Computational Techniques in Civil Engineering

1. Introduction
Physical problems represented via complex mathematical equations. The accurate solution of
complex engineering problems has been the aim of engineers and scientists. This has been
realized by analytical (exact) and numerical approximation techniques, or combination of the
two. It is difficult to have exact solution of higher order differential equations, so
computational technique is must to solve such problems.

Mathematical model Numerical model

Physical
Governed by differential e.g., finite element
Problem
equations model

1.1 Brief Description of Solution Techniques

The formulation for structural analysis is generally based on the three fundamental relations:
equilibrium, constitutive and compatibility. There are two major approaches to the analysis:
Analytical and Numerical. Analytical approach which leads to closed-form solutions is
effective in case of simple geometry, boundary conditions, loadings and material properties.
However, in reality, such simple cases may not arise. As a result, various numerical methods
are evolved for solving such problems which are complex in nature. For numerical approach,
the solutions will be approximate when any of these relations are only approximately satisfied.
The numerical method depends heavily on the processing power of computers and is more
applicable to structures of arbitrary size and complexity. It is common practice to use
approximate solutions of differential equations as the basis for structural analysis, which is
performed via numerical approximation techniques. Some numerical methods which are
commonly used to solve solid and fluid mechanics problems are:
1.1.1 Finite Element Method (FEM)
1.1.2 Finite Difference Method (FDM)
1.1.3 Boundary Element Method (BEM)
1.1.4 Discrete Element Method (DEM)
1.1.5 Smooth Particle Hydrodynamics (SHM)
1.1.1 Finite Element Method (FEM)
The Finite Element Method (FEM) is a numerical technique to find approximate solutions of
partial differential equations. It was originated from the need of solving complex elasticity and
structural analysis problems in Civil, Mechanical and Aerospace engineering. In FEM
continuum/domain is divided (discretized) into finite number of simple interconnected pieces
(meshing). These small pieces of finite dimension are called ‘Finite Elements’ (Fig. 1.1).
A field quantity in each element is allowed to have a simple spatial variation which can be
described by polynomial terms. Thus, the original domain is considered as an assemblage of
number of such small elements. These elements are connected through number of joints which
are called ‘Nodes’. While discretizing the structural system, it is assumed that the elements are
attached to the adjacent elements only at the nodal points. Each element contains the material
and geometrical properties. The material properties inside an element are assumed to be
constant. The elements may be 1D elements, 2D elements or 3D elements. The physical object
can be modeled by choosing appropriate element such as frame element, plate element, shell
element, solid element, etc. All elements are then assembled to obtain the solution of the entire
domain/structure under certain loading conditions. Nodes are assigned at a certain density
throughout the continuum depending on the anticipated stress levels of a particular domain.
Regions which will receive large amounts of stress variation usually have a higher node density
than those which experience little or no stress.
A structure can have infinite number of displacements. Approximation with a reasonable level
of accuracy can be achieved by assuming a limited number of displacements. This finite
number of displacements is the number of degrees of freedom (dof) of the structure. For
example, the truss member will undergo only axial deformation. Therefore, the degrees of
freedom of a truss member with respect to its own coordinate system will be one at each node.
If a two dimension structure is modeled by truss elements, then the deformation with respect
to structural coordinate system will be two and therefore degrees of freedom will also become
two. Here, represent displacement and rotation respectively. Here (u, v, w) and (θx, θy, θz)
represent displacement and rotation, respectively.
Here, field variable and their derivatives are the unknown quantities to be determined at the
nodes of the element. The variation of field quantities inside the element is evaluated using
interpolation functions or shape functions expressed in terms of nodal values. Finally, all
elements are assembled to represent continuum/structure to form global equations in terms of
unknown parameters. Global algebraic equations are solved to obtain nodal forces and
deformations.
 From Classic to Finite Element Analysis Solution
Various engineering problems like solid and fluid mechanics, heat transfer, electric and magnetic fields
can easily be solved by the concept of finite element technique.
Advantages of FEA
1. The physical properties, which are intractable and complex for any closed bound solution,
can be analyzed by this method.
2. It can take care of any geometry (may be regular or irregular).
3. It can take care of any boundary conditions.
4. Material anisotropy and non-homogeneity can be catered without much difficulty.
5. It can take care of any type of loading conditions.
6. This method is superior to other approximate methods like Galerkine and Rayleigh-Ritz
methods.
7. In this method approximations are confined to small sub domains.
8. In this method, the admissible functions are valid over the simple domain and have nothing to
do with boundary, however simple or complex it may be.
9. Enable to computer programming.

Disadvantages of FEA
1. Computational time involved in the solution of the problem is high.
2. For fluid dynamics problems some other methods of analysis may prove efficient than the
FEM.
Limitations of FEA
1. Proper engineering judgment is to be exercised to interpret results.
2. It requires large computer memory and computational time to obtain intend results.
3. There are certain categories of problems where other methods are more effective, e.g., fluid
problems having boundaries at infinity are better treated by the boundary element method.
4. For some problems, there may be a considerable amount of input data. Errors may creep up
in their preparation and the results thus obtained may also appear to be acceptable which
 indicates deceptive state of affairs. It is always desirable to make a visual check of the input
data.
5. In the FEM, many problems lead to round-off errors. Computer works with a limited number
of digits and solving the problem with restricted number of digits may not yield the desired
degree of accuracy or it may give total erroneous results in some cases. For many problems
the increase in the number of digits for the purpose of calculation improves the accuracy.
Errors and Accuracy in FEA
Every physical problem is formulated by simplifying certain assumptions. Solution to the problem,
classical or numerical, is to be viewed within the constraints imposed by these simplifications. The
material may be assumed to be homogeneous and isotropic; its behavior may be considered as
linearly elastic; the prediction of the exact load in any type of structure is next to impossible. As such
the true behavior of the structure is to be viewed with in these constraints and obvious errors creep in
engineering calculations.
1. The results will be erroneous if any mistake occurs in the input data. As such, preparation of
the input data should be made with great care.
2. When a continuum is discretized, an infinite degrees of freedom system is converted into a
model having finite number of degrees of freedom. In a continuum, functions which are
continuous are now replaced by ones which are piece-wise continuous within individual
elements. Thus the actual continuum is represented by a set of approximations.
3. The accuracy depends to a great extent on the mesh grading of the continuum. In regions of
high strain gradient, higher mesh grading is needed whereas in the regions of lower strain, the
mesh chosen may be coarser. As the element size decreases, the discretization error reduces.
4. Improper selection of shape of the element will lead to a considerable error in the solution.
Triangle elements in the shape of an equilateral or rectangular element in the shape of a
square will always perform better than those having unequal lengths of the sides. For very
long shapes, the attainment of convergence is extremely slow.
5. In the finite element analysis, the boundary conditions are imposed at the nodes of the
element whereas in an actual continuum, they are defined at the boundaries. Between the
nodes, the actual boundary conditions will depend on the shape functions of the element
forming the boundary.
6. Simplification of the boundary is another source of error. The domain may be reduced to the
shape of polygon. If the mesh is refined, then the error involved in the discretized boundary
may be reduced.
7. During arithmetic operations, the numbers would be constantly round-off to some fixed
working length. These round–off errors may go on accumulating and then resulting accuracy
of the solution may be greatly impaired.

For some problems, there may be a considerable amount of input data. Errors may creep up in
their preparation and the results thus obtained may also appear to be acceptable which indicates
deceptive state of affairs. It is always desirable to make a visual check of the input data.
In the FEM, many problems lead to round-off errors. Computer works with a limited number of
digits and solving the problem with restricted number of digits may not yield the desired degree
of accuracy or it may give total erroneous results in some cases. For many problems the increase
in the number of digits for the purpose of calculation improves the accuracy.

1.1.2 Finite Difference Method (FDM)

The domain is represented by an array of regularly spaced grid points (rows and columns). At
each grid points, the differential equations, representing physical phenomena, are
approximated by converting them into algebraic difference equation using Taylor’s series
expansion. This facilitates the pointwise approximation for the governing equation. A system
of linear equations are formed and solved for unknown values of function at all the grid points.
The application of finite difference method for engineering problems involves replacing the
governing differential equations and the boundary condition by suitable algebraic equations. For
Example, in the analysis of beam bending problem the differential equation is reduced to be
solution of algebraic equations written at every nodal point within the beam member. For example,
the beam equation can be expressed as:

To explain the concept of finite difference method let us consider a displacement function variable
namely w=f(x)
With the help of these equations finite difference method can be written as:

Thus, the displacement at node i of the beam member corresponds to uniformly distributed load
can be obtained from above equations with the help of boundary conditions. It may be interesting
to note that, the concept of node is used in the finite difference method. Basically, this method has
an array of grid points and is a point wise approximation, whereas, finite element method has an
array of small interconnecting sub-regions and is a piece wise approximation.
Each method has noteworthy advantages as well as limitations. However it is possible to solve
various problems by finite element method, even with highly complex geometry and loading
conditions, with the restriction that there is always some numerical errors. Therefore, effective and
reliable use of this method requires a solid understanding of its limitations.
Advantages of FEM
1. Simple to implement
2. Simple to implement via computer program in fluid dynamics related problems.
Disadvantage of FEM
1. Difficult to implement for irregular geometry
 1.1.3 Boundary Element Method (BEM)
In this method, only the boundary of the solution domain is discretized into elements. The
solution of differential equation is approximated by seeking the solution on the boundary and
then that information is used to get the solution inside the domain. Here, differential equation
is converted into an integral equation on the boundary of the domain. The solution at the
boundary is obtained by adding the contributions of the boundary elements describing the
boundary region, in which each boundary element is integrated over the boundary surface. The
solution inside the boundary is obtained using the solution obtained at boundary. The boundary
element method reduces the preprocessing time and dimension of the problem. The method is
applicable to structural analysis and fluid flow analysis.

