Moment Distribution Method - I.K. International Publishing ... The moment distribution method can be...
Transcript of Moment Distribution Method - I.K. International Publishing ... The moment distribution method can be...
Objectives:
Defi nition of stiffness, carry over factor, distribution factor. Analysis of continuous beams withoutsupport yielding – Analysis of continuous beams with support yielding – Analysis of portal frames – Naylor’s method of cantilever moment distribution – Analysis of inclined frames – Analysis of Gable frames.
2.1 INTRODUCTIONThe end moments of a redundant framed structure are determined by using the classical methods, viz. Clapeyron’s theorem of three moments, strain energy method and slope defl ection method. These methods of analysis require a solution of set of simultaneous equations. Solving equations is a laborious task if the unknown quantities are more than three in number. In such situations, the moment distribution method developed by Professor Hardy Cross is useful. This method is essentially balancing the moments at a joint or junction. It can be described as a method which gives solution by successive approximations of slope defl ection equations. In the moment distribution method, initially the structure is rigidly fixed at every joint or support. The fi xed end moments are calculated for any loading under consideration. Subsequently, one joint at a time is then released. When the moment is released at the joint, the joint moment becomes unbalanced. The equilibrium at this joint is maintained by distributing the unbalanced moment. This joint is temporarily fi xed again until all other joints have been released and restrained in the new position. This procedure of fi xing the moment and releasing them is repeated several times until the desired accuracy is obtained. The experience of designers points that about fi ve cycles of moment distribution lead to satisfactory converging results. Basically, in the slope defl ection method, the end moments are computed using the slopes and defl ection at the ends. Contrarily in the moment distribution method, as a fi rst step — the slopes at the ends are made zero. This is done by fi xing the joints. Then with successive release and balancing the joint moments, the state of equilibrium is obtained. The release-balance cycles can be carried out using the following theorems.
Moment Distribution Method2
Moment Distribution Method
130 Indeterminate Structural Analysis
In conclusion, when a positive moment M is applied to the hinged end of a beam
a positive moment of 12
Ê ˆÁ ˜Ë ¯
M will be transferred to the fi xed end.
2.2.2 Th eorem 2Consider a two span continuous beam ACB as shown in Fig. 2.2(a). A and B are fi xed supports with a prop at C. A moment is applied at C and it is required to know how much moment is distributed between spans AC and CB. Let this moment M be decomposed and distributed as M1 to CA and M2 to CB as shown in Fig. 2.2(b).
i.e. M1 + M2 = M (1)
l1
I1I2
l2
MCA B
FIG. 2.2(a) Continuous beam ACB with moment M
CA B
M1
M1
M2
2M2
2( )FIG. 2.2(b) Distribution of bending moments
The bending moment diagram is drawn by considering each span AC and CB respectively.
CA B
M1
M1
M2
21
++
−−
( )
M221( )
( )( )( )l13
2l13
2l23
l23
FIG. 2.2(c) Bending moment diagram
As the ends A and B are fi xed; the slope between A and B is zero. That is, the area of the bending moment diagram between A and B is zero.
133Moment Distribution Method
C
A B
A1M1
A2
M2
M1
x3
A3
x2
l13
2l13
x1
( )( )
( )
l2
2
FIG. 2.4(c)
The tangent drawn at A passes through C and B. Hence, from the above fi gure;
Ai xi /EIi = 0
i.e. A1 = (1/2) (l1/3) (M1/2) = 1 1
12M l
x1 = l2 + 11
2 2 ;3 3 3
llÊ ˆ+Á ˜Ë ¯ x1 = 2 189
l l+
A2 = (– 1/2) 1 11 1
23 3
M ll MÊ ˆ = -Á ˜Ë ¯
x2 = l2 + 11 23 3 3
lÊ ˆÊ ˆ Ê ˆÁ ˜ Á ˜ Á ˜Ë ¯ Ë ¯ Ë ¯
; x1 = 2 129
l l+
A3 = (1/2) (l2 )M2 = 2 2
2M l
x3 = 223
l
Thus, Aixi/EI gives
2
1 1 2 2 2
1 2
30
12 3M l l M lEI EI
-+ =
2
1 1 22 2
2 1
33 12
M l lM lEI EI
-=
2 2 1 1
2 1
34
l M M lI I
=
134 Indeterminate Structural Analysis
2 1
2 1
34
M Mk k
=
i.e., 22
2 1
(3/4 )kMk k
=
If one end of a member is not fi xed then the “stiffness” of that member should be multiplied by (3/4).
2.2.4 Th eorem 4Consider a fi xed beam AB as shown below. End B has settled by a distance . As the ends are fi xed, there must develop a fi xing moment M at the each end of the beam.
FIG. 2.5(a) Sinking of supports in a fi xed beam
A BA1
A2
l2l2
x2
x1
FIG. 2.5(b) Bending moment diagram
Taking moments about B and using the moment area theorem;
Ai xi /EIi = –
i.e. A1 = – 12 2 4
Mll M-Ê ˆ Ê ˆ =Á ˜ Á ˜Ë ¯ Ë ¯
A2 = 12 2 4
l MlMÊ ˆ Ê ˆ =Á ˜ Á ˜Ë ¯ Ë ¯
x1 = 2 5
2 3 2 6l l lÊ ˆ+ =Á ˜Ë ¯
x2 = 13 2 6
l lÊ ˆ =Á ˜Ë ¯
135Moment Distribution Method
1 54 6 4 6
Ml l Ml lEI
È ˘Ê ˆ Ê ˆ Ê ˆ Ê ˆ- + = -dÁ ˜ Á ˜ Á ˜ Á ˜Í ˙Ë ¯ Ë ¯ Ë ¯ Ë ¯Î ˚
2 25 124 24Ml Ml
EIÊ ˆ-
+ = -dÁ ˜Ë ¯
2( 5 1)
24 24Ml- +
= - d
M = 26El
ld
i.e., when a fi xed ended beam settles by an amount δ at one end, the moment required to make the ends horizontal = 6EI/l2. The above four theorems can be summarised as (1) When the member is fi xed at one end and a moment is applied at the other end
which is simply supported or hinged, the moment induced at the fi xed end is one half of the applied moment. The induced moment at the fi xed end is in the same direction as the applied moment.
(2) If a moment is applied in a stiff joint of a structure, the moment is resisted by various members in proportion to their respective stiffnesses (i.e., moment of inertia divided by the length). If the stiffness of the member is more; then it resists more bending moment and it absorbs a greater proportion of the applied moment.
(3) While distributing the moments in a rigid joint, if one end of the member is not restrained then its stiffness should be multiplied by (3/4).
(4) In a fi xed beam, if the support settles/subsides/sinks by an amount , the moment required to make the ends horizontal is 6EI/l2.
2.3 BASIC DEFINITIONS OF TERMS IN THE MOMENT DISTRIBUTION METHOD(a) StiffnessRotational stiffness can be defi ned as the moment required to rotate through a unit angle (radian) without translation of either end.
(b) Stiffness Factor (i) It is the moment that must be applied at one end of a constant section member
(which is unyielding supports at both ends) to produce a unit rotation of that end when the other end is fi xed, i.e. k = 4EI/l.
(ii) It is the moment required to rotate the near end of a prismatic member through a unit angle without translation, the far end being hinged is k = 3EI/l.
136 Indeterminate Structural Analysis
(c) Carry Over FactorIt is the ratio of induced moment to the applied moment (Theorem 1). The carry over factor is always (1/2) for members of constant moment of inertia (prismatic section). If the end is hinged/pin connected, the carry over factor is zero. It should be mentioned here that carry over factors values differ for non-prismatic members. For non-prismatic beams (beams with variable moment of inertia); the carry over factor is not half and is different for both ends.
(d) Distribution FactorsConsider a frame with members OA, OB, OC and OD rigidly connected at O as shown in Fig. 2.6. Let M be the applied moment at joint O in the clockwise direction. Let the joint rotate through an angle . The members OA,OB,OC and OD also rotate by the same angle θ.
OD
C
A
BM
FIG. 2.6
Let kOA, kOB, kOC and kOD be the stiffness values of the members OA, OB, OC and OD respectively; then
MOA = kOA (i)
MOB = kOB (ii)
MOC = kOC (iii)
MOD = kOD (iv)
A B
MBAMAB = 2EI
L= 4EIL
θA= 1
(a) Beam with far end fi xed
138 Indeterminate Structural Analysis
dOB = OBkkS = distribution factor for OB
dOC = OCkkS = distribution factor for OC
dOD = ODkkS
= distribution factor for OD
2.4 SIGN CONVENTIONClockwise moments are considered positive and anticlockwise moments negative.
2.5 BASIC STAGES IN THE MOMENT DISTRIBUTION METHODThe moment distribution method can be illustrated with the following example. It is desired to draw the bending moment diagram by computing the bending moments at salient points of the given beam as shown below.
CC'A
A B'I =
100 kN2.5 m
5 m6 m
50 kN/m
2I 3I
FIG. 2.8 A two span continuous beam
Step 1Determine the distribution factor at each joint A,B and C respectively. The distribution factor of a member is the ratio of the stiffness of the member divided by the total stiffness of all the members meeting at that joint. The distribution factor for the fi xed support A is determined by assuming an imaginary span AA (Fig. 2.8). The fl exural rigidity of AA is infi nity. Hence, the stiffness is infi nite. The stiffness of AB is 2I/5 = 0.4I. Hence, the total stiffness is infi nite. Thus, the distribution factor for AA is (infi nity/infi nity) = 1.0 and for AB the distribution factor kAB = (0.4I/infi nity), i.e. zero. In essence dAA = 1.0 and dAB = 0.0 The distribution factor at the support B is determined as follows. The stiffness of the member BA is (0, 4I). The stiffness of the member BC is taken as three-fourths
of its stiffness (refer Theorem 3). Hence, kBC = 3 34 6
IÊ ˆ Ê ˆÁ ˜ Á ˜Ë ¯ Ë ¯ = 0.375I. The sum of the
stiffness at the joint B is k = (kBA + kBC) = 0.775I. Therefore, the distribution factors are dBA = kBA/(kBA + kBC), i.e. dBA = 0.4I/(0.4I + 0.375I) = 0.52; similarly dBC = kBC/k, i.e. dBC
= (0.375I/0.775I) = 0.48. The sum of the stiffnesses is (0.52 + 0.48) = 1.0. The distribution factor for the simple support at C is determined by extending the span to CC. The rigidity of CC is zero and hence kCC = 0. On the other hand, the
139Moment Distribution Method
stiffness kCB = 0.375I. The total stiffness at C is kCB + kCC = 0.375I. The distribution factor
dCB = kCB/k = 1.0 and dCC = kCC/k = 0.
The above procedure is summarised in the following table for quick understanding.
Table 2.1 Distribution factor at joint B
Joint Members k = I/l k DF
BA2 0.45l I= 52
B 0.775I
BC 3 34 6
IÊ ˆÁ ˜Ë ¯ = 0.3/5I 0.48
Step 2Imagine all the three joints A, B and C are rigidly fi xed with horizontal tangents. Write down the fi xed end moments for the beam AB as if it were built in at A and B and also for the beam BC as if it were built in B and C.
CA Bl/m
100 kN2.5 m
5 m6 m
50 kN/m
62.562.5 150
FIG. 2.9 Fixed end moments
MFAB = 100 5
62.5 kNm8¥
- = -
MFBA = + 100 5
62.5 kNm8¥
= +
MFBC = 250 6
150.0 kNm12¥
- = -
MFCB = + 250 6
150.0 kNm12¥
= +
Step 3Each joint is released in turn and if B is released it will be out of balance. This unbalanced moment (– 150.0 + 62.5 = 87.5) is shown in Fig. 2.10.
140 Indeterminate Structural Analysis
A B C
62.5 kNm 87.5 kNm
150 kNm
FIG. 2.10 Out of balance moments
Step 4A moment is applied at B to balance this joint B and it will distribute itself according to the distribution factors. This is shown in the following fi gure.
A
1 0 1 0
B C
62.5 87.50.480.52
150 kNm
FIG. 2.11 Balancing moments
A B C45.5062.5 0 42.00 –150.00 0 kNm
FIG. 2.12 Distributed moments
Step 5Balance the joint C as its moment is zero (end support C is simple support). By balancing the moment – 150, half of it is carried over to B. By balancing the joint B, half of the moment is carried over to joint A, i.e. half of 45.50 kNm.
Step 6Again the joints become out of balance, and the above procedure is repeated until the moments to be distributed become negligible and can be ignored. This is illustrated in Table 2.2.
Table 2.2 Moment distribution table
Joint A B CMembers AB BA BC CBDF 0 0.52 0.48 1FEMSBal
– 62.50 +62.50+45.50
– 150.00+42.00
+150.00– 150.00
CoBal
+22.75+39.00
– 75.00+36.00
Co +19.50Total – 20.25 +147.00 – 147.00 0.00Nature O
141Moment Distribution Method
In the above moment distribution table a single vertical line is drawn between the members. Double lines are drawn at the end of each joint. The pure moment diagram can be drawn using the end moments in the moment distribution table. The pure moments are the values just to the left of double line. Thus, MA = – 20.25 kNm, MB = – 147.00 kNm, MC = 0.
CA B
−ve
147.00 kNm
20.25
FIG. 2.13 Pure moment diagram
The simple beam moment diagram is drawn by considering each span separately. The simple beam moment diagram is always positive. While the pure moment diagram is negative. The maximum positive bending moment for span AB is (wl/4) = 100 × 5/4 = 125 kNm. The maximum bending moment for a simple beam of BC is wl2/8 = (50 × 62/8) = 225 kNm.
CA B
− 225 kNm
125
+ +
D E
FIG. 2.14 Simple beam moment diagram
The net bending moment diagram is drawn by superimposing the pure moment diagram on the simple beam moment diagram. Thus, the net moment at D and E are
MD = 125 – 147.00 20.25
2+Ê ˆ
Á ˜Ë ¯ = +41.38 kNm
ME = 225 – 147.00 0.00
2+Ê ˆ
Á ˜Ë ¯ = +151.5 kNm
The fi nal net bending moment diagram is as follows.
142 Indeterminate Structural Analysis
CA B−−
151.5 kNm
41.4
++
D E
20.25
147
FIG. 2.15 Bending moment diagram
The net bending moment diagrams are preferable in design offi ces. The elastic curve is drawn using the net bending moment diagram.
