Energy Method CH-5
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Transcript of 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
q
cos2
sin2
lx
lx
B
B
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
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