BYDR. MAHDI DAMGHANI
Moment distribution method
Introduction
It was developed by Prof. Hardy Cross (one of America's most brilliant engineers) in 1932
It is also known as Hardy cross method
This method was originally though of to analyse reinforced concrete structures
A very powerful method of analysing indeterminate continuous beams and frames
Indeterminate structure
How many unknowns does the beam have?How many equations do we have (knowns) to
solve the unknowns?
15 kN/m 10 kN/m150 kN
8 m 6 m 8 m
AB C
DI I I
3 m
Example
Back to our problem
Fix all the joints (assume all the joints in the structure have no ability to rotate)
Calculate the Fixed End Moments (FEM)Allow the joints that were fixed artificially to
rotate freelyThe unbalanced moments created must be
balanced out based on the relative stiffness of members and carry over factor
Sum up the moments
See it in practice
15 kN/m 10 kN/m150 kN
8 m 6 m 8 m
AB C
DI I I
3 m
8 m
-80 kN.m +80 kN.m15 kN/m
A B6 m
-112.5kN.m 112.5 kN.m
B C 8 m
-53.33 kN.m10 kN/m
C D
150 kN53.33 kN.m
3 m
We are taking anti-clockwise moments as negative
How fixed end moment are calculated
In beam ABFixed end moment at A = -wl2/12 = - (15)(8)(8)/12 = - 80 kN.mFixed end moment at B = +wl2/12 = +(15)(8)(8)/12 = + 80 kN.m
In beam BCFixed end moment at B = - (Pab2)/l2 = - (150)(3)(3)2/62
= -112.5 kN.mFixed end moment at C = + (Pab2)/l2 = + (150)(3)(3)2/62
= + 112.5 kN.mIn beam ABFixed end moment at C = -wl2/12 = - (10)(8)(8)/12 = - 53.33 kN.mFixed end moment at D = +wl2/12 = +(10)(8)(8)/12 = + 53.33kN.m
You can get FEMs from next slide for various types of loadings
Fixed End Moment Table-
-
-
-
-
-
-
Fixed End Moment Table
Release the fixed joints
Now we allow those joints that were artificially fixed to rotate freely
Due to the joint release, the fixed end moments on either side of joints B, C and D act in the opposite direction now, and cause a net unbalanced moment to occur at the joint
15 kN/m 10 kN/m
8 m 6 m 8 m
A B C DI I I
3 m
150 kN
Released moments -80.0
-112.5 +53.33 -53.33+112.5
unbalanced moment +32.5 -59.17 -53.33
Unbalanced moment
The joint moments are distributed to either side of the joint B, C or D, according to their relative stiffnesses
These distributed moments also modify the moments at the opposite side of the beam span, i.e. at joint A in span AB, at joints B and C in span BC and at joints C and D in span CD.
Modification is dependent on the carry-over factor (which is equal to 0.5 in this case)15 kN/m 10 kN/m
8 m 6 m 8 m
A B C DI I I
3 m
150 kN
Released moments -80.0
-112.5 +53.33 -53.33+112.5
unbalanced moment +32.5 -59.17 -53.33
Essential concepts
Before continuing we need to know about Stiffness Distribution factor Carry-over factor
Stiffness
Stiffness = Resistance offered by member to a unit displacement or rotation at a point, for given support constraint conditions
Note: The above stiffness is obtained assuming that the opposite support is fixed, if it is not the case you may use
LEIK 4
Flexural stiffness of a member
LEI
LEIK 34
43
Distribution factor
For each member at each node is
Note: unbalanced moment is distributed between members based on DF of each member, i.e.
