Strength of Materials

Mechanics of Materials

Mechanics of Deformable Bodies

Engineering Mechanics of Deformable Bodies

Theory of Elasticity & Plasticity

Rigid Body – Deformation are negligible

No Rigid body on the face of the earth

Rigid Body Mechanics – Forces in members

Strength of Materials –

Design the members

Structural Actions

• Axial Force – Compression & Tension

• Bending (Flexure)

• Bearing

• Shear

Direct Shear

Bending (Flexural) Shear

Torsional Shear

12 kN

Fig. 2

CompressionBending

Shear

Double Shear

X

Y

ZF1 F2

F3F4Fn

R = Fxi + Fyj + Fzk

M = Mxi + Myj + Mzk

Y

Z

RX

Y

Z

Fx

Fy

Fz X

Y

Z

Fx

Fy

XFz

Stress at a point

Stress due to Normal force is called as Normal Stress ()

Stress due to Tangential force is called as Shear Stress ()

A

FLimStress

A

0

Units of Stress – ( FL-2) - N / Sq.m = Pa

Y

Z

M

XY

Z

X

Mx

My

Mz

M = Mxi + Myj + Mzk

Stress at a point Y

Z

X

x

xy

xz x

xy

xz

y

yx

yz

y

yx yz

z

zx

zy

Stress at a point - Stress Tensor

zzyzx

yzyyx

xzxyx

xy = xy; xz = zx; yz = zy

zyzxz

yzyxy

xzxyx

Strain at a point (a) Longitudinal Strain

PP

L

P P

L L

Longitudinal Strain = = L / L

Strain has no units

Strain at a point (b) Shear Strain

xy = d/h Tan =

Strain at a point - Strain Tensor

zzyzx

yzyyx

xzxyx

xy = xy; xz = zx; yz = zy

zyzxz

yzyxy

xzxyx

General Force system in Space

• Six Degrees of Freedom

• Six Static Equillibrium Equations

• Six Dynamic Equillibrium Equations

• Six Unknown Stresses viz., x, y, z, xy, yz, zx

• Six Unknown Strains viz., x, y, z, xy, yz, zx

State of the Stress in Two Dimensions

x

xy

x

xy

y

yx

y yx X

Y

yxy

xyx

Stress Tensor

State of the Strain in Two Dimensions

x

xy

x

xy

y

yx

y yx

X

Y

yxy

xyx

Strain Tensor

Co planar General Force system

• Three Degrees of Freedom

• Three Static Equillibrium Equations

• Three Dynamic Equillibrium Equations

• Three Unknown Stresses viz., x, y, xy

• Three Unknown Strains viz., x, y, xy

Idealisation of Materials

• Material is Isotropic and Homogenous

• Materials is within Elastic Limits

• Mechanical characteristics of material can be studied by Stress Strain Behaviour

Stress Vs Strain Behaviour

Hooke’s Law

Robert Hooke ( 1635 – 1703)

Stress is proportional to strain

Robert Hooke ( 1635 – 1703)

Within Elastic Limits Stress is proportional to Strain

Material 1

Material 2

Young’s Modulus of Elasticity

Thomas Young ( 1773 - 1829)

Young’s Modulus (E) = Stress / Strain

E1

E2

Material 1

Material 2

Tension Testing Machine – Recording Arrangement

Tension Test

Extensometer to Measure small deformations

Stress Strain Curve

Progression of a Fracture

Sequence of events in the necking and fracture of a tensile-test specimen: (a) early stage of necking; (b) small voids begin to form within the necked region; (c) voids coalesce, producing an internal crack; (d) the rest of the cross-section begins to fail at the periphery, by shearing; (e) the final fracture surfaces, known as cup- (top fracture surface) and cone- (bottom surface) fracture.

Fracture Types in Tension

Schematic illustration of the types of fracture in tension: (a) brittle fracture in polycrystalline metals; (b) shear fracture in ductile single crystals (c) ductile cup-and-cone fracture in polycrystalline metals; (d) complete ductile fracture in polycrystalline metals, with 100% reduction of area.

Material Failures

Schematic illustrations of types of failures in materials: (a) necking and fracture of ductile materials; (b) buckling of ductile materials under a compressive load; (c) fracture of brittle materials in compression; (d) cracking on the barreled surface of ductile materials in compression

Measure of Ductility

1. Percentage Increase in Length and

2. Percentage decrease in cross sectional area.