The Principle Of Virtual Work - MIT OpenCourseWare

2.092/2.093 — Finite Element Analysis of Solids & Fluids IFall ‘09Lecture 4 - The Principle of Virtual WorkProf. K. J. BatheMIT OpenCourseWareSu Surface on which displacements are prescribedSf Surface on which loads are appliedSu Sf S ; Sf Su Given the system geometry (V, Su , Sf ), loads (f B , f Sf ), and material laws, we calculate: Displacements u, v, w (or u1 , u2 , u3 ) Strains, stressesWe will perform a linear elastic analysis for solids. We want to obtain the equation KU R. Recall ourtruss example. There, we had element stiffness AELi . To calculate the stiffnesses, we could proceed this way:1

Lecture 4The Principle of Virtual WorkEvery differential element should satisfyEA2.092/2.093, Fall ‘092EA ddxu2d2 u 0dx2 0. To obtain F, we solve: ; u 1.0 ; u 0x 0x LiConsider a 2D analysis:In this case, the method used for the truss problem to get the stiffness matrix K would not work. In general3D analysis, we must satisfy (for the exact solution) Equilibrium:I. τij,j fiB 0 in V(i, j 1, 2, 3), where τij are the Cauchy stresses (forces per unit area in thedeformed geometry).SfII. τij nj fion Sf Compatibility: ui uSiu on Su and all displacements must be continuous. Stress-strain lawsThis is known as the differential formulation.ExampleReading assignment: Section 3.3.4 Equilibriumd2 u fB 0dx2 du EA Rdx EAx L2(a)(b)

Lecture 4The Principle of Virtual Work2.092/2.093, Fall ‘09 Compatibility u 0(c)dudx(d)x 0 Stress-strain lawτxx EIn a 1D problem, nodes are surfaces.In a 2D problem, we define line thickness surface, but one point can belong to both Sf and Su .Principle of Virtual Work (Virtual Displacements)Clearly, the exact solution u(x) must satisfy: d2 uEA 2 f B δu(x) 0dx(1)where δu(x) is continuous and zero at x 0. Otherwise, it is an arbitrary function. Hence, also, 0L d2 uBδu(x)dx 0EA 2 fdx3(2)

Lecture 4The Principle of Virtual Work2.092/2.093, Fall ‘09From Eq. (2):EAduδudxLZL 00dudδuEA dx dxdxLZf B δudx 0(A)0The equation above becomes:Internal virtual work External virtual work} {} {zzZ LZ LdδuduEA dx f B δudx dxdx00dδudxVirtual work due toboundary forcesz } {RδuLdudxare the virtual strains,are the real strains, and δu are the virtual displacements. We set δu 0where2on Su , since we do not know the external forces on Su . To solve EA ddxu2 f B 0, we look for a function u2where ddxu2 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:ZLZεxx EAεxx dx 0Luf B dx RuL(A’)0(the bar denotes ‘virtual’ quantities)In 3D vector form, the principle of virtual work now becomesR TRRε CεdV V uT f B dV Sf uSf T f Sf dSfV ε ε εxxεyyεzzγxyγyzγzx εxxεyyεzzγ xyγ yzγ zx ; εxx u x; εzz u z(B)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 is4

Lecture 4The Principle of Virtual Work2.092/2.093, Fall ‘09equal to the total external virtual work. Note that this variational formulation is equivalent to the differentialformulation, given earlier.5

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The principle of virtual work states that for any compatible virtual displacement field imposed on the body in its state of equilibrium, the total internal virtual work is 4. Lecture 4 The Principle of Virtual Work 2.092/2.093, Fall ‘09 equal to the total external virtual work. Note that this variational formulation is equivalent to the .