THE STRENGTH OF MATERIALS. The Strength of Materials. By J. A. ...
Strength Of Materials 1
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Transcript of Strength Of Materials 1
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.