moment distributiom

7/21/2019 Moment Distributiom

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Moment Distribution

Two‐Story Frame with side sway

Antoinette

Karam

MOMENT DISTRIBUTION




Moment Distribution


Antoinette

Karam

Find moments.

IAB=1200 in4 IDE=500 in4

IBC=1000 in4

IEF=600 in4

ICD=450 in4 IBE=600 in4



Moment Distribution


Antoinette

Karam

Stiffness Factor

E is constant.

4

4 1200

40 120

4 1000

25 160

4 450

30 60

4 500 25

80

4 600

40 60

4 600 30

80

Distribution Factor



Moment Distribution


Antoinette

Karam

160

120 80 160

4

9 0.444

80

120 80 160 2

9 0.222

160

60 160

8

11 0.727

60

60 160 3

11 0.273

60

60 80

3

7 0.429

80

60 80

4

7 0.571

80

80 80 60

4

11 0.364

80

80 80 60

4

11 0.364

60 3



Moment Distribution


Antoinette Karam

Frame Side Sway

Loading will create deflection of the frame at point B and C. We

apply an artificial support at B and another one at C. The frame will be

transformed in 3 frames.

We calculate reactions R1 and R2 at B and C respectively, due to

external loads.

We remove the support at B, and replace it by an opposite load

R1’. Then we calculate R1’ and R2’.

We remove the support at C, and replace it by an opposite load

R2’’. Then we calculate R1’’ and R2’’.

Frame 1



Moment Distribution


Antoinette Karam

Fixed End Moment

8

20 40

8 100 .

8

20 40

8 100 .

0

25 10 20

30 55.556 .

25 20 10

30 111.111 .

0

12

3 30

12 225 .

12

3 30

12 225 .

0



Moment Distribution


Antoinette Karam

Joint A F

Member AB BA BE BC CB CD DC DE ED EB EF FE

DF 0 0.333 0.222 0.444 0.727 0.273 0.429 0.571 0.364 0.364 0.273 0

FEM ‐100 100 ‐55.556 0 0 ‐225 225 0 0 111.11 0 0

‐14.7999 ‐9.86657 ‐19.7331 163.5750 61.4250 ‐96.5250 ‐128.475 ‐40.4440 ‐40.4440 ‐30.3330

‐7.399926 ‐20.2220 81.7875 ‐9.86657 ‐48.2625 30.7125 ‐20.2220 ‐64.2375 ‐4.9333 ‐15.1665

‐20.5013 ‐13.6675 ‐27.3351 42.2598 15.8692 ‐4.5004 ‐5.9901 25.1782 25.1782 18.8836

‐10.2507 12.5891 21.1299 ‐13.6675 ‐2.25 02 7 .93 46 12.5 891 ‐2.9950 ‐6.8338 9.4418

‐11.2284 ‐7.4856 ‐1 4.9 71 2 1 1.5 72 2 4 .3 45 5 ‐8.8047 ‐11.7190 3.5777 3.5777 2.6833

‐5.6142 1.7888 5.7861 ‐7.4856 ‐4.4023 2.1728 1.7888 ‐5.8595 ‐3.7428 1.3416

‐2.5225 ‐1.6816 ‐3.3633 8.6425 3.2454 ‐1.6995 ‐2.2621 3.4952 3.4952 2.6214

‐1.2612 1.7476 4.3213 ‐1.6816 ‐0.8498 1.6227 1.7476 ‐1.1310 ‐0.8408 1.3107

‐2.0209 ‐1.3473 ‐2.6946 1.8403 0.6911 ‐1.4459 ‐1.9245 0.7178 0.7178 0.5383

‐1.0105 0.3589 0.9202 ‐1.3473 ‐0.7229 0.3455 0.3589 ‐0.9622 ‐0.6736 0.2692

‐0.4259 ‐0.2839 ‐0.5679 1.5051 0.5652 ‐0.3022 ‐0.4022 0.5955 0.5955 0.4466

‐0.2130 0.2977 0.7525

‐0.2839

‐0.1511 0.2826 0.2977

‐0.2011

‐0.1420 0.2233

‐0.3497 ‐0.2332 ‐0.4663 0.3163 0.1188 ‐0.2490 ‐0.3314 0.1249 0.1249 0.0937

‐0.1749 0.0624 0.1581 ‐0.2332 ‐0.1245 0.0594 0.0624 ‐0.1657 ‐0.1166 0.0468

‐0.0735 ‐0.0490 ‐0.0979 0.2600 0.0976 ‐0.0523 ‐0.0696 0.1027 0.1027 0.0771

‐0.0367 0.0514 0.1300 ‐0.0490 ‐0.0261 0.0488 0.0514 ‐0.0348 ‐0.0245 0.0385

‐0.0604 ‐0.0403 ‐0.0805 0.0546 0.0205 ‐0.0430 ‐0.0572 0.0216 0.0216 0.0162

M1 ‐

125.9610 48.0175 ‐

93.5370 45.6756 195.411 ‐

195.411 154.557 ‐

154.557 ‐

82.2174 87.1721 ‐

4.9729 ‐

2.4945

D EB C



Moment Distribution


Antoinette Karam

Equilibrium

Member

AB

0 → 40 48.0175 125.961 20 20 0

11.95

Member EF

0 → 40 4.9729 2.4945 0

0.19

Member BC

0 → 25 45.6756 195.411 0

9.65

Member DE

0 → 25 154.557 82.2174 0

9.47

Equilibrium of the upper story

0 → 15 0



Moment Distribution


Antoinette Karam

Equilibrium of the whole frame

0 → 20 15 0

14.82 20 15 11.95 0.19 0

7.68

Frame 2

Let 100 .

assumed values for internal moments.

