This document discusses using the Laplace transform method to analyze various civil engineering structures. It provides examples of how the Laplace transform has been used to analyze beams, columns, beam-columns, and determine dead load deflections in bridges. The key benefits of the Laplace transform method are that it provides efficient solutions to differential equations governing the behavior of structures without finding the general solution, saving time and labor compared to classical methods.

1. 1

2. Presented by:
Group # 4
(1)MISBAHULLAH (G.LEADER)
(2) KAMRAN KHAN
(4) AZAZ DURANI
(5) SHAFIQ ABDUL MALIK
Presented to:
SIR NADEEM SHEIKH
Department of Civil Engineering
City University of Science and Information Technology

3. PRESENTATION
Outline
Brief History
Definition Of LT
Laplace Transform Tables
Properties of LT
Application of Laplace Transformation to
Civil engineering
Electrical engineering
Mechanical engineering
Environmental engineering
The field of economics
Medical field
Refrences

4. Introduction
Differential equations have wide applications in various engineering and
science disciplines. In general ,modeling variations of a physical quantity,
such as temperature, pressure, displacement, velocity, stress, strain, or
concentration of a pollutant, with the change of time t or location, such
as the coordinates (x, y, z), or both would require differential equations.
Similarly, studying the variation of a physical quantity on other physical
quantities would lead to differential equations. For example, the change
of strain on stress for some viscoelastic materials follows a differential
equation .It is important for engineers to be able to model physical
problems using mathematical equations ,and then solve these
equations so that the behavior of the systems concerned can be
studied. In this presentation, a few examples are presented to illustrate
how practical problems are modeled mathematically and how
differential equations arise in them and how these differential equations
can be solved by Laplace Transformation Method

5. The Laplace transformation was introduced by P. S. Laplace in1779 (4). It is a linear
integral transformation which enables one to solve many ordinary and partial linear
differential equations. The solution is easily obtained without finding the general
solution and then evaluating the arbitrary constants., as required by the classical
method.This results in a savings of time and labor.
The Laplace transformation is the best known form of operational mathematics. The
form as it is known today is the result of extensive research and development by
Doetsch ( 3) and others. Beginning in the late 19301-s., the Laplace. transformation
has been a powerful tool in the solution of linear· circuit problems in the electrical
engineering field, Only in the last ten to fifteen years has it gained usage in the
dynamics of mechanical and fluid systems. An area in which it has not been exploited
to the same degree is the analysis of statically loaded structural members.
HISTORY

6. The Laplace transform is defined in the following way. Let f(t) be defined for t >0: Then the
Laplace transform of f(T) which is denoted by L[f(t)] or by F(s), is defined by the following
equation.
DEFINITION OF LAPLACE TRANSFORM
 t is real, s: complex frequency
 Called “The One-sided or unilateral Laplace
Transform”.
 In the two-sided or bilateral LT, the lower limit is -.
We do not use this.




0
)()}({)( dtetftfLsF st

7. Laplace Transform
Definition
• The Common Notation of Laplace transform is
    
    st
st
GgL
FfL

    
   st
st
Gg
Ff



8. Laplace Transform Tables
Evaluating F(s) = L{f(t)}
1
1
)sin()(
sin)(
1
)(
)(
1
)10(
1
)(
1)(
2
0
0 0
)(
0









 



 




s
dttesF
ttf
as
dtedteesF
etf
ss
dtesF
tf
st
tasstat
at
st
let
let
(Integrate by parts)

9. Properties of Laplace Transforms
• Linearity
• Impulse function
• “frequency” or s-plane shift
• Multiplication by tn
• Step ,Unit function
• Differentiation

10. Properties: Linearity
)()()}()({ 22112211 sFcsFctfctfcL 
Example :
1
1
)
1
)1()1(
(
2
1
)
1
1
1
1
(
2
1
}{
2
1
}{
2
1
}
2
1
2
1
{
)}{sinh(
22













ss
ss
ss
eLeL
eey
tL
tt
tt
Proof :
)()(
)()(
)]()([
)}()({
2211
0
22
0
11
22
0
11
2211
sFcsFc
dtetfcdtetfc
dtetfctfc
tfctfcL
stst
st













