   # 16.8 through 16.14 Determine the reactions and draw the shear and bending moment diagrams for the beams shown in Figs. P16.8–P16.14 by using the moment-distribution method. FIG. P16.13

#### Solutions

Chapter
Section
Chapter 16, Problem 13P
Textbook Problem
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## 16.8 through 16.14 Determine the reactions and draw the shear and bending moment diagrams for the beams shown in Figs. P16.8–P16.14 by using the moment-distribution method.FIG. P16.13 To determine

Find the reaction and plot the shear and bending moment diagram.

### Explanation of Solution

Fixed end moment:

Formula to calculate the relative stiffness for fixed support IL and for roller support (34)(IL).

Formula to calculate the fixed moment for point load with equal length are PL8.

Formula to calculate the fixed moment for point load with unequal length are Pab2L2 and Pa2bL2.

Formula to calculate the fixed moment for UDL is WL212.

Formula to calculate the fixed moment for UVL are WL230 and WL220.

Formula to calculate the fixed moment for deflection is 6EIΔL2.

Calculation:

Consider the flexural rigidity of the beam EI is constant.

Show the free body diagram of the entire beam as in Figure 1.

Refer Figure 1,

The beam distance and load are symmetric. Hence, consider right half CE member only.

Calculate the relative stiffness KDC for member CD of the beam as below:

KDC=I6

Calculate the relative stiffness KDE for member DE of the beam as below:

KDE=(34)I6=I8

Calculate the distribution factor DFDC for member DC of the beam.

DFDC=KDCKDC+KDE

Substitute I6 for KDC and I8 for KCE.

DFDC=I6I6+I8=0.571

Calculate the distribution factor DFDE for member DE of the beam.

DFDE=KDEKDC+KDE

Substitute I6 for KDC and I8 for KDE.

DFDE=I8I6+I8=0.429

Check for sum of distribution factor as below:

DFDC+DFDE=1

Substitute 0.571 for DFDC and 0.429 for DFDE.

0.571+0.429=1

Hence, OK.

Calculate the fixed end moment for CD.

FEMCD=15×6212=45kNm

Calculate the fixed end moment for DC.

FEMDC=15×6212=45kNm

Calculate the fixed end moment for DE.

FEMDE=45kNm

Calculate the fixed end moment for ED.

FEMDE=45kNm

Show the calculation of final moments using moment distribution method as in Table 1.

Due to symmetry, the final moment of members AB and BC as follows;

MAB=0kNm

MBA=57.9kNm

MBC=57.9kNm

MCB=38.6kNm

Show the section free body diagram of the member CD and DE as in Figure 2.

Consider the member CD of the beam:

Calculate the vertical reaction at the left end of the joint D by taking moment about point C.

+MC=0Dy,L(6)15×(6)×(62)57.9+38.6=0Dy,L(6)=289.3Dy,L=289.36Dy,L=48.2kN

Calculate the vertical reaction at the right end of the joint C by resolving the vertical equilibrium.

+Fy=0Cy,R(15×6)+Dy,L=0Cy,R120+48.2=0Cy,R=41.8kN

Consider the member DE of the beam:

Calculate the vertical reaction at the right end of the joint D by taking moment about point E.

+ME=0Dy,R(6)+(15×6×62)+57

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