Energy Method CH-5

Advanced

Mechanics of Solids ME F312

Text Book: "Advanced Mechanics of Materials" -

Arthur P., Boresi and R.J. Schinid, John Wiley, 6th Ed.

• Shortcomings in elementary solid Mechanics formulations

• Materials & Mechanics

• Extension of topics

• Advanced topics

IC: Prof. M. S. Dasgupta

Principle of Stationary Potential Energy /

Principle of virtual work

Chapter 5

Ue=Ui

Work of External

Forces & Moments

Strain Energy

Internal Forces

Conservation of Energy (Elastic Material Behavior)

• When material is deformed by external loading, energy

is stored internally throughout its volume

• Internal energy is also referred to as strain energy

• Stress develops a force,

• Principle of virtual work - if a particle,

rigid body, or system of rigid bodies

which is in equilibrium under various

forces is given an arbitrary virtual

displacement, the net work done by the

external forces during that displacement

is zero.

• The principle of virtual work is particularly useful

when applied to the solution of problems involving

the equilibrium of machines or mechanisms consisting

of several connected members (system of connected rigid bodies)

As a result the deduction of exact force components in each

members are cumbersome and is made redundant by this method

as most of the components will have zero work and only a few

forces will produce non-zero work. These forces can be analyzed

in one go.

Forces which do no

work:

• Imagine the small virtual displacement of

particle which is acted upon by several forces.

• The corresponding virtual work,

rR

rFFFrFrFrFU

321321

Principle of Virtual Work:

• If a particle is in equilibrium, the total virtual work of forces

acting on the particle is zero for any virtual displacement.

• If a rigid body is in equilibrium, the total virtual work

of external forces acting on the body is zero for any

virtual displacement of the body.

• If a system of connected rigid bodies remains connected

during the virtual displacement, only the work of the

external forces need be considered.

Principle of Virtual Work

Applications of the Principle of Virtual Work • Determine the force of the vice on the

block for a given force P.

• Refer FBD, the work done by the

external forces for a virtual

displacement q. Only the forces P and

Q produce nonzero work. CBPQ yPxQUUU 0

qq

q

cos2

sin2

lx

lx

B

B

qq

q

sin

cos

ly

ly

C

C

q

qqqq

tan

sincos20

21 PQ

PlQl

• If the virtual displacement is consistent with the

constraints imposed by supports and connections,

only the work of loads, applied forces, and

friction forces need be considered.

Free Body Diagram

Real Machines. Mechanical Efficiency

q

qq

qq

cot1

sin

cos2

input work

koutput wor

Pl

Ql

q

qqqqqq

tan

cossincos20

0

21 PQ

PlPlQl

xFyPxQU BCB

machine ideal ofk output wor

machine actual ofk output wor

efficiency mechanical

• For an ideal machine without

friction, the output work is equal

to the input work.

• When the effect of friction is

considered, the output work is

reduced.

10

Applications of the Principle of Work and Energy

Q. Determine velocity of pendulum bob

at A2. Consider work & kinetic energy.

Force acts normal to path and

does no work: P

1 1 2 2

2

2

2

10

2

2

T U T

mgl mv

v gl

• Velocity found without determining acceleration and integrating.

• All quantities are scalars

• Forces which do no work are eliminated

Q. Find location of B*

L1

L2

From Geometry

2 2 21 1

2 2 22 2

2 2 2

1 1 1

2 2 2

2 2 2

L b h

L b h

L e b u h v

L e b u h v

Elongations:

L1 L2

2 2 2

1 1 1

2 2 2

2 2 2

2 2

1 1 1

2 2

2 2 2

L e b u h v

L e b u h v

e b u h v L

e b u h v L

21 11 1 1 1

1

22 22 2 2 2

2

1 11

1 1

2 22

2 2

1 E AU N e e

2 2L

1 E AU N e e

2 2L

N Le

E A

N Le

E A

Strain energy:

2 21 1 2 21 2 1 2

1 2

1 1 1 1 2 2 2 2

1 2

1 1 1 1 2 2 2 2

1 2

E A E AU U U e e

2L 2L

U E A e e E A e eP

u L u L u

U E A e e E A e eQ

v L v L v

Strain energy:

2 2

1 11 1 1

2 21

1

2 2

2 22 2 2

2 22

2

b h h v LE A b uP

L b h h v

b h h v LE A b u

L b h h v

2 2

1 11 1

2 21

1

2 2

2 22 2

2 22

2

b h h v LE A h vQ

L b h h v

b h h v LE A h v

L b h h v

1 11

1

2 22

2

1

2

E A NK 2.00mmL

E A NK 3.00mmL P 43.8N

b h 400mm Q 112.4 N

b 300mm

u 30mm

v 40mm

With numerical value from text book

1

2

1 1 1

2 2 2

e 49.54mm

e 16.24mm

N K e 99.08N

N K e 48.72 N

*

*

* *x 1 2

* *y 1 2

0.7739rad

0.5504rad

F 0 P N sin N sin P 43.8N

F 0 Q N cos N cos Q 112.4N

q

q

q

Castigliano’s Theorem on

Deflections

Based on Complimentary Energy Sometimes called “Principle of Complimentary Energy”

Castigliano’s Theorem on

Deflections If an elastic system is supported so that rigid-body displacements are prevented, and if certain concentrated forces of magnitudes (F1,F2,F3,…Fn) act on the system, in addition to distributed loads and thermal strains, the displacement component qi of the point of application of the force Fi, is determined by the equation:

n

i

i 1

C C

For Structure

Composed of

Many Members

For Linear Elastic material

Where P is a generalized force and q generalized displacement

Axial Force “N”

L 2

N

0

NU dz

2EA

Bending Moment “M”

L 2x

M

x0

MU dz

2EI

Shear “V” Due to Bending

2Ly

S

0

kVU dz

2GA

yzy

x

V Q

I b

Shear Correction Values

1.20

1.33

2.00

1.0

Cross Section k

Rectangle

Solid Circular

Thin Walled Circular

I-Section, Box, Channel

Torsion “T”

2

T

TU dz

2GJ