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

ExAmpLE 17.16
Determine the vertical deflection at joint A for the truss shown. For all
members, the cross-sectional area is A = 1,100 mm 2 and the elastic modulus
is E = 200 GPa.
Plan the Solution
The vertical deflection is to be determined at joint A. Since a vertical load
is already present at A on the truss, this load will be designated as P. Consequently,
instead of performing a truss analysis with the 75 kN load at A,
we will replace the 75 kN load with a variable load designated as P. With
this variable load P, we will then perform a truss analysis at joint A, using
an appropriate method, such as the method of joints. From the analysis,
an expression in terms of P and any additional constant forces that arise
from the 125 kN load applied at C will be obtained for each truss member.
Next, construct a table to organize the truss analysis results. For each truss member,
record the complete member-force expressions F in one column. Four additional operations
will then be performed:
(1) Differentiate the member-force expression F with respect to P, and record the partial
derivatives.
(2) Substitute the actual 75 kN value for P in the member-force expression F and calculate
the actual member force.
(3) Calculate the length of each truss member.
(4) Multiply the partial derivative ∂F/ ∂P
by the actual member-force value F and the
member length L.
The products (∂F/ ∂P)FL for all truss members will then be summed. Since the area
A and the elastic modulus E are the same for all members, these values will be introduced
after the summation. Finally, Equation (17.39) will be applied to determine the downward
deflection D of joint A.
SolutioN
We seek the vertical deflection of the truss at joint A. Because
there is already a load applied at this joint in the desired direction,
we will replace the 75 kN load with a variable load P. A freebody
diagram of the truss with load P applied at joint A is shown.
We perform a truss analysis, using the loadings shown in
the free-body diagram, to find the axial force in each truss
member. With these loads, the member forces F will each be
expressed as a unique function of P. Some member-force
functions F may also contain constant terms arising from the
125 kN load that acts at joint C.
The tabular format that follows is a convenient way to organize the calculation of the
truss deflection. The member name is shown in column (1), and the expression for the internal
member force F in terms of the variable P is listed in column (2). Differentiate the
function in column (2) with respect to P, and record the result in column (3). Next, substitute
the actual value of P = 75 kN into the member-force functions in column (2), and record
the result in column (4). These values are the actual member forces for the truss in
response to the 75 kN and 125 kN loads, and they will be used for the term FL/AE found
in Equation (17.39). Finally, calculate the member lengths and record them in column (5).
Castigliano’s second theorem applied to trusses is expressed by Equation (17.39).
For this particular truss, each member has a cross-sectional area A = 1,100 mm 2 and an
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