A B C
FIG. 2.16 Elastic curve
2.6 NUMERICAL EXAMPLES2.6.1 Analysis of Continuous Beams without Support SettlementExample 2.1 Analyse the continuous beam shown in Fig. 2.17 by the moment distribution method. Draw the bending moment diagram and shear force diagram. The beam is of uniform section.
A B C
6 m
10 kN/m30 kN/m
4 m
FIG. 2.17
SolutionStep 1The distribution factors at joint B are evaluated as follows.
143Moment Distribution Method
Table 2.3 Distribution factors
Joint Members Relative Stiffness (k) Sum k Distribution Factors (k/ k)BA I/6 = 0.167I 0.47
B 0.3555I
BC 34 4
I¥ = 0.188I 0.53
Step 2 Fixed End Moments
MFAB = 210 6
12- ¥
= – 30 kNm, MFBA = +30 kNm
MFBC = 230 4
12- ¥
= – 40 kNm, MFCB = +40 kNm
Step 3 Moment Distribution TableAs the joint C is a hinged end, the moment is zero. Hence, it is balanced fi rst. Then half of this moment is carried over. Then joint B is balanced. From the joint B, the moment is carried over to A.
Table 2.4 Moment distribution table
Joint A B CMembers AB BA BC CBDF 0 0.47 0.53 1FEMSBal
– 30.00 –
+30.00–
– 40.00 –
+40.00– 40.00
Co – – – 20.00TotalBal
– 30.00–
+30.00+14.10
– 60.00+15.90
0.00–
Co +7.05 – – –Final – 22.95 +44.10 – 44.10 0.00
A B C
22.95 kNm44.1 kNm
D E−−
++
11.48 kNm
37.95 kNm
FIG. 2.18 Bending moment diagram
144 Indeterminate Structural Analysis
MD = 210 6 22.95 44.1
11.48 kNm8 2¥ +Ê ˆ- =Á ˜Ë ¯
ME = 230 4 44.1 0
37.95 kNm8 2¥ +Ê ˆ- =Á ˜Ë ¯
Step 4 Shear Force DiagramsEquilibrium of span AB
A B6 m
10 kN/m22.95 kNm 44.1 kNm
FIG. 2.19
V = 0; VAB + VBA = 60 (1)
MA = 0; – 22.95 + 44.1 + 10 × 26
2 – 6VBA = 0 (2)
VBA = 33.5 kN
VAB = 26.5 kN Equilibrium of span BC
B C4 m
30 kN/m44.1 kNm
FIG. 2.20
V = 0; VBC + VCB = 120 (3)
MB = 0; – 44.1 + 30 × 24
2 – 4VCB = 0
VCB = 49 kN
VBC = 71 kN
26.5 kN71 kN
33.5 kN49 kN
x1
x2
x16x24
––
FIG. 2.21 Shear force diagram
145Moment Distribution Method
From similar triangles
1
1(6 )x
x- =
26.533.5
33.5x1 = 159 – 26.5x1
x1 = 2.65 m
2
2
71(4 ) 49
xx
=-
49x2 = 284 – 71x2
x2 = 2.37 m
Example 2.2 Analyse the continuous beam by moment distribution method. Draw the shear force diagram and bending moment diagram.
A B
EI = Constant
E
F
D
C
3 m 3 380 kN 80 kN 12 kN/m
12 m
FIG. 2.22
SolutionDistribution factors The distribution factors at joint B are obtained as follows.
Table 2.5 Distribution factors
Joint Members Relative Stiffness (k) Sum k Distribution Factors (k/ k)BA I/9 = 0.111I 0.64
B 0.174IBC 3
4 12I
¥ = 0.063I0.36
Fixed end moments
MFAB = – ¥ ¥
= - = -80 3 6
160 kNm9
Wabl
146 Indeterminate Structural Analysis
MFBA = +80 × 3 × 69
= +160 kNm
MFBC = – 12 × 212
12 = – 144 kNm
MFCB = +144 kNm
Table 2.6 Moment distribution table
Joint A B C
Members AB BA BC CB
DF 0 0.64 0.36 1
FEMSBal
– 160.00 +160.00 – 144.00 +144.00– 144.00
Co – 72.00
TotalBal
– 160.00 +160.00+35.80
– 216.00+20.20
0.00
Co +17.90
Total – 142.10 +195.80 – 195.80 0.00
Shear forces Equilibrium of span AB
A
ED
B
3 m 3 m 3 m80 kN 80 kN
195.8 kNm142.1 kNm
FIG. 2.23
V = 0; VAB + VBA = 160 (1)
MA = 0; – 142.1 + 195.8 + 80 × 3 + 80 × 6 – 9VBA = 0 (2)
VBA = 86 kN
VAB = 74 kN
MD = 74(3) – 142.1 = 80 kNm
ME = 86(3) – 195.8 = 62.2 kNm
147Moment Distribution Method
Equilibrium of span BC
12 kN/m195.8 kNm
12 mB CF
FIG. 2.24
V = 0; VBC + VCB = 144 (3)
MB = 0; – 195.8 + 12 × 212
12 – 12VCB = 0 (4)
VCB = 55.7 kN
VBC = 88.3 kN
MF = 55.7 × 6 – 12 × 26
2 = 118.2 kNm
A B C
142.1 kNm
195.8 kNm
D F
−
8062.2
118.2 kNm
E
FIG. 2.25 Bending moment diagram
74
88.3 kN
(12 − x)6 kN
8655.7 kN
x
FIG. 2.26 Shear force diagram
148 Indeterminate Structural Analysis
From similar ∆’s
55.7(12 ) 88.3
xx
=-
88.3x = 668.4 – 55.7x
x = 4.64 m
Example 2.3 Analyse the continuous beam by the moment distribution method. Draw the shear force diagram and bending moment diagram.
A B ED C
120 kN30 kN/m4 m4 m
8 m
FIG. 2.27
SolutionDistribution factors The distribution factors at joint B are evaluated as follows.
Table 2.7 Distribution factors
Joint Members Relative Stiffness (k) Sum k Distribution Factors (k/ k)
BA I/8 = 0.125I 0.5
B 0.25I
BC I/8 = 0.125I 0.5
Fixed end moments
MAB = – 30 × 28
12 = – 160 kNm; MBC = –
2120 88¥
= – 120 kNm
MBA = +160 kNm; MCB = + 120 kNm
149Moment Distribution Method
Moment Distribution
Table 2.8 Moment distribution table
Joint A B C
Members AB BA BC CB
DF 0 0.5 0.5 0
FEMSBal
– 160–
+160– 20
– 120– 20
+120–
Co – 10 – – – 10
Final – 170 +140 – 140 +110
Using the above end moments;
MD = 230 8 170 140
85 kNm8 2¥ +Ê ˆ- =Á ˜Ë ¯
ME = +Ê ˆ¥ - =Á ˜Ë ¯
140 1108120 115 kNm4 2
A B C
170 kNm140 kNm
110 kNm
D
− −−+
E+
85 115
FIG. 2.28 Bending moment diagram
Shear force diagrams Equilibrium of span AB
A B
30 kN/m
8 m
170 140
FIG. 2.29
V = 0; VAB + VBA = 240 (1)
150 Indeterminate Structural Analysis
MA = 0; – 170 + 140 + 30 × 28 8 0
2 BAV- =
(2)
VBA = 116.3 kN VAB = 123.7 kN
Equilibrium of span BC
B
120 kN44
8 m
140 kNm 110 kNm
C
FIG. 2.30
V = 0; VBC + VCB = 120 (3)
MB = 0; – 140 + 110 + 120(4) – 8VCB = 0
VCB = 56.3 kN
VBC = 63.7 kN
A B
63.7 kN
116.356.3 kN
123.7 kN
− −+ +
x1
FIG. 2.31 Shear force diagram
From similar ∆’s
1
1
123.7(8 ) 116.3
xx
=-
116.3x1 = 989.6 – 123.7x1
x1 = 4.12 m
Example 2.4 Determine the end moments for the continuous beam shown in Fig. 2.32. EI is constant. Use the moment distribution method.
151Moment Distribution Method
A B C D
22 m 250 kN 50 kN25 kN/m
4 m
75 kN/m
6 m
FIG. 2.32
SolutionDistribution factors The distribution factors at joints B and C are obtained as follows.
Table 2.9 Distribution factors
Joint Members Relative Stiffness Values (k) k Distribution Factors k/ k
BA I/6 0.50
B 2I/6
BC I/6 0.50
CB I/6 = 0.167 0.40
C 0.417
CD I/4 = 0.25 0.60
Fixed end moments
MFAB = – 25 × 26 75 kNm
12= -
MFBA = +25 × 26 75 kNm
12= +
MFBC = – 50 2 4
66.67 kNm6
¥ ¥= -
MFCB = +50 × 2 × 4 66.67 kNm6
= +
MFCD = – 75 × 42/30 = – 40.00 kNm
MFDC = +75 × 42/20 = +60.00 kNm
152 Indeterminate Structural Analysis
Table 2.10 Moment distribution table
Joint A B C D
Members AB BA BC CB CD DC
DF 0 0.5 0.5 0.4 0.6 0
FEMSBal
– 75.00 +75.00– 4.17
– 66.67– 4.16
+66.67– 10.67
– 40.00+16.00
+60.00–
Co
Bal
– 2.10 –
+2.67
– 5.34
+2.67
– 2.08
+0.83
–
+1.25
– 8.00
Co
Bal
+1.34
– 0.21
+0.42
– 0.21
+1.34
– 0.54 – 0.80
+0.63
Co
Bal
– 0.11
+0.13
– 0.27
+0.14
– 0.11
+0.04 +0.07
– 0.40
Co
Bal
+.07
– 0.01
+0.02
– 0.01
+0.07
– 0.03 – 0.04
+0.04
Final – 75.8 +73.41 – 73.41 +55.52 – 55.52 – 52.27
The end moments are
MAB = – 75.8 kNm, MB = – 73.41 kNm
Mc = – 55.52 kNm, Mc = – 52.27 kNm
Example 2.5 Analyse and sketch the bending moment diagram for the beam shown in Fig. 2.33. The values of the second moment area of each span are indicated along the members. Modulus of elasticity is constant.
A B C D
1.2540 kN30 kN/m
5 m6 m5 m
100 kN1.25 2.5 m2.5 m 2.5 m
80 kN
2I 3I 4I
FIG. 2.33
153Moment Distribution Method
Distribution factors
Table 2.11 Distribution factors
Joint Members Relative Stiffness Values (k) k Distribution Factors k/ k
BA 2I/5 = 0.4I 0.44
B 0.9I
BC 3I/6 = 0.5I 0.56
CB 3I/6 = 0.5I 0.45
C 1.1I
CD 3 4 0.64 5
I I¥ =0.55
Fixed end moments
MFAB = – 100 × 5 62.5 kNm8
= -
MFBA = +100 × 5 62.5 kNm8
= +
MFBC = – 30 × 26 90.0 kNm
12= -
MFCB = +30 × 26 90.0 kNm
12= +
MFCD = 2 2
2 2
1.25 80 3.75 3.75 40 1.2565.625 kNm
5 5¥ ¥ ¥ ¥
- - = -
MFDC = 2 2
2 2
1.25 40 3.75 3.75 80 1.2546.875 kNm
5 5¥ ¥ ¥ ¥
+ + = +
78.0
135.0 kNm
84.587.5
62.5 kNm
125
54.7
FIG. 2.34 Bending moment diagram
154 Indeterminate Structural Analysis
Table 2.12 Moment distribution table
Joint A B C D
Members AB BA BC CB CD DC
DF 0 0.44 0.56 0.45 0.55 1
FEMSBal
– 62.5 +62.50+12.10
– 90.00+15.40
+90.00– 10.97
– 65.63– 16.4
+46.88– 46.88
CoBal
+6.05+2.42
– 5.49+3.07
+7.70– 3.47 – 4.23
CoBal
+1.21+0.77
– 1.74+0.97
+1.54– 0.69 – 0.85
CoBal
+0.39+0.15
– 0.35+0.20
+0.49– 0.22 – 0.27
CoBal
+0.080.05
– 0.11+0.06
+0.10– 0.05 −0.05
CoBal
+0.03+0.01
– 0.03+0.02
+0.03– 0.01 – 0.02
Final – 54.74 +78.0 – 78.00 +84.45 – 84.45 0.00
2.6.2 Analysis of Continuous Beam with Support SettlementThe settlement of supports causes the bending moment in the members. The settlement of support is mainly due to soil subsidence. In a fi xed beam of AB, if the support B is lower by , the induced end moment is – 6EI/l2 where is the settlement of supports. If the support B is higher by , the end moment is +6EI/l2. (Refer Basic Structural Analysis – Chapter 11) The moments are computed using the given loading and due to settlement of supports separately and summed up.
Example 2.6 Analyse the beam shown in Fig. 2.35 by the moment distribution method. Support B sinks by 10 mm. E = 200 kN/mm2. I = 4000 × 104 mm4. Draw BMD and SFD.
A B CE D6 m
2470 kN 10 kN15 kN/m
4 m 1 m
FIG. 2.35
155Moment Distribution Method
SolutionDistribution factors
Table 2.13 Distribution factors
Joint Members Relative Stiffness Values (k) k Distribution Factors k/ kBA I/6 = 0.167I 0.47
B 0.355I
BC 34 4
IÊ ˆÁ ˜Ë ¯ = 0.188I 0.53
Fixed moments Due to applied loading
MAB = – 4(70) 2
226
= – 31.11 kNm
MBA = +2(70) 2
246
= +62.22 kNm
MBC = – 15 × 42/12 = – 20.00 kNm
MCB = +15 × 42/12 = +20.00 kNm
MCD = – 10(1) = – 10.00 kNm.
Due to sinking of supports
MAB = MBA = 2
2
6 6 8000 1013.33 kNm
1000 6B
AB
EIl
- D - ¥ ¥= = -
¥
MBC = MCB = 2
2
6 8000 106 30 kNm1000 4
B
CB
EIl
¥ ¥D+ = = +
¥
The actual fi xed end moment, is the addition of moment due to applied loading and due to the settlement of supports.
