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Analyze the beam by flexibility method or force method and draw BMD

Subject:- Structural Analysis II

Title:- Flexibility Method

Difficulty:- Hard

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1 Answer
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1) DSE = r-3 = 4-3 = 1

Treating VC as redunded rxn. Let us remove it and obtain statically determine as below :

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ΣMA = 0

5332VB38 =0

VB = 4.899KN

ΣFY = 0

VA(53)+4.833 =0

VA = 10.167KN

Now draw the diagram with no external load. and apply 1KN source which we have remove and find reaction. [M1 analysis]

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ΣMA = 0

-VB3 - 17 = 0

VB = -2.33KN

ΣFY 0

VA -2.33 +1 =0

VA = 1.33KN

2) Table:- Finding displacement of released structure

Part origin limit M M1
AB A 0-3 10.167x5xx2 1.33x
BD B 0-2 10.167(3+x)-5*3(x+1.5)+4.833x 1.33(3+x)-2.33x
DC C 0-2 0 1x

3) ^R1=l0M1.M2EIdx = $\int_{0}^{3} (10.167x - 2.5x^2)(1.33x)dx + \int_{0}^{2} (10.167(3+x) - 53(x+5) + 4.833x)(1.33(3+x) - 2 - 3.3x)dx + \int_{0}^{2} 0(1x)dx= 54.40 + 47.92 +0 ^R_1 = \frac{102.32}{EI}4)f_{11} = \int_{0}^{l} \frac{M_1.M}{EI}dx = \int_{0}^{l} \frac{(M_1)^2}{EI}dx=\int_{0}^{3} \frac{(1.33)^2}{EI} dx + \int_{0}^{2} \frac{(1.33(x+3)-2.33x^2 )^2}{EI}dx + \int_{0}^{2} \frac{(1x)^2}{EI}dx=15.92+18.55+2.67f_{11} = \frac{37.14}{EI}*5) Applying Compatibility Condition:-* [^I] = ^R +  [F][R] [^I_1] =^R_1[F_{11}][R_1] Due to roller support at D the net displacement at c in the direction of redundant is 0: i.e. ^I_1=00=\frac{102.32}{EI} + [\frac{37.14}{EI}][R_C]R_C=2.75KNR_C=2.75KN[V_C=2.75KN]![enterimagedescriptionhere][3]ΣM_A=053\frac{3}{2} - V_B3 - 8 - 2.757 = 0[V_B=11.25KN]ΣF_Y=0V_A(53)+11.252.75=0[V_A=6.5KN]![enterimagedescriptionhere][4]B.MCal:BM_A=0BM_B = 6.53 - 53*\frac{3}{2}=3KN.mBM_B=3KN.mB.M_C$=0

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