11. Properties: S-plane (frequency) shift
)()}({ asFtfeL at

Example :
22
)(
)}sin({





as
teL at Proof :
)(
)(
)(
)}({
0
)(
0
asF
dtetf
dtetfe
tfeL
tas
stat
at












12. Properties: Derivatives
(this is the big one)
)0()()}({ 
 fssFtDfL
Example :
)}sin({
1
1
1
)1(
1
1
)0(
1
)}cos({
2
2
22
2
2
2
2
tL
s
s
ss
s
s
f
s
s
tDL











Proof :
)()0(
)()]([
)(,)(
,
)()}({
0
0
0
ssFf
dtetfstfe
tfvdttf
dt
d
dv
sedueu
dtetf
dt
d
tDfL
stst
stst
st













let

13. ?)}({ 2
tfDL
)0(')0()()0('))0()((
)0(')}({)0()()}({
)0(')0()0(),()(
2
2
fsFsFsffssFs
ftDfsLgssGtgDL
fDfgtDftg


let
)0()0()0(')0()()}({ )'1()'2()2()1( 
 nnnnnn
fsffsfssFstfDL 
NOTE: to take
you need the value @ t=0 for
called initial conditions!
We will use this to solve differential equations!

)(),(),...(),( 21
tftDftfDtfD nn
)}({ tfDL n
Properties: Nth order derivatives

14. 14
Laplace Transform
(Unit, Impulse and exponential ftn
Determine the Laplace transform of each of the
following functions shown below:
 






11
00
u
t
t
t

15. Laplace Transform
Impulse Function
• The easiest transform is that of the impulse function:
    
1
δδL
0
0







s
st
e
dtett
  1δ  t
• Next is the unit step function
  



0
1
1)()(
s
dtesFtuL st

16. Application of Laplace Transformation
 One of the most easiest and simplest tool for the differential equations.
 The derivative theorem opens up the possibility of utilizing the Laplace transform
as a tool for solving ordinary differential equations. Numerous applications of the
Laplace transform to ODEs will be found in ensuing sections.
 Many practical problems involve mechanical or electrical system acted by
discontinues or impulsive forcing terms. The ODE used to model these process can
be a little awkward to solve by using ODE’s methods .we shall learn a new method
based on the concept of “integral transform “
How to use Laplace
 Find differential equations that describe system
 Obtain Laplace transform
 Perform algebra to solve for output or variable of interest
 Apply inverse transform to find solution

17. How to use Laplace
Taking Example
Solutios
Taking Laplace transform

18. How to use Laplace

19. How to use Laplace

20. How to use Laplace
Solutios

21. The more prominent American textbooks on the Laplace transformation e.g. Churchill (2) and
Thomson (13), apply the transform method to the static deflection of beams and columns.
APPLICATION TO THE FIELD OF CIVIL ENGINEERING
Strandhagen (12) applied the transform to the deflection of "beam columns", i.e. beams
subjected to axial loads as well as transverse loads. The general cases were:
1. Simple beam with unequal end moments and no transverse loads
2. Propped cantilever beam with a uniformly distributed load
3. Propped cantilever beam with a triangularly distributed load
4. Fixed beam with a triangularly distributed load
5. Fixed beam with a parabolic ally distributed load.
Wagner (16) investigated the stability of buckling members. The
types of columns covered were:
1. Hinged on both ends
2. One end fixed, and the other free
3. One end hinged,. the other guided and hinged
4. One end fixed, 'the other guided and hinged
5. One end fixed,. the other guided and fixed