MFAB = – 31.11 – 13.33 = – 44.44 kNm
MFBA = +62.22 – 13.33 = +48.89 kNm
MFBC = – 20.00 + 30.00 = +10.00 kNm
MFCB = +20.00 + 30.00 = +50.00 kNm
MFCD = – 10.00 kNm
156 Indeterminate Structural Analysis
Table 2.14 Moment distribution table
Joint A B CMembers AB BA BC CB CDDF 0 0.47 0.53 1 –
FEMSBal
– 44.44 +48.89– 27.61
+10.00– 31.22
+50.00– 40.00
– 10.00
CoBal
– 13.84+9.40
– 20.00+10.60
Co +4.70
Final – 53.58 +30.62 – 30.62 +10.00 – 10.00
−53.58
93.33 kNm
−30.6230 kNm
−10
A E B C D
FIG. 2.36 Bending moment diagram
Shear forces Equilibrium of span AB
A B
53.58 kNm 30.62 kNm70 kN4 2
FIG. 2.37
V = 0; VAB + VBA = 70 (1)
M = 0; – 53.58 + 30.62 + 70(4) – 6VBA = 0 (2)
VBA = 42.84 kN
VAB = 27.16 kN
157Moment Distribution Method
B
30.62 kNm
C
10 kNm
4 m
15 kN/m
FIG. 2.38
V = 0; VBC + VCB = 60 (3)
MB = 0; – 30.62 + 10 + 15 × 2
C4 4 02 BV- =
VCB = 24.85 kN
VBC = 35.15 kN
35.15
42.84 kN
10 kN
24.85 kN
27.16
A B (4 − x)
x
DC
FIG. 2.39 Shear force diagram
The location of zero shear force is located from similar triangles as
24.854 35.15
xx
=-
35.15x = 99.4 – 24.85x
x = 1.65 m
Example 2.7 Analyse the continuous beam by the moment distribution. The supports B and C settle by 8 mm and 4 mm respectively. EI = 30000 kNm2. Sketch the SFD and BMD.
BDAE6 m4 m
2 m 2 m240 kN 40 kN/m
C
FIG. 2.40
158 Indeterminate Structural Analysis
SolutionDistribution factors
Table 2.15 Distribution factors
Joint Members Relative Stiff Values (k) k k/ kBA I/4 = 0.25I 0.67
B 0.375I
BC3 0.1254 6
I I¥ = 0.33
Fixed end moments MFAB = MAB + MAB where MAB is the fi xed end moment due to loading and MAB is the fi xed end moment due to settlement of supports. Due to applied loading
MAB = – 240 × 4 120 kNm8
= -
MBA = + 240 × 4 120 kNm8
= +
MBC = – 40 × 26 120 kNm
12= -
MCB = + 40 × 26 120 kNm
12= +
Due to settlement of supports
A
B'
C
CB
' Δ2 = 4 mm
θB = −lBC
Δ2θA = lAB
Δ1
Δ1 = 8 mm
4 m 6 m
+
FIG. 2.41 Settlement diagram
MAB = MBA = 12 21
6 8 16 3000 90 kNm1000 4
EIl
- D= - ¥ ¥ ¥ = -
MBC = MCB = 22 22
6 ( ) 4 16 3000 20 kNm1000 6
EIl
- - D= + ¥ ¥ ¥ = +
159Moment Distribution Method
Hence, MFAB = – 120 – 90 = 210 kNm
MFBA = +120 – 90 = +30 kNm
MFBC = – 120 + 20 = – 100 kNm
MFCB = +120 + 20 = + 140 kNm
Moment distribution table
Table 2.16 Moment distribution table
Joint A B C
Members AB BA BC CB
DF 0 0.67 0.33 1
FEMSBal
– 210.00 +30.00+46.90
– 100.00+23.10
+140.00– 140.00
CoBal
+23.45 –+46.90
– 70.00+23.10
–
Co +23.45
Total – 163.10 – 123.8 – 123.80 0.00
Equilibrium of span AB
2 m
4 m
240 kN
163.10
2 m123.8 kNm
VBAVAB D
FIG. 2.42
V = 0; VAB + VBA = 240 (1)
MA = 0; – 163.10 + 123.8 + 240(2) – 4VBA = 0 (2)
VBA = 110.2 kN
VAB = 129.8 kN
MD = – 163.10 + 129.8(2) = 96.5 kNm
160 Indeterminate Structural Analysis
Equilibrium of span BC
6 m
123.8 kNm
VCBVBC E
40 kN/m
FIG. 2.43
V = 0; VBC + VCB = 240 (3)
MB = 0; – 123.8 – 6VCB + 40 × 26 0
2=
VCB = 99.4 kN
VBC = 140.6 kN
ME = – 123.8 + 140.6 × 3 – 40 × 23 118 kNm
2=
96.5
163.10 123.80
118 kNm
E CDBA
FIG. 2.44 Bending moment diagram
The reader can locate the point of contrafl exure and the maximum positive bending moment using basics.
129.8 kN 140.6
110.2 kN 99.4 kN
A D (6 x1)x1
FIG. 2.45
161Moment Distribution Method
The location of zero shear force can be obtained from similar triangles,
1
1
99.4(6 ) 140.6
xx
=-
and hence x1 = 2.5 m
Example 2.8 Analyse the continuous beam by the moment distribution method. Sketch the shear force and bending moment diagrams. Relative to the support A, the support B sinks by 1 mm and the support C raises by 1/2 mm, EI = 30000 kNm2.
4 m 4 m80 kN
2 EI EI
DA B C
20 kN/m
6 m
FIG. 2.46
Distribution factors
Table 2.17 Distribution factors
Joint Members Relative Stiffness Values (k) k k/ k
BA28I
0.67
B 3I/8
BC 34 6 8
I I¥ = 0.33
Fixed end moments Due to loading
MAB = – 80 × 8 80 kNm8
= -
MBA = + 80 kNm
MBC = – 20 × 26 60 kNm
12= -
MCB = +60 kNm
162 Indeterminate Structural Analysis
Due to settlement of supports
FIG. 2.47 Settlement angles diagram (R = ∆/l)
MAB = 12 21
6 (2 ) 6 30000 12 5.63 kNm
1000 8E I
l- D ¥ ¥
= - ¥ = -¥
MBA = 121
6 (2 )5.63 kNm
E Il
- D= -
MBC = 22 22
6 ( ) 6 30000 1.57.5 kNm
1000 6EI
l- - D ¥ ¥
= + =¥
It should be noted that the support B settles by 1 mm. In span AB, B settles and hence AB rotates in the clockwise direction. Therefore, (1/l1) is positive. In span BC, the support C raises by 1/2 mm from the original level AC. The total displacement is 1(1/2) mm with respect to the displaced position B. BC displaces to BC. Hence, the rotation angle (2/l2 ) is negative since the span BC rotates in the anticlockwise direction with respect to B. The fi xed end moment is the sum of the moments due to applied loading and due to the settlement of supports. MFAB = – 80 – 5.63 = – 85.63 kNm
MFBA = + 80 – 5.63 = + 74.37 kNm
MFBC = – 60 + 7.50 = – 52.50 kNm
MFCB = + 60 + 7.50 = + 67.50 kNm
Table 2.18 Moment distribution table
Joint A B C
Members AB BA BC CBDF 0 0.67 0.33 1FEMSBal
– 85.63 +74.37– 14.65
– 52.50– 7.22
+67.50−67.50
CoBal
– 7.33+22.61
– 33.75+11.14
Co +11.31Final – 81.65 +82.33 – 82.33 0.00
163Moment Distribution Method
Shear force diagrams Equilibrium of span AB
81.65 kNm 82.33 kNm80 kN
A BD
4 4 m
FIG. 2.48
V = 0, VAB + VBA = 80 (1)
MA = 0; – 81.65 + 82.33 + 80(4) – 8VBA = 0 (2)
VBA = 40.1 kN
VAB = 39.9 kN
MD = – 81.65 + 39.9 × 4 = 77.95 kNm
Equilibrium of span BC
B C
82.3320 kN/m
6 m
FIG. 2.49
V = 0; VBC + VCB = 120 (3)
MB = 0; – 82.33 + 20 × 26 6 0
2 CBV- =
VCB = 46.3 kN
VBC = 73.7 kN The zero shear force location in BC is obtained from
46.36 73.7
xx
=-
73.7x = 277.8 – 46.3x
x = 2.32 m
164 Indeterminate Structural Analysis
CA BD
39.9 kN
73.7 kN
46.3 kN
40.1
x
FIG. 2.50 Shear force diagram
Absolute maximum BM is 46.3 × 2.32 – 20 × 22.32 53.6 kNm
2=
C
A B
− −
+ +
D
81.65 kNm
77.95 kNm
53.6 kNm
2.32 m
88.33 kNm
FIG. 2.51 Bending moment diagram
Example 2.9 A continuous beam is loaded as shown in Fig. 2.52. During loading the support B sinks by 10 mm. Determine the bending moments at the supports. Sketch the BMD. Given that I = 1600(10)4 mm4; E = 200 kN/mm2. Use moment distribution method. Draw SFD also.
AB
C
8 m 4 m
3 kN/m 8 kN
2I I
2 2
FIG. 2.52
165Moment Distribution Method
SolutionDistribution factors
Table 2.19 Distribution factors
Joint Members Relative Stiffness Values (k) kSkk
BA 2 0.258I I= 0.57
B 0.438I
BC 3 0.1884 4
I I¥ = 0.43
Fixed end moments Due to loading
MFAB = – 3 × 28 16 kNm;
12= - MFBA = + 16 kNm
MFBC = – 8 × 4 4 kNm;8
= -
MFCB = + 4 kNm
Due to Settlement
A C8 m
10 mm
2 mθ
B
FIG. 2.53 Settlement diagram
MFAB = 11 12 2
1
6 10 16 2 3200 ( / is positive)1000 8
EI ll
D- = - ¥ ¥ ¥ ¥ D∵
MFAB = – 6 kNm
MFBA = +6 kNm
MFBA = 22 22 2
2
6 3200 ( 10)6 12 kNm ( / is negative)1000 4
EI ll
¥ ¥ -D- = - = + D
¥∵
MFCB = + 12 kNm
It is to be mentioned here that B settles by 10 mm with respect to A. The settlement angle is positive as AB rotates in clockwise direction with respect to A. The settlement
166 Indeterminate Structural Analysis
angle for CB is negative as CB rotates in the anticlockwise direction. With respect to C. Thus, the fi xed end moments at the joint is the sum of the moments due to loading and the settlement. They are as follows. Fixed end moments MFAB = – 16 – 6 = – 22 kNm
MFBA = 16 – 6 = 10 kNm
MFBC = – 4 + 12 = 8 kNm
MFCB = + 4 + 12 = 16 kNm
Table 2.20 Moment distribution table
Joint A B C
Members AB BA BC CB
DF 0 0.57 0.43 1
FEMSBal
– 22.00–
+10.00–
+8.00–
+16.00– 16.00
Co – – – 8.00 –
TotalBal
– 22.00–
+10.00– 5.70
0.00– 4.30
0.00–
Co – 2.85 – – –
Final – 24.85 +4.30 – 4.30 0.00
Shear forces Free body diagram
A B24.85 4.30
3 kN/m
8 m
FIG. 2.54(a)
V = 0; VAB + VBA = 24 (1)
MA = 0; – 24.85 + 4.30 + 3 × 28 8 0
2 BAV- =
VBA = 9.4 kN
VAB = 14.6 kN
167Moment Distribution Method
MD = – 24.85 + 14.6 × 4 – 23 4
9.55 kNm2¥
=
B E C
4.3 kNm8 kN
2 m 2 m
FIG. 2.54(b)
V = 0; VBC + VCB = 8
MB = 0; – 4.3 + 8(2) – 4VCB = 0
VCB = 2.9 kN
VBC = 5.1 kN
ME = 2.9(2) = 5.8 kNm
CA BD
14.6 kN
9.4
5.1 kN
2.9 kN
E
FIG. 2.55 Shear force diagram
− −
+ +
24.85 kNm4.30 kNm
9.555.8 kNm
FIG. 2.56 Bending moment diagram
168 Indeterminate Structural Analysis
2.6.3 Analysis of Beams with Variable Moment of InertiaExample 2.10 A horizontal beam ACB, three metres long is fi xed at both ends A and B. They are at the same level. The member has a change of section at the centre of the span C such that the second moments of area are I for a distance AC and 2I for distance CB. A concentrated load of 500 kN acts at the midpoint C. Determine the fi xing end moments at A and B.
A
B
1.5 m
500 kN
1.5 m
C
I
2I
FIG. 2.57
Solution
A B
C '
Δ6EIΔ1.52
6EIΔ1.52
6EΔ1.52
(2I)
(2I)
Δ1.52
6E
FIG. 2.58
Applying the moment distribution method,
Table 2.21 Distribution factors
Joint Members Relative Stiffness (k) Sum k Distribution Factor (k/ k)
CA I/1.5 0.33
C 2I
CB 2I/1.5 0.67
The point C is displaced vertically by an amount ∆ to the point C keeping the section at C clamped (i.e., preventing any rotation). If an arbitrary end moment MCA is given an arbitrary value of 100, the fi xed end moment of MCB will be 200.
169Moment Distribution Method
Table 2.22 Moment distribution table
Joint A C BMembers AC CA CB BC
DF 1 0.33 0.67 1
FEMSBal
– 100 – 100– 33
+200– 67
+200
Co – 16.5 – 33.5
Final – 116.5 – 133 +133 +166.5
Equilibrium of span AC
A C
116.5 kNm
1.5 m
133 kNm
VAC VCA
FIG. 2.59
MC = 0;
– 116.5 – 133 + 1.5VAC = 0
VAC = 166.33 kN
Equilibrium of span CB
C B
133 kNm
1.5 m
167.5 kNm
VCB VBC
FIG. 2.60
Taking moment about C;
133 + 167.5 – 1.5VBC = 0
VBC = 200.33 kN
Resolving the forces vertically
= 166.33 + 200.33 = 366.66 kN
170 Indeterminate Structural Analysis
i.e., if the concentrated load applied at the centre is 366.66 kN then this will yield the moments given in the table. But the applied load is 500 kN. Hence, the moments at A and B are obtained as
MA = – 116.5 × 500 158.9 kNm
366.66= -
MB = +167.5 × 500 228.4 kNm
366.66= +
Example 2.11 A vertical column of 8 m height is to carry a crane girder load of 50 kN applied at an eccentricity of 0.2 m. Calculate the moments at A and B due to this load assuming both ends are fi xed.
Solution: Evaluate the distribution factor at C.
