Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

17.7 Work–Energy method for Single Loads
As discussed in Section 17.2, the conservation-of-energy principle declares that energy in
a closed system is never created or destroyed—it is only transformed from one state to
another. For example, the work of an external load acting on a deformable body is transformed
into internal strain energy, and provided that no energy is lost in the form of heat,
the strain energy U is equal in magnitude to the external work W:
W
= U
(17.27)
Equation (17.27) can be used to determine the deflection or slope of a member or structure
under highly selective conditions—specifically, that the member or structure must be
loaded by a single external concentrated force or concentrated moment. Corresponding
displacements can be determined only at the location of the single load in the direction that
the load acts. But why is this approach limited to a single external load or moment? The
reason is that Equation (17.27) is the only equation available in that method. The strain energy
U of the structure will be a single number. The work W performed by an external force acting
on a deformable solid is one-half the product of the magnitude of the force and the
displacement through which the solid moves in the direction of the force. (See Section 17.2.)
Similarly, the work W performed by an external moment acting on a deformable solid is
one-half the product of the magnitude of the moment and the angle through which the solid
rotates. (See Section 17.5.) Consequently, if more than one external force or moment is
applied, then W in Equation (17.27) will have more than one unknown deflection or rotation
angle. Obviously, one equation cannot be solved for more than one unknown quantity.
Formulations for the strain energy were developed in Sections 17.3, 17.4, and 17.5 for
axial deformation, torsional deformation, and flexural deformation, respectively. To recapitulate,
the strain energy in prismatic axially loaded members can be determined from
Equation (17.12):
U
2
PL
=
2AE
For compound axial members and structures consisting of n prismatic axial members,
the total strain energy in the member or structure can be computed with Equation (17.14):
U =
n 2
∑
i=
1
Fi
Li
2AE
The strain energy in prismatic torsionally loaded members can be determined from
Equation (17.16):
U
i
T 2
=
L
2JG
The total strain energy in compound torsional members can be computed from
Equation (17.18):
U =
n 2
∑
i=
1
i
Ti
Li
2J G
For a flexural member, the strain energy stored in the beam can be determined from
Equation (17.20):
U =
L
∫ 0
i
i
2
M
2EI dx
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