Deflections of Elastic Beams - Solved Problems

The beam shown in the following figure is made out of wood (E = 104 N/mm2) and has a

rectangular cross-section with the dimensions of b = 15 cm and h = 30 cm.

a) determine the slope and the deflection at the mid-pan of the beam by using:

- the direct integration method;

- the moment area method;

- the conjugate beam method;

- the step function method

b) determine de maximum deflection

c) check up the stiffness requirement knowing that

The direct integration method:

According to the deflection curve differential equation of the second order (that considers only

the effect of the bending moment):

The relationships for the slope and the deflection are derived by means of tow consecutive

integrations as follows:

The two integration constants C1 and C2 can be determined by using the boundary conditions.

Thus, knowing w(0) = 0 and w(l) = 0 and substituting in the equation (7):

From equation (6), by substituting x = 0, it can be concluded that C1 is the rotation of the beam

at the origin, which in this case coincides with section A.

Knowing the constants C1 and C2, the slope and the deflection at the midspan can be easily

computed by substituting

The ‘–‘ sign in equation (9) for the section slope (rotation) signifies that the said section rotates

counterclockwise. On the other hand, the ‘+’ in equation (10) means that the beam moves downwards

(in the positive direction of “z” axis).

The moment area method:

The general expressions for the rotation and the deflection are as follows:

where y M

0x  is the area of the bending moment diagram between the sections of abscissa “0” and “x”

and M y

x S 0 is the first moment of area of the bending moment diagram between sections of abscissa “0”

and “x” with respect to the section of abscissa “x”.