1. Continuous Beam Analysis - Moment Distribution Method

Determine moment distribution factors and fixed-end moments for the frame members. The moment distribution procedure will be used to analyze the frame. Stiffness factors, carry over factors, and fixed-end moment factors for the beams and columns are determined as follows:

Determine the elastic bending moment diagrams for each of the load patterns per ACI and the maximum moment envelope values for all patterns as shown in Table 1.   

ACI 318-14 (6.4)

1.1. Load Combination

ACI 318-14 (Eq. 5.3.1b)

1.2. Flexural Stiffness of Beams and Columns Ends, K

Where K is referred to as stiffness factor at beam or column end and can be defined as the amount of moment required to rotate the end of the beam or column 1 rad.

ACI 318-14 (Eq. 19.2.2.1.a)

For member AB:

For column Ends:

1.3. Distribution Factor, DF

The distribution factor for a member that is connected to a fixed joint is defined as the fraction of the total resisting moment supplied by this member.

For member AB:

1.4. Carry Over Factor, COF

COF = 0.5

Where COF is the Carry-Over Factor that represents the fraction of the moment that is “carried over” from the joint to the beam end when the beam far end is fixed.

1.5. Fixed-End Moments, FEMs

For a beam with uniformly distributed load and fixed ends, FEM can be found using the following equation:

For member AB for Load Pattern I:

1.6. Beam Analysis Using Moment Distribution Method

Repeat the previous steps to all frame members to obtain the parameters necessary for the analysis. Moment distribution for the five loading conditions is shown in Table 1. Counter-clockwise rotational moments acting on member ends are taken as positive. Maximum positive span moments are determined from the following equation:

 

Where:

• Mmax+

=

Maximum positive moment in the span

• ML-

=

Negative moment in the left support

• MR-

=

Negative moment in the right support

• l1

=

The span length

For Load Pattern I:

Maximum positive moment in spans A-B.

 

Where:

ML- = 99.75 kip-ft                  MR- = 109.38 kip-ft

Maximum positive moment in spans B-C.

 

Where:

ML- = 52.44 kip-ft                  MR- = 35.74 kip-ft

Maximum positive moment in spans C-D.

 

Where:

ML- = 48.13 kip-ft                  MR- = 40.29 kip-ft

Table 1 - Moment Distribution‡

0.3

Joint

A

B

C

D

Member

AB

BA

BC

CB

CD

DC

DF

0.130

0.107

0.179

0.174

0.130

0.158

COF

0.500

0.500

0.500

0.500

0.500

0.500

Load Pattern I (S2*)

LP I

0.4

Joint

A

B

C

D

Member

AB

BA

BC

CB

CD

DC

FEM

110.44

-110.44

39.76

-39.76

46.68

-46.68

Dist.

-14.40

7.57

12.62

-1.20

-0.90

7.37

CO

3.79

-7.20

-0.60

6.31

3.69

-0.45

Dist.

-0.49

0.84

1.39

-1.74

-1.30

0.07

CO

0.42

-0.25

-0.87

0.70

0.04

-0.65

Dist.

-0.05

0.12

0.20

-0.13

-0.10

0.10

CO

0.06

-0.03

-0.06

0.10

0.05

-0.05

Dist.

-0.01

0.01

0.02

-0.03

-0.02

0.01

CO

0.00

0.00

-0.01

0.01

0.00

-0.01

Dist.

0.00

0.00

0.00

0.00

0.00

0.00

M-

(kip-ft)

99.75

-109.38

52.44

-35.74

48.13

-40.29

M+

(kip-ft)

61.13

15.84

25.86

Load Pattern II (Odd*)

LP II

0.5

Joint

A

B

C

D

Member

AB

BA

BC

CB

CD

DC

FEM

110.44

-110.44

26.26

-26.26

70.68

-70.68

Dist.

-14.40

9.02

15.03

-7.73

-5.79

11.16

CO

4.51

-7.20

-3.86

7.52

5.58

-2.90

Dist.

-0.59

1.19

1.98

-2.28

-1.71

0.46

CO

0.59

-0.29

-1.14

0.99

0.23

-0.85

Dist.

-0.08

0.15

0.26

-0.21

-0.16

0.13

CO

0.08

-0.04

-0.11

0.13

0.07

-0.08

Dist.

-0.01

0.02

0.03

-0.03

-0.03

0.01

CO

0.01

-0.01

-0.02

0.01

0.01

-0.01

Dist.

0.00

0.00

0.00

0.00

0.00

0.00

M-

(kip-ft)

100.54

-107.60

38.43

-27.86

68.87

-62.76

M+

(kip-ft)

61.60

6.42

40.23

Load Pattern III (S3*)

LP III

0.6

Joint

A

B

C

D

Member

AB

BA

BC

CB

CD

DC

FEM

72.94

-72.94

39.76

-39.76

70.68

-70.68

Dist.

-9.51

3.56

5.93

-5.38

-4.03

11.16

CO

1.78

-4.76

-2.69

2.96

5.58

-2.02

Dist.

-0.23

0.80

1.33

-1.49

-1.11

0.32

CO

0.40

-0.12

-0.74

0.66

0.16

-0.56

Dist.

-0.05

0.09

0.15

-0.14

-0.11

0.09

CO

0.05

-0.03

-0.07

0.08

0.04

-0.05

Dist.

-0.01

0.01

0.02

-0.02

-0.02

0.01

CO

0.01

0.00

-0.01

0.01

0.00

-0.01

Dist.

0.00

0.00

0.00

0.00

0.00

0.00

M-

(kip-ft)

65.36

-73.38

43.67

-43.07

71.20

-61.74

M+

(kip-ft)

40.07

16.26

39.60

Load Pattern IV (Even*)

LP IV

0.7

Joint

A

B

C

D

Member

AB

BA

BC

CB

CD

DC

FEM

72.94

-72.94

39.76

-39.76

46.68

-46.68

Dist.

-9.51

3.56

5.93

-1.20

-0.90

7.37

CO

1.78

-4.76

-0.60

2.96

3.69

-0.45

Dist.

-0.23

0.57

0.96

-1.16

-0.87

0.07

CO

0.29

-0.12

-0.58

0.48

0.04

-0.43

Dist.

-0.04

0.07

0.12

-0.09

-0.07

0.07

CO

0.04

-0.02

-0.04

0.06

0.03

-0.03

Dist.

0.00

0.01

0.01

-0.02

-0.01

0.01

CO

0.00

0.00

-0.01

0.01

0.00

-0.01

Dist.

0.00

0.00

0.00

0.00

0.00

0.00

M-

(kip-ft)

65.26

-73.62

45.54

-38.72

48.59

-40.09

M+

(kip-ft)

40.01

17.56

25.75

Load Pattern V (All*)

LP V

0.8

Joint

A

B

C

D

Member

AB

BA

BC

CB

CD

DC

FEM

110.44

-110.44

39.76

-39.76

70.68

-70.68

Dist.

-14.40

7.57

12.62

-5.38

-4.03

11.16

CO

3.79

-7.20

-2.69

6.31

5.58

-2.02

Dist.

-0.49

1.06

1.77

-2.07

-1.55

0.32

CO

0.53

-0.25

-1.03

0.88

0.16

-0.78

Dist.

-0.07

0.14

0.23

-0.18

-0.14

0.12

CO

0.07

-0.03

-0.09

0.11

0.06

-0.07

Dist.

-0.01

0.01

0.02

-0.03

-0.02

0.01

CO

0.01

0.00

-0.02

0.01

0.01

-0.01

Dist.

0.00

0.00

0.00

0.00

0.00

0.00

M-

(kip-ft)

99.85

-109.14

50.57

-40.10

70.74

-61.94

M+

(kip-ft)

61.19

14.42

39.73

Envelop of Maximum Moments

0.9

10

Span

AB

BC

CD

ML

MR

ML

MR

ML

MR

Max Mu-
(Column Center Line)

100.54

-109.38

52.44

-43.07

71.20

-62.76

Max Mu-
(Column Face)

83.53

-91.92

41.57

-32.97

57.22

-49.30

Max Mu+

61.60

17.56

40.23

‡ Moments units are kip-ft

* Live load pattern designation in spBeam