The document discusses the deflection of beams, specifically focusing on cantilever beams and the criteria for their design based on strength and stiffness. It outlines two methods for calculating slope and deflection: the double integration method and Macaulay's method, detailing their applicability for single and multiple loads. An example problem is provided to illustrate the calculation of maximum slope and deflection for a simply supported beam under a central point load.

1. Deflection of Beams

2. Introduction:
We see that whenever a cantilever or a beam
is loaded, it deflects from its original position. The
amount, by which a beam deflects, depends upon
its cross-section and the bending moment. In
modern design offices, following are the two
design criteria for the deflection of a cantilever or
a beam:
1. Strength 2. Stiffness.

3. As per the strength criterion of the beam
design, it should be strong enough to resist bending
moment and shear force. Or in other words, the
beam should be strong enough to resist the
bending stresses and shear stresses. And as per
the stiffness criterion of the beam design, which is
equally important, it should be stiff enough to resist
the deflection of the beam. Or in other words, the
beam should be stiff enough not to deflect more
than the permissible limit* under the action of the
loading. In actual practice, some specifications are
always laid to limit the maximum deflection of
cantilever or beam to a small fraction of its span.
In this chapter, we shall discuss the slope and
deflection of the entire line of beams under the
different types of loadings.

4. Methods for Slope and Deflection at a Section:
Though there are many methods to find out
the slope and deflection at a section in a loaded
beam, yet the following two methods are important
from the subject point of vi
1. Double integration method.
2. Macaulay's method.
It will be interesting to know that the first
method is suitable for a single load, whereas the
second method is suitable for several loads
Double Integration Method for Slope and
Deflection:
We have already discussed earlier that the
bending moment at a point,

5. and integrating the above equation once again,
It is thus obvious that after first integration the
original differential equation, we get the value slope
at any point. On further integrating, we get the value
of deflection at any point.

6. Note. While integrating twice the original
differential equation, we will get two constants C1, and
C2. The values of these constants may be found out
by using the end conditions.
Simply Supported Beam with a Central Point Load:

7. From the geometry of the figure, we find that the
reaction at A,

11. Example (1) : A simply supported beam of span 3 m is
subjected to a central load of 10 kN. Find the maximum
slope and deflection of the beam. Take I= 12× 106 mm4
and E = 200 GPa.
Solution: Given span (l) = 3 m = 3 × 103mm ; General
load(W) = 10 kN=10×103N.Moment of inertia (I)=12×106
mm4 and modulus of elasticity (E) = 200 GPa=200×103
N/mm2.

16. FIGURE (3) : Section X in AC

20. The equations (vii) and (ix) are the required equations for the slope and
deflection at any point in the section AC. A little consideration will show
that these equations are useful, only if the value of iC is known. Now to
obtain the value of iC, let us first out the deflection at C from the
equations for sections AC and CB.
Now substituting x = b in equation (iv) and equating the same with
equation (ix),

21. This is required equation for slope at any section in BC. We
know that the slope is maximum at B. Thus for maximum
slope, substituting x = 0 in equation (x),