Strain Energy Methods

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Strain Energy Methods
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Work and Energy
F
Consider a solid object acted upon by force, F,
at a point, O, as shown in the figure.
Let the deformation at the the point be
infinitesimal and be represented by vector dr,
as shown.
F
dr
The work done F dr
y
F
For the general case W Fx dx i.e., only the
force in the direction of the deformation does
work.
dx
x
z
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Amount of Work done
Constant Force If the Force is constant, the
work is simply the product of the force and the
displacement, W Fx
F
x
Displacement
Linear Force If the force is proportional to the
displacement, the work is
Fo
F
xo
x
Displacement
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Strain Energy
Consider a simple spring system, subjected to a
Force such that F is proportional to
displacement x Fkx. Now determine the work done
when F Fo, from before
This energy (work) is stored in the spring and is
released when the force is returned to zero
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Strain Energy Density
y
Consider a cube of material acted upon by a
force, Fx, creating stress sxFx/a2 causing an
elastic displacement, d in the x direction, and
strain exd/a
a
a
a
x
y
Fx
a
x
d
Where U is called the Strain Energy, and u is the
Strain Energy Density.
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Try it
A cube of SAE1045 steel is subjected to a uniform
uniaxial stress as shown Determine the strain
energy density in the cube when
(a) the stress is 300 MPa (b) the strain in the
x-direction is 0.004
y
sx
x
(a)
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(a) For a linear elastic material
u1/2(300)(0.0015) N.mm/mm3
0.225 N.mm/mm3
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(b) Consider elastic-perfectly plastic
u1/2(350)(0.0018) 350(0.0022)
1.085 N.mm/mm3
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Shear Strain Energy
y
Consider a cube of material acted upon by a shear
stress,txycausing an elastic shear strain gxy
a
a
a
x
y
txy
x
gxy
d gxya
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Total Strain Energy for a Generalized State of
Stress
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Strain Energy for axially loaded bar
D
L
A
F
F Axial Force (Newtons, N) A Cross-Sectional
Area Perpendicular to F (mm2) E Youngs
Modulus of Material, MPa L Original Length of
Bar, mm

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Comparison of Energy Stored in Straight and
Stepped bars
Db
L/2
L/2
Da
L
nA
A
F
A
F
(a)
(b)
Note for n2 case (b) has U which is
3/4 of case (a)
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