KKDF member
member
Summation of flexural stiffness of all connected members at a particular joint
memberunbalancedmember DFMM
Carry-over factor
If MA is applied to the beam below it causes;
Then, half of this moment goes to end B;
A
MAMB
A BA
RA RB
L
E, I – Member properties
EILM A
A 4
AB MM )21(
Carry over factor
Back to our problem
Calculate distribution factor for all members
00.1
4284.0500.0667.0
500.0
5716.0500.0667.0
667.0
5716.0667.05.0
667.0
4284.0667.05.0
5.0
0.0)(5.0
5.0
DC
DCDC
CDCB
CDCD
CDCB
CBCB
BCBA
BCBC
BCBA
BABA
wallBA
BAAB
K
KDF
EIEIEI
KK
KDF
EIEIEI
KK
KDF
EIEIEI
KK
KDF
EIEIEI
KK
KDF
stiffnesswallEI
KK
KDF
FEM
0.428 0.5710 1
A B C D0.571 0.428
-80 +80 -112.5
+112.5
-53.33
+53.33
Distribution of FEM
-32.5+13.91
+18.59
+59.17-
33.78-25.39 -53.33
Unbalancing moment
0.428 0.5710 1
A B C D0.571 0.428
-80 +80 -112.5
+112.5
-53.33
+53.33
Carry-over factor
DistributionCarry over (M*0.5) -
16.89+9.29
5-26.665 -12.695
-32.5+13.91
+18.59
+59.17-
33.78-25.39 -53.33
Unbalancing moment
0.428 0.5710 1
A B C D0.571 0.428
-80 +80 -112.5
+112.5
-53.33
+53.33
+6.955
Re-calculate unbalancing moment and redistribution
-16.89
-17.37+7.2
2+9.66
1+9.91
8+7.45 +12.6
95
DistributionCarry over (M*0.5) -
16.89+9.29
5-26.665 -12.695
-32.5+13.91
+18.59
+59.17-
33.78-25.39 -53.33
Unbalancing moment
0.428 0.5710 1
A B C D0.571 0.428
-80 +80 -112.5
+112.5
-53.33
+53.33
+6.955
Carry-over factor effect
-16.89
-17.37+7.2
2+9.66
1+9.91
8+7.45 +12.6
95
DistributionCarry over (M*0.5) -
16.89+9.29
5-26.665 -12.695
-32.5+13.91
+18.59
+59.17-
33.78-25.39 -53.33
Unbalancing moment
0.428 0.5710 1
A B C D0.571 0.428
-80 +80 -112.5
+112.5
-53.33
+53.33
+6.955
+3.61 +4.95 +4.8305
+6.347 +3.725
Calculate unbalancing moment and distribute it
-16.89
-17.37+7.2
2+9.66
1+9.91
8+7.45 +12.6
95
DistributionCarry over (M*0.5) -
16.89+9.29
5-26.665 -12.695
-32.5+13.91
+18.59
+59.17-
33.78-25.39 -53.33
Unbalancing moment
0.428 0.5710 1
A B C D0.571 0.428
-80 +80 -112.5
+112.5
-53.33
+53.33
+6.955
+3.61 +4.95 +4.8305
+6.347 +3.725+4.95 +11.1
77-2.1186
-2.8314
-6.382 -4.79 -3.725
Carry-over effect
-16.89
-17.37+7.2
2+9.66
1+9.91
8+7.45 +12.6
95
DistributionCarry over (M*0.5) -
16.89+9.29
5-26.665 -12.695
-32.5+13.91
+18.59
+59.17-
33.78-25.39 -53.33
Unbalancing moment
0.428 0.5710 1
A B C D0.571 0.428
-80 +80 -112.5
+112.5
-53.33
+53.33
+6.955
+3.61 +4.95 +4.8305
+6.347 +3.725+4.95 +11.1
77-2.1186
-2.8314
-6.382 -4.79 -3.725-
1.0593-3.191 -
1.4157-
1.8625-2.395
Re-calculate unbalancing moment
-16.89
-17.37+7.2
2+9.66
1+9.91
8+7.45 +12.6
95
DistributionCarry over (M*0.5) -
16.89+9.29
5-26.665 -12.695
-32.5+13.91
+18.59
+59.17-
33.78-25.39 -53.33
Unbalancing moment
0.428 0.5710 1
A B C D0.571 0.428
-80 +80 -112.5
+112.5
-53.33
+53.33
+6.955
+3.61 +4.95 +4.8305
+6.347 +3.725+4.95 +11.1
77-2.1186
-2.8314
-6.382 -4.79 -3.725-
1.0593-3.191 -
1.4157-
