Problem

Determine the reactions and the force in each member of the trusses shown in Figs. P13.3...

Determine the reactions and the force in each member of the trusses shown in Figs. P13.34–P13.36 using the method of consistent deformations.

Step-by-Step Solution

Solution 1

Step #1 of 24

Draw free body diagram of the frame.

$\\VIZAG-WINFILE01\Mech\Kavya\July\Authoring\35a.jpg$

Step #2 of 24

Calculate the degrees of indeterminacy of the beam.

Here, m is number of members in the truss, r is the number of reactions to be calculated, and j is the number of joints in the truss.

Substitute 8 for m, 3 for r, and 5 for j.

The truss is indeterminate to first degree.

Obtain the primary beam by removing the horizontal member BC from the given indeterminate truss.

Step #3 of 24

Draw the Primary beam subjected with the external loading.

$\\VIZAG-WINFILE01\Mech\Kavya\July\Authoring\35b.jpg$

Step #4 of 24

Calculate the support reactions.

Apply equilibrium conditions.

Consider moment about B

Consider forces along vertical direction.

Substitute for .

Consider forces along horizontal direction.

Step #5 of 24

Consider joint E.

$\\VIZAG-WINFILE01\Mech\Kavya\July\Authoring\1b.jpg$

Step #6 of 24

Calculate the angle made by the member ED with vertical.

Consider forces along horizontal direction.

Consider forces along vertical direction.

Step #7 of 24

Consider joint C.

$\\VIZAG-WINFILE01\Mech\Kavya\July\Authoring\1c.jpg$

Step #8 of 24

Consider forces along horizontal direction.

Consider forces along vertical direction.

Step #9 of 24

Consider joint B.

$\\VIZAG-WINFILE01\Mech\Kavya\July\Authoring\1d.jpg$

Step #10 of 24

Consider forces along horizontal direction.

Consider forces along vertical direction.

Step #11 of 24

Consider joint A.

$\\VIZAG-WINFILE01\Mech\Kavya\July\Authoring\1f.jpg$

Step #12 of 24

Calculate the angle made by the member ED with vertical.

Consider forces along vertical direction.

Step #13 of 24

Now, the primary truss is subjected to a change of temperature. Apply 1 kN tensile force in the redundant member AD.

$\\VIZAG-WINFILE01\Mech\Kavya\July\Authoring\34c.jpg$

Figure (3)

Step #14 of 24

Consider joint E.

$\\VIZAG-WINFILE01\Mech\Kavya\July\Authoring\35d.jpg$

Step #15 of 24

Consider forces along horizontal direction.

Consider forces along vertical direction.

Step #16 of 24

Consider joint C.

$\\VIZAG-WINFILE01\Mech\Kavya\July\Authoring\1.jpg$

Step #17 of 24

Consider forces along horizontal direction.

Consider forces along vertical direction.

Step #18 of 24

Consider joint B.

$\\VIZAG-WINFILE01\Mech\Kavya\July\Authoring\35f.jpg$

Step #19 of 24

Consider forces along horizontal direction.

Consider forces along vertical direction.

Step #20 of 24

Consider joint A.

$\\VIZAG-WINFILE01\Mech\Kavya\July\Authoring\3.jpg$

Consider forces along vertical direction.

Step #21 of 24

Calculate the forces in each member.

Member
AB	3	0	-0.6	0	1.08	71.9
BD	4	-266.67	-0.8	853.33	2.56	-170.8
AC	4	66.67	-0.8	-213.33	2.56	162.5
BC	5	0	1	0	5	-119.8
CD	3	-100	-0.6	180	1.08	-28.1
AD	5	250	1	1250	5	130.2
EC	4	66.67	0	0	0	66.7
ED	5	-83.33	0	0	0	-83.3
			SUM	2070	17.28

Step #22 of 24

Calculate force in the member BC by using the compatibility equation.

…… (1)

Here, is the deflection at D due to external loading, is the relative displacement between the same joints due to a unit value of the redundant.

Calculate the deflection at D due to external loading.

Substitute 2070 for .

Step #23 of 24

Calculate the deflection at D due to unit value.

Substitute 17.28 for

Step #24 of 24

Substitute

for

and

for

in equation (1)

Therefore, force in the member BC is .

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Solutions For Problems in Chapter 13

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