Transcript Lecture 13

Eng. 6002 Ship Structures 1
Lecture 13: Introduction to
Computer Methods of Structural
Analysis
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General Remarks
Computers have been widely used in structural
engineering for:
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Structural analysis
Computer-aided design and drafting (CADD)
Report preparation
Typical computer usage by an engineer:
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Word-processing
Preparation of tender documents and engineering drawings
Small and intermediate computations
Analysis of structures
Design work
Data reduction and storage
Software development
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Historical Development
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:
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In the 1940s and 1950s, structural engineers were
confronted with highly statically indeterminate systems:
high-rise tall buildings and large aircraft structures.
In 1940, Hardy Cross proposed the moment distribution
method, based on the relaxation concept, to solve large
systems of indeterminate frame structures.
Since the 1950s, digital computers have been rapidly
developed.
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Historical Development
In 1954, Professor J. Argyris and S. Kelsey formulated the
matrix method of structural analysis, which effectively
utilizes digital computers.
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.
Subsequently, a more general computer method—the
finite element method—was developed for conducting
structural analysis of a wide variety of structures.
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Historical Development
Advantages of Matrix Formulation:
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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
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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:
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Personal Computers
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Eg: Pentium 4: 3.6 GHz, etc.
Workstations
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Sun SPARC 20
HP Workstations
3 Computer Hardware
and Software
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Supercomputers
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Current trend: PC clusters (parallel processing):
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Vector machines: Cray 90, IBM, Convex
Parallel machines: CM-5, Intel Paragon, nCube, etc.
Cluster: group of PCs connected by a very fast network
Can outperform workstations or supercomputers of
equivalent price
Acenet (Atlantic Universities)
Operating systems:
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SUN: Workstation
Linux: Workstation, PC
Windows: PC
Mac OS X (Apple)
3 Computer Hardware
and Software
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Mathematical Software
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Excel (small-scale matrix work / optimization,
data storage & pre-processing, etc.)
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MatLab, MathCAD (general-purpose)
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Computer Algebra Systems (CAS): Mathematica,
Maple, Derive, etc.
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Handles numeric as well as symbolic work (e.g.
matrix inversion)
Small-to-medium scale work (inversion of 100100
numerical matrix on Mathematica: ~ 1 min.)
3 Computer Hardware
and Software
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Specialized Structural Analysis
Software
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Computer Aided Drafting Systems:
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ABAQUS, ADINA, ANSYS, ETABS, NASTRAN,
SAP2000, etc.
AutoCAD, MicroStation, I-DEAS (3-D modelling
& FEM), etc.
Application Areas:
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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.
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Force equilibrium or energy balance of a structure.
Consistent with support conditions.
The classical methods may
consist of the following:
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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 neqneq non-singular matrix with constant
coefficients, x and b are neq1 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.
5 Solution of
Linear Equations
Nowadays, for problems not too large (say, a matrix of size 2020),
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:
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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
5 Solution of
Linear Equations
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 “CTRLSHIFT-ENTER” at the same time.
7 Solution of
Linear Equations
An example for matrix inversion on a spreadsheet is as follows:
7 Solution of
Linear Equations
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.
7 Solution of
Linear Equations
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:
7 Solution of
Linear Equations
Gauss Elimination
The 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 Decomposition
For 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
7 Solution of
Linear Equations
The following Matlab code finds the Cholesky decomposition of the
matrix M:
7 Solution of
Linear Equations
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
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We will look at Matrix methods for frames