GATE Structural Analysis & Mechanics Book

8/10/2019 GATE Structural Analysis & Mechanics Book

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STRUCTURAL ANALYSIS

MECHANICS

for

Civil Engineering

By

www .the gate academy. com
http://www.thegateacademy.com/http://www.thegateacademy.com/http://www.thegateacademy.com/http://www.thegateacademy.com/http://www.thegateacademy.com/http://www.thegateacademy.com/http://www.thegateacademy.com/


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Syllabus Structural Analysis/Mechanics

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30 th Cross, 10 th Main, Jayanagar 4 th Block, Bangalore-11080 65700750 i f @ h d C i h d W b h d

Syllabus for

Structural Analysis

Analysis of statically determinate trusses, arches, beams cables & frames,displacement in statically determinate structure and analysis of staticallyindeterminate structures by force/energy methods, analysis by displacementmethods (slope deflection & moment distribution method) influence lines fordeterminate & indeterminate structure. Basic concept of matrix method of structuralanalysis.

MechanicsBending Moment & shear Force in statically determinate beams. Simple stress &strain relationship; stress & Strain in two dimensions, principal stress, stresstransformations, Mohrs cycle simple bending theory. Flexural & shear stresses, unsymmetrically 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


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Contents Structural Analysis/Mechanics

THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30 th Cross, 10 th Main, Jayanagar 4 th Block, Bangalore-11

080 65700750 i f @th t d C i ht d W b th t d i

C O N T E N T S

Chapters Page No.

STRUCTURAL ANALYSIS1. 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 Beam28 31

Macaulays Method 32 33 Conjugate Beam Method 33 Assignment 1 34 35 Assignment 2 36 38

Answer Keys & Explanations 39 41

4. Degree of Static Indeterminacy 42 - 51 Degree of Static Indeterminacy 42 Alternative Approach to find Degree of Static

Indeterminacy42

Solved Examples 43 44 Assignment 1 45 Assignment 2 46 48 Answer Keys & Explanations 49 51

5. Displacement Method 52-74 Displacement Methods 52 53 Carry Over Factor 53 56 Solved Examples 57 60 Slope Deflection Method 60 66

Assignment 1 67 68


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Assignment 2 69 71 Answer Keys & Explanations 72 74

6. Force / Energy Methods 75 - 91 Introduction 75 Maxwells Reciprocal Theorem 75 76 Bettis Theorem or Generalized Reciprocal Theorem 76 77 Castiglianos Theorem 77 78 Unit Load Method 79 82 Conjugate Beam Method 82 83 Assignment 1 84 Assignment 2 84 87 Answer Keys & Explanations 88 91

7. Matrix Method of Structural Analysis 92 105 Matrix Concept & Algebra 92 93 Flexibility & Stiffness 93 94 Solved Examples 95 97 Assignment 1 98 100 Assignment 2 100 101 Answer Keys & Explanations 102 105

MECHANICS

8. Simple Stress and Strain Relationship 106 134 Simple Stress 106 109 Compound Bars 109 114 Mohrs Circle 115 116 Solved Examples 117 128 Assignment 1 129 130 Assignment 2 130 132 Answer Keys & Explanations 133 134

9. Bending Moment and Shear Force Diagram 135 -165 Terminology 135 136 Bending Moment Diagram 136 - 141 Solved Examples 142 156 Assignment 1 157 158 Assignment 2 158 162 Answer Keys & Explanations 163 165


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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 Eulers Formula 173 174 Solved Examples 175 178 Assignment 1 179 180 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 Solved Examples 190 195 Assignment 1 196 Assignment 2 197 199 Answer Keys & Explanations 200 202

Module Test 203 - 217

Reference Books 218


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Chapter 1 Structural Analysis

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30 th Cross, 10 th Main, Jayanagar 4 th Block, Bangalore-11080 65700750 i f @ h d C i h d W b h d P 2

c) Graphical method

Analysis of Determinate Trusses.Methods of joints1. 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 22. In order to find internal reactions, following equations equilibrium may be used

, , M = 03. 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 one5. Tensile force are assumed to be (+) and compressive force are assumed to be (-).

Notes1. 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 oftruss 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 forceat 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 ifmembers 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

B

A


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Types of arches based on number of hinges1) Single hinged arch

2) Two hinged arch3) Third hinged arch4) Fixed arch or hinge less arch

A three hinged arch is a statically determinate structure whre the rest three aches arestatically indeterminate

In bridge construction, especially is railroad bridges, the more frequently used aches thetwo hinged and the fixed end ones.

Three Hinged Arches

A three hinged arch is a statically determinate structure, having a hinged at eachabutment or springing, and also at the crown.

In three hinged arch three equations are available from static equilibrium and oneadditional equation is available from the fact that B.M at the hinge at the crown is zero.

Standard casesCase1: A three hinged parabolic arch of span L rise h carries a udl of w over the whol

(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 zeroM = Beam moment H moment = 0

Case2: A three hinged semicircular ar ch 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 M max = ogging which occurs at = 3 from the horizontal and the distance of point of maximum bending moment fromthe crown is,

R cos 3 =

Case3: A three hinged arch consisting of two quadrant parts AC and CB of radii R and R . Thearch 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, whichmay be placed anywhere on the span

y h

AH

K

W

HB

x

L

H A

HB

C

W

R

H

W unit per run

CR

y

0

WR WR

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 thecrown 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 momentabout either end. The horizontal thrust at each support may be determined from the conditionthat the horizontal displacement of the either hinge with respect to other is zero

H = .

Where, M is beam moment

C

y h

BHH

A

L

x

H A

A

h

L

h

B H

L L


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GATE Structural Analysis & Mechanics Book

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