0.2 m
50 kN2 m
6 m
A
B
C
FIG. 2.61
Table 2.23 Distribution factors
Joint Members Relative Stiffness (k) k (k/ k)
CA I/6 0.25
C 0.667I
CB I/2 0.75
173Moment Distribution Method
Taking moment about C for the portion AC,
12 + 8 – 6RA = 0
RA = 3.33 kN
Resolving all the forces horizontally
RA + RB – Y = 0
3.33 + 18 – Y = 0
Y = 21.33 kN
The sway moments corresponding to the sway force 5 kN is obtained by multiplying the moments in the moment distribution table as
MAC = 5 8 1.88 kNm
21.33Ê ˆ =Á ˜Ë ¯
MCA = 5 12 2.81 kNm
21.33Ê ˆ =Á ˜Ë ¯
MCB = 5 12 2.81 kNm
21.33Ê ˆ ¥ - = -Á ˜Ë ¯
MBC = 5 24 5.63 kNm
21.33Ê ˆ ¥ - = -Á ˜Ë ¯
Then the fi nal moments is the sum of sway and nonsway moments as
MAC = MAC + MAC = 1.25 + 1.88 = 3.13 kNm
MBC = MBC + MBC = 3.75 – 5.63 = – 1.88 kNm
2.7 ANALYSIS OF RECTILINEAR FRAMESExample 2.12 Analyse the frame shown in Fig. 2.65 by moment distribution method. Draw the bending moment diagram.
FIG. 2.64
B
A
RA
RB
8.00
24.00
2 m
6 m
C − Y12.00
12.00
174 Indeterminate Structural Analysis
B C2 m
2 m
1 m
20 kN/m
EI
EI2
A
FIG. 2.65
SolutionDistribution factors
Table 2.25 Distribution factors
Joint Members Relative Stiffness Values (I/l) k k/ k
BA I/2 0.5B I
BC3 2 /24 3
I IÊ ˆ =Á ˜Ë ¯ 0.5
Fixed end moments
MBC = 2 22 2
2 20 0
( ) (20 ) (3 )13.33 kNm
3xwdx l x x dx x
l- -
= =Ú Ú
MCB = 2 22 2
20 0
( ) (3 ) 20 8.89 kNm9
x xI x wdx x dxl
- = - =Ú Ú Nonsway Analysis
Table 2.26 Moment distribution table
Joint A B CMembers AB BA BC CBDF 0 0.5 0.5 1.0FEMSBal +6.66
– 13.33+6.67
+8.89– 8.89
CoBal
+3.33+2.22
– 4.44+2.22
Co +1.11+4.44 +8.88 – 8.88 0.00
176 Indeterminate Structural Analysis
MBC = – 8.88 + 50(0.107) = – 3.530 kNm MCB = 0
BC
3.53 20 kN/m
1 m2 mD
FIG. 2.67
MB = 0
3VC + 3.53 = 220 2
2¥
VC = 12.157 kN
MD = 12.157(1) = 12.157 kNm
V = 0
VB + VC = 2(20)
VB = 40 – 12.157 = 27.843 kN (SF)X = 27.843 – 20x = 0
x = 1.392 m
Maximum +ve BM = 27.843(1.392) – 220(1.392) 3.53
2-
ME = 15.851 kNm
B CE D3.585
3.585
3.585 kNm
12.157 kNm15.851 kNm
1.392 m
−
−
+ve
AFIG. 2.68 Bending moment diagram
177Moment Distribution Method
Example 2.13 Analyse the frame shown in Fig. 2.69 by the moment distribution method. Draw the bending moment diagram.
E AG
B C
F
D
3 m2 m
3 m 3 m1 m
10 kN/m
25 kN
40 kN10 kN
I
2I
2I
3 m
FIG. 2.69
SolutionDistribution factors
Table 2.28 Distribution factors
Joint Members Relative Stiffness Values (I/l) k (k/ k)
BA 34 3 4
I I¥ = 0.26
B BD (2I/5) 0.98 0.41
BC 2I/6 0.33
Fixed end moments
MFAE = +10 × 1 = 10 kNm
MFAB = – 10 × 23 7.5 kNm
12= -
MFBA = +10 × 23 7.5 kNm
12= +
MFBC = – 40 × 6 30.0 kNm8
= -
178 Indeterminate Structural Analysis
MFCB = + 40 × 6 30.0 kNm8
= +
MFBD = +2 × 25 × 32/52 = +18.0 kNm
MFBE = – 3 × 25 × 22/52 = – 12.0 kNm
Table 2.29 Moment distribution table
Joint A B C D
Members AE AB BA BD BC CB DB
DF – 1.00 0.26 0.41 0.33 0.00 0.00
FEMSBal
+10.00 – 7.50– 2.50
+7.50+1.17
+18.00+1.85
– 30.00+1.48
+30.00 – 12.00
CoBal
– 1.25+0.33 +0.51 +0.41
+0.74 +0.93
Co 0.21
Total +10.00 – 10.00 +7.75 +20.36 – 28.11 +30.95 – 11.07
31.25
11.07
20.36 7.75
28.11 kNm 30.95 kNm
60–10.00
CGBA
F
D
FIG. 2.70 Bending moment diagram
179Moment Distribution Method
Example 2.14 Analyse the frame by the moment distribution method. Draw the bending moment diagram.
A E
D
BC3
3 m
3 m
2100 kN 20 kN/m
2I I
I
FIG. 2.71
SolutionDistribution factors
Table 2.30 Distribution factors
Joint Members Relative Stiffness (k) k DF = k/ k
BA 2I/5 = 0.4I 0.41
B BD I/3 = 0.33I 0.98I 0.34
BC 3 ( /3) 0.254
I I= 0.25
Fixed end moments
MFAB = – 2 × 100 × 2
23 72 kNm5
= -
MFBA = +3 × 100 × 2
22 48 kNm5
= +
MFBC = – 20 × = -23 15 kNm
12
MFCB = +20 × 23 15 kNm
12= +
180 Indeterminate Structural Analysis
Table 2.31 Moment distribution table
Joint A B C DMembers AB BA BD BC CB DBDF 0 0.41 0.34 0.25 1 0FEMSBal
– 72.00–
+48.00– 13.53
–– 11.22
– 15.00– 8.25
+15.00– 15.00
CoBal
– 6.77+3.08 +2.55
– 7.50+1.88
– 5.61
Co +1.54 +1.28Total – 77.23 +37.55 – 8.67 – 28.87 0.00 – 4.33
FIG. 2.72 Bending moment diagram
Example 2.15 Analyse the frame by the moment distribution method. Draw the bending moment diagram.
B
D
E
C
15 m
15 m
15 m
7 m1 kN/m8 kN
A
I
I I
I
15 m
21
FIG. 2.73
181Moment Distribution Method
SolutionDistribution factors
Table 2.32 Distribution factors
Joint Members Relative Stiffness (k) k k/ k
BBA
BC
I/15
I/152I/15
0.5
0.5
CCBCECD
I/15I/15I/15
3I/150.330.330.33
Fixed end moments
MFBC = – 1 × 152/12 = – 18.75 kNm
MFCB = +1 × 152/12 = +18.75 kNm
MFCD = – 8 × 15/8 = – 15.00 kNm
MFDC = +8 × 15/8 = +15.00 kNm
Table 2.33 Moment distribution table
Joint A B C D
Members AB BA BC CB CE CD DC EC
Distribution Factors 0 0.5 0.5 0.33 0.33 0.33 0 0FEMSBalance +9.38
– 18.75+9.37
+18.75– 1.25 – 1.25
– 15.00– 1.25
+15.00
Co +4.69 – 0.63 +4.69 – 0.63 – 0.63
Bal +0.32 +0.31 – 1.56 – 1.57 – 1.56
Co +0.16 – 0.78 +0.16 – 0.78 – 0.79
Bal +0.39 +0.39 – 0.05 – 0.05 – 0.06
Co +0.20 – 0.03 +0.20 – 0.03 – 0.03
Bal +0.02 +0.01 – 0.07 – 0.06 – 0.06
Final
Moments +5.05 +10.11 – 10.11 +20.87 – 2.93 – 17.93 +13.56 – 1.45
183Moment Distribution Method
Table 2.34 Distribution factors for symmetrical case
Joint Members Relative Stiffness k DF = k/ k
BA 5I
0.8
B 0.25I
BC 1 0.052 10
I IÊ ˆ =Á ˜Ë ¯ 0.2
Fixed end moment
MFBC = (10) 1.25
8 8WWl W- = - = -
Table 2.35 Moment distribution table
Joint A BMembers AB BA BCDF 0 0.8 0.2FEMBalCo +0.5W
+1.00W– 1.25W+0.25W
Final +0.5W +1.00W – 1.00W
Thus, from the above moment distribution table, joint moment 1.00W = 300
W = 300 kN
Hence, MAB = 0.5 × 300 = 150 kNm
450
150 150
300
300
300
300 kNm
A
B C
D
Elastic Curve
FIG 2.76 Bending moment diagram FIG 2.77
185Moment Distribution Method
Table 2.36 Moment distribution table
Joint A B
Members AB BA BC
DF 0.0 0.5 0.5
FEMSBalance +15.00
– 30.00+15.00
Co +7.50
Final Moments +7.50 +15.00 – 15.00
30
15 kNm15 kNm
7.5 kNm 7.5 kNm
−−− −
+
+ +
FIG. 2.80 Bending moment diagram
Example 2.18 The culvert shown in Fig. 2.81 is of constant section throughout and the top beam is subjected to a central concentrated load of 25 kN. Assume the base pressure is uniform throughout and analyse the box culvert. Draw the bending moment diagram.
5 m5 m
10 m
25 kN
2.5 kN/m
DA
B C
FIG. 2.81
186 Indeterminate Structural Analysis
Solution: The above frame is symmetrical and symmetrically loaded. This can be analysed using moment distribution in a straightforward and in a simpler manner by taking advantage of symmetry. The distribution factors are worked out by taking the central member BC as half of its value of stiffness. The process of moment distribution is carried out by considering only half of the frame.
Distribution factors
Table 2.37
Joint Members Relative Stiffness Values k k/ k
BA 0.110I I= 0.67
B 0.15I
BC1 0.052 10
I IÊ ˆ =Á ˜Ë ¯ 0.33
AB 0.110I I= 0.67
A 0.15I
AD1 0.052 10
I IÊ ˆ =Á ˜Ë ¯ 0.33
Fixed end moments
MFBC = 1025 31.25 kNm
8 8Wl- = - ¥ = -
MFAD = 2 2102.5 20.83 kNm
12 12wl+ = + ¥ = +
Moment Distribution Due to symmetry analyse half of the frame. The joints A, B, C and D are rigid. This frame is recognised as a continuous frame. It implies that when joint moments are balanced, it is being carried over to the neighbouring joints. This carry over moment is again balanced, the process is continued and all the joints are balanced. Hence, the joint moments are
MA = MD = 9.06 kNm
MB = MC = 16.95 kNm
187Moment Distribution Method
9.06
9.06 kNm
22.19
45.55
16.95 16.95 kNm
DA
B
− −
−−
− −
+
+
C
FIG. 2.82 Bending moment diagram
2.9 ANALYSIS OF UNSYMMETRICAL FRAMESIn the previous article, the frame was symmetrical and was symmetrically loaded. Half frame technique was used by taking beam stiffness as half of its original stiffness. It indicates that sways do not occur in the following cases (i) a structure which is held against lateral movement (Ref. Example 2.15) (ii) where both geometry of the frame and the loading are symmetrical (Ref. Examples 2.16 - 2.18).
Table 2.38 Moment distribution table
Joint A B
Members AD AB BA BC
DF 0.33 0.67 0.67 0.33
FEMSBal
+20.83– 6.94 – 13.89 +20.83
– 31.25+10.42
CoBal – 3.47
+10.42– 6.95
– 6.95+4.63 +2.32
CoBal – 0.77
+2.32– 1.55
– 3.48+2.32 +1.16
CoBal – 0.38
+1.16– 0.78
– 0.78+0.52 +0.26
CoBal – 0.17
+0.26– 0.08
– 0.39+0.26 +0.13
CoBal – 0.04
+0.13– 0.09
– 0.04+0.03 +.01
Final 9.06 – 9.06 16.95 – 16.95
188 Indeterminate Structural Analysis
In reality, the frames are unsymmetrical and the loading could also be unsymmetrical. In such frames, there will be rotation of joints as well as the lateral displacement joints. The lateral displacement causes the frame to sway either to the right or to the left with respect to the centre line of the frame. The sway (side sway) is caused by (i) Portal frames with unequal columns (Fig. 2.83a) (ii) Portal frames whose columns are of different moment of inertia (Fig. 2.83b) (iii) Unsymmetrical loading on the beams (Fig. 2.83c) (iv) Lateral loading (Fig. 2.83d)
(a)
I1
I2I3
Δ Δ Δ Δ
(b)
I1
I2
I3
(d)
I1
I2
P
I3
ΔΔΔ
Δ
(c)
I
2 I
W
I
FIG. 2.83 Examples of sway frames
Analysis of sway frames is done in the following way. (1) The moment distribution was carried out by assuming that the joints do not get
displaced. The moments obtained from the moment distribution table are called nonsway moments.
(2) The horizontal reactions at the base of the columns are found out which helps in fi nding the net out of balance force. A prop is assumed to act opposite to the direction of the above force.
(3) Allow the frame to sway in the direction of sway force which is equal and opposite to the prop force (which acts along the axis of the beam level). Let the actual prop force be X.
(4) Perform the sway moment distribution, by assuming arbitrary moments as per the joint moment ratios.
189Moment Distribution Method
(5) Calculate the displacement force (Y) from the sway moment distribution. (6) The correction factor/sway factor k = X/Y. The correction factor gives the
direction of sway of the frame. If the value of k is positive, then the frame sways in the direction of the sway solution. If the value of k is negative, then the frame sways in the direction opposite to that of assumed sway.
(7) The fi nal end moments are obtained as
Final end moments = Nonsway moments +‘Correction factor’ × Sway moments Expressing mathematically;
MAB = MAB + k MAB
where MAB is the end moment of the member AB, MAB is the moment obtained from nonsway moment distribution, MAB is the computed moment obtained from sway moment distribution k is the correction factor.
