Slope and Deflection

Relation between curvature, slope and deflection:

Eg: A simply supported beam of length L is subjected to couples as shown in figure determine maximum slope and deflection at the centre of the beam?

Method of determining slope and deflection:

• Double integration method:(Not suitable for objective type questions.)
• Area moment method:(for cantilevers, slopes and deflections can be determined very quickly.)
• Conjugate beam method:(very much suitable for beams of varying sections, subjected to couples, for cantilevers of S. S . Beams.)
• Macaulay ‘s method:(Also successive integration method.)

NOTE: In double integration or Macaulay’s method two constants of integration C1 and C2 will be obtained. These are determined using end conditions.

Mohr’s Theorem’s: Moment Area Method: 

• Theorem 1: The angle between tangents drawn at any two points on the deflected curve, is equal to the area of M / El diagram between the two points.

i.e.,  θ = area of M / EI diagram.

A = area of B.M.D.

• Theorem 2: The intercept on a vertical line made by two tangents drawn at the two points on the deflected curve, is equal to the moment of M / EI diagram between the two points about the vertical line.

Ax̅ / EI x̅ = distance of C.G. of B.M.D.

E.G: (Suitable for cantilevers) 

Step 1 : To determine slope and deflection at any point.

Step 2 : Draw (BMD) / (EI)

i.e., M / EI

Step 3 : Slope = area of (M / EI) diagram between fixed end point under consideration.

Step 4 : Deflection Ax̅ / EI

A = B.M.D area between fixed end and point under consideration.

x̅ = distance of C.G. of M / EI from point under consideration.

Ex.1

Ex.2

Maxwell’s Law of Reciprocal Deflections:

Consider cantilever beam AB. Let ‘C’ be intermediate point. Then the deflection at due to a point load ‘P’ at B say YCB, is equal deflection at ‘B’ due to a point load ‘P’ a

i.e., YBC

SLOPE & DEFLECTION FOR DIFFERENT LOADING OF BEAMS: