Example Problems

7.5.3 Beams on Elastic Foundations: Maximum Bending Stress of an Aluminum Beam

An aluminum alloy beam 3 inches wide, 2 inches deep and 60 inches long is manufactured with an initial concentrated angular deformation of 0.02 radians at midlength; this initial slope is shown in Fig. 7.11a.In use, the beam is placed on an elastic foundation that develops 500 lbf/in2 vertical upwardpressure for every 1 inch it is depressed.The beam is loaded by two concentrated loads of 4000 lbf each and a uniformly distributed load of 80 lbf/inch over the portion between the concentrated loads.The loading is shown in Fig. 7.11b.It is desired to determine the maximum bending stress in the aluminum beam.

Enter dimensions, properties and loading

Modulus of elasticity, E:

Fig. 7.11 a and b

Beam height, base, and length; bo, d,L:

Distance from the neutral axis, c:

Foundation modulus, ko:

Loads, W1, W2:

Unit load, w3:

Unit load, w4 (used to negate the effect of w3 past w3's end point ):

Angular deformation qo:

Calculation
procedure

First determine the beam and foundation parameters:

Moment of inertia of beam, Ib (Refer to Table 1 for moment of inertia calculations ):

Table 1

Reaction frequency, b:

A constant, reaction frequency multiplied
by the beam length,yields bL:

Table 7

An examination of Table 7 shows the loading conditions covered by the superposition of three cases in which both ends are free:
Case 1 used twice with W1 and a1,andW2 and a2;
Case 2 used twice with w3 anda3,and w4 and a4;
Case 5 used once with qo anda0 .

The loads and deformations at the left end are now evaluated by summing the values for the four different loads, which is done in the order in which the loads are mentioned.But before actually summing the end values, a set of constants involving the load positions must be determined for each case:

Constants and functions needed to derive the general equations for
slope, displacement, shear and moment

Distances to various loads, as shown in Figures:

Then, the functions from the table are:

The constants from the table are:

General equations for
shear, moment, slope
and deflection as a function
of beam length

Boundary values:

The equations for shear, moment, slope, and deflection as a
function of beam length are dependent upon the conditions
at an end of the beam (i.e. the boundaries). The values of shear, moment, slope and deflection at the left end of the beam are given by the following:

RA = 0 and MA = 0 because of the support structure.

So at point A:

The equations as a function of beam length for slope, deflection, moment and shear are:

Moment diagram:

Shear diagram:

An examination of the transverse shear V plot shows that the value of the shear passes through zero at or near x = 15, 30, and 45 in.

As is evident from the M plot as well as the V plot, the maximum positive bending moment occurs near 15 in. The exact value is calculated as follows: