Eng. 6002 Ship Structures 1 - Faculty of Engineering and ... · Eng. 6002 Ship Structures 1 ... Structural analysis Computer-aided design and drafting ... confronted with highly statically

Eng. 6002 Ship Structures 1

L E C T U R E 8 : I N T R O D U C T I O N T O C O M P U T E R M E T H O D S O F S T R U C T U R A L A N A L Y S I S

1 General Remarks

Computers have been widely used in structural engineering for: Structural analysis

Computer-aided design and drafting (CADD)

Report preparation

Typical computer usage by an engineer: Word-processing

Preparation of tender documents and engineering drawings

Small and intermediate computations

Analysis of structures

Design work

Data reduction and storage

Software development

2 Historical Development

1. In the 1940s and 1950s, structural engineers were confronted with highly statically indeterminate systems: high-rise tall buildings and large aircraft structures.

2. In 1940, Hardy Cross proposed the moment distribution method, based on the relaxation concept, to solve large systems of indeterminate frame structures.

3. Since the 1950s, digital computers have been rapidly developed.

The methods of structural analysis have been

dramatically revolutionalized by the advance in digital

computers and the demand in stringent design

requirements of airplanes. A number of significant

milestones are:


1. In 1954, Professor J. Argyris and S. Kelsey formulated the matrix method of structural analysis, which effectively utilizes digital computers.

2. In the 1950s, a group of structural engineers Turner, Clough, Martin and Topp at the Boeing Company also proposed the matrix formulation for structural analysis of airplanes.

3. Subsequently, a more general computer method—the finite element method—was developed for conducting structural analysis of a wide variety of structures.


Advantages of Matrix Formulation:

Convenient for computer programming.

It is difficult to analyze a complicated structure by hand calculation unless a great deal of simplification is made.

3 Computer Hardware and Software

Computers have evolved tremendously. The basic computer hardware has gone through several phase changes, from vacuum tubes to transistors, and then silicon chips. There are basically three classes of computers:

Personal Computers

Eg: Pentium 4: 3.6 GHz, etc.

Workstations

Sun SPARC 20

HP Workstations


Supercomputers Vector machines: Cray 90, IBM, Convex Parallel machines: CM-5, Intel Paragon, nCube, etc.

Current trend: PC clusters (parallel processing): Cluster: group of PCs connected by a very fast network Can outperform workstations or supercomputers of equivalent price Acenet (Atlantic Universities)

Operating systems: SUN: Workstation Linux: Workstation, PC Windows: PC Mac OS X (Apple)


Mathematical Software

Excel (small-scale matrix work / optimization, data storage & pre-processing, etc.)

MatLab, MathCAD (general-purpose)

Computer Algebra Systems (CAS): Mathematica, Maple, Derive, etc. Handles numeric as well as symbolic work (e.g. matrix

inversion) Small-to-medium scale work (inversion of 100100 numerical

matrix on Mathematica: ~ 1 min.)


Specialized Structural Analysis Software ABAQUS, ADINA, ANSYS, ETABS, NASTRAN,

SAP2000, etc.

Computer Aided Drafting Systems: AutoCAD, MicroStation, I-DEAS (3-D modelling &

FEM), etc.

Application Areas: Design of tall building and bridges Offshore platforms Aircraft and jet engine design Nuclear power plant design etc.

4 Computer Methods vs. Classical Methods

Both the computer and classical methods are established from the fundamental principles in mechanics, i.e.

Force equilibrium or energy balance of a structure.

Consistent with support conditions.

The classical methods may

consist of the following:

• Slope-deflection method

• Moment distribution

• Virtual displacements

• Unit load method

• Energy theorems, etc.

The computer methods (energy

principle) with the following

characteristics:

• The least amount of approximations

• For complex structures, the method

involves the solution of large systems of

linear equations.

• The method gives multiple results, e.g.

deflections of all joints, member forces.

• Computer does the routine calculations.

5 Flexibility and Stiffness Concepts

Fig. 1.1 An Elastic Spring Fig. 1.1 An Elastic Spring

We consider a linear spring, a one-degree of freedom system, as shown in Fig. 1.1.

Let the spring constant be k N/m while the spring is subjected to a force f. The

corresponding displacement is designated by d.

We have the following relationship

k · d = f (1)

The physical meaning of k, the spring constant, is

the amount of force required to stretch the spring by

a unit displacement. The inverse relation of Eq.(1) is

d = F · f (2)

where F is called the flexibility coefficient of the

spring, it is also the amount of displacement

induced by a unit force.


Let the deflection and rotation of the tip be denoted

by D and q, respectively. To find D and q, we may

consider the force and moment applied to the beam

separately.

Effects of force P:

(3)

where EI is the bending rigidity of the beam.

Effects of Moment M:

(4)

We consider next a cantilever beam subjected to a force P and a moment M at the

tip as shown in Fig. 1.2.

Fig. 1.2 A Cantilever Beam Deflected

by End Force and Moment

EI

PL

EI

PLPP

2 ,

3

23

EI

ML

EI

MLMM ,

2

2


EI

ML

EI

PLMP

23

23

EI

ML

EI

PLMP

2

2

M

P

EILEIL

EILEIL

/2/

2/3/2

23

The defection and rotation due to both P and M applied to the beam

simultaneously, then, can be obtained by using the principle of superposition, i.e.

