Engineering Mathematics - Civil Engineering Challenge

Engineering Mathematics 2
Civil Engineering Challenge
Ahmed Alharbi / 000908500
ABSTRACT
Computer modelling has two objectives, materializing the practical simulation and practical
application to the real time problem. Structural dynamics and structural analysis application is day to
day real time application in Civil Engineering. And exploring it deeper, Single Degree of Freedom
system enables the system in structural analysis, when implemented and demonstrated with computer
modelling, the internal structural dynamic procedures can be shown evident, pictorially.
RESEARCH
Structural analysis, which includes structural dynamics is the study of structure’s behaviour and it is
subjected to dynamic loading, which could be traffic, waves, blasts and earthquakes. Going further
and deeper, dynamic analysis is done to explore modal analysis, time history and displacements.
Structural analysis is associated with the study of physical structure behaviour, when subjected to
force.
SDOF or Single Degree of Freedom system stands as the basic structure of structural dynamics. It
respresents the seismic or dynamic behaviour of any structure and the way it can be approximated as a
SDoF or a lollipop model.
SDoF system has a stiffness k, damping coefficient c and mass m.
Here stiffness represents and accounts the restoration of the force that is caused by different structural
elements, during damping caused from loss of energy during the course of excitation. It is modelled as
viscous damping usually. Stiffness can be easily compared and represented by an element called
spring, which exhibits force, on the basis of the law called Hook’s law.
Fspring = ku spring
The equation of motion of the string with stiffness, k, damping c and mass m, of the spring has the
motion equation as,

MX + kx = F(t)
Here, x is displacement and X is the acceleration, represented as a the displacement double derivative.
Here, F(t) represents as sudden constant load application, in terms of Heaviside step function.
The equation of motion using single degree of freedom motion can be explained practically, when
compared to a free body diagram.
A free body diagram has the motion, as shown in the figure 1.
Figure 1
Let us consider a single degree of freedom system, for an object which is subjected to X g (t)
earthquake acceleration. When an object that undergoes relative displacement of x(t), relative velocity
of X(t) and relative acceleration of X(t), respectively.
The system would experience different forces that would be acting over the system, which can be
stiffness force, inertial force and damping force.
Then the Single Degree of Freedom system would have the displacement, velocity and acceleration as
the following.
Figure 2 – Single Degree of Freedom System
When the forces equilibrium act over the mass of the object, as in the free body, as shown in the
Figure 1,

The equilibrium of different forces is,
m(X(t) + xg (t)) + cx(t) + kx(t) = 0
or it can be written as,
mX(t) + cX(t) + kx(t) = -mXg(t)
here,
Xg(t) is ground acceleration of the earthquake
X(t) is mass relative mass acceleration with respect to the ground
X(t) is the mass relative velocity with respect to the ground
x(t) is the mass relative displacement with respect to ground
finally, motion of the single degree of freedom can be represented in the normalized form as,
X(t) + 2 ξ w0 X(t) + w02 x(t) = -Xg (t)
Here,
ξ = damping ration, which is c/2mw0
w0 = natural frequency, which is equal to square root of k/m
and SDoF system’s damped natural frequency is wd = w0 (square root (1- ξ2)
The linear motion equation that damped SDOF system would be constant coefficients consisting
second order differential equation.
The final solution of the linear motion equation for the acceleration would be the final SDOF system
response.
SDOF SYSTEM RESPONSE ANALYSIS
If earthquake is considered, for instance, for a given history of time, which is acceleration versus time
data, for earthquake ground motion, finally, the viscously damped SDOF system response is found
through Frequency Domain Analysis or Time Domain Analysis.
Frequency Domain Analysis
Time Domain Analysis

