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Mechanics of Solids- Formulae
Tension, Compression and Shear
Stress
Prismatic bar in tension: (a) free-body diagram of a segment of the bar, (b) segment of the bar before loading, (c) segment of the bar after loading, and
(d) normal stresses in the bar.
, for the analysis small parts
Strain
Hook’s Law
where, σ is the axial stress ϵ is the axial strain E is a constant of proportionality known as the modulus of elasticity for the material
Poisson’s ratio
Dilation (Unit Change)
Shear Stress
Hooke’s Law in shear
where, τ is the shear stress γ is the shear strain G is the shear modulus of elasticity
Relation between Modulus of elasticity in Tension and Shear
Bearing Pad in Shear
Fig. A bearing pad of the kind used to support machines and
bridge girders
where, d is the horizontal displacement due to shear h is the thickness V is the applied horizontal shear force a, b are the dimensions of plate
Factor of Safety
Margin of Safety
Allowable Stress and Allowable Load
Ultimate Stress and Ultimate Load
Axial Members
Fig. Elongation of an axially loaded spring Stiffness
where, k = Stiffness constant
Flexibility
where, f is flexibility
Relation between f and k
or
Elongation
For a System in rotation
Elongation in a tapered section
Fig. Change in length of a tapered bar of solid circular
cross-section
where, P = load applied on the section l = length of the section E = modulus of elasticity dA & dB = diameter of the ends For a prismatic bar,
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Composite Structures
Fig. Statically indeterminate composite structure
Load acting on Material s
Load acting on Material c
Net Load acting on structure
Net Elongation
Thermal Effect Thermal Strain (ϵT)
where, α = coefficient of thermal (depends upon the properties of the material) ΔT = change in temperature Axial Stress
Temperature Displacement Relation
Fig. Increase in the length of a prismatic bar due to a uniform change in
temperature
Sleeve and Bolt assembly with uniform temperature increase ΔT
Assumption: The coefficient of thermal expansion of sleeve αs is greater than the coefficient of thermal expansion of bolt αb i.e. (αs > αb)
1. Equation of Compatibility
2. Equation of Equilibrium
3. Stresses in the Sleeve and Bolt
4. Net Elongation of the assembly
Stresses on Inclined Plane
Fig. Prismatic bar in tension showing the stresses acting on an inclined section pq
1. Normal Stress on a cross-section
where, P is the axial load acting on the centroid of the cross-sectional area A is the cross-sectional area
2. Stresses acting on the section inclined at an angle θ i. Normal Stress
ii. Shear Stress
Strain Energy
Load Displacement Diagram
Strain Energy = Area under the load displacement curve
∫
SI Unit: J (Joules)
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Some Cases of Strain Energy 1. Linearly Elastic Behavior
Strain Energy stored in bar which follows Hooke’s law is
2. Linearly Elastic Spring Replacing the stiffness ⁄ of the prismatic bar by the stiffness k of the spring
or,
3. Non-uniform Bars i. Bar consisting of several segments
Total Strain Energy = sum of strain energies of individual segments
∑
where, Ni is the axial force acting in segment i and Li , Ei , and Ai are properties of segment i
ii. Bar with uniformly varying cross-section
∫[ ]
where, N(x) and A(x) are the axial force and cross-sectional area at distance x from the end of the bar.
Strain-Energy Density (u) Strain Energy per unit volume
Case 1. Strain Energy of a prismatic bar suspended from its upper end
Fig. (a) Bar hanging under its own weight, and (b) bar hanging under its own weight
and also supporting a load P
Considering i. The weight of the bar itself
where, γ is the weight density of the material and A is the cross sectional area of the bar
ii. The weight of the bar plus a load P at the lower end
Vertical displacement of the joint B of the truss
Fig. Displacement of a truss supporting a single load P
Down displacement of the joint B (δB)
Impact Loading
Fig. Impact load on a prismatic bar AB due to a falling object of mass M
Maximum Elongation of the bar (δmax)
[(
)
(
)]
⁄
where, W is the weight of the collar L is the length of the bar A is the cross-sectional area of the bar h is the height from which the collar falls
[ (
)
⁄
]
√
√
where,
is the elongation of the bar due to the weight of the collar under static loading conditions
Maximum Stress in the Bar due to Impact Loading
[ (
)
⁄
] √
√
Impact Factor (IF)
Transformation of Stress Stresses on inclined sections
Fig. Wedge-shaped stress element in plane stress: (a) stresses acting on the element, and
(b) forces acting on the element (free-body diagram)
Transformation Equations for Plane Stress
where, & are the stresses acting on the x and y planes
Also,
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Special Cases for Plane Stress 1. Uniaxial Stress
2. Pure Shear
3. Biaxial Stress
Principal Stresses
The maximum and minimum normal stresses, called the principal stresses.
where the angle θP defines the orientation of the principal planes
√(
)
√(
)
Maximum Shear Stresses
gives maximum shear stress
The plane of maximum shear stress occurs at 45° with the principal plane.
√(
)
Mohr’s Circle
Equation of Mohr’s circle
( )
where,
R = Radius of the Mohr’s circle = √(
)
Centre of the Circle is at
Hooke’s Law for Plane Stress
fig. Element of material in plane fig. Element of material subjected fig. Shear strain γxy
stress ( 0) to normal strains ϵx , ϵy and ϵz
The resultant strains in the x, y and z direction are
( )
( )
The Stresses are
( )
( )
Volume Change(ΔV)
where V1 = Final Volume = Vo(1 + ϵx + ϵy + ϵz) and Vo = Initial Volume
Dilation(e)
Volume per unit change =
Torsion
Rate of twist
Shear Strain
fig. Deformation of an element of length dx cut from a bar in torsion
fig. Shear stresses in a circular bar in torsion
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Torsion Formula Polar Moment of Inertia(IP)
∫
for circle,
Angle of twist
Torsional Stiffness and Flexibility
Circular Tube in Torsion
fig. Circular Tube in Torsion
Power Transmitted by Shafts
Non-Uniform Torsion Case 1. Bar with prismatic section
∑
∑
Case 2. Bar with continuously varying cross-section and constant torque
∫
∫
Case 3. Bar with varying cross-section with varying torque
∫
Strains in pure shear
Strain Energy in Torsion and Pure Shear
Strain Energy Density
Thin walled Pressure Vessel 1. Tensile stresses in the wall of a spherical shell
2. Stresses at outer surfaces
3. Stresses at inner surfaces
(
)
Cylindrical Pressure Vessels 1. Circumferential Stress
2. Longitudinal Stress
3. Stresses at the outer surface
4. Stresses at the inner surface
Maximum Stresses in Beams
Stresses in Beams Curvature of a Beam
where κ is curvature and ρ is radius of curvature
Longitudinal Strains in Beams Strain Curvature Relation
Normal Stresses in beams
fig. Normal Stresses in a beam- Side view
Moment Curvature Relationship
∫
Flexure Formula
fig.. Normal Stresses in a beam- Cross-Section
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Section Moduli
Maximum Normal Stresses
For doubly symmetric cross-sections
Required Section Modulus
Ideal Cross-sectional Shape
for standard wide
Shear Stress in beams Shear Formula
Shear Stress in a Rectangular Beam
Shear Stress in Beams of circular cross-section
Shear Stresses in the webs of beams with Flanges
[
]
Deflection of Beams
Differential Equations of the deflection curve Curvature
Slope of the Deflection Curve
since tan θ ≈ θ, therefore,
(
)
Differential Equation
Columns
Critical loads, effective lengths, and effective-length factors for ideal columns
Pinned-pinned column Fixed-free column Fixed-fixed column Fixed-pinned column