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).

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.

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.

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

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)

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)

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)

NON LINEARITY

• MATERIAL

ELASTIC, ELASTO PLASTIC

VISCO PLASTIC, TEMPERATURE DEPENDENT

• GEOMETRICAL

VERY LARGE, MODERATELY LARGE

SMALL STRAINS, SMALL ROTATIONS

LARGE DEFLECTIONS

NON LINEAR VIBRATION

MECHANICAL

LOADING

THERMAL

LOADING

NONAXISYMMETRIC

PROBLEMS

AXISYMMETRIC

PROBLEMS

(ζ = 0)

TIME DOMAIN

EXPLICIT

EULER

RUNGE KUTTA

DIFFERENCE METHODS

ETC.

IMPLICIT

NEWMARK – β

WILSON – θ

HOUBOLT

PARK STUFFY

ETC.

RESPONSE STABILITY

BIFURCATION

CHAOS

Virtual Work Principle

The displacements are called virtual because they are imagined

to take place.

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)

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.

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)

(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)

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.

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)

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)

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 .

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

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

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

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

26

MODE-1

MODE-4 MODE-3

MODE-2

MODE-5 MODE-6

Mode shapes of Longitudinal stiffened plate subjected to partial

edge loading

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.

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

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.

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

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

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

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)

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

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

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

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

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/

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)

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

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

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

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.