2.9.1 Joint Moment RatiosThe various moment ratios for rectangular frames are given below. Portal frames with both ends fi xed
I1
I3L
W
D
C
A
B
I2H2
H1
FIG. 2.84(a)
D
C
A
B Δ Δ
FIG. 2.84(b)
190 Indeterminate Structural Analysis
MAB = MBA = 121
6EIH
D-
and MCD = MDC = 222
6EIH
D-
2
1 22
2 1
AB
DC
M I HM I H
=
Portal frame with end fi xed and other hinged
I1
I3
L
D
C
A
B
I2H2
H1
FIG. 2.85(a)
D
C
A
B Δ Δ
FIG. 2.85(b)
MAB = 0 ( Hinged)
MBA = 121
3EIH
D-
MDC = MCD = 222
6EIH
D-
193Moment Distribution Method
Nonsway Analysis
Table 2.40 Moment distribution table
Joint A B C D
Members AB BA BC CB CD DC
DF 0 0.45 0.55 0.55 0.45 0
FEMSBal +12.96
– 28.8015.84
+19.20– 10.56 – 8.64
CoBal
6.48+2.38
– 5.28+2.90
+7.92– 4.36 – 3.56
– 4.32
CoBal
1.19+0.98
– 2.18+1.20
+1.45– 0.80 – 0.65
– 1.78
CoBal
0.49+0.18
– 0.40+0.22
+0.60– 0.33 – 0.27
– 0.33
CoBal
0.09+0.08
– 0.17+0.09
+0.11– 0.06 – 0.05
– 0.14
CoBal
0.04+0.01
– 0.03+0.02
+0.05– 0.03 – 0.02
– 0.03
Total 8.29 16.59 – 16.59 13.19 – 13.19 – 6.60
MAB MBA MBC MCB MCD MDC
Sway Analysis The assumed sway moments are in the joint ratio
2
2
6 /36 /3
BA
CD
EIMM EI
- D=
- D
i.e. MBA : MCD = – 1 : – 1
The moments are arbitrarily assumed in the columns as – 100 kNm. Final moments These are obtained by adding the nonsway and sway moments as
MAB = 8.29 – 82.33 k
MBA = 16.59 – 64.73 k
MBC = – 16.59 + 64.73 k
MCB = 13.19 + 64.73 k
MCD = – 13.19 – 64.73 k
MDC = – 6.60 – 82.33 k
195Moment Distribution Method
D
C
A 8.006.89 kNm
B
15.5014.29 kNm
32.89
FIG. 2.88 Bending moment diagram
D
C
A
B
FIG. 2.89 Elastic curve
Example 2.20 Analyse the given frame and draw the bending moment diagram
2I
D
C
A
BI
2I
3 6 m100 kN
50 kN4
4
FIG. 2.90
196 Indeterminate Structural Analysis
SolutionDistribution factors
Table 2.42
Joint Members Relative Stiff Values (I/l) k k/ k
BA 2I/8 = 0.25I 0.7B 0.36I
BC I/9 = 0.11I 0.3CB I/9 = 0.11I 0.3
C 0.36ICD 2I/8 = 0.25I 0.7
Fixed end moments
MFAB = 850 50 kNm
8 8Wl- = - ¥ = -
MFBA = 850 50 kNm
8 8Wl+ = + ¥ = +
MFBC = 2 2
2 26100 3 133.33 kNm9
Wabl
- = - ¥ ¥ = -
MFCB = 2
22 2
6100 3 66.67 kNm9
Wa bl
+ = + ¥ ¥ = +
Nonsway Analysis
Table 2.43 Moment distribution table
Joint A B C DMembers AB BA BC CB CD DCDF 0 0.70 0.30 0.30 0.70 0FEMS Bal
– 50.00 +50.00 +58.33
– 133.33+25.00
+66.67– 20.00 – 46.67
CoBal
+29.20+7.00
– 10.00 +3.00
+12.50– 3.75 – 8.75
– 23.34
CoBal
+3.50+1.32
– 1.88+0.56
+1.50– 0.45 – 1.05
– 4.38
CoBal
+0.66+0.16
– 0.23+0.07
+0.28– 0.08 – 0.20
– 0.53
CoBal
+0.08+0.03
– 0.04+0.01
+0.04– 0.01 – 0.03
– 0.10
– 16.56 +116.84 – 116.84 56.70 – 56.70 – 28.35
MAB MBA MBC MCB MCD MDC
197Moment Distribution Method
Sway Analysis
Table 2.44 Moment distribution table
Joint A B C D
Members AB BA BC CB CD DC
DF 0 0.70 0.30 0.30 0.70 0
FEMSBal
+100.00 +100.00– 70.00 – 30.00 – 30.00
+100.00– 70.00
+100.00
COBal
– 35.00+10.50
– 15.00+4.50
– 15.00+4.50 +10.50
– 35.00
CoBal
+5.25– 1.57
+2.25– 0.68
+2.25– 0.68 – 1.57
+5.25
COBal
– 0.79+0.24
– 0.34+0.10
– 0.34+0.10 +0.24
– 0.79
COBal
+0.12– 0.04
+0.05– 0.01
+0.05– 0.02 – 0.03
+0.12
Final 69.58 39.13 – 39.13 – 39.13 +39.13 +69.58
MAB MBA MBC MCB MCD MDC
The fi nal moments are the addition of nonsway moments and a constant times sway moments; for example, MAB = MAB + k MAB. Therefore, End moments
MAB = – 16.56 + 69.58 k
MBA = +116.84 + 39.13 k
MBC = – 116.84 – 39.13 k
MCB = +56.70 – 39.13 k
MCD = – 56.70 + 39.13 k
MDC = – 28.35 + 69.58 k
The value of k is determined from the column shear condition as follows. Column shear condition Considering the free body diagram of column AB and the value of HA isdetermined as
199Moment Distribution Method
23.096 – 27.18 k = 50
k = – 0.99 Hence, Final moments MAB = – 16.56 + 69.58(– 0.99) = – 85.4 kNm
MBA = +116.84 + 39.13(– 0.99) = +78.1 kNm
MBC = – 116.84 – 39.13(– 0.99) = – 78.1 kNm
MCB = +56.70 – 39.13(– 0.99) = +95.4 kNm
MCD = – 56.70 + 39.13(– 0.99) = – 95.4 kNm
MDC = – 28.35 + 69.58(– 0.99) = – 97.2 kNm
D
C
A 97.285.4
78.1B
+ 95.4 kNm
200
+
− −
100
FIG. 2.93 Bending moment diagram
Example 2.21 Analyse the portal frame shown in Fig. 2.94 by the moment distribution method. Sketch the bending moment diagram.
3 m
60 kN/m
4 m
A
B C
D
I
I2I
FIG. 2.94
200 Indeterminate Structural Analysis
SolutionDistribution factors
Table 2.45
Joint Members Relative Stiffness Values k k/ k
BBA
BC
2I/4
I/30.83I
0.60
0.40
CCB
CD
I/3
I/40.58I
0.57
0.33
Fixed end moments
MFBC = – 60 × = -23 45 kNm
12
MFCB = +60 × 23 45 kNm
12= +
Though the boundary conditions and the loading are symmetrical, the frame sways as the columns AB and CD are having different moments of inertia of the cross–section. Hence, we have to do nonsway and sway analysis to determine the joint moments. Nonsway Analysis
Table 2.46 Moment distribution table
Joint A B C DMembers AB BA BC CB CD DCDF 0 0.60 0.40 0.57 0.33 0FEMSBal +27.00
– 45.00+18.00
+45.00– 25.65 – 19.35
CoBal
+13.50+7.70
– 12.83+5.13
+9.00– 5.13 – 3.87
– 9.68
CoBal
+3.85+1.54
– 2.56+1.02
+2.56– 1.43 – 1.13
– 1.94
CoBal
+0.77+0.43
– 0.72+0.29
+0.51– 0.29 – 0.22
– 0.57
CoBal
+0.22+0.09
– 0.15+0.06
+0.15– 0.08 – 0.06
– 0.11
CoBal
+0.05+0.02
– 0.04+0.02
+0.03– 0.02 – 0.01
– 0.03
Final +18.39 +36.78 – 36.78 +24.65 – 24.65 – 12.33
MAB MBA MBC MCB MCD MDC
202 Indeterminate Structural Analysis
Column shear condition In the given portal frame, the columns are fi xed and there is no horizontal load acting externally. Hence, the horizontal equilibrium gives
MAB + MBA + MCD + MDC = 0
(18.39 + 36.78 – 24.65 – 12.33) – (14.57 + 9.09 + 7.26 + 8.63) k = 0
18.19 – 39.55 k = 0
k = 0.46
Substituting this value of k in the above equations; End moments
MAB = 18.39 – 14.57(0.46) = 11.7 kNm
MBA = 36.78 – 9.09(0.46) = 32.6 kNm
MBC = – 36.78 + 9.09(0.46) = – 32.6 kNm
MCB = 24.65 + 7.26(0.46) = +28.0 kNm
MCD = – 24.65 – 7.26(0.46) = – 28.0 kNm
MDC = – 12.33 – 8.63(0.46) = +16.3 kNm
D
C
A 16.311.7
B32.6
28.0 kNm
37.2
FIG. 2.95 Bending moment diagram
203Moment Distribution Method
D
C
A
B
FIG. 2.96 Elastic curve
Example 2.22 Analyse the portal frame shown in Fig. 2.97 by the moment distribution method. Draw the bending moment diagram.
2EI
D
C
A
B
EIEI
6 3 3 m
6 m
60 kN 60 kN
E F
FIG. 2.97
Solution: The frame is subjected to asymmetrical loading. Hence, nonsway and sway analyses were carried out separately and the results are summed up. Distribution factors
Table 2.48
Joint Members Relative Stiff Values (k) k DF
BBA
BC
I/6
2I/12 = I/6I/3
0.5
0.5
CCB
CD
2I/12 = I/6
I/6I/3
0.5
0.5
204 Indeterminate Structural Analysis
Fixed end moments
MFBC = 22
2 2
60 9 31260 123.75 kNm8 8 12
Wl Wabl
¥ ¥- - = - ¥ - = -
MFCB = 22
2 2
60 3 91260 191.25 kNm8 8 12
Wl Wa bl
¥ ¥+ + = ¥ + = -
Nonsway Analysis
Table 2.49 Moment distribution table
Joint A B C D
Members AB BA BC CB CD DC
DF 0.0 0.5 0.5 0.5 0.5 0.0
FEMSBal +61.88
– 123.75+61.87
+191.25– 95.63 – 95.62
CoBal
+30.94+23.91
– 47.82+23.91
+30.94– 15.47 – 15.47
– 47.81
CoBal
+11.96+3.87
– 7.74+3.87
+11.96– 5.98 – 5.98
– 7.74
CoBal
+1.94+1.50
2.99+1.50
+1.94– 0.97 – 0.97
– 2.99
CoBal
+0.75+0.24
– 0.49+0.25
+0.75– 0.37 – 0.38
– 0.49
CoBal
+0.12+0.10
– 0.19+0.09
+0.13– 0.06 – 0.07
– 0.19
Final 45.71 91.50 – 91.50 118.49 – 118.49 – 59.22
MAB MBA MBC MCB MCD MDC
Sway moments The sway moments are assumed in the joint ratio of
2
2
6 /6 16 /6 1
BA
DC
EIMM EI
- D -= =
- D -
MBA : MDC = – 100.00 : – 100.00
207Moment Distribution Method
Example 2.23 Analyse the given frame by the moment distribution method. Draw the bending moment diagram and shear force diagram.
1.5I
D
C
A
B2I
I
3 m2 m
3 m
6 m
80 kN
10 kN/m
20 kN
FIG. 2.100 Distribution factors
Table 2.51
Joint Members Relative Stiffness k k/ k
BA 1.5 0.256
I I= 0.38
B 0.65I
BC2 0.45I I= 0.62
CB 2 0.45I I= 0.55
C 0.73I
CD 0.333I I= 0.45
Fixed end moments
MFAB = – 10 × 26 30 kNm
12= -
MFBA = +10 × 26 30 kNm
12= +
MFBC = 2
2
2(90)3 64.8 kNm5
- = -
208 Indeterminate Structural Analysis
MFCB = +3(90) 2
23 43.2 kNm5
= +
Nonsway Analysis
Table 2.52 Moment distribution table
Joint A B C DMembers AB BA BC CB CD DCDF 0 0.38 0.62 0.55 0.45 0
FEMSBal
– 30.00 +30.00+13.22
– 64.80+21.58
+43.20– 23.76
–– 19.44
–
CoBal
+6.61+4.51
– 11.88+7.37
+10.79– 5.93 – 4.86
– 9.72
CoBal
+2.26+1.13
– 2.97+1.84
3.69– 2.03 – 1.66
– 2.43
CoBal
+0.57+0.39
– 1.02+0.63
+0.92– 0.51 – 0.41
– 0.83
CoBal
+0.20+0.10
– 0.26+0.16
0.34– 0.19 – 0.15
– 0.21
CoBal
+0.05+0.04
– 0.10+0.06
+0.08– 0.04 – 0.04
– 0.08
Final – 20.31 +49.39 – 49.39 +26.56 – 26.56 – 12.44
MAB MBA MBC MCB MCD MDC
Sway Analysis
MBA : MCD = – 6E(1.5I) 2 2: 6 ( )
6 3E ID D-
MBA : MCD = – 15.00 : – 40.00 kNm
Correction factor MAB = – 20.31 – 14.15 k
MBA = +49.39 – 13.34 k
MBC = – 49.39 + 13.34 k
MCB = +26.56 + 22.63 k
MCD = – 26.56 – 22.63 k
MDC = – 12.44 – 31.30 k
209Moment Distribution Method
The value of k is determined from the column shear condition as Column shear condition MB = 0
MAB + MBA + 6HA = 10 × 26
2 and hence
HA = 30 – ( )
6AB BAM M+
Table 2.53 Moment distribution ta ble
Joint A B C D
Members AB BA BC CB CD DC
DF 0 0.38 0.62 0.55 0.45 0
FEMSBal
– 15.00 – 15.00+5.70
–+9.30
–+22.00
– 40.00+18.00
– 40.00
CoBal
+2.85– 4.18
+11.00– 6.82
+4.65– 2.56 – 2.09
+9.00
CoBal
– 2.09+0.49
– 1.28+0.79
– 3.41+1.88 +1.53
– 1.05
CoBal
+0.25– 0.36
+0.94– 0.58
+0.40– 0.22 – 0.18
+0.77
CoBal
– 0.18+0.04
– 0.11+0.07
– 0.29+0.16 +0.13
– 0.09
CoBal
+0.02– 0.03
+0.08– 0.05
+0.04– 0.02 – 0.02
+0.07
Final – 14.15 – 13.34 +13.34 +22.63 – 22.63 – 31.30
MAB MBA MBC MCB MCD MDC
FIG. 2.101
BMBA
MAB
VA
HA
6 m
20
10 kNm
210 Indeterminate Structural Analysis
MC = 0
MCD + MDC + 3HD = 0
HD = ( )
3CD DCM M+
-
HA + HD = 6(10) + 20
( ) ( )30 80
6 3AB BA CD DCM M M M+ +
- - =
Substituting the values of MAB, MBA, MCD and MDC from the above equations; and solving
k = 1.843
Final moments
MAB = – 20.31 – 14.15(1.843) = – 46.4
MBA = +49.39 – 13.34(1.843) = +24.8
MBC = – 49.39 + 13.34(1.843) = – 24.8
MCB = +26.56 + 22.63(1.843) = +68.3
MCD = – 26.56 – 22.63(1.843) = – 68.3
MDC = – 12.44 – 31.30(1.843) = – 70.1
FIG. 2.102
CMCD
MDC
VD
HD
3 m
211Moment Distribution Method
D
C
A
70.1 kNm
46.4
B
24.8 kNm68.3 kNm
69.8
9.4
108
FIG. 2.103 Bending moment diagram
46.4 kN
46.4 kN
44.7
56.4
33.6
45.3 kN
FIG. 2.104 Shear force diagram
Example 2.24 Analyse the rigid portal frame by moment distribution method and hence draw the bending moment diagram (Fig. 2.105).
