The work done by the load in straining the body is stored within the strained material in the form of strain energy.
Strain energy,
U = 1 / 2 P(Al)
U = P^{2}L / 2AE
Put
P = AEAℓ / L
Or
U = σ^{2 }/ 2E x V
Strain Energy Diagram
The maximum strain energy that can be stored in a material is known as proof resilience.
U = σ^{2 }/ 2E^{1}
Where,
σ = p / A, u = 1 / 2 Ee^{2}
Strain energy of prismatic bar with varying sectionsPrismatic bar
Strain energy of nonprismatic bar with varying axial forceA_{x} = Crosssection of differential section.
NonPrismatic Bar
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
.....(b)
By equating, stress will be
and If h is very small then
For solid shaft,
U = τ^{2 }/ 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
[Where M is function of W (load)]
Deflection:
Slope:
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,
σ_{1 = }σ_{y} or σ_{3} = σ^{r}_{y}
Where, σ_{y} = Yield stress point in simple tension, and σ_{y} = 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 e_{y} = Yield point strain tensile σ_{y }/ E
e^{r}_{y = }Yield point strain compressive σ^{r}_{y }/ E
According to theory, e_{1} = e_{y}
Yield criteria:
And
For 2D system,
Rhombus
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:
Yielding criteria,
τ_{ma}x = 1 / 2 (σ_{1}  σ_{3}) = σ_{y }/ 2
In case of 2D: σ_{1} – σ_{3} = σ_{y}
Yielding criteria, σ_{1}  σ_{2} = σ_{y}
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/MisesHenky theory
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