22. ANALYSIS OF ELEMENT ARY BEAMS
BY THE LAPLACE TRANSFORM METHOD
The fundamental basis for the analysis of these members is the equation of their elastic curves.
When these equations are known, the other pertinent design data can be readily obtained, i. e. :
a. The maximum deflection b. The support reactions
c. The restraining moments (if any) d. The slopes at the supports
e. The distribution of the bending moment and shearing force along the member
f. The maximum bending moment.
In normal design practice the bending moment and shearing force distribution are perhaps the
most important of the above data.
Beam Equation
The equation of the elastic curve can be obtained from the basic. beam equation
El y(4) = F(x)
where
E = Young's modulus of elasticity. I = Moment of inertia of area.
y(4) = Fourth derivative. with respect to x. of the transverse deflection.
F(x) = · Load function.
E q 1-1

23. ANALYSIS OF ELEMENT ARY BEAMS
BY THE LAPLACE TRANSFORM METHOD
Transformation of the Beam Equation If y(s) and f(s) denote the Laplace transforms of Y(x) and
F(x) respectively, then the Laplace transformation of Eq. (1-2) gives
E q 1-2
Taking inverse transformation on the E q 1-2
E q 1-3
This expression is then solved for the subsidiary equation, y(s),
where the initial boundary conditions are(These boundary conditions can be determine
from
the beam supports)

24. ANALYSIS OF ELEMENT ARY BEAMS
BY THE LAPLACE TRANSFORM METHOD
Load Function The load function F(x). is defined as the system of loads acting on the beam.
Distributed loads will be represented by the unit step function S k(x). Concentrated loads by the
unit impulse function S’k. (x), and applied moments by the unit doublet function S ‘'k.. (x) •
Sign Convention
A right-handed system of rectangular co-ordinates X, Y, Z
will be used. The origin will be taken at the left end of the
beam with the X-axis coinciding with the neutral surface and
the Y- and z- axes taken along the centroidal principal axes
of the cross section.

25. ANALYSIS OF ELEMENT ARY BEAMS
BY THE LAPLACE TRANSFORM METHOD
General Solution of a Simple Beam
A. Boundary Conditions
Y(0) = Y'' (0) = Y''' (0) = 0, = 0
B. Load Function
C. Laplace Transform of Load Function
D. Inverse Transform of

26. ANALYSIS OF ELEMENT ARY BEAMS
BY THE LAPLACE TRANSFORM METHOD
E. Substituting (A) and (D) into Eq. (1-3)
F. Evaluation of Unknown BoundaryConditions and Reactions
The reaction R1 can be determined by statics.
G. General Elastic/Deflection Curve Equation

27. ANALYSIS OF ELEMENT ARY BEAMS
BY THE LAPLACE TRANSFORM METHOD
H. Slope, Bending Moment, and Shearing Force Equations
a. Slope
b. Bending Moment
c. Shearing Force

28. ANALYSIS OF VARIUS CIVIL ENGINEERING STRUCTURES
BY THE LAPLACE TRANSFORM METHOD
• COLUMNS
Thomson (13) and Wagner (16) have made extensive use of the Laplace transform in the
analysis of columns. They have considered the centrally loaded, constant and multiple cross
section column with the usual types of end conditions.
• However, in the related area of 11beam column" analysis,. The Laplace transform can be
used to develop an efficient and powerful procedure of analysis. Strandhagen (12) has laid
the basis for this method. He applied the transform in determining the deflections of single
span, constant cross section members that were subjected to various kinds of distributed
transverse loads and to axial loads. The development of a method similar to that for
elementary beams should be possible, i.e., a set of generalized solutions for the various types
of beam columns loaded by any system of distributed loads, concentrated loads, and applied
moments over any portion of the span.
• Dead Load Deflections
• Bridge beams and similar structures are usually cambered to compensate for dead load
deflections. To determine the amount of camber, it is necessary to know the deflection
values at many points along, the span or spans.
• In the case of a constant cross section beam,. regardless of the number of spans, it was
found that the method developed for elementary beams is ideally suited, for this task, over
the classical methods. Be sides knowing the kind and amount of dead load, the only
additional information required is either the moment or reaction values at one support as
determined by other methods, Knowing either of themsevalues, the deflection can be easily
computed at any point in any span of the beam.

