Energy Methods Notes | EduRev

Solid Mechanics

Mechanical Engineering : Energy Methods Notes | EduRev

The document Energy Methods Notes | EduRev is a part of the Mechanical Engineering Course Solid Mechanics.
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Strain Energy

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 = P2L / 2AE
Put
P = AEAℓ / L

Or

U = σ/ 2E x V

Energy Methods Notes | EduRevStrain Energy Diagram

Proof Resilience

The maximum strain energy that can be stored in a material is known as proof resilience.

Energy Methods Notes | EduRev

U = σ/ 2E1

Where,

σ = p / A, u = 1 / 2 Ee2

Strain energy of prismatic bar with varying sections

Energy Methods Notes | EduRev

Energy Methods Notes | EduRevPrismatic bar

Strain energy of non-prismatic bar with varying axial force

Energy Methods Notes | EduRev

Ax = Cross-section of differential section.

Energy Methods Notes | EduRevNon-Prismatic 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

Energy Methods Notes | EduRev

Energy Methods Notes | EduRev
Work stored in the bar

Energy Methods Notes | EduRev.....(b)
By equating, stress will be
Energy Methods Notes | EduRev

Energy Methods Notes | EduRev
and If h is very small thenEnergy Methods Notes | EduRev

Strain Energy in Torsion

Energy Methods Notes | EduRev
For solid shaft,

U = τ/ 4G x Volume of Shaft

For hollow shaft,
Energy Methods Notes | EduRevx Volume of Shaft

Castigliano’s First Theorem

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

Energy Methods Notes | EduRev
[Where M is function of W (load)]
Deflection:
Energy Methods Notes | EduRev
Slope:
Energy Methods Notes | EduRev

Theories of Failure

Theories of failure are defined as following groups:

1. Maximum Principal Stress Theory (Rankine theory)
  • According to this theory, permanent set takes place under a state of complex stress, when the value of maximum principal stress is equal to that of yield point stress as found in a simple tensile test.
  • For design, critical maximum principal stress (σ1) must not exceed the working stress (s1) for the material.
    σ< σy

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.

Energy Methods Notes | EduRevStresses 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,

Energy Methods Notes | EduRev

If ey = Yield point strain tensile σ/ E
ery = Yield point strain compressive σr/ E

According to theory, e1 = ey

Yield criteria: Energy Methods Notes | EduRev
And Energy Methods Notes | EduRev
For 2D system, Energy Methods Notes | EduRev

Energy Methods Notes | EduRev

Energy Methods Notes | EduRevRhombus

Note: This theory can estimate the elastic strength of ductile material.

3. Maximum Shear Stress Theory (Guest & Tresca’s theory)

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,
τmax = 1 / 2 (σ1 - σ3) = σ/ 2

In case of 2D: σ1 – σ3 = σy

Yielding criteria, σ1 - σ2 = σy
This theory gives well estimation for ductile material.

4. Maximum Strain Energy Theory (Haigh’s theory)
  • According to this theory, a body under complex stress fails when the total strain energy on the body is equal to the strain energy at elastic limit in simple tension. For 3D stress system yield criteria,
    Energy Methods Notes | EduRev

For 2D stress system,

Energy Methods Notes | EduRev

Energy Methods Notes | EduRevEllipseThis 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
  • It states that inelastic action at any point in a body, under any combination of stress begins, when the strain energy of distortion per unit volume absorbed at the point is equal to the strain energy of distortion absorbed per unit volume at any point an a bar stressed to the elastic limit under the state of uniaxial stress as occurs in a simple tension/compression test.
    1/2[(σ1 - σ2)2 + (σ1 - σ3)2 + (σ3 - σ1)2] ≤ σ2For no failure
    1/2[(σ1 - σ2)2 + (σ2 - σ3)2 + (σ3 - σ1)2] ≤ (σ/ FOS)2 For no failure

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