METHOD OF CONSISTENT DEFORMATIONDetermine the reactions at the support using the method of consistent deformation.

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SOLMABA0-5BCB 0-15CDD0-30

SOLMACA0-30CDC0-30

Using method of consistent deformation, determine the reactions for each member of the truss shown.

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F:

@ jt. A

T

@ jt. C

T

T

@ jt. D

C

@ jt. A

C

@ jt. C

C

@ jt. D

T

MFULFuLLAB- 1-50.5AD- 1-50.5BC- 1-50.5BD-10-10AC00CD- 1-50.5-34.14214.8284

TAB = 2.07 KN TAD = 2.07 KN T

BC = 2.07 KN TBD = 2.9289 KN CAC = 7.0711 KN TCD = 2.07 KN T

Sample Problems1. Determine the reactions at A and C using Force Method

B as redundant

Deflection at B due to applied loading

Deflection at B due to Redundant R

R = 37.5 KNEquations of equilibrium:

0

2. Determine the moment at B Deflection at A due to appplied loading

Deflection at A due to redundant force

R= 15 KN

DOUBLE INTEGRATION METHOD1. Determine the reactions at A and B using Double Integration method.

@

ANALYSIS OFINDETERMINATEBEAMS

2. Using Double Integration Method, Determine the Reactions and Moment at A and Reactions at B.

4.54.5M = RcX + 15(X-4.5) () 15(X) ()-75.94(X-4.5)0EIy = Rcx + 7.5(x-4.5)2 7.5x2 75.94 (x-4.5)0 EIy = EIy= At x=0 ; y=0C2=0At x= 9 y=00=C1= 1936.4175- At x=9 y=00=0=Rc= 52.73 Kn Ra= 14.77 Kn 9(52.73)+ Ma = 75.94 + 15(4.5)(6.75)Ma= 57 Kn.m

3. Find the reactions at A and B and the Moment at B using Double Integration Method.

BOUNDARY CONDITION:x=0, y=0 C1=0x=0, y=0 C2=0x=0, y=0

DEFLECTION EQUATION:

When x=6, y=0

SAMPLE PROBLEMS1.Find the reaction at the simple support of the propped beam shown in Fig. P-705 and sketch the shear and moment diagrams.

yx=woL y=woxL

Moment at x:

M=RAx12xy(13x)M=RAx16x2yM=RAxx26(woxL)M=RAxwox36L

Thus,EIy=RAxwox36LEIy=RAx22wox424L+C1EIy=RAx36wox5120L+C1x+C2 At x = 0, y = 0, thus C2 = 0At x = L, y = 00=RAL22woL424L+C1C1=woL324RAL22 Thus, the deflection equation isEIy=RAx36wox5120L+(woL324RAL22)x At x = L, y = 00=RAL36woL5120L+(woL324RAL22)L0=RAL36woL4120+woL424RAL320=RAL33+woL430RAL33=woL430

RA=woL10

2. A couple M is applied at the propped end of the beam shown in Fig. P-707. Compute R at the propped end and also the wall restraining moment.

The moment at any point point on the beam which is at distance x from the left support isMx=MRx

By double integration methodEIy=MxEIy=MRxEIy=Mx12Rx2+C1EIy=12Mx216Rx3+C1x+C2 Boundary conditionsAt x = 0, y = 0; C2 = 0 At x = L, y = 0;0=12ML216RL3+C1LC1=16RL212ML At x = L, y' = 0;0=ML12RL2+(16RL212ML)0=M12RL+16RL12M13RL=12M

R=3M2L

3.The propped beam shown in Fig. P -706 is loaded by decreasing triangular load varying from wo from the simple end to zero at the fixed end. Find the support reactions and sketch the shear and moment diagrams.

yLx=woLy=woL(Lx)

Solving for moment equationM=RAx12(x)(wo)(23x)12(x)(y)(13x)M=RAx13wox216x2yM=RAx13wox216x2[woL(Lx)]M=RAxwo3x2wo6L(Lx2x3)

Doing the double integrationEIy=RAxwo3x2wo6L(Lx2x3)EIy=RA2x2wo9x3wo6L(Lx33x44)+C1EIy=RA6x3wo36x4wo6L(Lx412x520)+C1x+C2

Boundary conditionsAt x = 0, y = 0, C2 = 0At x = L, y = 00=RAL36woL436wo6L(L412L420)+C1L+00=RAL36woL436woL4180+C1LC1=woL330RAL26 At x = L, y' = 00=RAL22woL39wo6L(L43L44)+(woL330RAL26)0=RAL22woL39woL372+woL330RAL26RAL23=11woL3120

RA=11woL40

CONJUGATE BEAM METHOD1. I= 360X106mm4I= 180X104mm4 E=200 GPa

0=0=36RA -18MA -1620

MA+30(6)(3) = 25(4)+6RA0 = 6RA MA 440 RA= 87.5 KNMA = 85 KNm

87.5 + RB = 25 + 30(6)RB = 117.52.

Va = Va=

-7Va +-171.5Ra + 1100.4581 +0= Ra= 7.7 kN

7.7 + Rb =

7(30.8)= Mb+Mb= 35.93 Kn.mAREA-MOMENT METHOD1.

0=4Rd- Md -4 -8(2)+10=-MD+4Rb-19tb/d=00=-MD(4)(2) -8 +0=-8MD +

Rd= 5.75 kNMd= 4 Kn.m-2(5) + Rb+5.75=0RB= 4.25 k

2.

ta/c = 00= +12/5)0=Ra= 0.717 kN

0.717 +Rc = Rc= 5.2836 kN

`Mc= 3.132 kN.m

3.

6Ra-Ma -5(6) (3) +10(1) =06Ra Ma -80=0

tb/a=0 Ra= 16.25 kNMa= 17.5 kN.m

16.25+RB= 5(8)Rb= 23.75 kN

4. ta/c=0tc/a=0

ta/c=00= 10(6)(3) +0=180 +

tc/a=00=

Rc= 1.44 kNMc= 1.2 kN.m

Ra +1.44 =0Ra= 1.44 kN

Ma +10+1.2= 10(1.44)Ma = 3.2 kN.m

5.

ta/c =0= 0= 8000/3 Ra +200Ma -48960

tc/a = 0 =)(3)=

Rc +30.24 = 10(12) Rc= 89.76 kN

158.4 +89.76(20) = Mc +10(12)(6+8)Mc= 273.6

3.SAMPLE PROBLEMS: