Lecture 1

CE 5109
Computer Aided Design and
Analysis of Structures
Chapter 1 INTRODUCTION

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
 Email
 Etc.

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.
4. In 1954, Professor J. Argyris and S. Kelsey formulated the matrix method of
structural analysis, which effectively utilizes digital computers.
5. 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.
6. Subsequently, a more general computer method—the finite element method—was
developed for conducting structural analysis of a wide variety of structures.
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:

2 Historical Development
 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
 486: 25-50 MHz
 586/Pentium: 100-500 MHz
 Pentium 4: 3.6 GHz,
 Dual Core processors,
 Core i3, i5 and i7,
 First, second, third and fourth generation is etc.

3 Computer Hardware
and Software
 Workstations (some obsolete now)
 Sun SPARC 20
 HP Workstations
 Appollo
 DEC Stations
 IBM Risk 6000
 DELL Systems
 Apple MAC systems etc.

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 : Workstation, PC

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)

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

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

3 Computer Hardware
and Software
 NASTRAN – NASA (National Aerospace and Space Administration)
Structural analysis Program
 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
 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

3 Computer Hardware
and Software
 ADINA – MIT (Automatic Dynamic Incremental Non-linear Analysis)
 Time integration
 Sub-structuring
 Solution to frequencies and mode shapes
 Mode superposition
 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

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.

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.
 Compatibility in deformation.
 Consistent with support conditions.
The classical methods may
consist of the following:
• Slope-deflection method
• Moment distribution
• Virtual displacements
• Unit load method
• Castigliano’s theorem
• Energy theorems, etc.
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.

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

5 Flexibility and
Stiffness Concepts
Fig. 1.1 An Elastic SpringFig. 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.

5 Flexibility and
Stiffness Concepts
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
PL
PP
2
,
3
23
 
EI
ML
EI
ML
MM
  ,
2
2

5 Flexibility and
Stiffness Concepts
EI
ML
EI
PL
MP
23
23
 
EI
ML
EI
PL
MP

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)

5 Flexibility and
Stiffness Concepts
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
/6/12
2
23
(10)
This matrix inversion can be
performed efficiently on
Mathematica as shown :
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.

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.
s
Normal 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 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-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

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, 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 EliminationThe 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 Mathematica and MatLab. You may utilize the
CholeskyDecomposition command, which is built into Mathematica’s linear
algebra package, as shown in the following:

Appendix
I. Computer algorithm for forward reduction:
II. Computer algorithm for back-substitution:

Appendix
III. Computer algorithm for LU decomposition:

Lecture 1

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Lecture 1