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
𝐾𝐴𝐵 =
EI= constant
4 𝐸𝐼
𝐿
𝐾𝐴𝐵 =
3 𝐸𝐼
𝐿
Thus, it may summarize that the stiffness factor of a member ij at the end i is given
by
𝐾𝑖𝑗 =
𝐾𝑖𝑗 =
4 𝐸𝐼
𝐿
3 𝐸𝐼
𝐿
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
𝐂𝐎𝐅 = { 2
0
𝑖𝑓 𝑡ℎ𝑒 𝑓𝑎𝑟 𝑒𝑛𝑑 𝐵 𝑖𝑠 𝑓𝑖𝑥𝑒𝑑
𝑖𝑓 𝑡ℎ𝑒 𝑓𝑎𝑟 𝑒𝑛𝑑 𝐵 𝑖𝑠 ℎ𝑖𝑛𝑔𝑒𝑑
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.
𝑑𝑖𝑗 =
𝐾𝑖𝑗
∑𝐾
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
𝐹
𝑀12
−𝑤𝐿2
−18 ∗ (6)2
=
=
= −54 𝑘𝑁. 𝑚
12
12
𝐹
𝑀21
𝑤𝐿2
18 ∗ (6)2
=
=
= +54 𝑘𝑁. 𝑚
12
12
𝐹
𝑀23
=
−𝑃𝐿
−38 ∗ 4
=
= −19 𝑘𝑁. 𝑚
8
8
𝐹
𝑀32
=
+𝑃𝐿
38 ∗ 4
=
= +19 𝑘𝑁. 𝑚
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