   # 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. P.16.14

#### Solutions

Chapter
Section
Chapter 16, Problem 14P
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. P.16.14 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 modulus of elasticity E is constant.

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

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

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

KED=2I20=I10

Calculate the relative stiffness KEG for member EG of the beam as below:

KEG=I20

Calculate the distribution factor DFED for member ED of the beam.

DFED=KEDKED+KEG

Substitute I10 for KED and I20 for KEG.

DFED=I10I10+I20=23

Calculate the distribution factor DFEG for member EG of the beam.

DFEG=KEGKED+KEG

Substitute I10 for KED and I20 for KEG.

DFEG=I20I10+I20=13

Check for sum of distribution factor as below:

DFED+DFEG=1

Substitute 23 for DFED and 13 for DFEG.

23+13=1

Hence, OK.

Calculate the fixed end moment for DE.

FEMDE=1.5×20212=50kft

Calculate the fixed end moment for ED.

FEMED=1.5×20212=50kft

Calculate the fixed end moment for EG.

FEMEG=40×208=100kft

Calculate the fixed end moment for GE.

FEMGE=100kft

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;

MAC=108.3kft

MCA=83.3kft

MCD=83.3kft

MCB=33.3kft

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

Consider the member DE of the beam:

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

+MD=0Ey,L(20)1.5×(20)×(202)+33.383.3=0Ey,L(20)=350Ey,L=35020Ey,L=17.5k

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

+Fy=0Dy,R(1.5×20)+Ey,L=0Dy,R30+17.5=0Dy,R=12.5k

Consider the member EG of the beam:

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

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