1.8625-2.395-3.191 -3.278 +2.39
5
Distribute the unbalancing moment
-16.89
-17.37+7.2
2+9.66
1+9.91
8+7.45 +12.6
95
DistributionCarry over (M*0.5) -
16.89+9.29
5-26.665 -12.695
-32.5+13.91
+18.59
+59.17-
33.78-25.39 -53.33
Unbalancing moment
0.428 0.5710 1
A B C D0.571 0.428
-80 +80 -112.5
+112.5
-53.33
+53.33
+6.955
+3.61 +4.95 +4.8305
+6.347 +3.725+4.95 +11.1
77-2.1186
-2.8314
-6.382 -4.79 -3.725-
1.0593-3.191 -
1.4157-
1.8625-2.395-3.191 -3.278 +2.39
5+1.365
+1.825
+1.871
+1.406
-2.395
Carry-over effect
-16.89
-17.37+7.2
2+9.66
1+9.91
8+7.45 +12.6
95
DistributionCarry over (M*0.5) -
16.89+9.29
5-26.665 -12.695
-32.5+13.91
+18.59
+59.17-
33.78-25.39 -53.33
Unbalancing moment
0.428 0.5710 1
A B C D0.571 0.428
-80 +80 -112.5
+112.5
-53.33
+53.33
+6.955
+3.61 +4.95 +4.8305
+6.347 +3.725+4.95 +11.1
77-2.1186
-2.8314
-6.382 -4.79 -3.725-
1.0593-3.191 -
1.4157-
1.8625-2.395-3.191 -3.278 +2.39
5+1.365
+1.825
+1.871
+1.406
-2.395+0.68
2+0.93
5+0.91
2-1.197 0.703
Unbalancing moment
-16.89
-17.37+7.2
2+9.66
1+9.91
8+7.45 +12.6
95
DistributionCarry over (M*0.5) -
16.89+9.29
5-26.665 -12.695
-32.5+13.91
+18.59
+59.17-
33.78-25.39 -53.33
Unbalancing moment
0.428 0.5710 1
A B C D0.571 0.428
-80 +80 -112.5
+112.5
-53.33
+53.33
+6.955
+3.61 +4.95 +4.8305
+6.347 +3.725+4.95 +11.1
77-2.1186
-2.8314
-6.382 -4.79 -3.725-
1.0593-3.191 -
1.4157-
1.8625-2.395-3.191 -3.278 +2.39
5+1.365
+1.825
+1.871
+1.406
-2.395+0.68
2+0.93
5+0.91
2-1.197 0.703
+0.935
-0.285
Distribute the unbalancing moment
-16.89
-17.37+7.2
2+9.66
1+9.91
8+7.45 +12.6
95
DistributionCarry over (M*0.5) -
16.89+9.29
5-26.665 -12.695
-32.5+13.91
+18.59
+59.17-
33.78-25.39 -53.33
Unbalancing moment
0.428 0.5710 1
A B C D0.571 0.428
-80 +80 -112.5
+112.5
-53.33
+53.33
+6.955
+3.61 +4.95 +4.8305
+6.347 +3.725+4.95 +11.1
77-2.1186
-2.8314
-6.382 -4.79 -3.725-
1.0593-3.191 -
1.4157-
1.8625-2.395-3.191 -3.278 +2.39
5+1.365
+1.825
+1.871
+1.406
-2.395+0.68
2+0.93
5+0.91
2-1.197 0.703
+0.935
-0.285-0.4 -0.535 +0.16 +0.12
5-0.703
Sum up the moments
-16.89
-17.37+7.2
2+9.66
1+9.91
8+7.45 +12.6
95
DistributionCarry over (M*0.5) -
16.89+9.29
5-26.665 -12.695
-32.5+13.91
+18.59
+59.17-
33.78-25.39 -53.33
Unbalancing moment
0.428 0.5710 1
A B C D0.571 0.428
-80 +80 -112.5
+112.5
-53.33
+53.33
+6.955
+3.61 +4.95 +4.8305
+6.347 +3.725+4.95 +11.1
77-2.1186
-2.8314
-6.382 -4.79 -3.725-
1.0593-3.191 -
1.4157-
1.8625-2.395-3.191 -3.278 +2.39
5+1.365
+1.825
+1.871
+1.406
-2.395+0.68
2+0.93
5+0.91
2-1.197 0.703
+0.935
-0.285-0.4 -0.535 +0.16 +0.12
5-0.703
81.69MM A
81.10197.99
BR
BL
MM
90.9790.97
BR
BL
MM 0DM
Example 2
Draw the moment diagram for the beam below
Answer 2
Example 3
The beam below is simply supported at A, cantilevered at D. IAB=8500cm4, IBC=6500cm4 and ICE=5500cm4
Solution 3
Distribution factors
FEM calculation
Table
Bending moment diagram
Example 4
Example 5