0

Δ′

6

100

6

’ ’

6

6



Moment Distribution


Antoinette Karam

Joint A F


DF 0 0.333 0.222 0.444 0.727 0.273 0.429 0.571 0.364 0.364 0.273 0

FEM ‐100 ‐100 0 0 0 0 0 0 0 0 ‐50 ‐50

33.333 22.222 44.444 0.000 0.000 0.000 0.000 18.182 18.182 13.636

16.66667 9.0909 0.0000 22.2222 0.0000 0.0000 9.0909 0.0000 11.1111 6.8182

‐3.0303 ‐2.0202 ‐4.0404 ‐16.1616 ‐6.0606 ‐3.8961 ‐5.1948 ‐4.0404 ‐4.0404 ‐3.0303

‐1.5152 ‐2.0202 ‐8.0808 ‐2.0202 ‐1.9481 ‐3.0303 ‐2.0202 ‐2.5974 ‐1.0101 ‐1.5152

3.3670 2.2447 4.4893 2.8860 1.0823 2.1645 2.8860 1.3118 1.3118 0.9839

1.6835 0.6559 1.4430 2.2447 1.0823 0.5411 0.6559 1.4430 1.1223 0.4919

‐0.6996 ‐0.4664 ‐0.9328 ‐2.4196 ‐0.9073 ‐0.5130 ‐0.6840 ‐0.9328 ‐0.9328 ‐0.6996

‐0.3498 ‐0.4664 ‐1.2098 ‐0.4664 ‐0.2565 ‐0.4537 ‐0.4664 ‐0.3420 ‐0.2332 ‐0.3498

0.5587 0.3725 0.7450 0.5258 0.1972 0.3943 0.5258 0.2092 0.2092 0.1569

0.2794 0.1046 0.2629 0.3725 0.1972 0.0986 0.1046 0.2629 0.1862 0.0784

‐0.1225 ‐0.0817 ‐0.1633 ‐0.4143 ‐0.1554 ‐0.0871 ‐0.1161 ‐0.1633 ‐0.1633 ‐0.1225

‐0.0612 ‐0.0817 ‐0.2071 ‐0.0817 ‐0.0435 ‐0.0777 ‐0.0817 ‐0.0580 ‐0.0408 ‐0.0612

0.0963 0.0642 0.1284 0.0911 0.0341 0.0683 0.0911 0.0360 0.0360 0.0270

0.0481 0.0180 0.0455 0.0642 0.0341 0.0171 0.0180 0.0455 0.0321 0.0135

‐0.0212 ‐0.0141 ‐0.0282 ‐0.0715 ‐0.0268 ‐0.0150 ‐0.0200 ‐0.0282 ‐0.0282 ‐0.0212

‐0.0106 ‐0.0141 ‐0.0358 ‐0.0141 ‐0.0075 ‐0.0134 ‐0.0141 ‐0.0100 ‐0.0071 ‐0.0106

0.0166 0.0111 0.0222 0.0157 0.0059 0.0118 0.0157 0.0062 0.0062 0.0047

M2 ‐

83.2591 ‐

66 .5 01 6 2 9.6 192 36 .8 82 4 6 .7 72 7 ‐

6.7727 ‐

4.7 90 6 4 .7 90 6 1 3.3 24 1 2 5.7 40 8 ‐

39.0649 ‐

44.5348

B C D E



Moment Distribution


Antoinette Karam

Equilibrium

Member

AB

0 → 40 83.2591 66.5016 0

3.74

Member

BC

0 → 25 36.8824 6.7727 0

1.75

Member DE

0 → 25 4.7906 13.3241 0

0.72

Member EF

0 → 40 39.0649 44.5348 0

2.09


0 → ′ 0



Moment Distribution


Antoinette Karam


0 → ′ ′ 0

′ 1.26 3.74 2.09 0

′ 7.09

Frame 3

Let 100 .

assumed values for internal moments.

0

Δ′′

6

100

6

′′ ′′

100 600

1200 50 .



