STRUCTURAL ANALYSIS & MECHANICS for
Civil Engineering By
www.thegateacademy.com
Syllabus
Structural Analysis/Mechanics
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
2013
19
2012
10
2011
19
2010
12
2009
14
2008
15.33
2007
14
2006
18
2005
12.3
2004
14.6
2003
16
Overall Percentage
14.93%
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Contents
Structural Analysis/Mechanics
CONTENTS Chapters #1.
STRUCTURAL ANALYSIS
Trusses and Arches
1-16
1–2 2–8 9 – 14 15 16
Trusses Arches Solved Examples Assignment 1 Answer Keys&Explanations
#2.
#3.
Influence Line Diagram and Rolling loads Assignment 1 Assignment 2 Answer Keys & Explanations
17 - 27 17 – 20 21 22 – 25 26 – 27
28 – 41
28 – 31
Various Methods for Determining Slope and Deflection at any Section of a Beam Macaulay’s Method Conjugate Beam Method Assignment 1 Assignment 2 Answer Keys & Explanations
32 – 33 33 34 – 35 36 – 38 39 – 41
Degree of Static Indeterminacy
42 - 51
42 42
#5.
Influence Line Diagram and Rolling Loads
Slope and Deflection Method
#4.
Page No.
Degree of Static Indeterminacy Alternative Approach to find Degree of Static Indeterminacy Solved Examples Assignment 1 Assignment 2 Answer Keys & Explanations
43 – 44 45 46 – 48 49 – 51
Displacement Method
52-74
52 – 53 53 – 56 57 – 60 60 – 66 67 – 68
Displacement Methods Carry Over Factor Solved Examples Slope Deflection Method Assignment 1
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i
Contents
#6.
#7.
Structural Analysis/Mechanics
69 – 71 72 – 74
Assignment 2 Answer Keys & Explanations
Force / Energy Methods
75 - 91
75 75 – 76 76 – 77 77 – 78 79 – 82 82 – 83 84 84 – 87 88 – 91
Introduction Maxwell’s Reciprocal Theorem Betti’s Theorem or Generalized Reciprocal Theorem Castigliano’s Theorem Unit Load Method Conjugate Beam Method Assignment 1 Assignment 2 Answer Keys & Explanations
Matrix Method of Structural Analysis
92 – 105
92 – 93 93 – 94 95 – 97 98 –100 100 – 101 102 – 105
Matrix Concept & Algebra Flexibility & Stiffness Solved Examples Assignment 1 Assignment 2 Answer Keys & Explanations
MECHANICS #8. Simple Stress and Strain Relationship
Simple Stress Compound Bars Mohr’s Circle Solved Examples Assignment 1 Assignment 2 Answer Keys & Explanations
#9. Bending Moment and Shear Force Diagram
Terminology Bending Moment Diagram Solved Examples Assignment 1 Assignment 2 Answer Keys & Explanations
106 – 134 106 – 109 109 – 114 115 – 116 117 – 128 129 – 130 130 – 132 133 – 134
135 -165 135 – 136 136 - 141 142 – 156 157 – 158 158 – 162 163 – 165
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ii
Contents
#10. Thin Walled Pressure Vessel
Thin Walled Pressure Vessels Tension of Circular Bars or Shafts Limitations of Euler’s Formula Solved Examples Assignment 1 Assignment 2 Answer Keys & Explanations
#11. Simple Bending Theory
Simple Bending Theory Bending of Built up Section Shear Stresses in Beams Solved Examples Assignment 1 Assignment 2 Answer Keys & Explanations
Structural Analysis/Mechanics
166–181 166 – 168 168 – 173 173 – 174 175 – 178 179 – 180 180 181
182-202 182 – 183 183 – 184 184 – 189 190 – 195 196 197 – 199 200 – 202
Module Test
203 - 217
Reference Books
218
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iii
Chapter 1
Structural Analysis
CHAPTER 1 Trusses and Arches
Trusses Classification of trusses Trusses
Plane Trusses (2D)
Determinate trusses ( )
Where
Space Trusses (3D)
Indeterminate trusses ( > 0)
- Degree of static indeterminacy
Assumption involved in analysis of trusses 1. 2. 3. 4.
All the joints are pin connected and free from friction. Members will be subjected to only axial forces Self wt of members is negligible The loading is such that force in the member are within elastic limit.
Procedure Of Analysis 1. Find degree of static indeterminacy using m = No. of members r = No. of external reasons j = No. of joints if = 0, Truss is determinate and stable.
= m + r – 2j
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 THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, info@thegateacademy.com © Copyright reserved. Web: www.thegateacademy.com Page 1
Chapter 1
Structural Analysis
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 C
A
= m + r -2j = x = 4-3 = 1 = -1 No. of equilibrium conditions
B
=0 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.
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Chapter 1
Structural Analysis
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)
(c)
Single Hinged arch
(b)
Two hinged arch
Two hinged arch
(d)
Fixed 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 W unit per run
h H
H
L (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.
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Chapter 1
Structural Analysis
W unit per run X
C R y
H
θ 0
WR
WR
(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 W C
H
R
A
B
=
H
=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 W K y
h
H
H
A
B
x L
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Chapter 1
Structural Analysis
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
h H A
h L B
A
L
H
L 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
C y
h
x
H
B
A
H
L
H=
∫
.
∫
Where, M is beam moment
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