Mumbai university > MECH > SEM 3 > Strength Of Materials

Marks: 5M

Year: Dec 2013

When the loads on the beam do not conform to standard cases, the solution for slope and deflection must be found from first principles. Macaulay developed a method for making the integration simpler.

The basic equation governing the slope and deflection of beams is,

$ EI \frac{d^2y}{dx^2} = M$ , Where M is a function of x.

When a beam has a variety of loads it is difficult to apply this theory because some loads may be within the limits of x during the derivation but not during the solution at a particular point. Macaulay’s method makes it possible to do the integration necessary by placing all the terms containing x within bracket “( )” and integrating the bracket, not x. Every term with a different bracket is separated by means of a separator “|”. During the evaluation, any bracket with a negative value is ignored because a negative value means that the load it refers to is not within the limit ofx.

The general method of solution is conducted as follows:

• Write down the bending moment equation by placing x on the extreme right hand end of the beam so that it contains all the loads. Write all terms containing x in brackets “( )” as shown below:

  $EI \frac{d^2y}{dx^2} = M = R_1x |F_1 (x - a)|F_2 (x - b)|F_3 (x - c)$

• Integrating once treating the square brackets as the variables.

  $EI \frac{dy}{dx} = R_1\frac{x^2}{2} |F_1 \frac{(x - a)^2}{2}|F_2 \frac{(x - b)^2}{2}|F_3 \frac{(x - c)^2}{2} + c_1$

• Integrating again in the same way,

  $EI_y = R_1\frac{x^3}{6} | F_1\frac{(x - a)^3}{6} | F_2 \frac{(x - b)^3}{6} | F_3 \frac{(x - c)^3}{6} + C_1x + C_2$

• Use boundary conditions to solve $C_1 and C_2$

• Solve slope and deflection by putting the approximate value of x. IGNORE any brackets containing negative values.