# Moment Distribution Method

```Dr. Abdulkhaliq A. Jaafer
Misan University
Theory of Structures
Third Stage
Lecture 18
Moment-Distribution Method
The moment-distribution method of analysis provides a means of
analyzing statically indeterminate beams and frames by relatively simple
hand calculations. It is basically an iteration procedure.
Definitions:
Member Stiffness Factor (kij)
It is the moment required to rotate the end, while acting on it, through a unit
angle, without translation of the far end. It is denoted by the letter k. the stiffness
factor for a beam AB at A is given by:
= carryover
moment
= applied moment
= applied moment
B
A
A
B
L
L
EI= constant
πΎπ΄π΅ =
EI= constant
4 πΈπΌ
πΏ
πΎπ΄π΅ =
3 πΈπΌ
πΏ
Thus, it may summarize that the stiffness factor of a member ij at the end i is given
by
πΎππ =
πΎππ =
4 πΈπΌ
πΏ
3 πΈπΌ
πΏ
if far end j of member is fixed
if far end j of member is hinged
Carryover Factor (C.O.F.)
1
Dr. Abdulkhaliq A. Jaafer
Misan University
Theory of Structures
Third Stage
It is the ratio of moment produced at a joint to the moment applied at the other joint,
without displacing it. The carryover factor from point A to point B is given by:
1
πππ = { 2
0
ππ π‘βπ πππ πππ π΅ ππ  πππ₯ππ
ππ π‘βπ πππ πππ π΅ ππ  βπππππ
Distribution Factor (D.F.)
The distribution factor for a member is equal to the stiffness of the member divided
by the sum of the stiffnesses of all members meeting at the joint. It is denoted by the
letter d.
πππ =
πΎππ
∑πΎ
Notes:
1. The distribution factor is zero for a fixed end and one for an end roller or
hinged support.
2. The sum of all the distribution factors at a joint must equal one.
2
Lecture 18
Dr. Abdulkhaliq A. Jaafer
Misan University
Theory of Structures
Third Stage
Fixed-End Moments (F.E.M.)
The moments at the fixed end of a loaded member are called fixed-end
moments. It can be determined from Table below.
3
Lecture 18
Dr. Abdulkhaliq A. Jaafer
Misan University
Theory of Structures
Third Stage
Example: Draw the bending moment diagram for the beam shown in Fig.
by using the moment-distribution method.
18 kN/m
1
38 kN
2
2I
3
I
6m
2m
2m
Solution: Fixed -End Moment
πΉ
π12
−π€πΏ2
−18 ∗ (6)2
=
=
= −54 ππ. π
12
12
πΉ
π21
π€πΏ2
18 ∗ (6)2
=
=
= +54 ππ. π
12
12
πΉ
π23
=
−ππΏ
−38 ∗ 4
=
= −19 ππ. π
8
8
πΉ
π32
=
+ππΏ
38 ∗ 4
=
= +19 ππ. π
8
8
Stiffness Factor (Kij)
Distribution Factor (dij)
4
Lecture 18
Dr. Abdulkhaliq A. Jaafer
Misan University
Theory of Structures
Third Stage
18 kN/m
1
2I
38 kN
2
3
I
6m
2m
5
2m
Lecture 18
Dr. Abdulkhaliq A. Jaafer
Misan University
Theory of Structures
Third Stage
Example: Draw the bending moment diagram for the beam shown in Fig.
by using the moment-distribution method.
80 kN
1
2I
6m
24 kN/m
2
2m
3I
6m
3
I
6m
4
Solution: F. E. M.
Stiffness factors (Kij)
Distribution Factors
6
Lecture 18
Dr. Abdulkhaliq A. Jaafer
Misan University
Theory of Structures
Third Stage
80 kN
1
2I
6m
24 kN/m
2
2m
3I
6m
7
3
I
6m
4
Lecture 18
Dr. Abdulkhaliq A. Jaafer
Misan University
Theory of Structures
Third Stage
Example: Draw the bending moment diagram for the beam shown in Fig.
by using the moment-distribution method.
24 kN/m
1
80 kN
3
2
I
1.25 m
I
5m
Solution: Fixed End Moments:
1.25 m
Stiffness Factors
Distribution Factors:
8
Lecture 18
Dr. Abdulkhaliq A. Jaafer
Misan University
Theory of Structures
Third Stage
24 kN/m
1
80 kN
3
2
I
1.25 m
I
5m
9
1.25 m
Lecture 18
Dr. Abdulkhaliq A. Jaafer
Misan University
Theory of Structures
Third Stage
Example: Draw the bending moment diagram for the beam shown in Fig.
by using the moment-distribution method.
10 kN/m
1
I
6m
2
30 kN
3
I
9m
4
4m
Solution:
Treating overhang portion
10 kN/m
1
I
6m
2
I
-120 kN.m
3
9m
Fixed End Moments:
Stiffness Factors:
Distribution Factors:
10
Lecture 18
Dr. Abdulkhaliq A. Jaafer
Misan University
Theory of Structures
Third Stage
10 kN/m
1
I
6m
2
I
9m
11
30 kN
3
4
4m
Lecture 18
Dr. Abdulkhaliq A. Jaafer
Misan University
Theory of Structures
Third Stage
Example: Draw the bending moment diagram for the frame shown in Fig.
below by using the moment-distribution method.
2 kN/m
3
40 kN
2I
5
4
2I
10 m
I
I
10 m
2
1
30 m
30 m
Solution:
Fixed End Moments:
Stiffness Factors:
12
Lecture 18
Dr. Abdulkhaliq A. Jaafer
Misan University
Theory of Structures
Third Stage
Distribution Factors:
13
Lecture 18
Dr. Abdulkhaliq A. Jaafer
Misan University
Theory of Structures
Third Stage
Example: Draw the bending moment diagram for the frame shown in Fig.
below by using the moment-distribution method.
36 kN
64.8 kN/m
1
2
2I
5m
I
4
1.5 m
3
5m
Solution:
Fixed End Moments:
14
Lecture 18
Dr. Abdulkhaliq A. Jaafer
Misan University
Theory of Structures
Third Stage
Stiffness Factors:
Distribution Factors:
15
Lecture 18
```