   # Determine the member end moments and reactions for the frames shown in Figs. P16.17–P16.20 by using the moment-distribution method. FIG. P16.20

#### Solutions

Chapter
Section
Chapter 16, Problem 20P
Textbook Problem
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## Determine the member end moments and reactions for the frames shown in Figs. P16.17–P16.20 by using the moment-distribution method. FIG. P16.20

To determine

Find the member end moments and reactions for the frames.

### 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 EI of the frame is constant.

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

Refer Figure (1),

Calculate the relative stiffness KCA for member AC of the frame as below:

KCA=I8

Calculate the relative stiffness KCD for member CD of the frame as below:

KCD=I10

Calculate the relative stiffness KDC for member DC of the frame as below:

KDC=I10

Calculate the relative stiffness KDB for member DB of the frame as below:

KBD=I8

Calculate the distribution factor DFCA for member CA of the frame.

DFCA=KCAKCA+KCD

Substitute I8 for KCA and I10 for KCD.

DFCA=I8I8+I10=59

Calculate the distribution factor DFCD for member CD of the frame.

DFCD=KCDKCA+KCD

Substitute I8 for KCA and I10 for KCD.

DFCD=I10I8+I10=49

Check for sum of distribution factor as below:

DFCA+DFCD=1

Substitute 59 for DFCA and 49 for DFCD.

59+49=1

Hence, OK.

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

DFDC=KDCKDC+KDB

Substitute I8 for KDB and I10 for KDC.

DFDC=I10I8+I10=49

Calculate the distribution factor DFDB for member DB of the frame.

DFDB=KDBKDC+KDB

Substitute I8 for KDB and I10 for KDC.

DFDB=I8I8+I10=59

Check for sum of distribution factor as below:

DFDC+DFDB=1

Substitute 59 for DFDB and 49 for DFDC

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