Determine the reactions and draw the shear and bending moment diagrams for the beams shown in Figs. P16.1–P16.5 by using the moment-distribution method. FIG. P16.3

Structural Analysis

6th Edition
KASSIMALI + 1 other
Publisher: Cengage,
ISBN: 9781337630931

Chapter
Section

Structural Analysis

6th Edition
KASSIMALI + 1 other
Publisher: Cengage,
ISBN: 9781337630931
Chapter 16, Problem 3P
Textbook Problem
58 views

Determine the reactions and draw the shear and bending moment diagrams for the beams shown in Figs. P16.1–P16.5 by using the moment-distribution method.FIG. P16.3

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.

Calculation:

Consider the elastic modulus E of the beam is constant.

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

Refer Figure 1,

Calculate the relative stiffness KBA for member AB of the beam as below:

KBA=I9

Calculate the relative stiffness KBC for member BC of the beam as below:

KBC=(2I18)=I9

In the above beam, only joint B is free to rotate. Hence, calculate the distribution factor at joint C.

Calculate the distribution factor DFCA for member AC of the beam.

DFBA=KBAKBA+KBC

Substitute I9 for KBA and I9 for KBC.

DFBA=I9I9+I9=0.5

Calculate the distribution factor DFBC for member BC of the beam.

DFBC=KBCKBA+KBC

Substitute I9 for KBA and I9 for KBC.

DFBC=I9I9+I9=0.5

Check for sum of distribution factor as below:

DFBA+DFBC=1

Substitute 0.5 for DFAB and 0.5 for DFBC.

0.5+0.5=1

Hence, OK.

Calculate the fixed end moment for AB.

FEMAB=0

Calculate the fixed end moment for BA.

FEMBA=0

Calculate the fixed end moment for BE.

FEMBE=100×6×122182+100×12×62182=400kNm

Calculate the fixed end moment for EB.

FEMEB=100×12×62182100×6×122182=400kNm

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

Show the section free body diagram of the member AB and BE as in Figure 2.

Consider the member AB of the beam:

Calculate the vertical reaction at the left end of the joint B by taking moment about point A.

+MA=0By,L(9)+100+200=0By,L=3009By,L=33.33kN

Calculate the horizontal reaction at point A by resolving the horizontal equilibrium.

+Fx=0Ax=0

Calculate the vertical reaction at point A by resolving the vertical equilibrium.

+Fy=0Ay+By,L=0Ay+33

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