Building Design and Construction Handbook - Merritt - Ventech!

5.44 SECTION FIVE
y � t � xt
AD AB
where t AD is the tangential deviation of D from the tangent at A and t AB is the
tangential deviation of B from that tangent. This equation, which is perfectly general
for the deflection of any point of a simple beam, no matter how loaded, may be
rewritten to give the deflection directly:
y � xt � t (5.65)
AB AD
But t AB is the moment of the area of the M/EI diagram for the whole beam about
support B. And t AD is the moment about D of the area of the M/EI diagram included
between ordinates at A and D. Hence
� �
3
1 PL L 2 1 1 PLx xL PL 2
y � x � L � xL � x(3 � 4x )
24EI 2 3 3 2 2EI 3 48EI
It is also noteworthy that, since the tangential deviations are very small distances,
the slope of the elastic curve at A is given by
t AB
�A � (5.66)
L
This holds, in general, for all simple beams regardless of the type of loading.
The procedure followed in applying Eq. (5.65) to the deflection of the loaded
beam in Fig. 5.28 is equivalent to finding the bending moment at D with the M/
EI diagram serving as the load diagram. The technique of applying the M/EI diagram
as a load and determining the deflection as a bending moment is known as
the conjugate-beam method.
The conjugate beam must have the same length as the given beam; it must be
in equilibrium with the M/EI load and the reactions produced by the load; and the
bending moment at any section must be equal to the deflection of the given beam
at the corresponding section. The last requirement is equivalent to requiring that
the shear at any section of the conjugate beam with the M/EI load be equal to the
slope of the elastic curve at the corresponding section of the given beam. Figure
5.29 shows the conjugates for various types of beams.
Deflections for several types of loading on simple beams are given in Figs. 5.30
to 5.35 and for overhanging beams and cantilevers in Figs. 5.36 to 5.41.
When a beam carries a number of loads of different types, the most convenient
method of computing its deflection generally is to find the deflections separately
for the uniform and concentrated loads and add them up.
For several concentrated loads, the easiest solution is to apply the reciprocal
theorem (Art. 5.10.5). According to this theorem, if a concentrated load is applied
to a beam at a point A, the deflection it produces at point B is equal to the deflection
at A for the same load applied at B(d AB � d BA).
Suppose, for example, the midspan deflection is to be computed. Then, assume
each load in turn applied at the center of the beam and compute the deflection at
the point where it originally was applied from the equation of the elastic curve
given in Fig. 5.33. The sum of these deflections is the total midspan deflection.
Another method for computing deflections of beams is presented in Art. 5.10.4.
This method may also be applied to determining the deflection of a beam due to
shear.