Civil Engineering : Structural analysis & mechanics, THE GATE ACADEMY

STRUCTURAL ANALYSIS
&
MECHANICS
for
Civil Engineering
By
www.thegateacademy.com

Syllabus Structural Analysis/Mechanics
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Syllabus for
Structural Analysis
Analysis of statically determinate trusses, arches, beams cables & frames,
displacement in statically determinate structure and analysis of statically
indeterminate structures by force/energy methods, analysis by displacement
methods (slope deflection & moment distribution method) influence lines for
determinate & indeterminate structure. Basic concept of matrix method of structural
analysis.
Mechanics
Bending Moment & shear Force in statically determinate beams. Simple stress &
strain relationship; stress & Strain in two dimensions, principal stress, stress
transformations, Mohr’s cycle simple bending theory. Flexural & shear stresses, un
symmetrically bending, shear center, Thin walled pressure vessels, uniform torsion,
buckling of column, combined & Direct bending tresses.
Analysis of GATE Papers
(Mechanics & Structural Analysis)
Year Percentage of marks Overall Percentage
2013 19
14.93%
2012 10
2011 19
2010 12
2009 14
2008 15.33
2007 14
2006 18
2005 12.3
2004 14.6
2003 16

Contents Structural Analysis/Mechanics
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C O N T E N T S
Chapters Page No.
STRUCTURAL ANALYSIS
#1. Trusses and Arches 1-16
 Trusses 1 – 2
 Arches 2 – 8
 Solved Examples 9 – 14
 Assignment 1 15
 Answer Keys&Explanations 16
#2. Influence Line Diagram and Rolling Loads 17 - 27
 Influence Line Diagram and Rolling loads 17 – 20
 Assignment 1 21
 Assignment 2 22 – 25
 Answer Keys & Explanations 26 – 27
#3. Slope and Deflection Method 28 – 41
 Various Methods for Determining Slope and Deflection at
any Section of a Beam
28 – 31
 Macaulay’s Method 32 – 33
 Conjugate Beam Method 33
Answer Keys & Explanations 39 – 41
#4. Degree of Static Indeterminacy 42 - 51
 Degree of Static Indeterminacy 42
 Alternative Approach to find Degree of Static
Indeterminacy
42
 Assignment 1 45
#5. Displacement Method 52-74
 Displacement Methods 52 – 53
 Carry Over Factor 53 – 56
 Slope Deflection Method 60 – 66

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#6. Force / Energy Methods 75 - 91
 Introduction 75
 Maxwell’s Reciprocal Theorem 75 – 76
 Betti’s Theorem or Generalized Reciprocal Theorem 76 – 77
 Castigliano’s Theorem 77 – 78
 Unit Load Method 79 – 82
 Conjugate Beam Method 82 – 83
 Assignment 1 84
#7. Matrix Method of Structural Analysis 92 – 105
 Matrix Concept & Algebra 92 – 93
 Flexibility & Stiffness 93 – 94
 Assignment 1 98 –100
MECHANICS
#8. Simple Stress and Strain Relationship 106 – 134
 Simple Stress 106 – 109
 Compound Bars 109 – 114
 Mohr’s Circle 115 – 116
#9. Bending Moment and Shear Force Diagram 135 -165
 Terminology 135 – 136
 Bending Moment Diagram 136 - 141

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#10. Thin Walled Pressure Vessel 166–181
 Thin Walled Pressure Vessels 166 – 168
 Tension of Circular Bars or Shafts 168 – 173
 Limitations of Euler’s Formula 173 – 174
 Assignment 2 180
 Answer Keys & Explanations 181
#11. Simple Bending Theory 182-202
 Simple Bending Theory 182 – 183
 Bending of Built up Section 183 – 184
 Shear Stresses in Beams 184 – 189
 Assignment 1 196
Module Test 203 - 217
Reference Books 218

Chapter 1 Structural Analysis
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CHAPTER 1
Trusses and Arches
Trusses
Classification of trusses
Where - Degree of static indeterminacy
Assumption involved in analysis of trusses
1. All the joints are pin connected and free from friction.
2. Members will be subjected to only axial forces
3. Self wt of members is negligible
4. The loading is such that force in the member are within elastic limit.
Procedure Of Analysis
1. Find degree of static indeterminacy using = m + r – 2j
m = No. of members
r = No. of external reasons
j = No. of joints
if = 0, Truss is determinate and stable.
Determinate truss can be analyzed by methods
a) Method of joints
b) Method of sections
c) Graphical method or Williot Mohr diagram
2. If > 0, then truss is indeterminate.
In terminate truss can be analyzed by methods
a) Unit load method
b) Maxwell’s method
Trusses
Plane Trusses (2D) Space Trusses (3D)
Determinate trusses
( )
Indeterminate trusses
( > 0)

