (NDPL(I), NOLF, ALFA, BETA, NFACT,
NOECO, NUECO, NSTXCO,
NOBC, DGP, NELM, NXD, NYD, NNODE, NBH,
NOD, 'ALFA, BETA EXFRP EXFRN
PLATE XL,YL
BEAM E,L, LOADING MAGNITUDE
STIFFENED NSTX, NSTY,NSSSTY,NESSTY, AX,(I),FAX,(I),
SAX,(I), TCX,(I), PAX,(I),
NSNSTXCO, NSTYCO
LAMINATED
COMPOSITE
NLAYER,
THETAX,THETAY,BXX(10),HXX(10),NLYRXX(10),T
HXX(10,20),DSTX,(THTHETA(I),I=1,NLAYER)
E1,E2,G12,G13,G23,ANU1,T,RO,
DX(I),BXX(I),HXX(I),TCX(I),NLYRXX(I)
SHELL NSHELL1, NSHELL2, NSHELL3, NSHELL4,
NSHELL5
PARABOL, HYPER, CYL, SYM, ANTIS, ANTC,
ANTSY
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40
GEOMETRY AND MESH GENERATION
CALL GEN
(DGP1, DGP, WF, DGP 2, WF 2,NXD, NELM, NNODE, NT, NBH, NOD,
XL, YL, XG, YG)
DGP1(1) =- 1.0
DGP2(1) =- 0.577350269189626
DGP3(1) =- 0.774596669241483
WF(1) = 0.556555555555556
WF(2) = 0.888888888888889
WF(3) = WF(1)
WF2(1) = 1.0
WF2(2) = WF2(1)
INDEXING FOR ELEMENT
NOECO, NUECO NSTX, NSTY, NSTXCO, NSTYCO
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41
FOR PLATE, BEAM, COMPOSITE PLATE,
COMPOSITE SHELL)
(1) Step
CALL LSMP (Stiffness Matrix of Plate and Beam)
(DGP, XG, YG, E, ANU, T, RO, GSM, GMM, RGD)
CALL LSMX (FOR SHELL, COMPOSITE PLATE, COMPOSITE SHELL
( NSTY, DY, AY, FAY, SAY, TCY, PAY, DGP, WF)
SUBROUTINE BBP (ZI,ETA,BB,X,Y,AJAC,C)
SUBROUTINE SHAPE
HERE IT CALCULATES:
RGD , STIFFNESS MATRIX , MASS MATRIX
For 3 x 3 /2 x2 GAUSS POINT AS BTDB(I,J), CTFC, EMM, ESM,
AT ELEMENT LEVEL
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42
BOUNDARY CONDITIONS
FOR PRE BUCKLING (BDRY)
DATA INBD12 /0,0,1,0,1, 0,0,1,1,1, 0,0,0,0,0, 1,0,0,1,0/
DATA INBD34 /0,0,1,1,0, 0,0,1,1,1, 0,0,0,0,0, 0,1,0,0,1/
FOR POST BUCKLING
DATA INBD12 /0,1,1,0,1, 1,1,1,1,1, 0,0,0,0,0, 1,0,0,1,0/
DATA INBD34 /1,0,1,1,0, 1,1,1,1,1, 0,0,0,0,0, 0,1,0,0,1/
FOR STIFFENED, COMPOSITE, SHELL
FOR BDRY 1
SUBROUTINE BDRY1(NBC,NXD,NYD,KINDX,GSM)
DATA INBD12 /0,1,1,0, 1, 1,1,1,1,1, 0,0,0,0,0, 1,0,0,1,0/
DATA INBD34 /1,0,1,1, 0, 1,1,1,1,1, 0,0,0,0,0, 0,1,0,0,1/
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43
FOR BENDING
IG=1,3OR 1,2 OR 1,4
JG=1,3OR 1,2 OR 1,4
CALL DECOM (IER,GSM,KINDX,NT)
CALL FOR (GLOAD,GLOAD,GSM,KINDX,NT)
CALL BAC(GLOAD,GLOAD,GSM,KINDX,NT)
CALL STRESS
(IELM,DGP(IG),DGP(JG),GLOAD,NOD,XG,YG,D,ANX,ANY,ANXY)
FOR BUCKLING, FREE VIBRATION, VIBRATIION WITH LOAD
CALL GSMP
(DGP,WF,IELM,NBH,NOD,XG,YG,T,GGM,ANXG,ANYG,ANXYG)
CALL STRESSX
(IELM,DGP(IG),ETA,GLOAD,NOD,XG,YG,ANX,AX(ISTX),FAX(IST
X)
CALL GSMX
