Slide 1
SNPIT & RC,UMRAKHGuided By:- Zalak P. Shah
Subject:- Structural Analysis 2
Topic:- Energy PrincipleNagma Modi130490106065
Prepared by1
CASTIGLIANOS FIRST THEOREMWhen forces act on elastic systems
subject to small displacements, the displacement corresponding to
any force, in the direction of the force, is equal to the partial
derivative of the total strain energy w.r.t. that force.
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Where:Fi = Force at i-th application pointi = Displacement at
i-th point in the direction of FiU = Total strain energy
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CASTIGLIANOS THEOREM We can also use this method to find the
angle of rotation ().
3Where:Mi = Moment at i-th application pointi = Slope at i-th
point resulting from MiU = Total strain energy
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CASTIGLIANOS THEOREM
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U stored in structureF1F2F3Fn123n
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EXAMPLE:CALCULATE AND FOR THE BEAM SHOWN IN FIGURE. TAKE E= AND
I=
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For deflection at BSegmentOriginLimitsMxBCB0 to 1-P.x-xCAC0 to
3-P(x+1)-40x-(x + 1)
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Now apply fictitious moment M at BSegmentOriginLimitsMxBCB0 to
1-M-30x-1CAC0 to 3-M-30(x + 1)-40x-1
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EXAMPLE:COMPUTE HORIZONTAL DISPLACEMENT OF SUPPORT D OF A SIMPLE
FRAME USING CASTIGLIANOS FIRST THEOREM. TAKE EI=60000kN.m2
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Apply Vertical force P = 40kN at D
SegmentOriginLimitsMxDCD0 to 3-xCBC0 to 3 -3P-90-3BAB0 to 3
-3P-90-20x-3
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For horizontal displacement at D:Apply fictitious horizontal
force P at DSegmentOriginLimitsMxDCD0 to 30CBC0 to
3-120-90-Px-xBAB0 to 3-120-90-P(x + 3)-20x-(x + 3)
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CASTIGLIANOS SECOND THEOREM: If U is the total strain energy
stored up in a frame work in equillibrium under an external system
of force its magnitude Is always a minimum.
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For member in bending :
For the members in axial loading:
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EXAMPLE:Determine reaction at prop for a propped cantilever beam
using castiglianos second theorem. As shown in figure
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SegmentOriginLimitsMxBCB0 to 3xCAC0 to 2X +3
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Let S.F. = 0 at x from B22.91 10x = 0X = 2.29m
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UNIT LOAD METHOD: FOR DETERMINANT STRUCTURE:
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Find and for a beam shown in figure by unit load method EI=
SegmentOriginLimitsMM1M2IBCB0 TO 2-60x00ICAC 0 to 2-1.x-12I
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FOR INDETERMINANT STRUCTURE:
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EXAMPLE:Determine reaction at prop B using unit load method for
a beam shown in figure
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SegmentOriginLimitsMmIBCB0 to 31.XICAC0 to 21.(x + 3)I
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THANK YOU38
