Introduction & Absolute and Relative Stiffness - Displacement Method: Moment Distribution Method | Strength of Material Notes - Agricultural Engg - Agricultural Engineering PDF Download

In slope-deflection method the unknown displacements/rotations are obtained by solving a set of algebraic equations. This becomes cumbersome for structures with large number of members. In such cases the Moment distribution method, also known as the Hardy Cross method (named after Prof. Hardy Cross), provides a convenient means for analyzing the structures in an iterative way. In this lesson we will formulate the basic ingredients of Moment distribution method. Illustration of the general procedure and examples will be discussed in the subsequent lessons.

Sign Convention
For the development and application of Moment Dsitribution Method we will use similar sign convention as in the case of Slope-Deflection Method.
Moment: At the end of a member clockwise moment is positive.
Transverse displacement: Transverse displacement in upward direction is positive.
Rotation: Rotation in anti-clockwise direction is positive.The above sign convention is depicted in Figure 15.1.

Fig. 15.1.1 Absolute and Relative Stiffness

Stiffness of a member may be defined as the force/moment required to cause unit displacement/rotation. The central idea of Moment distribution method is to distribute moment at any joint, among the connenting members (members meeting at that joint) according to their rotational stiffnesses. In this section we will derive the expressions of rotational stifness of member with different support conditions.

1.1 Beam Hinged at Both Ends

Fig. 15.2.Slope deflection equation at A and B (δ = 0),

 (15.1)      
(15.2)
Now, at B, equilibrium equation is,MBA = 0 . Therefore form equaition (2), we have,

Substituting, θB = -θA/2   in equation (1), we have,

1.2 Beam Hinged at one End and Fixed at other EndFig. 15.3.Slope deflection equation at A and B
                                     
(15.3)

   (15.4)
From equations (3) and (4), we have, (15.5)
Absolute Stiffness  

1.3 Several members meeting at a joint          Fig. 15.4.

Figure 15.4 shows, four members (for illustration purpose only four members are taken, but the theory is applicable for any number of members), AO, BO, CO and DO meeting at O. L_OA, L_OB, L_OC, and L_OD are the length and  I_OA, I_OB, I_OC, and I_OD are the second moment of area of the respective members. Support A, C are fixed and B, D are hinged. An external moment M is applied at O. The moment M will be distributed among all the members meeting at O. Suppose M_OA, M_OB, M_OC, and M_OD are the corresponding distribution.Compatibility condition at O,
(15.6)

Equilibrium condition at O,

(15.7)
Now from the previous tow cases, we may express MOA, MOB, MOC, and MOD as,


From Equations (8a) – (8b),

(15.30)

From equations (7) and (9),




Therefore, moment acting at a joint will be divided amongst the connecting members in proportion to their stiffness.
The factors
 
are called distribution factor (DF) and moments are called distributed moments.

1.4 Carry Over Factor
Consider a fixed beam AB as shown bellow. Suppose the rotational constraint of joint A is released and a balancing moment M_AB is applied at A. Then M_AB will cause a moment M_BA at B.

 Fig. 15.5.

The carry over factor is defined as,

From 15.1.2 we have,

Here, M_BA is called caried over momnet at B due to M_AB at A.