Lecture-2 (Introduction of Solid Mechanics)

7/27/2019 Lecture-2 (Introduction of Solid Mechanics)

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CE 251 (Solid Mechanics)

Lecture-2

Introduction of Solid Mechanics

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What do we mean a solid?

A solid can sustain shear.

A body is and remains CONTINUOUS under the action

of external forces.

Consisting of continuous material points

Neighboring points remain neighbors

Neglecting its atomistic structure

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What is Solid Mechanics?

Mechanics is a branch of physics that is concerned with

the analysis of the action of forces on matter or materialsystems.

Solid Mechanics is mechanics about deformable solids.

The solid bodies considered in this subject include bars

with axial loads, shafts in torsion, beams in bending, and

columns in compression.

Principal objective: To determine the stresses, strains,

and displacements in structures and their components due

to the loads acting on them.

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If we can find these quantities for all values of the loads

up to the loads that cause failure, we will have a complete

picture of the mechanical behavior of these structures.

An understanding of mechanical behavior is essential for

the safe design of all types of structures, whether airplanes

and antennas, buildings and bridges, machines and

motors, or ships and spacecraft.

Examine the stresses and strains inside real bodies, that

is, bodies of finite dimensions that deform under loads.

To determine the stresses and strains, we use the

physical properties of the materials as well as numerous

theoretical laws and concepts.

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Theoretical analyses and experimental results have

equally important roles.

We use theories to derive formulas and equations forpredicting mechanical behavior, but these expressions

cannot be used in practical design unless the physical

properties of the materials are known.

Such properties are available only after carefulexperiments have been carried out in the laboratory.

All practical problems are not amenable to theoretical

analysis alone, and in such cases physical testing is anecessity.

To determine the stresses and strains, we use the

physical properties of the materials as well as numerous

theoretical laws and concepts.

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Length scales where solid mechanics is valid

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What is Solid Mechanics about?

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Problems:

1. Understanding the logical

development of the concepts

Your Efforts

2. Applying concepts to

practical situations

By studying the derivations,

discussions, and examples

in each lecture

By solving the

problems at the

ends of the lectures

and chapter

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Solid Mechanics

Fundamental

Concepts

Treats general

concepts that pertainto the entire solid

mechanics, theories of

stress and strain,

linear stress strain-

temperature relations,

yield criteria for multi

axial stress states.

These topics are

intended to be read

sequentially, more or

less.

Classical applications

of the methods of

mechanics of

materials, namely,torsion, principle

stress-strain, SF, BM,

SFD & BMD,

symmetrical-

nonsymmetrical

bending and

deflection of beams,

shear center for thin-

wall beam cross

sections, and

buckling of columns.

Introduces

chapters on

selected

advancedtopics, namely,

fracture

mechanics,

energy

formulation fordeformable

body (elastic

strain energy,

virtual work

method, virtualforce)

Part-1 Part-3Part-2

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Load-stress and load-deflection relations:

For most of the members, we derive relations, in terms

of known loads and known dimensions of the member,for either the distributions of normal and shear

stresses on a cross section of the member or for stress

components that act at a point in the member.

For a given member subjected to prescribed loads, the

derivation of load-stress relations depends on

satisfaction of the following requirements:

1. The equations of equilibrium (or equations of motion

for bodies not in equilibrium)

2. The compatibility conditions (continuity conditions)

that require deformed volume elements in the member

to fit together without overlap or tearing.

3. The constitutive relationsAnil Mandariya 10


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Two different methods are used to satisfy requirements

1 and 2: the method of mechanics of materials and the

method of general continuum mechanics.

1. Method of Mechanics of Materials:

Assumptions:

Plane sections before loading remain plane after

loading (requirement 2).application: axially loaded members of uniform cross

sections, for slender straight torsion members having

uniform circular cross sections, and for slender

straight beams of uniform cross sections subjected to

pure bending The method of mechanics of materials is used in solid

mechanics to treat several advanced topics likely,

Symmetrical-nonsymmetrical Bending of Straight

Beams, torsion. Anil Mandariya 11


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Example: A simple member such as a circular shaft of

uniform cross section may be subjected to complex

loads that produce a multi axial state of stress in the

shaft. However, such complex loads can be reduced to

several simple types of load, such as axial, bending,

and torsion. Each type of load, when acting alone,

produces mainly one stress component, which is

distributed over the cross section of the shaft. The

method of mechanics of materials can be used to obtainload-stress relations for each type of load.

