Short Notes: Displacement Method of Analysis (Moment Distribution Method) | Short Notes for Civil Engineering - Civil Engineering (CE) PDF Download

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DISPLACEMENT METHOD OF ANALYSIS (MOMENT DISTRIBUTION METHOD) S 
1. INTRODUCTION 
It is a displacement method for analysis of statically indeterminate beams and frames 
developed by Hardy Cross. The method only accounts for flexural effects and ignores axial and 
shear effects. In this method it is assumed in the beginning that all joints of the structure are 
fixed. Then by locking and unlocking each joint in succession, the internal moments are 
distributed such that each joint attains its final position. 
2. IMPORTANT DEFINITIONS   
2.1. Stiffness Factor 
Stiffness factor can be defined as the moment required to produce unit rotation in the beam. 
Stiffness factor for various cases is defined as follows. 
Case 1: Far end is fixed 
 
==
4EI
Stiffness factor s
l
 
Case 2: Far end is hinged 
 
??????????????????  ????????????   =  ??   =
3????
?? 
2.2. Relative stiffness (k) 
Relative stiffness is the relative value of the stiffness factor. It is value for various cases can be 
expressed as follows. 
 
Case 1: Far end is fixed 
=
I
k
L
 
 
 
Case 2: Far end is hinged 
=
3I
k
4L
 
Case 3: Far end is free 
K = 0 
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DISPLACEMENT METHOD OF ANALYSIS (MOMENT DISTRIBUTION METHOD) S 
1. INTRODUCTION 
It is a displacement method for analysis of statically indeterminate beams and frames 
developed by Hardy Cross. The method only accounts for flexural effects and ignores axial and 
shear effects. In this method it is assumed in the beginning that all joints of the structure are 
fixed. Then by locking and unlocking each joint in succession, the internal moments are 
distributed such that each joint attains its final position. 
2. IMPORTANT DEFINITIONS   
2.1. Stiffness Factor 
Stiffness factor can be defined as the moment required to produce unit rotation in the beam. 
Stiffness factor for various cases is defined as follows. 
Case 1: Far end is fixed 
 
==
4EI
Stiffness factor s
l
 
Case 2: Far end is hinged 
 
??????????????????  ????????????   =  ??   =
3????
?? 
2.2. Relative stiffness (k) 
Relative stiffness is the relative value of the stiffness factor. It is value for various cases can be 
expressed as follows. 
 
Case 1: Far end is fixed 
=
I
k
L
 
 
 
Case 2: Far end is hinged 
=
3I
k
4L
 
Case 3: Far end is free 
K = 0 
 
 
2.3. Distribution Factor 
It is the ratio in which the applied moment is distributed to various members meeting at 
a rigid point. Sum of distribution factor of all members meeting at a rigid joint is one. If 
far end is free, its D, k and distribution factor is zero. 
???? =
?? S?? 
Where, 
K = Relative stiffness of the member 
?K = Summation of relative stiffness of all members meeting at a joint 
2.4. Carry over moment 
It is the moment developed at one end due to applied moment at the other end. It is 
developed to make the slope zero. It is exerted by the fixed support on the beam. It is 
developed to make slope zero not to keep the structure in equilibrium. Various case for 
carry over moment are as follows. 
Case 1: Far end is fixed 
=
M
COM
2
 
Case 2: Far end is hinged 
COM = 0 
2.5. Carry Over Factor 
Carry over factor can be defined as the ratio of carry over moment and applied moment. 
Carry over factor for various cases can be given as follows. 
Case 1: Far end is fixed 
==
M1
COF
2 2
M
 
Case 2: Far end is hinged 
?????? =
0
?? = 0 
3. ANALYSIS OF BEAMS USING MOMENT DISTRIBUTION METHOD 
3.1. Sign convention 
(i) Clockwise end moments and clockwise rotations are taken as positive. Anti-clockwise 
end moments and anti-clockwise rotations are taken as negative. 
(ii) Bending moment which is sagging in nature are taken as positive and hogging 
bending moment is taken as negative. 
3.2. Procedure 
Step 1: Find out Distribution factors and fixed end moments. 
Step 2: Assume all joints to be initially locked. Then Determine the moment needed to 
bring each joint in equilibrium. Release the joints and distribute the counterbalancing 
moment into the connecting span at each joint. Carry these moments in each span over 
to its other end.  
Repeat the same cycle until the moment equilibrium at the joint achieved. 
.