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8/17/2019 Bhairav Thakkar Mos

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Mechanics of Solids

BH IR V TH KK R

ASSOCIATE PROFESSOR,NAVRACHANA UNIVERSITY

If I have seen further than others, it is by standing upon the shoulders of giants – Sir I


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Mechanics of Solids

Solid mechanics is the branch of continuum mechanics that studies the behavior ofmaterials, especially their motion and deformation under the action of forces, tempchanges, phase changes, and other external or internal agents.*

* Wikipedia

Build Structures that do NOT fail under applied


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Force and Moment

A force arises from the action (or reaction) of one body on a

The SI unit of force is the Newton (N).

The moment of a force about a point is equal to the producmagnitude of the force and the perpendicular distance frompoint to the line of action of the force.


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Body Under System of Forces

Stress: Respo

material to ap

forces on a giv

in a given dire


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Stress at a Point

3D State of Stress

sx

sy

sy

tyx

txy

txy

tyx


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Transformation of Stress


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Effect of ForceWhen a force is applied to an elastic body, the body deforms. The way in which the deforms depends upon the type of force applied to it.

Compression force makes the body shorter.

A tensile force makes the body longer


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Bar Under Axial ForceThe usual case – Normal Stress or Direct Stress

A

F

Area

ForceStress s


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Shear Stress

The stress that tends to CUT a member

tXY

tXY

tXY

tXY

tXY

tXY

Complimentary Stre

Appl


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Strain

P

Original length Change in lengthShear St


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Stress-Strain Relationship

Proportionality Limit

Elasticity LimitLower Yield Point

Ruptur

Upper Yield Point

Ultimate Stress Point

S t r e s s

Strain


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Hooke’s Law

Proportionality Limit

Elasticity LimitLower Yield Point

Rupture

Upper Yield Point

Ultimate Stress Point

S t r e s s

Strain

ds

de


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Hooke’s Law - RelevanceRelates stress with strain

Load carrying capacity with deformation

P

Aluminum

Steel

If Areas are equal f

and Aluminum, whmaterial will take m

load? Steel or Alum

ESteel = 2x105 MEAluminum = 70,0


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Hooke’s Law to Rescue

Since the top will move equally for both materials Deformations can be equated


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Stress Components


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Principal Planes and Principal StressePlanes on which shear stress component vanishes Principal Planes

Principal Stresses are the stresses on Principal Planes Principal Stresses

sx

sx

sy

sy

tXY

tXY

tXY

tXY


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Hydrostatic (Mean) Stress

Mean of three Principal Stresses

“Hydrostatic stress is simply the average of thenormal stress components of any stress tenso

Remaining stress is “Deviatoric Stress”

* Wikipedia


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Failure Envelope


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Example: Stress at crack tip


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Examples: von Mises Stresses


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Forces

Self WeightLive Loads

Wind

Earthquakes

Etc…


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Idealization


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1D StructuresBeamsMembers under bending

ColumnsMembers under axial compression

Beam-ColumnsMembers under bending and axial comp

TieMembers under axial tension


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Equilibrium

Every action has an

equal and opposite

reaction

Sir Isaac Newton


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Conditions of Equilibrium (2D)

Sum of all the forces acting in x-direction MUST be zero, i.e. ∑Fx = 0.

Sum of all the forces acting in y-direction MUST be zero, i.e. ∑Fy = 0.

Sum of all the moments about any point must be zero, i.e. ∑Mz = 0.


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Beams

We determine reactions and internal actions usingconditions of static equilibrium

P


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Euler-Bernoulli Theory of Bending

Plane section before bending remains plane after bMaterial is homogeneous and isotropic

Beam bends in an arc of circle

Radius of curvature is large compared to beam dim


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Strain Deformations


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Beam Under Bending


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TorsionMoment is applied about the longitudinal axis of a member Torsio

Deformation Twist

Extremely common in Civil Engineering structures

Usually considered as equivalent moment and shear acting additiona


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Effect of Torsion

• The existence of the axial shear components is

demonstrated by considering a shaft made up

of axial slats.

The slats slide with respect to each other whenequal and opposite torques are applied to the

ends of the shaft.


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SummaryWe discussed the following

Basic effect of loads on bodies

Stresses and Strains

Hooke’s Law

Failure Theories

Euler-Bernoulli’s Theory of Pure bending of beams

Torsion


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Thank You

Questions?

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