29. ANALYSIS OF VARIUS CIVIL ENGINEERING STRUCTURES
BY THE LAPLACE TRANSFORM METHOD
Frames, Grid Structures,. and End Fixity
In general, these topics are quite similar to section discuss previously; that is, once an end
condition (reaction or moment) is determined by other methods, the procedures of
elementary beam analysis can be used to determine the values of deflection, slope, moment,
or shear.
IMPACT ANALYSIS USING THE LAPLACE TRANSFORM METHOD
When an item, be it a machine, household appliances, or guided missile is to be shipped via a
commercial or military mode of transportation, one of the primary factors in.its design criteria
is the "G"factor.This "G'' factor is the maximum or peak acceleration to which the. item may be
subjected to while in transit or during handling and loading operations. It is used in the design
procedure either as a "factor of safety" or as a "limiting factor". As a factor of safety it is used
to determine the design load or working stress for the structural elements of the item. In those
cases where the strength of the elements is limited by space, size, materials,. and other
criteria, the "G" factor is used in the design of a shock mitigating system which will limit the
maximum acceleration to the required safe level.
Different methods is used for determinig of G values however the Laplace transform method
can be used in analytically determining a 'G''' factor which can be verified by instrumented
drop tests.

30. SUMMARY AND CONCLUSIONS
Utilizing the Laplace. transform method, the formulation of a generalized procedure
of analysis for elementary static beam systems has been the primary objective of
this presentation.This objective has been achieved. The procedure developed is
applicable to any single span, constant cross section beam that may be subjected to
a static transverse system of distributed loads, concentrated loads, and/or applied
moments .
The principal advantages of the transform method are found in the ease with which
complex load functions can be dispatched, the capability of being able to write the
solution as one equation for the entire span, and the reduction in solution time over
classical methods.
 In the field of impact analysis the Laplace transformation is an important tool. It
vast used now a days is done in the field of dynamics , specially used in the field
of fluid mechanics and earthquake engineering.
In conclusion, the Laplace transformation is considered to be an efficient and
powerful method of analysis in the static and dynamic structures field where Ws
development and applications are far from having been exhausted.

31. APPLICATION TO SOME OTHER FIELD OF ENGINEERING
Pollutant Transport and Diffusion - Introduction

32. APPLICATION TO SOME OTHER FIELD OF ENGINEERING

34. APPLICATION IN MECHANICAL ENGINEERING
Vibrating Mechanical Systems: In examining the suspension system of the car the
important elements in the system are the mass of the car and the springs and damper used
to connect to the body of the car to the suspension links. Mechanical translational systems
may be used to model many situations, and involve three basic elements: masses (having
mass M, measured in kg), springs (having spring stiffness K, measured in Nm−1) and
dampers (having damping coefficient B, measured in Nsm−1). The associated variables are
displacement x(t) (measured in m) and force F(t) (measured in N). Consider Mass–spring–
damper system
Mass–spring–damper system can be modeled using
Newton’s and Hooke’s law. Therefore the differential
equation representing to the above system is given by

35. Taking inverse Laplace transforms gives the required response
APPLICATION IN ECONOMIC PROBLEMS
Consider case of investment project. For various alternatives one wishes to calculate the
present value of series of cash receipts and transactions. The Present value of a series of
payments given by,
Writing in exponential form ,taking Laplace transform ,

36. Example An insurance company has just launched a security that will pay Rs.500
indefinitely, starting the first payment next year. How much should this security be worth
today if the appropriate return is 10% ? We solve this example by using the time line,
APPLICATION IN ECONOMIC PROBLEMS
SOME REAL LIFE USES INTO VARIOUS FILDS

37. Various
applications of
laplace
transforms
Nuclear Physics
Control systems
Digital Signal Processing
Electrical
circuit
analysis
Deflection
in beams

38. Application in digital signal
processing
A simple Laplace transform is conducted while
sending signals over two-way communication
medium (FM/AM stereos, 2 way radio sets,
cellular phones.)
When information is sent over medium such as
cellular phones, they are first converted into
time-varying wave and then it is super imposed
on the medium. In this way, the information
propagates. Now, at the receiving end, to
decipher the information being sent, medium
wave’s time functions are converted to
frequency functions. This is a simple example of
Laplace transform in real life.

