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CIVL 337

Computer Methods of Structural Analysis Chapter 1 INTRODUCTION

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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 Email Etc.

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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:

1.

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3. 4.

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6.

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. 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

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

Advantages of Matrix Formulation:

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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 486: 25-50 MHz 586/Pentium: 100-500 MHz Pentium 4: 3.6 GHz, etc. Workstations (some obsolete now) Sun SPARC 20 HP Workstations Appollo DEC Stations IBM Risk 6000

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3 Computer Hardware and Software

Supercomputers Vector machines: Cray 90, IBM, Convex Parallel machines: CM-5, Intel Paragon, nCube, etc. Current trend: PC clusters (parallel processing): Cluster: group of (e.g. 8) PCs connected by a very fast network Can outperform workstations or supercomputers of equivalent price Operating systems: Oracle: Workstation, PC Linux: Workstation, PC MS Windows 2000: PC

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3 Computer Hardware and Software

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 100×100 numerical matrix on Mathematica: ~ 1 min.) Many numerical schemes built-in (e.g. LUDecomposition, RowReduce)

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3 Computer Hardware and Software

Specialized Structural Analysis Software: ABAQUS, ADINA, ANSYS, ETABS, NASTRAN, SAP2000, etc. SAP2000 Integrated software for structural analysis & design (e.g. for bridges) Will be discussed in this class ETABS Analysis, design and drafting of building systems Will be discussed in this class

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3 Computer Hardware and Software

ANSYS (Engineering analysis system) Special features of the package Linear time history Nonlinear time history Sub-structuring Nonlinear transient dynamic Types of analysis Linear elastic analysis Materially-nonlinear analysis Large deformation analysis Fracture mechanics Element library Bar, beam, pipe, elbow and tee element Two-dimensional membrane element Three-dimensional solid element Two-dimensional bending element Shell element 4 node shell 8 node curved shell 16 node thick shell

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3 Computer Hardware and Software

NASTRAN NASA (National Aerospace and Space Administration) Structural analysis Program Special features of the package Direct and modal complex eigenvalue analysis Direct and modal transient analysis Aeroelastic response Aeroelastic flutter Sparse matrix solutions Generalized dynamic reduction Multi-level super-elements Automatic re-sequencing Automatic singularity suppression Types of analysis Linear elastic analysis Material non-linearity and geometric non-linearity Complex eigenvalue analysis Response spectral analysis Element library Truss and beam elements Two-dimensional inplane and bending elements Three-dimensional solid element Constraint elements (rigid and interpolating) Curved shell element

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3 Computer Hardware and Software

ADINA MIT (Automatic Dynamic Incremental Non-linear Analysis) Special features of the package Time integration Sub-structuring Solution to frequencies and mode shapes Mode superposition Types of analysis Linear elastic analysis Materially-nonlinear analysis Large deformation formulation Element library Truss and cable Two-dimensional solid and fluid element Three-dimensional solid and fluid element Two-node beam element Isoparametric beam element Three-node plate/shell element Isoparametric shell element

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3 Computer Hardware and Software

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.

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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. Compatibility in deformation. Consistent with support conditions. The computer methods are actually formulated on the basis of the energy principle with the following characteristics: · The least amount of approximations is involved. · 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.

The classical methods may consist of the following: · · · · · · Slope-deflection method Moment distribution Virtual displacements Unit load method Castigliano's theorem Energy theorems, etc.

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4 Computer Methods vs. Classical Methods

Scope of the Course:

Structures: beam, continuous beam, plane truss, space truss, plane frame, space frame, grid, etc. Materials: linearly elastic Deformation: small Analysis: static and dynamic Support conditions: arbitrary

Expectation from the Course:

1. 2. 3. 4. 5. 6.

Basic theory behind the computer methods of structural analysis How to model a structure for computer analysis How to form the stiffness and mass matrices by hand calculation How to form the loading in matrix form How to use mathematical software to assist in (1) (4) How to solve practical problems using a structural software

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Flexibility and Stiffness Concepts

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)

Fig. 1.1 An Elastic Spring

where F is called the flexibility coefficient of the spring, it is also the amount of displacement induced by a unit force.

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Flexibility and Stiffness Concepts

We consider next a cantilever beam subjected to a force P and a moment M at the tip as shown in Fig. 1.2.

Let the deflection and rotation of the tip be denoted by and , respectively. To find and , we may consider the force and moment applied to the beam separately. Effects of force P:

PL3 PL2 , P = P = 3EI 2 EI

Effects of Moment M:

(3)

where EI is the bending rigidity of the beam.

