Handbook of Civil Engineering Calculations

FIGURE 16

Express the axial force P in each bar in terms of R 1 because both reactions are as-

sumed to be directed toward the left. Use subscripts corresponding to the bar numbers

(Fig. 16). Thus, P 1 =R 1 P 2 = - 30; P 3 =R 1 - 40.

2. Express the deformation of each bar in terms of the reaction
   and modulus of elasticity
   Thus, A/! = RL(16)/(2.QE) = 18RL/E; M 2 = (R 1 - 30)(48)/(1.6£) = (3QRL - 900)/£; A/ 3 =
   (RL -4Q)24/(l.2E) = (2QRL - 800)/£.


3. Solve for the reaction
   Since the ends of the member are stationary, equate the total deformation to zero, and
   solve for R 1. Thus A/, = (68RL - 1700)/£ = O; RL = 25 kips (111 kN). The positive result
   confirms the assumption that RL is directed to the left.


4. Compute the displacement of the points
   Substitute the computed value ofRL in the first two equations of step 2 and solve for the
   displacement of the points A and B. Thus AZ 1 = 18(25)72000 = 0.225 in (5.715 mm); A/ 2 =
   [30(25) - 900]/2000 = -0.075 in (-1.905 mm).
   Combining these results, we find the displacement of A = 0.225 in (5.715 mm) to the
   right; the displacement of B = 0.225 - 0.075 = 0.150 in (3.81 mm) to the right.
   5. Verify the computed results
   To verify this result, compute RR and determine the deformation of bar 3. Thus H*FH =

• R 1 + 30 + 10 - RR = O; RR = 15 kips (67 kN). Since bar 3 is in compression,
  AZ 3 = -15(24)/[1.2(2000)] = -0.150 in (-3.81 mm). Therefore, B is displaced 0.150 in
  (3.81 mm) to the right. This verifies the result obtained in step 4.

REACTIONS AT ELASTIC SUPPORTS

The rigid bar in Fig. YIa is subjected to a load of 20,000 Ib (88,960 N) applied at D, It is
supported by three steel rods, 1, 2, and 3 (Fig. 17a). These rods have the following rela-
tive cross-sectional areas: A 1 = 1.25, A 2 = 1.20, A 3 = 1.00. Determine the tension in each
rod caused by this load, and locate the center of rotation of the bar.

Calculation Procedure:

1. Draw/ a free-body diagram; apply the equations of equilibrium
   Draw the free-body diagram (Fig. 176) of the bar. Apply the equations of equilibrium:

Area, in^2