SolutionDistribution factorsTable 2.54
Joint Members Relative Stiffness (I/l) k k/ k
BBA
BC
2I/5
2I/4
0.9I0.44
0.56
CCB
CD
2I/4
3 1.54 3
I¥
0.875I0.57
0.43
214 Indeterminate Structural Analysis
Column shear equation
MB = 0
MAB + MBA + 5HA – 20 × 25 0
2=
HA = 50 – 0.2(MAB + MBA)
MC = 0
MCD + 3HD = 0
HD = – 0.33MCD
H = 0; HA + HD = 20(5)
50 − 0.2(MAB + MBA) – 0.33 MCD = 100 (1)
FIG. 2.106
BMBA
MAB
A HA
5 m20 kN/m
FIG. 2.107
CMCD
VD
VC
D HD
HC
3 m
215Moment Distribution Method
The end moments are obtained using the sway and nonsway moment distribution table as
MAB = – 45.49 – 86.00 k
MBA = +34.05 – 64.09 k
MBC = – 34.05 + 64.09 k
MCB = 6.50 + 44.11 k
MCD = – 6.50 – 44.11 k MDC = 0.00
Substituting the above end moments in Eq. (1);
k = 1.022
Hence, the end moments are obtained by back substitution as
MAB = – 133.4 kNm; MBA = – 31.5 kNm; MBC = +31.5 kNm;
MCB = +51.6 kNm; MCD = – 51.6 kNm; MDC = 0
133.4 kNm
51.6 kNm40
62.5
FIG. 2.108 Bending moment diagram
Example 2.25 The frame shown in Fig. 2.109 is hinged at A. The end D is fi xed and the joints B and C are rigid. Column CD is subjected to horizontal loading of 30 kN/m. A concentrated load of 90 kN acts on BC at 1 m from B. Analyse the frame and sketch the BMD?
216 Indeterminate Structural Analysis
D
C
A
B
1.5I
I
I
1 m1 m
34 m
90 kN
30 kN/m
FIG. 2.109
SolutionDistribution factors
Table 2.57
Joint Members Relative stiffness k k k/ k
B
BA
BC
¥ =3 1.5 0.3754 3
l I
I/2 = 0.5I0.875I
0.43
0.57
CCB
CD
I/2 = 0.5I
I/4 = 0.25I0.75I
0.67
0.33
Fixed end moments
MFBC = – 90 × 2 22.5 kNm8
= -
MFCB = +90 × 2 22.5 kNm8
= +
MFCD = – 30 × 24 40 kNm
12= -
MFDC = +30 × 24 40 kNm
12= +
217Moment Distribution Method
Nonsway Analysis
Table 2.58 Moment distribution table
Joint A B C D
Members AB BA BC CB CD DC
DF 1 0.43 0.57 0.67 0.33 0
FEMSBal +9.68
– 22.50+12.82
+22.50+11.73
– 40.00+5.77
+40.00
CoBal – 2.52
+5.87– 3.35
+6.41– 4.30 – 2.11
+2.89
CoBal +0.92
– 2.15+1.23
– 1.68+1.12 +0.56
– 1.06
CoBal – 0.24
+0.56– 0.32
+0.62– 0.42 – 0.20
+0.28
CoBal +0.09
– 0.21+0.12
– 0.16+0.11 +0.05
– 0.10
CoBal – 0.03
+0.06– 0.03
+0.06– 0.04 – 0.02
+0.03
Final 0 +7.90 – 7.90 +35.95 – 35.95 +42.04
Members MAB MBA MBC MCB MCD MDC
Sway AnalysisThe sway moments are assumed in the following ratios
2
2
3 (1.5 ) /36 ( ) /4
BA
CD
E IMM E I
D=
D
43
BA
CD
MM
=
MBA : MCD = +40 : +30 kNm
Final moments MAB = 0
MBA = 7.90 + 25.80 k
MBC = – 7.90 – 25.80 k
MCB = +35.95 – 23.22 k MCD = – 35.95 + 23.22 k
MDC = +42.04 + 26.58 k
219Moment Distribution Method
Equilibrium of column CD
MC = 0;
MCD + MDC + 30 × 24 4 0
2 DH- =
HD = 60 + 1 ( )4 CD DCM M+
Considering the horizontal equilibrium of the whole structure;
HA + HD = 4(30)
i.e. ++ + =
( )60 120
3 4CD DCBA M MM
Substituting the values of the moments from the fi nal moment equations;
k = 2.65 The end moments were calculated as
MAB = 0
MBA = 7.90 + 25.80 × 2.65 = 76.3 kNm
MBC = – 7.90 – 25.80 × 2.65 = −76.3 kNm
MCB = +35.95 – 23.22 × 2.65 = – 25.6 kNm
MCD = – 35.95 + 23.22 × 2.65 = +25.6 kNm
MDC = +42.04 + 26.58 × 2.65 = +112.5 kNm
FIG. 2.111
CMCD
MDC
HDD
HC
4 m 30 kN/m
222 Indeterminate Structural Analysis
FIG. 2.114
BMBA
MAB
HAA
1 m
3 m20 kN/m
Sway Analysis
Table 2.62 Moment distribution table
Joint A B C D
Members AB BA BC CB CD DC
DF 0 0.5 0.5 0.5 0.5 0
FEMSBal
– 100.00 – 100.00+50.00 +50.00 +50.00
– 100.00+50.00
– 100.00
CoBal
+25.00
– 12.50
+25.00
– 12.50
+25.00
– 12.50 – 12.50
+25.00
CoBal
– 6.25+3.12
– 6.25+3.13
– 6.25+3.13 +3.12
– 6.25
CoBal
+1.56
– 0.78
+1.56
– 0.78
+1.57
– 0.79 – 0.78
+1.56
CoBal
– 0.39+0.20
– 0.40+0.20
– 0.39+0.20 +0.20
– 0.69
CoBal
+0.10
– 0.05
+0.10
– 0.05
+0.10
– 0.05 – 0.05
+0.10
Final – 79.98 – 60.01 +60.01 60.02 – 60.02 – 79.98
MAB MBA MBC MCB MCD MDC
Column shear condition Equilibrium of column AB
MB = 0;
MAB + MBA – 20 × 23 4 0
2 AH+ =
HA = 90 ( )
4AB BAM M- +
223Moment Distribution Method
MC = 0;
MCD + MDC + 4HD = 0
HD = ( )
4CD DCM M- +
Considering the horizontal equilibrium Substituting the values in terms of moments and simplifying HA + HD = 20 × 3 MAB + MBA + MCD + MDC = – 150
(– 26.41 + 11.81 + 3.38 + 1.68) – (79.98 + 60.01 + 60.02 + 79.98)k = – 150 k = 0.502 End moments MAB = – 26.41 – 79.98(0.502) = 66.6 kNm MBA = 11.81 – 60.01(0.502) = −18.3 kNm MBC = – 11.81 + 60.01(0.502) = +18.3 kNm MCB = – 3.38 + 60.02(0.502) = +26.8 kNm MCD = +3.38 – 60.02(0.502) = – 26.8 kNm MDC = 1.68 – 79.98(0.502) = – 38.5 kNm
D
C
A 38.566.6
B
18.3
26.8 kNm
26.8 kNm18.3
FIG. 2.116 Bending moment diagram
FIG. 2.115
224 Indeterminate Structural Analysis
Example 2.27 Analyse the rigid frame shown in Fig. 2.117 by the moment distribution method. Draw the bending moment diagram.
D
C
A
B
1.5EI
EIEI4 m
6 m
5 m
100 kN
FIG. 2.117
Solution: In the above problem, a lack of symmetry makes the frame to sway to the right. In the sway analysis, it is assumed that the joints B and C have no movement of rotation but there is only lateral translation. In the fi rst instant, an external force necessary to prevent the lateral translation is assumed. The moments at the supports and joints are computed for the above external force. This is followed by a redistribution of moments allowing for the side sway by removing the assumed external force. The frame is analysed by considering the sway moments only. Distribution factors
Table 2.63
Joint B CMembers BA BC CB CD
k3 0.1884 4
I IÊ ˆ =Á ˜Ë ¯1.5 0.25
6I I=
1.5 0.256
I I= 0.205I I=
k 0.438I 0.45I
DF = k/k 0.43 0.57 0.56 0.44
Sway moments The sway moments are assumed in the following ratio
2
2
3 /46 /5
BA
CD
EIMM EI
- D=
- D
MBA : MCD = – 25.00 : – 32.00 kNm
225Moment Distribution Method
Moment distribution table
Table 2.64
Joint A B C D
Members AB BA BC CB CD DC
DF 0 0.43 0.57 0.56 0.44 1
FEMSBal
0 – 25.00+10.75 +14.25 +17.92
– 32.00+14.08
– 32.00
CoBal – 3.85
+8.96– 5.11
+7.13– 4.00 – 3.13
+7.04
CoBal +0.86
– 2.00+1.14
– 2.56+1.43 +1.13
– 1.57
CoBal – 0.31
+0.72– 0.41
+0.72– 0.40 – 0.32
+0.57
CoBal +0.08
– 0.20+0.12
– 0.21+0.12 +0.09
– 0.16
CoBal – 0.03
+0.06– 0.03
+0.06– 0.03 – 0.03
+0.05
Final – 17.50 +17.50 +20.18 – 20.18 – 26.07
Moment 0
Column shear condition
MB = 0 MC = 0
4HA + MBA = 0 MCD + MDC + 5HD = 0
HA = 4
BAM-
HD =
( )5
CD DCM M- +
FIG. 2.118 FIG. 2.119
226 Indeterminate Structural Analysis
Considering the horizontal equilibrium, substituting the values of moments, HA and HD are calculated as
HA = ( 17.50)
4.38 kN4
- -=
HD = 1
( 20.18 26.07) 9.25 kN5
-- - =
The addition of HA and HD gives the lateral load which produces the assumed moments. Thus, HA + HD gives 13.63 kN but the applied load is 100 kN. Hence, the end moments in the table are to be multiplied by the ratio (100/13.63 = 7.34). The fi nal moments are
MAB = 0 × 7.34 = 0
MBA = – 17.50 × 7.34 = – 128.5 kNm
MBC = +17.50 × 7.34 = +128.5 kNm
MCB = +20.18 × 7.34 = +148.1 kNm
MCD = – 20.18 × 7.34 = – 148.1 kNm
MDC = – 26.07 × 7.34 = – 191.4 kNm
D
C
A 191.4
B
148.1 kNm
148.1 kNm128.5
128.5
−−
+
+
FIG. 2.120 Bending moment diagram
D
C
A
B θθ
FIG. 2.121 Elastic curve
227Moment Distribution Method
Example 2.28 Analyse the frame by moment distribution method and draw the shear force diagram and bending moment diagram.
FIG. 2.122
SolutionDistribution factors
Table 2.65
Joint Members Relative Stiff (I/l) k k/ k
BBA
BC
I/4
2I/43I/4
0.33
0.67
CCB
CD
2I/4
3 0.1884 4
I IÊ ˆ =Á ˜Ë ¯
0.688I0.73
0.27
Sway moments
The sway moments are assumed in the following ratio:
2
2
6 /4 23 /4 1
BA
CD
EIMM EI
- D -= =
- D -
Hence, MBA : MCD = – 20 : – 10 kNm
228 Indeterminate Structural Analysis
Table 2.66 Sway moment distribution table
Joint A B C D
Members AB BA BC CB CD DC
DF 1 0.33 0.67 0.73 0.27 0
FEMSBal
– 20.00 – 20.00+6.60 +13.40 +7.30
– 10.00+2.70
CoBal
+3.30– 1.21
+3.65– 2.44
+6.70– 4.89 – 1.81
CoBal
– 0.61+0.81
– 2.45+1.64
– 1.22+0.89 +0.33
CoBal
+0.41– 0.15
+0.45– 0.30
+0.82– 0.60 – 0.22
CoBal
– 0.08+0.10
– 0.30+0.20
– 0.15+0.11 +0.04
CoBal
+0.05– 0.02
+0.06– 0.04
+0.10– 0.07 – 0.03
– 16.93 – 13.87 +13.87 8.99 – 8.99 0.00
MAB MBA MBC MCB MCD MDC
Column shear equation
MB = 0 MC = 0
MAB + MBA + 4HA = 0 MCD + 4HD = 0
HA = 1 ( )4 AB BAM M- + HD =
4CDM
FIG. 2.123
BMBA
A
4 m
MAB
HA
HB
VA
FIG. 2.124
C MCD
D
4 m
HC
HD
VC
VD
230 Indeterminate Structural Analysis
Example 2.29 Analyse the frame shown in Fig. 2.127 by the moment distribution method. Draw the bending moment diagram.