(DGP,WF,IELM,NBH,NOD,XG,YG,AX(ISTX),SAX(ISTX),GGM,ANX
G, ETA,AJAC)
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44
FOR UDL, PARTIAL, TRIANGULAR, PARABOLIC, ELLIPTIC
BY TAKING LOADING = 1,2,3,4,5,6
CALL CRUNCH 1(GSM,KINDX,NBH,NT, UDL)
CALL CRUNCH 2(GMM,MIND,NBH,NT, PARTIAL)
CALL CRUNCH 3(GGM,JIND,NBH,NT, TRIANGULAR)
CALL CRUNCH 4(GGM,JIND,NBH,NT, PARABOLIC)
CALL CRUNCH 5(GGM,JIND,NBH,NT, VARYING)
CALL BDRY1-6 (NBC,NXD,NYD,KINDX,GSM, PARABOLIC)
CALL R8USIV (NT,NRQD,GSM,KINDX,GGM,JIND,BD)
PCR=(BD(1))**2
BPARA(II)=((BD(II)**2)*YL)/RGD
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45
DYNAMIC STABILITY FOR PARTIAL LOAD '
' GDMP(I)=GSMB(I)-ALFA(J)*PCR*GGMB(I)+
CALL CRUNCH(GDMP,KINDP,NBH,NT)
CALL BDRY1(NBC,NXD,NYD,KINDP,GDMP)
SUBROUTINE R8USIV (N,NRQD,AL,LD,G,NGD,BD)
CALCULATE : Excitation Frequency
SOLVE EIGEN VALUE ,
FIND
BUCKLING LOAD PARAMETER)
EXCITATION FREQUENCY PARAMETER
FREQUENCY PARAMETER
DYNAMIC STABILITY RANGE AND FREQUENCY
SIMPLE RESONANCE
COMBINATION RESONANCE
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46
FOR VIBRATION FOR PARTIAL LOAD
GSM(I)=GSMB(I)-FACT(IFACT)*PCR*GGMB(I)
CALL CRUNCH(GSM,KINDX,NBH,NT)
CALL BDRY1(NBC,NXD,NYD,KINDX,GSM)
IT NEEDED
SUBROUTINE BBX(ZI,ETA,BB,X,Y,AN)
SUBROUTINE STRESS (ZI,ETA, ANX, ANY, ANXY)
SUBROUTINE SHAPE (ZI,ETA,X,Y,ANZ,ANE,DZDX,DZDY, AJAC)
SUBROUTINE R8USIV(N,NRQD,AL,LD,G,NGD,BD)
CALCULATE FPARA(IK) IN VARIOUS MODES
SOLVE EIGEN VALUE ,
FIND
BUCKLING LOAD PARAMETER
FREE VIBRATION FREQUENCY
FREQUENCY PARAMETER WITH LOAD
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165
A number of subroutines are called in turn for the execution of R8USIV
Subroutine R8URED : Decompose the stiffness matrix into upper and lower
triangular matrix.
Subroutine R8URAN : Generate random initial eigenvectors.
Subroutine R8UORT : Orthonormalise the trial vector by Schmidt
decomposition
Subroutine R8UBAC : Back substitution in linear equation solution
Subroutine R8UPRE : Premultiply a matrix by a vector
Subroutine R8UFOR : Forward substitution in linear equation solution
Subroutine R8UERR :Estimate vector error in successive trial.
Postprocessor
In this final stage of programming, all the input data are echoed to
check their accuracy. The output sets the desired data in the form of
displacements, stresses, strains, eigen values etc. depending on the type
of analysis carried out. Then the results are stored in separate files for
presentation in the form of graph or tables.