(5)

and

(6)

The above equations can be rearranged in the form similar to Eq.(2),

(7)

We may also express the above relationship in matrix notation

D =F · F (8)


where D is the “displacement vector”; F is the “flexibility matrix” of the beam; F is

the “force vector”. The inverse of Eq.(8) gives

KD = F (9)

where K = F -1 is the stiffness matrix of the beam, namely

K =

LEILEI

LEILEI

/4/6

/6/122

23

(10)

This matrix inversion can be

performed efficiently on

Mathematica as shown :

6 Symbols and Notations

In this section, we will list the definitions of frequently used symbols and notations. Note that bold-faced letters such as D or F represent either vectors or matrices.

sNormal stress

t Shear stress

e Normal strain

g Shear strain

Deflection

Angle or rotation

E Young’s modulus

A Cross sectional area

I Bending moment of inertia

J Polar moment of inertia

Notations:

x A position vector (or coordinate vector of a point)

k Member stiffness matrix

F Member flexibility matrix

Joint displacements of a member

f Joint force vector of a member

K Structural stiffness matrix

D Structural nodal displacement vector

F Structural nodal force vector

B Matrix relating nodal displacements to element strains

N Matrix of shape functions

Note: In the above, notations with no overbar represent quantities defined in the

“global” coordinate system, whereas (¯) indicates the quantity is defined in a “local”

(or member) coordinate system. These terms will be made clear in the subsequent

chapters.

Symbols:

7 Solution of Linear Equations

We consider a system of linear equations of the form

Ax = b (1)

where A is an neqneq non-singular matrix with constant coefficients, x and b are neq1 vectors

with x being the unknown. Matrix formulation of structural problems often leads to a large

system of such simultaneous equations. Efficient ways of solving such equations have been

the major concern of numerical analysts.

Nowadays, for problems are not too large (say, a matrix of size 2020), we may simply use a

spreadsheet or even a calculator to invert (1) for a direct solution x = A-1b. For example, the

following Excel commands (to be entered with Ctrl-Shift-Enter) can be helpful:

• To multiply matrices and vectors: MMULT

• To transpose a matrix: TRANSPOSE

• To invert a matrix: MINVERSE

• To obtain the determinant of a matrix: MDETERM

• To retrieve the (r, c) component of a matrix M: INDEX(M,r,c)

It is a good practice to name arrays for convenient selection

You may press Ctrl-* to select a matrix


An example for matrix inversion on a spreadsheet is as follows:


To tackle problems of a large size, traditionally there has been basically two

different solution approaches: direct and iterative methods. The direct methods

successively decouple the simultaneous equations so that the unknowns can be

readily calculated. Most are some kind of variation of the Gaussian elimination

method, such as the Cholesky and Gauss-Jordan methods.

Iterative methods give approximate solutions that can be improved by successive

iterations. They usually consume less memory than direct methods, but the solution

convergence and accuracy are difficult to control. Therefore, direct methods are

most preferred.

In solving the linear system of simultaneous equations arising in structural analysis,

the following special characteristics can be utilized in coding:

• The matrices are usually symmetric and positive definite

(xTAx > 0 for all nonzero x).

• The matrices are often sparse (avoids multiplications by 0’s and 1’s).

8 Gaussian Elimination

The basic idea of Gauss elimination is to suitably combine the rows of Eq.(1) to transform the coefficient matrix

into upper triangular form. This is called a forward reduction process. Then, the resulting equations become

sufficiently uncoupled. All unknowns x can be found by back-substitution, starting from the last row. To illustrate

this procedure, we consider a 4×4 matrix equation with 4 unknowns:

8 Gaussian Elimination

Summary of Procedures:

We considered the above simple example for illustration of the Gauss elimination

procedures. In reality, the number of equations in Eq. (1) can be fairly large. Then,

Gauss elimination may be used in two phases as follows.

Phase 1: Forward Reduction

Eq.(1) is reduced into upper triangular form

Ux = c

Where

Phase 2: Back-Substitution to determine x

Computer algorithms for forward reduction and back-substitution are given in the Appendix.

9 Cholesky Decomposition

For a large system of linear equations, the Cholesky decomposition is often a

preferred and efficient direct method. We consider the equation of the form

Ax = b (4)

Fact: if A is symmetric and positive definite, then A can be decomposed into two

parts as

A = LU (5)

where

• L is a lower triangular square matrix (i.e. all 0’s above the diagonal),

• U is an upper triangular square matrix (i.e all 0’s below the diagonal), and

• L = UT

Substituting (5) into (4), we have

LUx = b (6)

In the above, we define

Ux = y (7)

So we have

Ly = b (8)

Obviously, we can efficiently solve for y from Eq.(8) using forward-substitution, then

x can be readily determined from Eq.(7) using back-substitution.

9 Cholesky Decomposition

• The detailed procedures for obtaining L and U are given in the

Appendix.

• Nowadays, such algorithms are well implemented on various

mathematical software packages such as Maple and MatLab.

• You may utilize the Cholesky Decomposition command, which is built

into Maple’s linear algebra package: LinearAlgebra[LUDecomposition] ,

or Matlab: chol

Appendix

I. Computer algorithm for forward reduction:

II. Computer algorithm for back-substitution:

AppendixIII. Computer algorithm for LU decomposition:

Eng. 6002 Ship Structures 1 - Faculty of Engineering and ... · Eng. 6002 Ship Structures 1 ... Structural analysis Computer-aided design and drafting ... confronted with highly statically

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