D
C
A
B
3I
I I
4 m
4
5 m
20 kN50 kN/m
40 kN
FIG. 2.127
SolutionDistribution factors
Table 2.67
Joint Members Relative Stiffness (I/l) k k/ k
B
BA
BC
3 1 .0944 8
Ê ˆ =Á ˜Ë ¯
3 0.65I I=
0.694I
0.14
0.86
CCB
CD
3 0.65I I=
Ê ˆ =Á ˜Ë ¯3 .0944 8
I I
0.694I0.86
0.14
Fixed end moments
MFAB = – 40 × 8 40.0 kNm8
= -
MFBA = +40.0 kNm
MFBC = 250 5
104.17 kNm12
- ¥= -
MFCB = +50 × 25 104.17 kNm
12= +
231Moment Distribution Method
End moments
MAB = 0
MBA = 75.18 – 90.27 k
MBC = – 75.18 + 90.27 k
MCB = +21.07 + 90.27 k
MCD = – 21.07 – 90.27 k
MDC = 0
Nonsway Analysis
Table 2.68
Joint A B C D
Members AB BA BC CB CD DC
DF 1 0.14 0.86 0.86 0.14 1
FEMSBal
– 40.00+40.00
+40.00+8.98
– 104.17+55.19
+104.17– 89.58
0– 14.59
0
CoBal
+20.00+3.47
– 44.79+21.32
+27.60– 23.74 – 3.86
CoBal +1.66
– 11.87+10.21
+10.66– 9.16 – 1.50
CoBal
0+0.64
– 4.58+3.94
+5.11– 4.40 – 0.71
0
CoBal +0.31
– 2.20+1.89
+1.97– 1.69 – 0.28
CoBal +0.12
– 0.85+0.73
+0.95– 0.82 – 0.13
75.18 – 75.18 +21.07 – 21.07
MAB MBA MBC MCB MCD MDC
233Moment Distribution Method
Hence, MBA : MCD = – 100.00 : – 100.00
20 – 0.125MBA – 0.125MCD = 60
20 – 0.125(75.18 – 90.27 k) – 0.125(– 21.07 – 90.27 k) = 60
Solving for k = 2.07
Substituting the values of k by back substitution:
MAB = 0, MBA = – 111.86, MBC = +111.86,
MCB = 207.9, MCD = – 207.9 kNm, MDC = 0
Table 2.69 Moment distribution table
Joint A B C D
Members AB BA BC CB CD DC
DF 1 0.14 0.86 0.86 0.14 1
FEMSBal
0 – 100.00+14.00
0+86.00
0+86.00
– 100.00+14.00
0
CoBal – 6.02
+43.00– 36.98
+43.00– 36.98 – 6.02
CoBal +2.59
– 18.49+15.90
– 18.49+15.90 +2.59
CoBal – 1.11
+7.95– 6.84
+7.95– 6.84 – 1.11
CoBal +0.48
– 3.42+2.94
– 3.42+2.94 +0.48
CoBal – 0.21
+1.47– 1.26
+1.47– 1.26 – 0.21
0 – 90.27 +90.27 +90.27 – 90.27 0
MAB MBA MBC MCB MCD MDC
236 Indeterminate Structural Analysis
FIG. 2.133
Decomposition of the structure taking advantage of symmetry. Fixed end moments in symmetrical case
MFBC = – 20 × 2 × 3 24 kNm5
= -
MFCB = +20 × 2 × 3 24 kNm5
= +
Table 2.71 Moment distribution for symmetric case
Joint A B
Members AB BA BC
DF 0 0.62 0.38
FEMBal +14.88
– 24.00+9.12
Co 7.44
Final 7.44 14.88 – 14.88
Skew–symmetric case The cantilever stiffness for the column AB is taken as (I/3). While the skew stiffness of the beam BC is taken as 6(2I/5). The derivation of the cantilever stiffness and skew stiffness are available in standard references.
237Moment Distribution Method
FIG. 2.134
A
B
20 kN2 m
(a) Symmetric case (b) Skew symmetric case
21 2I
5( ) I3
A
B
20 kN2 m
2I5( )
I3
6Axisofsymmetry
Axisofsymmetry
Table 2.72 Distribution factors for skews symmetric case
Joint Members Relative Stiffness k DF = k/ k
BA I/3 = 0.33I 0.12
B 2.73I
BC 6(2I)/5 = 2.4I 0.88
Fixed end moments in skew symmetric case
MFBC = 2 2
2 2
2 20 3 3 20 24.8 kNm
5 5¥ ¥ ¥ ¥
- + = -
MFCB = – 4.8 kNm
Table 2.73 Moment distribution for skew symmetric case
Joint A BMembers AB BA BC
0 0.12 0.88FEMS – 4.80Bal +0.58 +4.22Co – 0.58Final – 0.58 +0.5 – 0.58
240 Indeterminate Structural Analysis
FIG
. 2.1
37
Dec
ompo
sitio
n of
giv
en lo
ad s
yste
m t
o sy
mm
etri
c an
d sk
ew s
ymm
etri
c lo
adin
g sy
stem
s
C
AD
B6
63
36
m6
33
++
=EI
EIEI
EIEI
EI
2EI
2EI
2EI
6 m
33 EI
EI
2EI
6060
6030
3030
30 k
N
Cas
e (i)
Cas
e (ii
)C
ase
(iii)
241Moment Distribution Method
Table 2.76
Joint A B
Members AB BA BC
DF 0.0 0.67 0.33
FEMSBal +45.0
– 67.5+22.5
CO +22.5
Total 22.5 45.0 – 45.0
Case (iii) Skew symmetric caseThe distribution factors at joint B are calculated by considering the cantilever stiffness and taking the skew stiffness of the beam. MFBC = (3 × 30 × 92/122 – 9 × 30 × 32/122) = + 33.75 kNm. The cantilever stiffness is taken as (I/6) and for the beam (6 × 2I/12). Hence,
dBA : dBC = (I/6) : (6 × 2I/12), i.e. dBA : dBC = 1 : 6
i.e. dBA = (1/7) and dBC = (6/7).
Table 2.77
Joint A B
Members AB BA BC
DF 0 0.14 0.86
FEMSBal – 4.72
+33.75– 29.03
CO +4.72
Total +4.72 – 4.72 +4.72
The fi nal moments are obtained by adding:
Members AB BA BC CB CD DC
Case (i) 30.00 60.00 – 60.00 +60.00 – 60.00 – 30.00
Case (ii) 22.50 45.00 – 45.00 +45.00 – 45.00 – 22.50
Case (iii) 4.72 – 4.72 +4.72 +4.72 – 4.72 +4.72
Total 57.22 100.28 – 100.28 109.72 – 109.72 – 47.78
The above moments tally with the moments obtained by classical moment distribution (Ref. Ex. 2.22). The Naylor’s method is very much useful if the reader identifi es the application of the same to specifi c problems.
243Moment Distribution Method
The distribution factors are obtained as follows.
B
BA BC
2I
4I
4I
0.5 0.5
Table 2.78 Member end moments for symmetrical loading
Members AB BA BC
DF 0 0.5 0.5
FEMSBal
– 26.67 +26.67– 13.33 – 13.34
CO – 6.67
Bal 0.00 0.00 0.00
Total – 33.34 13.34 – 13.34
The analysis for antisymmetrical distribution is carried out using cantilever moment distribution. The stiffness for the column is taken as (EI/4) and for the beam it is taken as six times the original stiffness. Hence, 6E(2I)/4 = EI. The fi xed end moments for the columns are taken as
MFAB = 2 2 2420 106.67 kNm
12 2 2 3 3wl wl l wlÊ ˆ- - = - = - ¥ = -Á ˜Ë ¯
MFBA = 22 2 20 4
53.33 kNm12 2 2 6 6wl wl l wl ¥Ê ˆ+ - = - = - = -Á ˜Ë ¯
The carryover factor to the column is (– 1) in antisymmetrical loading. The distribution factors for the antisymmetrical loading is
k
k
DF = kkS
244 Indeterminate Structural Analysis
B
BA BC
4I
46(2I)
0.077 0.923
13
1 12
Table 2.79 Member end moments for symmetrical loading
Members AB BA BC
DF 0 0.77 0.923
FEMSBal
– 106.67 – 53.33+4.17 +49.16
CO 4.17
Bal 0.00 0.00 0.00
Final – 110.84 – 49.16 +49.16
The end moments of the members were obtained by adding the moments due to symmetric loading and antisymmetric loading.
Members AB BA BC CB CD DC
Symmetric Loading – 33.34 +13.34 – 13.34 +13.34 – 13.34 +33.34
Antisymmetric Loading – 110.84 – 49.16 +49.16 +49.16 – 49.16 – 110.84
Total – 144.18 – 35.82 +35.82 +62.5 – 62.50 – 77.50
Joint Memberkkk
kkS
247Moment Distribution Method
C
A
B
E
F
18.47
18.47 18.47 kNm
66.07
66.0711.53 54.56 kNm
54.56
65.4465.44 kNm
FIG. 2.142 Bending moment diagram
2.11 ANALYSIS OF FRAMES WITH INCLINED LEGSExample 2.34 Analyse the rigid frame shown in fi gure by the moment distribution method and draw the BM diagram Support A is hinged and support D is fi xed support.
I
2I
D
C
A
B
2I
6 m 3 m
4 m
10 kN
O
FIG. 2.143
SolutionDetermination of sway (∆) of members
248 Indeterminate Structural Analysis
FIG. 2.144 Sway displacement diagram
Δ
'
C C1
C
O
FIG. 2.145
Consider the above triangle, CC = ∆ cot = 0.75∆ CC = ∆ cosec = 1.25∆ Sway of AB = 1 = ∆AB = BB = +∆
Sway of BC = 2 = ∆BC = – CC = – ∆ cot = – 0.75∆ Sway of CD = 3 = ∆CD = +CC = ∆ cosec = +1.25∆
Sway fi xed end moments MFAB = 0
MFAB = – 3EI11/l21 = 2 33 /4
16EI EI- D = - D
MFBC = MFCB = – 6EI22/l22 = 26 (2 )( 0.75 )/6
4EIE I D- - D =
MFCD = MFDC = – 6EI33/l23 = 2 36 (2 )( 0.25 )/5
5EIE I D- - D =
249Moment Distribution Method
MFBA : MFBC : MFCD = 3 1 3: :
16 4 5- -
– 15 : +20 : – 48
Relative Stiffness k Values and Distribution Factors
Table 2.82 Distribution factors
Joint Members Relative Stiffness Values I/l k DF = k/ k
BA3 34 4 16
I IÊ ˆ =Á ˜Ë ¯ 0.36
B = 0.1875I = 0.5175I
BC2 0.336I I= 0.64
CB 2 0.336I I= 0.45
C 0.73I
CD 2 0.45I I= 0.55
Table 2.83 Sway moment distribution table
Joint A B C DMembers AB BA BC CB CD DCDF 1 0.36 0.64 0.45 0.55 0FEMSBal
– 15.00– 1.80
+20.00– 3.20
+20.00+12.60
– 48.00+15.40
– 48.00
COBal – 2.27
+6.30– 4.03
– 1.60+0.72 +0.88
+7.70
COBal +0.13
+0.36– 0.23
– 2.02+0.91 +1.11
+0.44
COBal – 0.17
+0.46– 0.29
– 0.12+0.05 +0.07
+0.56
COBal – 0.01
+0.03– 0.02
– 0.15+0.07 +0.08
+0.04
COBal – 0.01
+0.04– 0.03
– 0.01+0.005 +0.005
+0.04
– 19.39 19.39 30.46 – 30.46 – 39.22
252 Indeterminate Structural Analysis
10 kN
D
B C
C
C1
A
3 mΔ Δ
'
FIG. 2.148 Elastic curve
Example 2.35 Determine the end moments and draw the bending moment diagram.
6 kN/m
12 kN
2 m
C
D
B
A
3 m
1.5 m
1.5 m
FIG. 2.149
Solution: In inclined frames, the sway of the members is to be determined from geometry. Let the column AB sway by an amount ∆. The beam member BC moves horizontally to BC as shown in Fig. 2.150. The displacement CC is related to sway ∆ and the angle of the inclined member CD. Thus, using the geometry of the defl ected shape the sway of the members are related. Fixed end moments due to loading
MFAB = – 12 × 3 4.5 kNm;8
= -
MFBA = +12 × 310 = +4.5 kNm
MFBC = – 6 23 4.5 kNm
12¥ = -
MFCB = +6 × 23 4.5 kNm
12= +
253Moment Distribution Method
Determination of sway (∆) of the member
6 kN/m
C
CB
C1
D
RCD
RAB
B
A
Δ Δ
''
φ
FIG. 2.150 Sway displacement diagram
C
C1C
'
φ
FIG. 2.151
From ∆CC1C; cot = 1C C ¢D
C1C = cot
cosec = CC ¢
D
CC = cosec
Referring to the above given frame
cot = 2/3 = 0.67
cosec = 13/3 1.20=
Sway of AB = ∆
Sway of BC = – 0.67∆
Sway of CD = +1.20∆
254 Indeterminate Structural Analysis
Sway moment distribution The sway moments are assumed as
MFAB = MFBA = 12 21
6 6 0.6673
EI EI EIl
D- = - = - D
MFBC = MFCB = 2 22 22
6 ( 0.7 )6 0.4473
EIEI EIl
- DD- = = + D
MFCD = MFDC = 3 323
6 (1.2 )6 0.55413
EIEI EIl
DD- = - = - D
MFBA : MFBC : MFCD = – 0.667EI∆ : +0.447EI∆ : – 0.554EI∆
= – 66.70 : +44.70 – 55.40
Table 2.84 Moment distribution table
Joint A B C D
Members AB BA BC CB CD DC
0 0.5 0.5 0.55 0.45 0
FEMSBal
– 66.70 – 66.70+11.00
+44.70+11.00
+44.70+5.89
– 55.40+4.82
– 55.40
COBal
+5.50– 1.48
+2.95– 1.48
+5.50– 3.03 – 2.47
+2.41
COBal
– 0.74+0.76
– 1.52+0.76
– 0.74+0.41 +0.33
– 1.24
COBal
+0.38– 0.10
+0.21– 0.11
+0.38– 0.21 – 0.17
+0.17
COBal
– 0.05+0.06
– 0.11+0.05
– 0.06+0.03 +0.03
– 0.09
COBal
+0.03– 0.01
+0.02– 0.01
+0.03– 0.02 – 0.01
+0.02
Total – 61.58 – 56.47 +56.47 +52.88 – 52.88 – 54.13
MAB MBA MBC MCB MCD MDC
255Moment Distribution Method
Distribution factors
Table 2.85
Joint Members Relative Stiff Values (k) k k/ k
BBA
BC
I/3
I/32I/3
0.5
0.5
CCB
CD
I/3
/ 3I2I/3
0.55
0.45
Table 2.86 Nonsway member distribution table
Joint A B C D
Members AB BA BC CB CD DC
DF 0 0.5 0.5 0.55 0.45 0
FEMSBal
– 4.50 +4.500
– 4.500
+4.50– 2.48
0– 2.02
0
COBal
0+0.62
– 1.24+0.62
00 0
– 1.01
COBal
+0.310
00
+0.31– 0.17 – 0.14
0
COBal
0+0.05
– 0.09+0.04
00 0
– 0.07
COBal
+0.030
00
+0.02– 0.01 – 0.01
0
Total – 4.16MAB
+5.17MBA
– 5.17MBC
2.17MCB
– 2.17MCD
– 108MDC
– 61.58MAB
– 56.47MBA
+56.47MBC
52.88MCB
– 52.88MCD
– 54.13MDC
End moments The end moments are the sum of the moments of the nonsway and sway moments.