ML2 ML M = , M = 2 EI EI

(4)

Fig. 1.2 A Cantilever Beam Deflected by End Force and Moment

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Flexibility and Stiffness Concepts

PL3 ML2 = P + M = + 3EI 2 EI

The deflection 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 PL2 ML = P + M = + 2 EI EI

(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

L3 / 3EI = 2 L / 2 EI

=F · F

L2 / 2 EI P L / EI M

(8)

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Flexibility and Stiffness Concepts

K = F (9)

where is the "displacement vector"; F is the "flexibility matrix" of the beam; F is the "force vector". The inverse of Eq.(8) gives

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

12 EI / L3 K= 2 - 6 EI / L

This matrix inversion can be performed efficiently on Mathematica as shown :

- 6 EI / L2 4 EI / L

(10)

Note that both the flexibility and stiffness matrices are symmetric, and this property is related to Maxwell and Betti's law, or the reciprocal theorem.

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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 or F represent either vectors or matrices. Symbols:

E A I J Normal stress Shear stress Normal strain Shear strain Deflection Angle or rotation Young's modulus Cross sectional area Bending moment of inertia Polar moment of inertia N

Notations:

x k A position vector (or coordinate vector of a point) Member stiffness matrix Member flexibility matrix Joint displacements of a member Joint force vector of a member Structural stiffness matrix Structural nodal displacement vector Structural nodal force vector Matrix relating nodal displacements to element strains Matrix of shape functions

F

f K

F B

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.

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7 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. Nowadays, for problems are 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-ShiftEnter) 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)

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It is a good practice to name arrays for convenient selection You may press Ctrl-* to select a matrix

7 Solution of Linear Equations

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

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7 Solution of Linear Equations

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).

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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:

5 -4 1 0 x1 0 -4 6 -4 1 x 1 2 = 1 -4 6 -4 x3 0 0 1 -4 5 x4 0

Step 5. Make the last two elements of column 3 to be "1": (Row 3)(-112/75), (Row 4)(-7/20)

0 x1 0 1 -4 / 5 1/ 5 0 1 -8 / 7 5 /14 x2 5 / 14 = 0 0 1 -4 / 3 x3 8 / 15 0 1 -13 / 8 x4 1/ 8 0

Step 1. To make the first column to be 1 or 0: (Row 1)/5, (Row 2)/(-4)

0 x1 0 1 -4 / 5 1/ 5 1 -6 / 4 1 -1/ 4 x -1/ 4 2 = 1 -4 6 -4 x3 0 1 -4 5 x4 0 0

Step 6. Uncouple the last equation: Row 4 Row 3

0 x1 0 1 -4 / 5 1/ 5 0 1 -8 / 7 5 /14 x2 5 /14 = 0 0 1 -4 / 3 x3 8 /15 0 0 -7 / 24 x4 -49 / 120 0

Step 2. Make the first column to be 0's, except the first entry being "1": Row 2 Row 1, Row 3 Row 1

0 x1 0 1 -4 / 5 1/ 5 0 -7 /10 4 / 5 -1/ 4 x -1/ 4 2 = 0 -16 / 5 29 / 5 -4 x3 0 1 -4 5 x4 0 0

Step 7. Further simplification of the last equation: (Row 4)(-24/7)

0 x1 0 1 -4 / 5 1/ 5 0 1 -8 / 7 5 /14 x2 5 /14 = 0 0 1 -4 / 3 x3 8 /15 0 0 1 x4 7 / 5 0

Step 3. We uncouple equations 2-3: (Row 2)/(-10/7), (Row 3)/(-5/16)

1/ 5 0 x1 0 1 -4 / 5 0 1 -8 / 7 5 /14 x2 5 /14 = 0 1 -29 / 16 5 / 4 x3 0 1 -4 5 x4 0 0

Therefore

x4 = x3 = x2 =

7 5

Step 4. (Row 3 Row 2), (Row 4 Row 2)

1/ 5 0 x1 0 1 -4 / 5 0 1 5 /14 x2 5 /14 -8 / 7 = 0 0 -75 /112 25 / 28 x3 -5 /14 0 -20 / 7 65 / 14 x4 -5 /14 0

8 4 12 + x4 = 15 3 5

5 8 5 13 + x3 - x4 = 14 7 14 5 4 1 8 x2 - x3 = 5 5 5

x1 =

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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 (2)

Phase 2: Back-Substitution to determine x Computer algorithms for forward reduction and back-substitution are given in the Appendix. 24

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 (3) Fact: if A is symmetric and positive definite, then A can be decomposed into two parts as A = LU (4) 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 (4) into (3), we have LUx = b In the above, we define Ux = y So we have Ly = b

(5) (6) (7)

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Obviously, we can efficiently solve for y from Eq.(7) using forward-substitution, then x can be readily determined from Eq.(6) 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 Mathematica and MatLab. You may utilize the CholeskyDecomposition command, which is built into Mathematica's linear algebra package, as shown in the following:

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Appendix

I. Computer algorithm for forward reduction:

II. Computer algorithm for back-substitution:

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Appendix

III. Computer algorithm for LU decomposition:

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