Handbook of Civil Engineering Calculations

DEFLECTION OF A BEAM UNDER

MOVING LOADS

The moving-load system in Fig. 53a traverses a beam on a simple span of 40 ft (12.2
m). What disposition of the system will cause the maximum deflection at midspan?

Calculation Procedure:

1. Develop the equations for the
   midspan deflection under a unit load
   The maximum deflection will manifestly occur
   when the two loads lie on opposite sides of the cen-
   terline of the span. In calculating the deflection at
   midspan caused by a load applied at any point on
   the span, it is advantageous to apply Maxwell's
   { b j theorem of reciprocal deflections, which states
   following: The deflection at A caused by a load at
   FIGURE 53 B equals the deflection at B caused by this load at
   A.
   In Fig. 536, consider the beam on a simple span
   L to carry a unit load applied at a distance a from the left-hand support. By referring to
   case 7 of the AISC Manual and applying the principle of reciprocal deflections, derive the
   following equations for the midspan deflection under the unit load: When a < Lf2, y =
   (3L^2 a - 4a^3 )/(48EI). When a < L/2, y = [3L^2 (L - a) - 4(L - a)^3 ]/(48EI).


2. Position the system for purposes of analysis
   Position the system in such a manner that the 20-kip (89.0-kN) load lies to the left of cen-
   ter and the 12-kip (53.4-kN) load to the right of center. For the 20-kip (89.0-kN) load, set
   a = x. For the 12-kip (53.4-kN) load, a = x + 7; L - a = 40 - (jc + 7) = 33 - x.


3. Express the total midspan deflection in terms of x
   Substitute in the preceding equations. Combining all constants into a single term &, we
   find ky = 20(3 ) x 402 - 4^3 ) + 12(3 x 402 (33 - Jt) - 4(33 - *)^3 ].


4. Solve for the unknown distance
   Set dyldx = O and solve for x. Thus, x = 17.46 ft (5.321 m).
   For maximum deflection, position the load system with the 20-kip (89.0-kN) load
   17.46 ft (5.321 m) from the left-hand support.

Riveted and Welded Connections

In the design of riveted and welded connections in this handbook, the American Institute
of Steel Construction Specification for the Design, Fabrication and Erection of Structural
Steel for Buildings is applied. This is presented in Part 5 of the Manual of Steel Construc-
tion.
The structural members considered here are made of ASTM A3 6 steel having a yield-
point stress of 36,000 lb/in^2 (248,220 kPa). (The yield-point stress is denoted by Fy in the
Specification.) All connections considered here are made with A141 hot-driven rivets or
fillet welds of A233 class E60 series electrodes.