MAB = – 4.16 – 61.58 k
MBA = +5.17 – 56.47 k
MBC = – 5.17 + 56.47 k
MCB = +2.17 + 52.88 k
256 Indeterminate Structural Analysis
MCD = – 2.17 – 52.88 k
MDC = – 1.08 – 54.13 k
Column shear equation
FIG. 2.152(a)
BMBA
SBA
A
12 kN
MAB
SAB
NA
NB
1.5 m
1.5 m
FIG. 2.152(b) FIG. 2.152(c)
259Moment Distribution Method
FIG. 2.156
cosec = 5/4 = 1.25
cot = 3/4 = 0.75
From the displacement diagram
cot = 1B B¢D
B1B = cot
cosec = 1
BBBB
¢
BB = BB1 cosec
BB = 1.25∆
CC = 1.25∆
Vertical displacement between B and C
= BB1 + C1C = 2cot = 1.5∆
Sway of AB = AB = BB = cosec = +1.25∆
Sway of BC = ∆BC = – 1.5∆
Sway of CD = ∆CD = cosec = +1.25∆
Nonsway Analysis
MFBC = – 100 × = -8 100 kNm8
MFCB = +100 × 8 100 kNm8
= +
260 Indeterminate Structural Analysis
A
MAB
S ABNA
MBA
M CD
MDC
SCD
SDC
SBA
NBNB
ND
4 m
5 m
3 m
5 m
25 kN
B
D
C
(a) (b)
FIG. 2.157
Considering the above free body diagram,
MB = 0;
MAB + MBA + 5 SAB = 0
i.e. SAB = ( )
5AB BAM M+
-
(1)
MC = 0;
MCD + MDC – 5 SDC = 0
SDC = ( )
5CD DCM M+
- (2)
On simplifying
(MAB + MDC) + 1.75(MBA + MCD) + 100 = 0
The values of the moments are obtained from the above nonsway moment distribution and sway moment distribution case as
{(38.27 – 26.39 k) – (38.27 + 24.78 k)} + 1.75{(76.55 – 22.74 k) + (– 76.55 – 24.78 k)} + 100 = 0
263Moment Distribution Method
– 51.17 k – 83.16 k + 100 = 0
k = 0.74
Final moments MAB = 38.27 – 26.39 k(0.74) = 18.74 kNm
MBA = 76.55 – 22.74 k(0.74) = 59.72 kNm
MBC = – 76.55 + 22.74(0.74) = – 59.72 kNm
MCB = +76.55 + 24.78(0.74) = 94.88 kNm
MCD = – 76.55 – 24.78(0.74) = – 94.88 kNm
MDC = – 38.27 – 27.40(0.74) = – 58.55 kNm
CB
59.7294.88 kNm122.7
18.74 kNm 58.7A D
FIG. 2.159 Bending moment diagram
ME = 8 1100 (59.72 94.88) 122.7 kNm4 2
¥ - + =
FIG. 2.160 Elastic curve
2.12 ANALYSIS OF GABLE FRAMESExample 2.37 Analyse the rigid frame shown in Fig. 2.161 by the moment distribution method and sketch the bending moment diagram.
264 Indeterminate Structural Analysis
C
DB
A E
200 kN
3 m
8 m
4 m4 m
1.5I1.5I
II
φ
FIG. 2.161
SolutionSway (∆) values of the members The Gable frame sways equally on both sides (either side) at B and D by virtue of symmetry on the frame and the loading on YY axis.
Sway of AB = ∆AB = – BB = – ∆ (negative sign because AB sways in theanticlockwise direction with respect to AB)
Sway of BC = ∆BC = +C1C = +cosec
∆BC = 53
+ D
Sway of CD = ∆CD = – C2C = – cosec
∆CD = 53
- D
Sway of DE = ∆DE = DD
∆DE = +∆
265Moment Distribution Method
FIG. 2.162 Sway diagram FIG. 2.163
Sway moment values(Fixed end moments due to sway)
MFAB = MFBA = 1 12 2
6 6 ( )8 64 64EI EIEI EI- D - D
- = - = - = +
MFBC = MFCB = 2 22 22
6 (1.5 ) (5/3 )6 155 25
EEI EIl
- D DD D- = - = -
MFCD = MFDC = 3 32 23
6 (1.5 ) ( 5/3 )6 155 25
E ZEI EIl
- - DD D- = - = +
MFDE = MFED = 4 42 24
66 68 64EIEI EI
l- DD D= - = -
6 /64 10
15 / 25 64FAB
FBC
EIMM EI
+ D += =
- D -
Assume the sway moments in the above proportion as follows:
MFAB = MFBA = +10.00 kNm
MFBC = MFCB = – 64.00 kNm
MFCD = MFDC = +64.00 kNm
MFDE = MFED = – 10.00 kNm
Distribution factorsDue to symmetry, consider joints B and C only.
267Moment Distribution Method
C
B
MBC
200 kN
VBC
HCB
MCB
VCB
4 m
B
AMAB
100 kN
100 kN
HBC
MBA
HAB
HBA
8 m 3 m
Column AB Member BC
FIG. 2.164(b) FIG. 2.164(c)
In Fig. 2.164(b);
MB = 0;
– 8HAB + MBA + MAB = 0
HAB = ( )
8AB BAM M+
HBC = HBA = HAB = (MAB + MBA)/8
VAB = VBC = VBA = 100 kN
Referring to Fig. 2.164(c) and taking moment about (c),
4 VBC – 3 HBC + MBC + MCB = 0
Substituting the value of HBC in terms of moments,
( )4(100) 3 ( ) 0
8AB BA
BC CBM M
M M+
- + + =
3200 – 3(MAB + MBA) + 8(MBC + MCB) = 0
Substituting the end moments from moment distribution table with a multiplier ‘k’;
3200 – 3(18.1 k + 26.2 k) + 8(– 26.2 k – 45.1 k) = 0
k = 4.55
269Moment Distribution Method
Solution: The Gable frame sways equally on either side B and D by virtue of symmetry of the frame and loading.
A
C
C
E
D DBB
Sway Diagram
θ
θ θ
θ
ΔΔ
ΔC1 C2
′′′
FIG. 2.167 Sway diagram
FIG. 2.168
In Fig. 2.166 length BC = 2 26 3 6.71 m+ =
cosec θ = 6.71/3 = 2.237
In Fig 2.167 cosec θ = 1C C ¢D
C1C = cosec θ = 2.237∆
In Fig. 2.167 Sway of AB = BB = – ∆
Sway of BC =C1C = +2.237∆
Sway of CD =C2C = – 2.237∆
Sway of ED = DD = +∆
270 Indeterminate Structural Analysis
Fixed end moments
Due to loading
6 m
3 m
B
C
25 kN/m
FIG. 2.169
MFAB = MFBA = MFDE = MFED = 0
MFBC = – 25 × 26 75 kNm
12 FCDM= - =
MFCB = +25 × 26 75 kNm
12 FDCM= + =
Sway moment values
MFAB = MFBA = 1 12 2
6 (2 )( ) 126 36
E IEI EI-D- = - =
MFBC = MFCB = 2 22 22
40.276 (3 )(2.237 )66.71 45
EIE IEIl
- DDD- = - =
15
40.27FAB
FBC
MM
+=
-
Assume the sway moments in the above proportion as
MFAB = MFBA = +15.00 kNm
MFBC = MFCB = – 40.27 kNm
Distribution factors Due to symmetry, consider joints B and C only.
272 Indeterminate Structural Analysis
6 m
3 m
HCB
MCB
MBC
BHBC
VBC = 150 kN
C25 kN/m
FIG. 2.171
MC = 0
150(6) + MBC + MCB – 3HBC – 25 × 26 0
2=
As HBC = HBA = HAB and replacing in terms of moments as above.
150 × 6 + MBC + MCB – 3 ( ) 450 06 AB BAM M+ - =
900 + (64.13 – 58.94 k) – 0.5(48.38 + 46.3 k) – 450 = 0
489.94 – 82.09 k = 0
k = 5.97
End momentsThe end moments are obtained by substituting the value of k in the above equations as
MAB = 16.13 + 20.43(5.97) = 138.1 kNm
MBA = 32.25 + 25.87(5.97) = 186.7 kNm
MBC = – 32.25 – 25.87(5.97) = – 186.7 kNm
MCB = +96.38 – 33.07(5.97) = – 101.0 kNm
HAB = 54.13 kN6
AB BAM M+=
From geometry; cos θ = 6/6.71 = 0.894 and
sin θ = 3/6.71 = 0.447
273Moment Distribution Method
25 kN/m
6 m
150 kN186.7
186.7109.9
54.13
101.0
54.13 kN
24.2
24.2
101.048.39
101.0
115.44
(6.71 − x)
x
C
θ
θ
SFD
FIG. 2.172 Bending moment diagram (BMD)
54.13 kN
54.13 kN
−
SFD
(b)
186.7 kNm
138.1 kNm
−
+
BMD
(c)
6 m
138.11 kNm
54.13 kN
54.13
186.67
B
A
(a)
FIG. 2.173
Referring to shear force diagram,
109.96.71 24.2
xx
=-
x = 5.5 m
186.67
275Moment Distribution Method
MFCD = MFDC = 2 22 22
6 6 34 8
EI EI EIl
D D- = - = D
MEF = MFE = 3 32 23
6 6 63 9
EI EI EIl
D D D- = - =
MFBC = MFCB = 0
MFCE = MFEC = 0
MFBA : MFCD : MFEF = 12 3 6: :
25 8 9EI EI EID D D- - -
MFAB : MFCD : MFEF = – 0.48∆ : – 0.375∆ : – 0.67∆
Hence, MFAB : MFCD : MFEF = – 48.00 : – 37.50 : – 66.7 kNm
54
A
B C
D
E
F
3I I
I
2I
2I
Δ Δ Δ
FIG. 2.177
Distribution factors
Table 2.93
Joint Members Relative Stiffness Values (k) I/l k/ k
BBA
BC
2I/5
2I/54I/5
0.50
0.50
CCBCDCE
2I/5I/4I/4
0.9I0.440.280.28
EEC
EF
I/4
I/30.58I
0.43
0.57
277Moment Distribution Method
Column shear equation
MB = 0
MAB + MBA + 5HA = 0
HA = – 0.2(MAB + MBA)
MC = 0
MCD + MDC + 4HD = 0
HD = – 0.25(MCD + MDC)
ME = 0
MEF + MFE + 3HF = 0
HF = – 0.33(MEF + MFE)
Considering the horizontal equilibrium 80 – HB – HD – HF = 0
80 + 0.2(MAB + MBA) + 0.25(MCD + MDC) + 0.33(MEF + MFE) = 0
FIG. 2.178
5 m
MBA
MAB
HA
VA
B
A
FIG. 2.179
4 m
MCD
MDC
HD
HC
VD
VC
C
D
FIG. 2.180
3 m
MEF
MFE
HF
E
F
278 Indeterminate Structural Analysis
80 + {0.2(– 36.66 – 25.36) + 0.25(– 34.09 – 35.77) + 0.33(– 29.68 – 48.18)}k = 0
k = 1.44 Final momentsThe moments obtained at the end of moment distribution table is multiplied by k and hence
MAB = – 52.8 kNm, MBA = – 36.5 kNm, MBC = – 36.5 kNm
MCB = +24.1 kNm, MCD = – 49.1 kNm, MDC = – 51.5 kNm
MCE = +35.4 kNm, MEC = +42.7 kNm, MEF = – 42.7 kNm
MFE = – 69.4 kNm
35.4 kNm
42.7 kNm
69.4
51.5
52.79
36.5 ECB
A
D
F
FIG. 2.181 Bending moment diagram
REVIEW QUESTIONSRemembrance 2.1. Who has developed the moment distribution method? 2.2. Is the moment distribution a stiffness method or fl exibility method? 2.3. List the advantages of the moment distribution method? 2.4. List the important steps in the moment distribution method? 2.5. Does the axial deformation considered in the development of the moment
distribution method? 2.6. Defi ne rotational stiffness? 2.7. Explain distribution factor? 2.8. Defi ne carry over factor? 2.9. Which moment distribution is preferable for symmetric frames subjected to lateral
loads as storey heights?
279Moment Distribution Method
2.10. What are the other names for cantilever moment distribution method? 2.11. What are the advantages of Naylor’s method? 2.12. What is the rotational stiffness of a cantilever? 2.13. What is the magnitude of a stiffness of a member under antisymmetric bending
in Naylor’s method? 2.14. List the values of symmetric stiffness factor, skew symmetric stiffness factor and
cantilever stiffness factor?
Understanding 2.1. In a member AB, if a moment of 10 kNm is applied at A, what is the moment
carried over to the fi xed end B? 2.2. What is sinking of supports? What is its effect on the end moments of the
member? 2.3. Calculate the MFBA and MFBC for the beam shown in fi gure below due to sinking
of support by 2 mm. E = 6000 N/mm2 and I = 1.6(10)8 mm4.
2 m
B CA
4 m
EI 2EI
2.4. Calculate the MFBC and MFCB for the beam shown in fi gure below EI = 10 × 1011 Nmm2. Support B and C sinks by 2 mm and 3 mm respectively.
3 m 3 mB DA
4 mC
2.5. Is it possible to determine the beam defl ections in a continuous beam by moment distribution method?
2.6. List the reasons for sidesway of the portal frame. 2.7. What is balancing at a joint? 2.8. While applying the moment distribution method, a designer remembers that
“nothing comes back from the fi xed end”. Justify. 2.9. What we can understand from the computed sway correction factor, i.e., if it is
positive what does it indicate as well as if it is negative what does it point out? 2.10. Why sway correction is required when we analyse unsymmetrical frames? 2.11. Does the carry over factor is half for non-prismatic members? 2.12. When a symmetrical structure is subjected to symmetrical loading, it does not
sway. Does this statement applicable to Gable frames? If not, why?