Slope and deflection of beams || part - 21

 Deflection of Beams

◾ Introduction:

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

1. Strength.
2. Stiffness.

 

    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 stress 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 a cantilever or a beam to a small friction of its span.

◾ Check the video for more details:

◾ Deflection of beams:

◾ Macaulay's method for slope and deflection:

      We have seen in the previous articles and examples that the problems of deflections in Beams are bit tedious and laborious, specially when the beam is carrying some point loads. Mr. W.H. Macaulay devised a method, a continuous experience, for bending moment and integrating in such a way, that the constant of integration are valid for all sections of the beam; even though the law of bending moment varies from section to section. Now we shall discuss the application of Macaulay's method for for finding out the slopes and deflections of a few types of beams:

Note. The following rules are observed while using Macaulay's method:

1. Always take origin on the extreme left of the beam.
2. Take left clockwise moment as negetive and left anticlockwise moment as positive.
3. While calculating the slopes and deflections, it is convenient to use the values first in terms of kN and metres. 

◾ Beams of composite section:

    It is a beam made up of two or more different materials, joined together in such a manner, that they behave like a single piece, and the deflection of each piece is equal.

The slope and deflection of such a beam, is find out by algebraically adding the flexural rigidities of the two or more different materials, in the application of the respective relation.

 Mathematically,

Note. The moment of inertia of the composite section is to be found out about the c.g. of the section.

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