## LESSON 17. Displacement Method: Moment Distribution Method – 3

17.1 Introduction: In this lesson the application of the Moment Distribution Method in continusous beam is illustrated via two examples.

Example 1

Draw the bending moment diagram for the following continuous beam. All spans have constant EI. Fig. 17.1.

From lesson 15.1.3 we have,

${k_{BA}} = {{4E{I_{BA}}} \over {{L_{BA}}}} = {{4EI} \over 5}$  and  ${k_{BC}} = {{3E{I_{BC}}} \over {{L_{BC}}}} = {{3EI} \over 5}$

Distribution factors for BA and BC are,

$D{F_{BA}} = {4 \over 7}$  and  $D{F_{BC}} = {3 \over 7}$

End A is fixed and therefore no moment will be carrid over to B from A. Carry over factors for other joints,

${C_{BA}} = {1 \over 2}$ ,  ${C_{BC}} = {1 \over 2}$  ,  ${C_{CB}} = {1 \over 2}$

Fixed end moments are,

$M{}_{FAB}=-{{3 \times {5^2}} \over {12}}=-6.25{\rm{ kNm}}$ ;  $M{}_{FBA} = {{3 \times {5^2}} \over {12}} = 6.25{\rm{ kNm}}$

$M{}_{FBC}=-{{10 \times 2 \times {3^2}} \over {{5^2}}}=-7.2{\rm{ kNm}}$ ;  $M{}_{FCB} = {{10 \times 3 \times {2^2}} \over {{5^2}}} = 4.8{\rm{ kNm}}$

Calculations are performed in the following table.  Fig. 17.2: Bending moment diagram (in kNm).

Example 2

Replace the fixed support at A by a hinge in the continuous beam shown in Example 1 and determine the  bending moments. Fig. 17.3 .

From lesson 15.1.3 we have,

${k_{BA}} = {{4E{I_{BA}}} \over {{L_{BA}}}} = {{4EI} \over 5}$  and  ${k_{BC}} = {{3E{I_{BC}}} \over {{L_{BC}}}} = {{3EI} \over 5}$

Distribution factors for BA and BC are,

$D{F_{BA}} = {4 \over 7}$  and  $D{F_{BA}} = {3 \over 7}$

Carry over factors,

${C_{AB}} = {1 \over 2}$ ,  ${C_{BA}} = {1 \over 2}$ ,  ${C_{BC}} = {1 \over 2}$ , ${C_{CB}} = {1 \over 2}$

Fixed end moments are,

$M{}_{FAB}=-{{3 \times {5^2}} \over {12}}=-6.25{\rm{ kNm}}$ ;  $M{}_{FBA} = {{3 \times {5^2}} \over {12}} = 6.25{\rm{ kNm}}$

$M{}_{FBC}=-{{10 \times 2 \times {3^2}} \over {{5^2}}}=-7.2{\rm{ kNm}}$ ;  $M{}_{FCB}=-{{10 \times 3 \times {2^2}} \over {{5^2}}} = 4.8{\rm{ kNm}}$

Calculations are performed in the following table.  Fig. 17.4. Bending moment diagram (in kNm).