Deflection: Castiglianos TheoremTheory of Structure - I

ContentsCastiglianos TheoremTrussesBeams and Frames

Castiglianos TheoremEnergy method derived by Italian engineer Alberto Castigliano in 1879.

Allows the computation of a defection at any point in a structure.

StatementAccording to Castiglianos First Theoremthe deflection of any point in a beam or truss subjected to any set of loads is equal to the first partial derivative of the strain energy, with respect to a load at the point acting in the direction of deflection

Castiglianos Second TheoremThis theorem is also termed asTheorem of Least WorkTheorem of Minimum Strain EnergyIt states:The redundant reaction components of a statically indeterminate structure are such as to make the total strain energy stored up a minimum

Castiglianos Second TheoremIf an indeterminate axial force or bending moment acts on the structure, the magnitude of that force or moment must be such so that the total strain energy, including the effect of unknown force or moment, is a minimum.If then an expression be determined for the total strain energy in the structure, the first partial derivative of this expression, with respect to redundant must be zero.

DerivationLet the body be linear elastic; and let Di be the component of deflection in the direction of the ith force, Fi.The total work done is then:U =F1D1+ F2D2+ F3D3+. FnDnF1Consider a solid object acted upon by n forces, Fi=1 to n, as shown in the figure.F2F3Fn

Increase force Fn by an amount dFThis changes the state of deformation and increases the total strain energy slightly:

Hence, the total strain energy after the increase in the nth force is:

Now suppose, the order of this process is reversed; i.e., Apply a small force dFn to this same body and observe a deformation dDn; then apply the forces, Fi=1 to n.

As these forces are being applied, dFn goes through displacement Dn.(Note dFnis constant) and does work:dU = dFnDn

Hence the total work done is:U+ dFnDn

The end results are equal Since the body is linear elastic, all work is recoverable, and the two systems are identical and contain the same stored energy:

Castiglianos TheoremU = U*dU = dU*Ui = f (P1, P2,, Pn)

Castiglianos Second TheoremThe term force may be used in its most fundamental sense and can refer for example to a Moment, M, producing a rotation, q, in the body.Mq

Applications:Castiglianos 2nd theorem can be used to determine the deflections in structures (eg, trusses, beams, frames, shells) and we are not limited to applications in which only 1 external force or moment acts. Furthermore, we can determine the deflection or rotation at any point, even where no force or moment is applied externally.

Axial Load Bending Shear Displacement :WhereD = external displacement of the truss, beam or frameP = external force applied to the truss, beam or frame in the direction of DN = internal axial force in the member caused by both the force P and the loads on the truss, beam or frameM = internal moment in the beam or frame, expressed as a function of x and caused by both the force P and the real loads on the beamV = internal moment in the beam or frame caused by both the force P and the real loads on the beam

BendingSlope :Whereq = external slope of the beam or frameMi = external moment applied to the beam or frame in the direction of qM = internal moment in the beam or frame, expressed as a function of x and caused by both the force P and the real loads on the beam

Castiglianos Theorem : TrussWhere:D = external joint displacement of the trussP = external force applied to the truss joint in the direction of DN = internal force in a member cause by both the force P and the loads on the trussL = length of a memberA = cross-sectional area of a memberE = modulus of elasticity of a member

Example 8-25

Determine the vertical displacement of joint C of the truss shown in the figure below. The cross-sectional area of each member of the truss shown in the figure is A = 400 mm2 and E = 200 GPa.

SOLUTION10.656-10.4110.412+2.5-2.50.667P-0.833P-0.833P+

Example 8-26

Determine the vertical displacement of joint C of the steel truss shown. The cross-section area of each member is A = 400 mm2 and E = 200 GPa.

SOLUTION4-5.6570-5.6574444-40.667P-0.471P-0.471P-0.943P0.667P0.333P0.333P1P-0.333P10.6715.07030.1810.675.335.33165.33+

Example 8-27

Determine the vertical displacement of joint C of the steel truss shown. The cross-section area of each member is A = 400 mm2 and E = 200 GPa.

SOLUTION31.1260104.12406023.3330-24.03620200.667P0-1.2P01P+

Castiglianos Theorem : Beams and Frames DisplacementWhere:D = external displacement of the point caused by the real loads acting on the beam or frameP = external force applied to the beam or frame in the direction of DM = internal moment in beam or frame , expressed as a function of x and cause by both the force P and the loads on the beam or frame

SlopeWhere:D = external displacement of the point caused by the real loads acting on the beam or frameM = external moment applied to the beam or frame in the direction of qM = internal moment in beam or frame , expressed as a function of x and cause by both the force P and the loads on the beam or frame

Example 8-28

The beam shown is subjected to a load P at its end. Determine the slope and displacement at C. EI is constant.

SOLUTIONDisplacement at C

Slope at C

Example 8-29

Determine the slope and displacement of point B of the steel beam shown in the figure below. Take E = 200 GPa, I = 250(106) mm4.

SOLUTIONDisplacement at BDB = 0.00469 m = 4.69mm,

slope at BqB = 0.00125 rad,

Example 8-30

Determine the slope and displacement of point B of the steel beam shown in the figure below. Take E = 200 GPa, I = 60(106) mm4.

SOLUTIONDisplacement at B

SOLUTIONSlope at BqB = 0.000194 rad,

Example 8-31

Determine the displacement of point B of the steel beam shown in the figure below. Take E = 200 GPa, I = 200(106) mm4.

= (22.5 + P)x3 - (75 + 6P) = -(2.5 + P)x1 = 10 - 2.5x1SOLUTION

= (22.5 + P)x3 - (75 + 6P) = -(2.5 + P)x2 = 10 - 2.5x1= 7.875 mm,

Example 8-32

Determine the slope and the horizontal displacement of point C on the frame. Take E = 200 GPa, I = 200(106) mm4

Horizontal Displacement at CSOLUTION

Slope C= + 0.00125 rad ,