Time domain analysis enables to get the SDOF response, both in non-linear along with linear range.
These differential equations can get numerical solutions through Numerical schemes and Duhamel
integration, like Runge-Kutta, Newmark and some more methods.
The numerical solution, Duhamel Integral is specific to the SDOF system, having subjected to ground
motion of earthquake. And the SDOF system motion equation is subjected to the acceleration of
ground motion. The final solution would be split into particular part and homogeneous part as,
X(t) = xh(t) + xp (t)
Here,
xp(t) is particular solution
xh(t) is homogenous solution
And complimentary or homogeneous solution would be the vibration response, which are damped
free, as shown as below...
xh(t) = g(t) x0 + h(t) X(0)
here, X0 is the SDOF system velocity and
x0 is the SDOF initial displacement
In this context, even minor changes in displacement and velocity that result during certain time
interval would result in momentum change with negligible contribution.
Numerical Methods for SDOF System’s Seismic Analysis
Initial boundary value problems can be solved using various numerical methods available. And the
most common methods used are Newmark’s Beta method, which is also known to be linear
acceleration method.
Newmark’s Beta Method
Velocity, acceleration and displacement at a specific duration of next unit of time can be obtained as a
function of displacement, velocity and acceleration at the specific instant of time, when linear
acceleration is assumed during this small step of time.
The method implements integration of step by step numerical integration to obtain SDOF system
response, for the given time history.

This method has a limitation. Since it implements time stepping methods, it has certain limitation, in
terms of resulting in errors, while accumulating while proceeding with the calculation. Hence, when
above equality satisfies, the method stands as conditionally stable, on the other hands it blows up,
while giving results illogically.
Frequency Domain Analysis
Frequency domain analysis method helps to obtain linear systems response, which is subjected to
excitations that are irregular, especially, like forces of earthquake. The basic requisite for this method
is the knowledge of function of complex frequency response for certain specific application. This
method would be much superior compared to the time domain, when the SDOF system’s damping c
and stiffness k are dependent on the frequency.
The SDOF system response can be given in frequency domain analysis, as,
Here, Xg(w) is [-xg (t)]’s Fourier Transform and
H(w) stands as the frequency response complex function.
When SDOF system is subjected to the eiwt forcing function, displacement response is produced as,
x(t) = H(w)eiwt
Figure: Complex Frequency Response Function
When SDOF motion equation is considered, the complex function of frequency response can be
expressed as,
H(w) = 1/ ((w02 – w2) + i2ξww0)
After Fourier transform of [-xg(t) ] is implemented with the properties of Fourier transform and
substituted in the motion equation,

MODELLNG
Having researched the Single Degree of Freedom system and deriving equations for motion,
acceleration, through the basic inputs, mass, stiffness and damping, the further step is to model the
system by taking a specific case and simulating with certain assumed input values.
Modelling is performed in two parts, Part A and Part B.
Part A
Part A is the solution performed through Fourier Analysis for the Single Degree of Freedom object.
Initially, a closed indoor construction, a cafe is considered. The column is constructed with concrete
specific bulk density. The column is taken with Young’s Modulus of 210*10^9.
Part A is implemented with the following steps.
1. For the construction, length, breadth, depth, beam length and column length are considered as
the input values and given as the inputs for the simulation in MATLAB. Other important
inputs are Young’s modulus of column, second moment of inertia are also taken as input
constant values.
2. Slab area is calculated.
3. Slab load, beam load, column load are calculated and then total weight of the structure,
eurocode for cafe, mass of the structure, total SDoF structure stiffness, natural period of SDoF
structure and SDoF structure natural frequency are calculated.
4. Then based on the inputs and intermediary calculated values, Fourier analysis is performed.
The basic time duration is taken 0.01 seconds, initial time is considered and taken as 0.
Thereby, sampling frequency, maximum frequency, frequency step and frequency range are
calculated.
a. By F(w) vector is attempted to obtain through a loop, which repeats for number of
units, divided on the basis of time duration. Then finally, excitation factor is
determined.
b. Again in another loop, transfer function is calculated by iterating for total time units.
c. Shifting of transform function is performed to certain and specified units.