40. • Laplace transformation is crucial for the study of
control systems, hence they are used for the analysis
of HVAC(Heating, Ventilation and Air Conditioning )
control systems, which are used in all modern building
and constructions.

41. Use of Laplace transforms in
nuclear physics
• In order to get the true form of radioactive decay, a Laplace
transform is used.
• It makes studying analytic part of Nuclear Physics possible.

42. Mathematical Modelling of the Road Bumps Using
Laplace Transform
• Traffic engineering is the application of Laplace Transform to the
quantification of speed control in the modelling of road bumps
with hollow rectangular shape. In many countries the current
practice used for lowering the vehicle speed is to raise road
bumps above the road surface. If a hollow bump is used it may
be and offers other advantages over road bumps raised above
the road surfaces. The method models the vehicle as the
classical one-degree-of-freedom system whose base follows the
road profile, approximated by Laplace Transform.
 Laplace transforms can be used in areas such as medical field for
blood-velocity/time wave form over cardiac cycle from common
femoral artery.

43. Consider the suspension bridge as shown, which consists of the main cable, the
hangers, and the deck. The self-weight of the deck and the loads applied on the
deck are transferred to the cable through the hangers
Set up the Cartesian coordinate system by placing the origin O at the lowest point
of the cable. The cable can be modeled as subjected to a distributed load w(x). The
equation governing the shape of the cable is given by
where H is the tension in the cable at the
lowest point O. This is a second-order
ordinary differential equation

44. 1
A sawtooth function
Laplace transforms are particularly effective
on differential equations with forcing functions
that are piecewise, like the Heaviside function,
and other functions that turn on and off.

45. An inverse Laplace transform is an improper
contour integral, a creature from the world
of complex variables.
That’s why you don’t see them naked very often.
You usually just see what they yield, the output.
In practice, Laplace transforms and inverse
Laplace transforms are obtained using tables
and computer algebra systems.
Laplace Transforms in real life

46. Other Applications
Semiconductor
mobility
Call completion in
wireless networks
Vehicle vibrations on
compressed rails
Behavior of magnetic
and electric fields
above the atmosphere

47.  Johnson, William B. Transform method for semiconductor mobility, Journal of
Applied Physics 99 (2006).
 Laplace Transform solved problems Pavel Pyrih (May 24, 2012)
 International Journal of Trend in Research and Development, Volume 3(1), ISSN:
2394-9333 www.ijtrd.com Applications of Laplace Transforms in Engineering and
Economics Lecture Notes for Laplace Transform Wen Shen (April 2009)
 Differential Equations for Engineers Wei-Chau Xie University of Waterloo
 ANALYSIS AND APPLICATIONS OF LAPLACE /FOURIER TRANSFORMATIONS IN
ELECTRIC CIRCUIT BY M. C. Anumaka
 Laplace Transforms: Theory, Problems, and Solutions Marcel B. Fina Arkansas
Tech University
 LAPLACE TRANSFORMS AND ITS APPLICATIONS Sarina Adhikari Department of
Electrical Engineering and Computer Science, University of Tennessee.
 APPLICATION OF THE LAPLACE TRANSFORM METHOD(TO THE ANALYSIS OF LOAD
CARRYING.MEMBERS BY KENNETH HORACE KOERNER JR.I(Bachelor of Science
Oklahoma State University Stillwater,. Oklahoma 1957
References :