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

3

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.

4

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

linearmaterial

nonlinear material

linearmaterial

nonlinear material

linearmaterial

Energy methods

5

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

6

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: …

linearmaterial

7

Elastic strain energy for torsional deformations

Thus,

For uniform shaft subjected to axial end torques:

Problem 41: …

linearmaterial

Strain energy density – Torsional deformation

8

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

linearmaterial

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:

9

v

PC

△C

Problem 43:

10

Energy methods

Determine the deflection at section C. (Can we neglect the contribution of the shear strain energy?)

Problem 44:

11

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:

12

Energy methods

Determine the deflections at sections B and C (Neglect the contribution of the shear strain energy)

Problem 45 (cont.):

13

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!

Any questions?

14

Energy methods