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

The external work done by a force acting on an axial member that deforms is
W
=
1 2
Pδ
747
wORk–ENERgy METHOd FOR
SINgLE LOAdS
where δ is the distance (which equals the axial deformation) that the member moves in the
direction of the force. The external work of a torque that acts on a shaft is
W
=
1 2
Tφ
where φ is the rotation angle (in radians) through which the shaft rotates. For a beam subjected
to a single external force, the work of the load is
W
1
= 2
Pv
where v is the beam deflection at the location of the external force in the direction of the
load. If the beam is subjected to a single external concentrated moment, the work of the
external moment is
W
=
1 2
Mθ
where θ is the beam slope (i.e., dv/dx) at the location of the external concentrated moment.
Another common use for the work–energy method involves the determination of deflections
for simple trusses and for other assemblies of axial members. The work of a single
external load acting on such a structure is
W
= 1 2
PD
where D is the deflection of the structure in the direction that the force acts at the location
of the external load. To reiterate, the method described here and in the example that follows
can be used only for structures subjected to a single external load, and only the deflection
in the direction of the load can be determined.
While the work–energy method has limited application, it serves as a useful introduction
to more powerful energy methods that will be developed in subsequent sections. These
other energy methods can be used to perform a completely general deflection analysis on a
member or structure.
ExAmpLE 17.9
A tie rod (1) and a pipe strut (2) are used to support a 50 kN
load as shown. The cross-sectional areas are A 1 = 650 mm 2 for
the tie rod and A 2 = 925 mm 2 for the pipe strut. Both members
are made of structural steel that has an elastic modulus E =
200 GPa. Determine the vertical deflection of the two-member
assembly at B.
y
A
1.25 m
(1)
B
x
Plan the Solution
From a free-body diagram of joint B, the internal axial forces in
members (1) and (2) can be calculated. From Equation (17.12),
the strain energy of each member can be computed. The total
strain energy in the assembly is found from the sum of the two
strain energies. The total strain energy is then set equal to the
work done by the 50 kN load as it deflects downward at B.
From this conservation-of-energy equation, the unknown
downward deflection of joint B can be determined.
C
(2)
50 kN
1.15 m