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

(b) Member Deformations
The deformations in the members are determined from either Equation (5.1) or Equation
(5.2). The elongation in tie rod (1) is
δ
1
2
s 1L1
(83.63 N/mm )(1.25 m)(1,000 mm/m)
= = = 0.5227 mm
2
E
200,000 N/mm
1
Ans.
The length of inclined pipe strut (2) is
2 2
L = (1.25 m) + (1.15 m) = 1.70 m
2
and its deformation is
2
s 2L2
( 79.84 N/mm )(1.70 m)(1,000 mm/m)
δ 2 = = − =− 0.6786 mm Ans.
2
E
200,000 N/mm
2
The negative sign indicates that member (2) contracts (i.e., gets shorter).
42.61°
B 2
δ 2 = – 0.6782 mm
B
δ 1 = 0.5227 mm
(c) Deformation Diagram
B 1
Step 1: To determine the displaced position of joint B, let us first imagine that
the pin at joint B is temporarily removed, allowing members (1) and (2) to deform
freely by the amounts computed in part (b). Since joint A of the tie rod is fixed to
a support, it remains stationary. Thus, when tie rod (1) elongates by 0.5227 mm,
joint B moves to the right, away from joint A to the displaced position B 1 .
Similarly, joint C of the pipe strut remains stationary. When member (2) contracts by
0.6782 mm, joint B of the pipe strut moves toward joint C, ending up in displaced position
B 2 . These deformations are shown in the accompanying figure.
42.61°
B 2
δ 2 = – 0.6782 mm
B
δ 1 = 0.5227 mm
Step 2: In the previous step, we imagined removing the pin at B and allowing each member
to deform freely, either elongating or contracting, as dictated by the internal forces
acting in each member. In actuality, however, the two members are connected by pin B. The
second step of this process requires finding the displaced position B′ of the pin connecting
tie rod (1) and pipe strut (2) such that B′ is consistent with member elongations δ 1 and δ 2 .
Because of the axial deformations, both tie rod (1) and pipe strut (2) must rotate
slightly if they are to remain connected at pin B. Tie rod (1) will pivot about stationary end A,
and pipe strut (2) will pivot about stationary end C. If the rotation angles are
small, the circular arcs that describe possible displaced positions of joint B can
B 1 be replaced by straight lines that are perpendicular to the unloaded orientations
of the members.
Consider the figure shown. As tie rod (1) rotates clockwise about stationary
end A, joint B 1 moves downward. If the rotation angle is small, the circular arc
describing the possible displaced positions of joint B 1 can be approximated by a
line that is perpendicular to the original orientation of tie rod (1).
Similarly, as pipe strut (2) rotates clockwise about stationary end C,
the circular arc describing the possible displaced positions of joint B 2 can be
approximated by a line that is perpendicular to the original orientation of
member (2).
The intersection of these two perpendiculars at B′ marks the final displaced
position of joint B.
B
Step 3: For the two-member structure considered here, the deformation diagram
forms a quadrilateral shape. The angle between member (2) and the x axis is
42.61°; therefore, the obtuse angle at B must equal 180° − 42.61° = 137.39°.
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