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2.092/2.093FiniteElementAnalysisofSolids&FluidsI Fall 09Lecture4- ThePrincipleofVirtualWork

Prof. K.J.Bathe MITOpenCourseWare

Su =SurfaceonwhichdisplacementsareprescribedSf =SurfaceonwhichloadsareappliedSuSf =S ; SfSu =

Giventhesystemgeometry(V, Su, Sf),loads(fB,fSf),andmaterial laws,wecalculate:Displacementsu,v,w (oru1, u2, u3)Strains,stresses

We will perform a linear elastic analysis for solids. We want to obtain the equationKU =R. Recall ourtrussexample. There,wehadelementstiffness AE. Tocalculatethestiffnesses,wecouldproceedthisway:

Li

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Lecture4 ThePrincipleofVirtualWork 2.092/2.093,Fall09uEverydifferentialelementshouldsatisfyEAd2 =0. ToobtainF,wesolve:

dx2

; u = 1.0 ; ud2u = 0 = 0EAdx2 x=0 x=Li

Considera2Danalysis:

Inthiscase,themethodusedforthetrussproblemtogetthestiffnessmatrixK wouldnotwork. Ingeneral3Danalysis,wemustsatisfy(fortheexact solution)

Equilibrium:I. ij,j +fiB =0 inV(i,j = 1,2,3), whereij are the Cauchy stresses (forces per unit area in the

deformedgeometry).II. ijnj =fiSf onSf

Compatibility: ui =uSiu onSu andalldisplacementsmustbecontinuous.Stress-strain laws

This isknownasthedifferentialformulation.ExampleReadingassignment: Section3.3.4

Equilibriumd2u

EA +fB =0 (a)dx2du

EA =R (b)dx x=L2

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Lecture4 ThePrincipleofVirtualWork 2.092/2.093,Fall09

Compatibilityu =0 (c)

x=0Stress-strain law

duxx =E (d)dx

Ina1Dproblem,nodesaresurfaces.

Ina2Dproblem,wedefine linethickness=surface,butonepointcanbelongtobothSf andSu.PrincipleofVirtualWork(VirtualDisplacements)Clearly,theexactsolutionu(x)mustsatisfy:

d2u+fB u(x) = 0 (1)EA

dx2

whereu(x)iscontinuousandzeroatx=0. Otherwise,itisanarbitraryfunction. Hence,also,

L d2u+fBEA

dx20 u(x)dx=0 (2)

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Lecture 4 The Principle of Virtual Work 2.092/2.093, Fall 09

From Eq. (2):

EAdu

dxu

L

0

L0

du

dxEA

du

dxdx+

L0

fBudx= 0 (A)

The equation above becomes:

Internal virtual work L0

du

dxEA

du

dxdx =

External virtual work L0

fBudx +

Virtual work due toboundary forces

RuL

where dudx

are the virtual strains, dudx

are the real strains, andu are the virtual displacements. We setu = 0

on Su, since we do not know the external forces on Su. To solve EAd2udx2

+fB = 0, we look for a function u

where d2udx2

exists (dudx

should be continuous). In order to calculate the virtual work, we look for the solutionswhere only u is continuous.

(A) can be written as:

L

0

xxEAxxdx= L

0

ufBdx+RuL (A)

(the bar denotes virtual quantities)

In 3D vector form, the principle of virtual work now becomes

VTCdV =

VuTfBdV +

Sf

uSfTfSfdSf (B)

=

xxyyzzxyyzzx

; xx=u

x

=

xxyyzzxyyzzx

; zz =u

z

We see that (B) is the generalized form of (A). The principle of virtual work states that for any compatiblevirtual displacement field imposed on the body in its state of equilibrium, the total internal virtual work is

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Lecture4 ThePrincipleofVirtualWork 2.092/2.093,Fall09equaltothetotalexternalvirtualwork. Notethatthisvariationalformulationisequivalenttothedifferentialformulation,givenearlier.

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MIT OpenCourseWarehttp://ocw.mit.edu

2.092 / 2.093 Finite Element Analysis of Solids and Fluids I

Fall 2009

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