Lecture 23 – Energy methods
Instructor: Prof. Marcial Gonzalez
Fall, 2021ME 323 – Mechanics of Materials
Reading assignment: 11.1 – 11.4
Last modified: 8/16/21 9:22:05 AM
Strain energy density
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Strain energy density for linear elastic bodies
Q: Combined loading?
Under combined loading, each point in the bodymay experience a different state of stress, thusa different amount of elastic energy will be storedat different differential volumes in the body.
Strain energy density:
Goal: use the strain energy density to determine an expressionfor the energy stored in an elastic body under axial loads,torsional loads, and bending loads.
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Load
-def
orm
atio
n cu
rves
Work done by external force
Work done by the force: Work done by the torque: Work done by the moment:
ElasticSolids
Strain energy density
ElasticSolids
ElasticSolids
F
v
nonlinear material
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nonlinear material
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nonlinear material
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Energy methods
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Theory of deformable bodies- Geometry of the solid body- Kinematic assumptions- Material behavior- Equilibrium
Work-energy principle:
For an elastic body, the work, , done on the body by external loads (forces and moments) is stored as elastic strain energy, .
AA
A
AA
A
nonlinear material
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Elastic strain energy for axial deformations
(STATICS and Lecture 8)
Thus,
For uniform rods subjected to axial end loads:
Strain energy density – Axial deformation
Problem 40: …
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Elastic strain energy for torsional deformations
Thus,
For uniform shaft subjected to axial end torques:
Problem 41: …
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Strain energy density – Torsional deformation
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Elastic strain energy for bending deformations
Thus,
Bending momentand shear force
diagrams
is a form factor for shearthat depends on the shapeof the cross section (see last slide).
Q: Can we use an ‘energy approach’ to determine deflections in linear elastic beams?
Problem 42: …
v
PC
△C
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Strain energy density – Bending deformation
Energy methods
Work and elastic strain energy
Work done by the force: Work done by the torque: Work done by the moment:
AA
A
AA
A
Stored elastic strain energy: Stored elastic strain energy: Stored elastic strain energy:
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v
PC
△C
Problem 43:
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Energy methods
Determine the deflection at section C. (Can we neglect the contribution of the shear strain energy?)
Problem 44:
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Energy methods
Determine the deflection at section C. (Can we neglect the contribution of the shear strain energy? Yes …)
Work-energy principle:
Table with values of:
Problem 45:
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Energy methods
Determine the deflections at sections B and C (Neglect the contribution of the shear strain energy)
Problem 45 (cont.):
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Energy methods
Determine the deflections at sections B and C (Neglect the contribution of the shear strain energy)
We cannot solve it with the Work-Energy Principle!