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# Solid Mechanics notes.docx

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MECE2420U – Solid Mechanics
Notes
Winter 2014 Semester
UOIT
Riddhesh Soneji 2
Table of Contents
Shear and Bending Moment Diagrams……………………………. 2
Stress and Strain ……………..…………………………................. 3
Material Properties ………………………………………………….. 6
Axial Load…………………………………………………………….. 9
Torsion………………………………………………………………. 11
Bending…………………………………………………………….... 14
Combined Loading…………………………………………………. 17
Transverse Shear…………………………………………………... 18 3
Solid Mechanics – MECE2420U
14/01/14
Shear Force and Bending Moment
Procedure for Analysis
Relation between shear force/moment and distribution load =
16/01/14 – review of shear and bending moment diagrams 4
21/01/14
Stress and Strain
Mechanics of Materials studies external loads to deformable bodies, and also intensity
of internal forces
Deformation and stability depend on internal loadings and type of material
Force Intensity
ΔF / ΔA = Stress
Describes intensity of internal force on a specific plane passing through a
point
Stress
Normal Stress (normal to ΔA) and sh ear stress (tangent to ΔA) 5
First subscript specifies orientation of area, x & y are directions (seen below)
Average Normal Stress
Assumptions: uniform deformation & isotropic (same properties in all directions)
P , the internal resultant force must be passing through the centroid so moment
effect of force distribution will be zero
Average Shear Stress
Lap Joints (Single Shear) 6
Shear Equilibrium
23/01/14
Allowable Stress
Stress that is less than what material can support
Normal Strain
Dimensionless
Shear Strain
Radians (rad) 7
If θ’ < pi/2 , then positive shear
If θ’ > pi/2 , then negative shear
Cartesian Strain Components
Undeformed vs. deformed (see below)
28/01/14
Material Properties 8
Conventional stress-strain diagram
Ductile Materials
Percent elongation
Percent reduction of Area 9
Yield strength can be found by taking line parallel to elastic line and intersecting with
graph
Hooke’s Law
Larger E = stiffer material
Strain Hardening
When ductile material is stretched into plastic region and then unloaded 10
Elastic strain is recovered and plastic strain is retained: Permanent set
Mechanical Hysteresis
May needed if dealing with vibrations
Strain Energy
when material stores energy internally as material is deformed by an external
loading
Modulus of Resilience 11
When stress reaches proportional limit strain energy, it gives the area under the
graph which is the modulus of resilience
Modulus of Toughness
ENTIRE Area under stress-strain curve
Poisson’s Ratio 12
Shear Stress-Strain Diagram
30/01/14
Axial Load 13
Elastic deformation of axially loaded member
This equation comes from relating stress and strain through the modulus of elasticity
Uniform Cross-Section If Area OR load P are NOT changing
If segment is under tension, then positive
If segment is under compression, then negative
Statically Indeterminate Axially Loaded Member 14
requires you to solve two equations to determine desired values (see below)
Thermal Stress
Increased temp. = expansion
Decreased temp. = contraction
o o o
α = coefficient of thermal expansion (1/ F or 1/ C or 1/ K)
04/02/14
Torsion
Length and radius are unchanged because small rotations are considered
Angle of twist, Φ(x), changes with x.
Shear strain 15
For fixed shaft,
Small torsional element
Since small circular element, assume dΦ/dx = constant
Equation will give you shear strength at any point on shaft
Torsion Formula
Assumptions**
Circular shaft
Homogeneous material
Linear-Elastic Region
Polar Moment of Iner

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