If the deformations of the shaft that result from one type

of load do not influence the magnitudes of the othertypes of loads and if the material remains linearly

elastic for the combined loads, the stress components

due to each type of toad can be added together (i.e., the

method of superposition may be used).

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Example: Derivation of the flexure formula:

Consider a symmetrically loaded straight beam of uniform

cross section subjected to a moment M that produces

pure bending.

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The plane of loads lies in a plane of symmetry of every

cross section of the beam.

We assume that () is the major stress component and,

hence, ignore other effects.

Pass a section through the beam at the specified cross

section so that the beam is cut into two parts.

Consider a free-body diagram of one part.

The applied moment M for this part of the beam is in

equilibrium with internal forces represented by thesum of the forces that result from the normal stress ()

that acts over the area of the cut section.

Equations of equilibrium (requirement 1) relate the

applied moment to internal forces.

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The plane of loads lies in a plane of symmetry of every

cross section of the beam.

Since no axial external force acts, two integrals are

obtained as follows: *dA = 0 and *y*dA = M, where

M is the applied external moment and y is the

perpendicular distance from the neutral axis to the

element of area dA.

Since the stress distribution is not known, it is

determined indirectly through a strain distribution

obtained by requirement 2.

The continuity condition (requirement 2) is examined

by consideration of two cross sections of the un-

deformed beam separated by an infinitesimal distance

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Under the assumption that plane sections remain plane,

the cross sections must rotate with respect to each

other as the moment M is applied.

There is a straight line in each cross section called the

neutral axis along which the strains remain zero.

Since plane sections remain plane, the strain

distribution must vary linearly with the distance y asmeasured from this neutral axis.

Requirement 3 is now employed to obtain the relation

between the assumed strain distribution and the stressdistribution.

Tension and compression stress-strain diagrams

represent the response for the material in the beam.Anil Mandariya 16


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For sufficiently small strains, these diagrams indicate

that the stresses and strains are linearly related.

Their constant ratio, / = E, is the modulus of elasticity

for the material.

In the linear range the modulus of elasticity is the samefor tension and for compression for many engineering

materials.

Hence, the stress-strain relation for the beam is = *E.

Therefore, both the stress and strain vary linearly

with the distance y as measured from the neutral axis of

the beam.

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Hence, the equations of equilibrium can be integrated to

obtain the flexure formula = M*y/I, where M is the

applied moment at the given cross section of the beam

and I is the moment of inertia of the beam cross

section.

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2. Method of Continuum Mechanics,

Theory of Elasticity:

Many of the problems likely noncircular torsion, thick-

wall cylinders, and stress concentration have multi-

axial states of stress of such complexity that the

method of mechanics of materials cannot be employedto derive load-stress and load-deflection relations.

In such cases, the method of continuum mechanics is

used.

We consider small displacements and linear elastic

material behavior only, continuum mechanics reduces

to the method of the theory of linear elasticity.

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In the derivation of load-stress and load-deflection

relations by the theory of linear elasticity, an

infinitesimal volume element at a point in a body with

faces normal to the coordinate axes is often employed.

Requirement 1 is represented by the differential

equations of equilibrium.

Requirement 2 is represented by the differential

equations of compatibility.

The material response (requirement 3) for linearly

elastic behavior- is determined by one or moreexperimental tests that define the required elastic

coefficients for the material.

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Methods in Solid Mechanics

1. Experiments:

1. Tension test

2. Shear test

3. Impact test4. Torsion test

5.

2. Theory:1. Continuum mechanics

2. Micromechanics

3. Constitutive modelAnil Mandariya 21


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3. Computation:

1. Finite element methods

2. Boundary element methods3. Molecular dynamics simulations

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1. Stress and Strain

2. Uniaxial Stress

3. Stress-Strain relationship

Topics to be Cover:

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4. Torsion

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5. Beams

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6. Bending in beam

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7. Shear stresses in beam

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8. Principle stress & strain

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9. Deflection of beam

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10. Column

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11. Failure criteria

12. Energy Formulation for deformable body

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Lecture-2 (Introduction of Solid Mechanics)

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