Moment Distribution


Antoinette Karam

Joint A F


DF 0 0.333 0.222 0.444 0.727 0.273 0.429 0.571 0.364 0.364 0.273 0

FEM 0 0 0 ‐100 ‐100 0 0 ‐50 ‐50 0 0 0

3 3.3 33 3 2 2.2 22 22 4 4.4 44 4 7 2.7 27 3 2 7.2 72 7 2 1.4 28 6 2 8.5 71 4 1 8.1 81 8 1 8.1 81 8 1 3.6 36 4

16.66667 9.0909 36.3636 22.2222 10.7143 13.6364 9.0909 14.2857 11.1111 6.8182

‐15.1515 ‐10.1010 ‐20.2020 ‐23.9538 ‐8.9827 ‐9.7403 ‐12.9870 ‐9.2352 ‐9.2352 ‐6.9264

‐7.5758 ‐4.6176 ‐11.9769 ‐10.1010 ‐4.8701 ‐4.4913 ‐4.6176 ‐6.4935 ‐5.0505 ‐3.4632

5.5315 3.6877 7.3753 10.8881 4.0830 3.9038 5.2051 4.1978 4.1978 3.1484

2.7658 2.0989 5.4441 3 .6877 1.9519 2.0415 2.0989 2.6026 1.8438 1.5742

‐2.5143 ‐1.6762 ‐3.3524 ‐4.1015 ‐1.5381 ‐1.7745 ‐2.3660 ‐1.6169 ‐1.6169 ‐1.2127

‐1.2572 ‐0.8084 ‐2.0508 ‐1.6762 ‐0.8872 ‐0.7690 ‐0.8084 ‐1.1830 ‐0.8381 ‐0.6063

0.9531 0.6354 1.2708 1.8643 0.6991 0.6761 0.9014 0.7349 0.7349 0.5512

0.4765 0.3675 0.9322 0 .6354 0.3380 0.3496 0.3675 0.4507 0.3177 0.2756

‐0.4332 ‐0.2888 ‐0.5776 ‐0.7079 ‐0.2655 ‐0.3073 ‐0.4097 ‐0.2794 ‐0.2794 ‐0.2096

‐0.2166 ‐0.1397 ‐0.3540 ‐0.2888 ‐0.1536 ‐0.1327 ‐0.1397 ‐0.2049 ‐0.1444 ‐0.1048

0.1646 0.1097 0.2194 0.3218 0.1207 0.1168 0.1557 0.1270 0.1270 0.0953

0.0823 0.0635 0.1609 0 .1097 0.0584 0.0603 0.0635 0.0778 0.0549 0.0476

‐0.0748 ‐0.0499 ‐0.0997 ‐0.1222 ‐0.0458 ‐0.0531 ‐0.0708 ‐0.0483 ‐0.0483 ‐0.0362

‐0.0374 ‐0.0241 ‐0.0611 ‐0.0499 ‐0.0265 ‐0.0229 ‐0.0241 ‐0.0354 ‐0.0249 ‐0.0181

0.0284 0.0189 0.0379 0.0556 0.0208 0.0202 0.0269 0.0219 0.0219 0.0164

M3 1 0.9 04 3 2 1.8 37 0 2 0.5 88 9 ‐

42.4260 ‐

28.4894 28.4894 24.9420 ‐

24.9420 ‐

2 8.4 16 1 1 9.3 53 3 9 .0 62 8 4 .5 23 2

B C D E



Moment Distribution


Antoinette Karam

Equilibrium

Member

AB

0 → 40 10.9043 21.8370 0

0.82

Member BC

0 → 25 42.4260 28.4894 0

2.84

Member DE

0 → 25 24.9420 28.4161 0

2.13

Member EF

0 → 40 9.0628 4.5232 0

0.34


0 → ′′ 0



Moment Distribution


Antoinette Karam


0 → ′′ ′′ 0

′′ 4.97 0.82 0.34 0

′ 6.13

Correction Factors

′ ′′ 0

′ ′′ 0

7.68 7.09

6.13

0

14.82 1.26 4.97 0

3 2.2



Moment Distribution


Antoinette Karam

We multiply each moment value of Frame 2 by C'. It results in the following table:

Joint A F


M2 ‐83.2591 ‐66.5016 29.6192 36.8824 6.7727 ‐6.7727 ‐4.7906 4.7906 13.3241 25.7408 ‐39.0649 ‐44.5348

M2 x C' ‐249.7774 ‐199.505 88.8577 110.647 20.3181 ‐20.3181 ‐14.3717 14.3717 39.9723 77.2223 ‐117.195 ‐133.604

We multiply each moment value of Frame 3 by C''. It results in the following table:

Joint A F


M3 10.9043 21.8370 20.5889 ‐42.4260 ‐28.4894 28.4894 24.9420 ‐24.9420 ‐28.4161 19.3533 9.0628 4.5232

M3

x

C'' ‐23.9895 ‐48.0415 ‐45.2957 93.3371 62.6767 ‐62.6767 ‐54.8725 54.8725 62.5155 ‐42.5773 ‐19.9382 ‐9.9510

We calculate MF as for free moment by the following equation:

MF = M1 + M2 x C' + M3 x C''

It reslults in the following table:

Joint A F


MF ‐

399.7279 ‐

199.529 ‐

49.9750 249.6600 278.4059 ‐

278.4059 85.3128 ‐

85.3128 20.2704 121.8171 ‐

142.106 ‐

146.050

B C D E

B C D E

B C D E

moment distributiom

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