Eng. 6002 Ship Structures 1 Lecture 13: Introduction to Computer Methods of Structural Analysis

Eng. 6002 Ship Structures 1

Lecture 13: Introduction to Computer Methods of Structural

Analysis

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 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 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)

You may press Ctrl-* to select a matrix


How to Invert a Matrix in Excel:

1. Type in the matrix you wish to invert. Each cell should correspond to an element of the matrix.

2. Select a set of empty cells corresponding to the size of the inverted matrix.

3. From the Insert menu select “Function>Math & Trig >MINVERSE function and click OK

4. For Array, select the matrix that you wish to invert and then click OK.

5. Go to the formula bar and select its contents. Hold “CTRL-SHIFT-ENTER” at the same time.


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


To tackle problems of a large size, 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:


Gauss EliminationThe basic idea of Gauss elimination is to suitably combine the rows to transform the coefficient matrix into upper triangular form.

All unknowns are then found by back-substitution, starting from the last row.

Cholesky DecompositionFor a large system of linear equations, the Cholesky decomposition is often a preferred and efficient direct method. Nowadays, such algorithms are well implemented on various mathematical software packages such as Mathematica and MatLab


The following Matlab code finds the Cholesky decomposition of the matrix M:


In Maple, we have:

M := Matrix([[9, 0.6, -0.3, 1.5], [0.6, 16.04, 1.1800, -1.5], [-0.3, 1.18, 4.1, -0.57], [1.5, -1.5, -0.57, 25.45]] );LinearAlgebra:-LUDecomposition( M, method = Cholesky );

For more help on either of these routines or on the LinearAlgebra package, enter:

?LinearAlgebra?LinearAlgebra,LUDecomposition?LinearAlgebra,LinearSolve

Maple Output

Next Class We will look at Matrix methods for frames

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Eng. 6002 Ship Structures 1 Lecture 13: Introduction to Computer Methods of Structural Analysis