Civil Engineering : Structural analysis & mechanics, THE GATE ACADEMY

STRUCTURAL ANALYSIS

&

MECHANICS

for

Civil Engineering

By

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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


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 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)



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

B

A



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



(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

WR WR

X



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 H H

A

L

x

H A

A

h

L

h

B H

L

L

Civil Engineering : Structural analysis & mechanics, THE GATE ACADEMY

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Transcript of Civil Engineering : Structural analysis & mechanics, THE GATE ACADEMY