Moment Area Theorems: Theorem 1: When a beam is subjected to external loading, it under goes deformation. Then the intersection angle between tangents drawn at any two points on the elastic curve is given by the area of bending moment diagram divided by its flexural rigidity.

Moment Area Theorems: Theorem 2: The vertical distance between any point on the elastic curve and intersection of a vertical line through that point and tangent drawn at some other point on the elastic curve is given by the moment of area of bending moment diagram between two points taken about first point divided by flexural rigidity.

Fixed end moment due to a point load at the mid span:

Both moments are negative and hence they produce hogging bending moment.

Stiffness coefficients a) When far end is simply supported

b) When far end is fixed

Substituting in (1)

Fixed end moments due to yielding of support.

Hence sagging BM

Fixed end moment for various types of loading

Assumptions made in slope deflection method: All joints of the frame are rigid 2) Distortions due to axial loads, shear stresses being small are neglected. 3) When beams or frames are deflected the rigid joints are considered to rotate as a whole.

Sign conventions: Moments: All the clockwise moments at the ends of members are taken as positive. Rotations: Clockwise rotations of a tangent drawn on to an elastic curve at any joint is taken as positive. Sinking of support: When right support sinks with respect to left support, the end moments will be anticlockwise and are taken as negative.

Development of Slope Deflection Equation Span AB after deformation Effect of loading Effect of rotation at A

Effect of rotation at B Effect of yielding of support B

Slope Deflection Equations

EXAMPLES

Example: Analyze the propped cantilever shown by using slope deflection method. Then draw Bending moment and shear force diagram. Solution:

Slope deflection equations

Boundary condition at B MBA=0 Substituting in equations (1) and (2)

Free body diagram

Example: Analyze two span continuous beam ABC by slope deflection method. Then draw Bending moment & Shear force diagram. Take EI constant

Solution:

Slope deflection equations

Boundary conditions i. -MBA-MBC=0 MBA+MBC=0 ii. MCB=0 Now Solving

Free body diagram Span AB: Span BC:

BM and SF diagram

Example: Analyze continuous beam ABCD by slope deflection method and then draw bending moment diagram. Take EI constant. Solution:

Slope deflection equations:

Boundary conditions Solving

Substituting

Example: Analyse the continuous beam ABCD shown in figure by slope deflection method. The support B sinks by 15mm. Take Solution:

FEM due to yielding of support B For span AB: For span BC:

Slope deflection equation

Boundary conditions Now Solving

Final moments