Examples of Continuous Beam - Moment Distribution Method - Displacement Method, Strength of Material | Strength of Material Notes - Agricultural Engg - Agricultural Engineering PDF Download

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).