Strength of Material

LESSON 14. Displacement Method: Slope Deflection Equation – 4

14.1 Example

Draw the bending moment diagram for the follwing frame. EI is constant for all members.

Fig.14.1

Step 1: Fixed end Moments

\[M{}_{FBC} =-{{7.5 \times {{10}^2}} \over {12}} =-62.5{\rm{kNm}}\]  ;   \[M{}_{FCB} = {{7.5 \times {{10}^2}} \over {12}} = 62.5{\rm{kNm}}\]

\[M{}_{FAB} = M{}_{FBA} = M{}_{FCD} = M{}_{FDC} = 0\]

Step 2: Slope-Deflection Equaitons

Fig.14.2

Since A and D are fixed ends, θ_A = θ_D = 0

Since axial deformation is neglected, δ_B = δ_C = δ

For span AB,

\[{M_{AB}} = {M_{FAB}} + {{2EI} \over {{L_{AB}}}}\left( {2{\theta _A} + {\theta _B} - {{3\delta } \over {{L_{AB}}}}} \right) = 0.4EI{\theta _B} - 0.24EI\delta \]       (14.1)

\[{M_{BA}} = {M_{FBA}} + {{2EI} \over {{L_{AB}}}}\left( {{\theta _A} + 2{\theta _B} - {{3\delta } \over {{L_{AB}}}}} \right) = 0.8EI{\theta _B} - 0.24EI\delta \]       (14.2)

For span BC,

\[{M_{BC}} = {M_{FBC}} + {{2EI} \over {{L_{BC}}}}\left( {2{\theta _B} + {\theta _C} - {{3\delta } \over {{L_{BC}}}}} \right) =-62.5 + 0.2EI\left( {2{\theta _B} + {\theta _C}} \right)\]            (14.3)

\[{M_{CB}} = {M_{FCB}} + {{2EI} \over {{L_{BC}}}}\left( {2{\theta _C} + {\theta _B} - {{3\delta } \over {{L_{BC}}}}} \right) = 62.5 + 0.2EI\left( {{\theta _B} + 2{\theta _C}} \right)\]            (14.4)

For span CD,

\[{M_{CD}} = {M_{FCD}} + {{2EI} \over {{L_{CD}}}}\left( {2{\theta _C} + {\theta _D} - {{3\delta } \over {{L_{CD}}}}} \right) = 0.8EI{\theta _C} - 0.24EI\delta \]        (14.5)

\[{M_{DC}} = {M_{FDC}} + {{2EI} \over {{L_{CD}}}}\left( {{\theta _C} + 2{\theta _D} - {{3\delta } \over {{L_{CD}}}}} \right) = 0.4EI{\theta _C} - 0.24EI\delta \]        (14.6)

Step 3: Equilibrium Equaitons

At B,

\[{M_{BA}} + {M_{BC}} = 0 \Rightarrow 0.8EI{\theta _B} - 0.24EI\delta-62.5+0.2EI\left( {2{\theta _B} + {\theta _C}} \right) = 0\]     

\[\Rightarrow 1.2EI{\theta _B} + 0.2EI{\theta _C}-0.24EI\delta-62.5=0\]                 (14.7)

At C,

\[{M_{CB}} + {M_{CD}} = 0 \Rightarrow 62.5 + 0.2EI\left( {{\theta _B} + 2{\theta _C}} \right) + 0.8EI{\theta _C}--0.24EI\delta= 0\]        

\[\Rightarrow 0.2EI{\theta _B} + 1.2EI{\theta _C}-0.24EI\delta+62.5=0\]                (14.8)

Step 3: Additional Shear Equation

Free body diagram of each member are shown bellow.

Fig.14.3

Summation of force in horizontal direction is zero \[ \Rightarrow \sum {{F_x} = 0}\]

Here, horizontal forces are, external horizontal force of 10 kN and shear forces V_A and V_B respectively at support A and D.

From the above FBD, V_A and V_B may be expressed as,

\[{V_A} = {{{M_{AB}} + {M_{BA}}} \over {{L_{AB}}}} = {{0.4EI{\theta _B} - 0.24EI\delta+0.8EI{\theta _B}-0.24EI\delta }\over 5}={{1.2EI{\theta _B}-0.48EI\delta } \over 5}\]

\[{V_B} = {{{M_{AB}} + {M_{BA}}} \over {{L_{AB}}}} = {{0.8EI{\theta _C} - 0.24EI\delta+0.4EI{\theta _C}-0.24EI\delta }\over 5}={{1.2EI{\theta _C}-0.48EI\delta } \over 5}\]

Now,

\[\sum {{F_x} = 0}\Rightarrow {V_A} + {V_B} + 10 = 0\]

\[{{1.2EI{\theta _B} - 0.48EI\delta } \over 5} + {{1.2EI{\theta _C} - 0.48EI\delta } \over 5} + 10 = 0\]

\[1.2EI{\theta _B} + 1.2EI{\theta _C}-0.96EI\delta+50= 0\]                 (14.9)

Solving equations (7) – (9), we have,

\[{\theta _B} = {{78.125} \over {EI}}\]  ,   \[{\theta _C} =-{{46.875} \over {EI}}\]  ,    \[\delta=-{{91.1458} \over {EI}}\]

Step 4: End Moment calculation

Substituting,  θ_B,  θ_C and δ into equations (1) – (6), we have,

\[{M_{AB}} = 0.4EI{\theta _B} - 0.24EI\delta= 9.375{\rm{ kNm}}\]

\[{M_{BA}} = 0.8EI{\theta _B} - 0.24EI\delta= 40.625{\rm{ kNm}}\]

\[{M_{BC}} =-62.5 + 0.2EI\left( {2{\theta _B} + {\theta _C}} \right) =-40.625{\rm{kNm}}\]

\[{M_{CB}} = 62.5 + 0.2EI\left( {{\theta _B} + 2{\theta _C}} \right) =59.375{\rm{ kNm}}\]

\[{M_{CD}} = 0.8EI{\theta _C} - 0.24EI\delta=-59.375{\rm{kNm}}\]

\[{M_{DC}} = 0.4EI{\theta _C} - 0.24EI\delta=-40.625{\rm{ kNm}}\]

 Fig. 14.4. Bending Moment Diagram.

 

Suggested Readings

Hbbeler, R. C. (2002). Structural Analysis, Pearson Education (Singapore) Pte. Ltd.,Delhi.

Jain, A.K., Punmia, B.C., Jain, A.K., (2004). Theory of Structures. Twelfth Edition, Laxmi Publications. 

Menon, D.,  (2008), Structural Analysis, Narosa Publishing House Pvt. Ltd., New Delhi.

Hsieh, Y.Y., (1987), Elementry Theory of Structures , Third Ddition, Prentrice Hall.