The work done by the load in straining the body is stored within the strained material in the form of strain energy.
U = 1 / 2 P(Al)
U = P2L / 2AE
P = AEAℓ / L
U = σ2 / 2E x V
Strain Energy Diagram
The maximum strain energy that can be stored in a material is known as proof resilience.
U = σ2 / 2E1
σ = p / A, u = 1 / 2 Ee2Strain energy of prismatic bar with varying sections
Prismatic barStrain energy of non-prismatic bar with varying axial force
Ax = Cross-section of differential section.
Stresses due to
(i) Gradual Loading: σ = F / A
(ii) Sudden Loading: σ = 2F / A
(iii) Impact Loading: Work done by falling weigth P is
Work stored in the bar
By equating, stress will be
Strain Energy in Torsion
and If h is very small then
For solid shaft,
U = τ2 / 4G x Volume of Shaft
For hollow shaft,
x Volume of Shaft
It the strain energy of an elastic structure can be expressed as a function of generalized displacement, then the partial derivative of the strain energy with respect to generalized displacement gives the generalized force
Theories of Failure
[Where M is function of W (load)]
Theories of failure are defined as following groups:1. Maximum Principal Stress Theory (Rankine theory)
Note: For bittle material, it gives satisfactory result. Yield criteria for 3D stress system,
σ1 = σy or |σ3| = σry
Where, σy = Yield stress point in simple tension, and σy = Yield stress point in simple compression.
Stresses on rectangular Section
2. Maximum Principal Strain Theory (St. Venant’s theory)
According to this theory, a ductile material begins to yield when the maximum principal strain at which yielding occurs in simple tension.
For 3D stress system,
If ey = Yield point strain tensile σy / E
ery = Yield point strain compressive σry / E
According to theory, e1 = ey
For 2D system,
3. Maximum Shear Stress Theory (Guest & Tresca’s theory)
Note: This theory can estimate the elastic strength of ductile material.
According to this theory, failure of specimen subjected to any combination of loads when the maximum shearing stress at any point reaches the failure value equal to that developed at the yielding in an axial tensile or compressive test of the same material.
For 3D system:
τmax = 1 / 2 (σ1 - σ3) = σy / 2
In case of 2D: σ1 – σ3 = σy
Yielding criteria, σ1 - σ2 = σy
This theory gives well estimation for ductile material.
For 2D stress system,
EllipseThis theory does not apply to brittle material for which elastic limit stress in tension and in compression are different.5. Maximum shear strain energy/Distortion energy theory/Mises-Henky theory