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

Structures with a Rotating Rigid Bar
Problems involving a rotating rigid element can be particularly difficult to solve. For these
structures, a deformation diagram should be drawn at the outset. This diagram is essential
to obtaining the correct geometry-of-deformation equation. The assumed internal force and
the assumed deformation must be consistent. Thus, if elongation is assumed for a member
in the deformation diagram, then the positive internal force for the member is tensile.
MecMovies Example M5.7 illustrates problems of this type.
111
STATICALLy INdETERMINATE
AxIALLy LOAdEd MEMbERS
mecmovies
ExAmpLE
m5.7 Rigid bar AD is pinned at A and supported by bars (1)
and (2) at B and C, respectively. Bar (1) is aluminum and bar
(2) is brass. A concentrated load P = 36 kN is applied to the
rigid bar at D. Compute the normal stress in each bar and the
downward deflection of the rigid bar at D.
Some structures with rotating rigid bars have opposing members; that is, one member
is elongated, while the other member is compressed. Figure 5.11 illustrates the subtle difference
between these two types of configuration.
For the structure with two tension members (Figure 5.11a), the geometry of deformations
in terms of vertical joint deflections v B and v C is found by similar triangles
(Figure 5.11b):
v
x
B
B
=
v
x
C
C
From Figure 5.11c, the member deformations δ 1 and δ 2 are related to the joint deflections
v B and v C by
and
δ
= L − L = ( L + v ) − L = v ∴ v = δ
1 final initial 1 B 1 B B 1
δ2 = Lfinal − Linitial = ( L2 + vC)
− L2 = vC ∴ vC
= δ2 (5.6)