Boundary
domain

1.1.4 Discrete Element Method (DEM)

Used to model granular materials as an assembly of separate discrete particles with different
shapes and properties. In this method, stresses and displacements in a volume of discrete
particles (aggregates, sand etc.) are computed via simulation. Here, all particles are assumed
to be oriented spatially and an initial velocity assigned. The forces acting in each particle are
computed from the initial data applying relevant physical laws and contact models. All force
are added up to find the total force acting on each particle during a certain time step from
Newton’s laws of motion. Then, new positions are used to compute the forces during the next
step and this loop is repeated until the simulation ends. It is computationally intensive, since it
contains large number of equations covering all particles. The method is applicable to analyze
the mechanical behavior of soils, aggregates, concrete.
1.1.5 Smooth Particle hydrodynamics (SPH)
SPH is a mesh free method based on Lagrangian concept of fluid dynamics. It is a particle
method, where system domain is represented by a set of arbitrary distributed particles.
Connectivity between particles in not required. The physical quantity of any particle is taken
by computing average over neighboring particles which is called smoothing.
In grid based methods, mesh generation becomes difficult for complex geometry. Grid based
methods are not suitable for free surface, deformable boundaries, discontinuities and moving
interface.

1.2 History of Computational Techniques

The finite element analysis can be traced back to the work by Alexander Hrennikoff (1941) and
Richard Courant (1942). Hrenikoff introduced the framework method, in which a plane elastic
medium was represented as collections of bars and beams. These pioneers share one essential
characteristic: mesh discretization of a continuous domain into a set of discrete sub-domains,
usually called elements.
 In 1950s, solution of large number of simultaneous equations became possible
because of the digital computer.
 In 1960, Ray W. Clough first published a paper using term “Finite Element Method”.
 In 1965, First conference on “finite elements” was held.
  In 1967, the first book on the “Finite Element Method” was published by Zienkiewicz
and Chung.
 In the late 1960s and early 1970s, the FEM was applied to a wide variety of
engineering problems.
 In the 1970s, most commercial FEM software packages (ABAQUS, NASTRAN,
ANSYS, SAP) originated. Interactive FE programs on supercomputer lead to rapid
growth of CAD systems.
 In the 1980s, algorithm on electromagnetic applications, fluid flow and thermal
analysis were developed with the use of FE program.
In the 1980s,CAD progressed from a 2D drafting tool to a 3D surfacing tool, and then to a 3D
sIn the 1980s, the use of FEA and CAD on the same workstation with developing geometry
olid modeling system. Design engineers began to seriously consider incorporating FEA into
the general product design process.
Engineers can evaluate ways to control the vibrations and extend the use of flexible, deployable
structures in space using FE and other methods in the 1990s. Trends to solve fully coupled
solution of fluid flows with structural interactions, bio-mechanics related problems with a
higher level of accuracy were observed in this decade.
As the 1990s draw to a place, the PC platform has become a major force in high end analysis.
The technology has become to accessible that it is actually being “hidden” inside CAD
packages.
With the development of finite element method, together with tremendous increases in computing
power and convenience, today it is possible to understand structural behavior with levels of
accuracy. This was in fact the beyond of imagination before the computer age.

1.3 Review of Programming Methods ( FORTRAN or C or C++ or MATLAB)

Old versions of most FEM books include programming samples or even complete programs
in Fortran. Those face an uncertain future. Since the mid-1990s, Fortran is gradually
disappearing as a programming language taught engineering undergraduate programs. It still
survived because of large amounts of legacy code. At present, scientific programming is
moving to C and C++, Java, Perl and Python, Matlab,

Software Packages for FEA

 ABAQUS
 ANSYS
 SAP2000/ETABS
 PATRAN
 NISA/DISPLAY III
 LS DYNA
 CATIA
 SOLID WORKS
 COSMOS
 HYPERMESH
Programming Languages
 MATLAB/OCTAVE
 C, C+
 Fortran
 Python

Problem: Find the perimeter L of a circle of diameter d. Since L = π d, this is equivalent to

obtaining a numerical value for π. Draw a circle of radius r and diameter d = 2r as in Figure
1.1(a). Inscribe a regular polygon of n sides, where n = 8 in Figure 1.1(b). Rename polygon
sides as elements and vertices as nodes. Label nodes with integers 1, . . 8. Extract a typical
element, say that joining nodes 4–5, as shown in Figure 1.1(c). This is an instance of the
generic element i– j pictured in Figure 1.1(d). The element length is Li j = 2r sin(π/n). Since all
elements have the same length, the polygon perimeter is Ln = nLi j , whence the approximation to
π is πn = Ln/d = n sin(π/n).

Figure 1