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c) Graphical method
Analysis of Determinate Trusses.
Methods of joints
1. Every joints there are two equation of equilibrium i.e. ∑ and ∑ = 0,
This method is not applicable for numbers of unknowns at any point are more than 2
2. In order to find internal reactions, following equations equilibrium may be used
∑ , ∑ , ∑M = 0
3. Not more than three equilibrium in general cannot be obtained until and unless additional
equations of equilibrium are provided.
4. After finding the external equations, apply joint equilibrium at each joint one by one
5. Tensile force are assumed to be (+) and compressive force are assumed to be (-).
Notes
1. If truss is externally indeterminate by degree 1 but internally, determinate by -1, then = 0.
In such case in order to find external reactions, there must be conditions such that a part of
truss may be rotated about a practical joint.
Example
= m + r -2j = x = 0
= 4-3 = 1
= -1
No. of equilibrium conditions 3 condition obtained through rotation at C i.e. ∑ M = 0
2. If at a joint three members meet and two of them are collinear and there is no external force
at that joint, then the third members carries zero force always.
3. If at a joint only two members meet and there is no external force at that joint and if
members are not collinearly then both member will carry zero forces.
Arches
Arches are a structure which eliminates tensile stresses in spanning great amount of open space.
All the forces are resolved into compressive stresses.
C
BA

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Types of arches based on number of hinges
1) Single hinged arch
2) Two hinged arch
3) Third hinged arch
4) Fixed arch or hinge less arch
 A three hinged arch is a statically determinate structure whre the rest three aches are
statically indeterminate
 In bridge construction, especially is railroad bridges, the more frequently used aches the
two hinged and the fixed end ones.
Three Hinged Arches
 A three hinged arch is a statically determinate structure, having a hinged at each
abutment or springing, and also at the crown.
 In three hinged arch three equations are available from static equilibrium and one
additional equation is available from the fact that B.M at the hinge at the crown is zero.
Standard cases
Case1: A three hinged parabolic arch of span L rise ‘h’ carries a udl of w over the whole span
(a) The horizontal reaction at each support is H =
(b) The net bending moment and shear force at any section on the parabolic three hinged
arch is zero
M = Beam moment – H moment = 0
Case2: A three hinged semicircular arch of radius ‘R’ carries a udl of w even the whole span.
H H
h
L
W unit per run
(a) Single Hinged arch (b) Two hinged arch
(c) Two hinged arch (d) Fixed arch

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(a) The horizontal thrust at each end; H =
(b) The maximum bending moment for the arch is Mmax = ( ogging) which occurs at θ
= 3 from the horizontal and the distance of point of maximum bending moment from
the crown is,
R cos 3 =
√
Case3: A three hinged arch consisting of two quadrant parts AC and CB of radii R and R . The
arch carries a concentrated load of W on the crown
= = H =
Case4: A symmetrical three- hinged parabolic arch of span land rise carries a point load w, which
may be placed anywhere on the span
y h
A
H
K
W
H
B
x
L
H A
H
B
C
W
R
H
W unit per run
C
R
y
θ
0
WRWR
X

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H =
The absolute maximum bending moment occurs at a distance of
√
on either side of the crown
Case5: A three hinged parabolic arch of span l has its abutments at depth h and h below the
crown the arch carries a ude of w per unit length over the whole span
The horizontal thrust at each support is give by H =
(√ √ )
Case6: A three hinged parabolic arch of span l has its abutments A and B at depth h and h
below the crown C. The arch carries a concentrated load W at the crown.
The horizontal thrust at each support is given by
H =
(√ √ )
Hinged Arches:
Two hinged arch is an indeterminate structure. and can be determined by taking moment
about either end. The horizontal thrust at each support may be determined from the condition
that the horizontal displacement of the either hinge with respect to other is zero
H =
∫
.
∫
Where, M is beam moment
C
y h
B
HH
A
L
x
H A
A
h
L
h
B
HL
L

Civil Engineering : Structural analysis & mechanics, THE GATE ACADEMY

Civil Engineering : Structural analysis & mechanics, THE GATE ACADEMY

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Civil Engineering : Structural analysis & mechanics, THE GATE ACADEMY