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,

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)

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,

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

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

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