1 Chapter 6 Virtual Work Principles Principle of virtual Displacement = stiffness method = Displacement method Principle of virtual force = flexibility method = force method The principle of virtual displacements finds its most powerful application in the development of approximate solutions. Advantages: Without the force-equilibrium, the governing equation ()Ku= P can be obtained, assuming a displacement function. Even a displacement function not satisfying the equilibrium can be used to obtain an approximate solution. Principle of Virtual Displacement δδW= U ( or δ Wext = δ Wint ) δW = virtual external work δU = virtual internal work (virtual strain energy) 2 δδW= uP ⋅ δU=∫= δ Udν δ U δε⋅ σ =∫⋅δε σd ν 6.1 Principle of virtual Displacements – Rigid Bodies δδWU= = 0 For a particle subjected to a system of force in equilibrium, the work due to a virtual displacement is zero. ddvv1 and 2 does not induce internal work because of the rigid body motion. 3 δWFv=+−yy11 δδδ Fv 2 2 Pv y 33 xx δv=−+1 δδ vv12 (satisfy the compatibility for rigid body) LL xx33 δv3=−+1 δδ vv 12 LL xx33 δWFP=−yy1310 − δδ v 1 +− FPy 23 y v 2 = LL // // 0 0 A particle is in equilibrium under the action of a system of forces if the virtual work is zero for every independent virtual displacement. x3 FPyy13=1 − L Px F = y33 y2 L 4 Example 6.1 After releasing member force F36− , lateral displacement δu5 cause a rigid body motion which does not cause internal deformation and energy. δWF=0 ⇒ solve 36− 1.5 δ= δδ +− δ = W2()() Pu5 Pu5 F 36− ()0 u5 3.25 F36− =2 3.25 P 5 6.2 Principle of virtual Displacements – Deformable bodies Principle of virtual displacement for deformable bodies δδWU= For a deformable structure in equilibrium under the action of a system of applied force, the external virtual work due to an admissible virtual displaced state is equal to the internal virtual work due to the same virtual displacements. δδWU= ⇒= Ku P General Mechanics Energy Principle force – equilibrium principle of virtual displacement Displacement – compatibility ⇒ Displacement - compatibility force – Displ. Relationship force – Displ. Relationship Since energy is expressed as displacement ｘ force, in order to satisfy δδWU= , at least internal force ≈ external force in average sense if the 6 displacement-compatibility is satisfied. Internal force is defined as the function of the assumed displacement. 6.3 Virtual Displacement analysis procedure and Detailed Expressions 6.3.1 General procedure The principle of virtual displacements finds its most powerful application in the development of approximate solutions. Only displacement functions, which satisfy b/c, are required while force- equilibrium is assumed to be satisfied by using the condition of energy conservation. 6.3.2 Internal virtual work 1) Axial force member δU (virtual energy density) Internal work is defined as the function of =δεxx σ displacement. 7 δU (virtual energy) =∫δUdV δε σ =∫ xxdV L =dδε σ x ∫o xx LLdδ u du = δεEA ε dx =( ) EA ( ) dx ∫∫ooxx dx dx 2) Torsional Member (pure torsion) δU = δε ⋅= σ δγ ⋅ τ dθ γβ=rr = ⋅ dx τγ= G =⋅=τβ2 = β Mx ∫∫ rdA G r dA GJ δU=∫ δγ ⋅ τ dV LA = δγ⋅ τ dAdx ∫∫oo =∫∫δβGr2 β dAdx =∫δβGJ β dx ddδθ θ =(xx )GJ ( ) dx ∫ dx dx dθ =δβ ⋅=M dx ( M GJx ) ∫ xxdx 8 3) flexural member δU = δε ⋅= σ δε σ xx dv2δ δε =−=−yyδφ x dx2 σεxx=E = − Ey φ δ= δε σ U∫ xx dV =δε σ dA dx ∫∫LA xx =δφEy2 φ dA dx ∫∫A δφ φ 2 = =∫∫EIzz dx ( y dA I ) 22 L dδ v dv =∫∫22 EIz dx o dx dx L =δφM dx ( M= EI φ ) ∫o z zz 6.3.3 External virtual work δδ=∑+⋅ δ W uii P∫ u b dV (or δu⋅ b dx) ∫ Example 6.2 9 6.4 Construction of Analytical Solutions by the principle of virtual displacements 6.4.1 Exact solutions x at xu=0 = 0 select u= u2 which satisfy the B/C’s: L at xLuu= = 2 x du u u= u ε = = 2 L 2 x dx L x 1 δδuu= δε= δu L 2 x L 2 δU=∫ δε⋅ σ dV L = δεEA ε dx ∫0 xx L δuu = 22EA dx ∫0 LL EA = ()δuu L 22 10 δδW= uF22x EA δδUW= ⇒= F u x22L L d = EA x Why is uu= exact displacement function? L 2 The exact displacement function is the one that satisfies the force- equilibrium. By equilibrium, dF+= bx dx 0 dF b = − x dx dF When b =0, = 0 x dx du F=⋅=σε A EA = EA dx d du ⇒=(EA )0 dx dx 11 When E and A are constant, du2 =0 ⇒ the condition for the exact displacement function dx2 x du2 u=u satisfies (= 0) L 2 dx2 Virtual displacement with different B/C displacement shape yields the same equilibrium equation δU= ∫ δεσ dV xx For virtual displacement δu=−+(1 ) δδ uu Eq. 1 v LL1 2 - Virtual displacement is applied to the equilibrium system and is not related to the actual displ. B/C. Thus δuv in Eq. 1 is valid. EA - FF=−=− u ⇒ Reaction can be calculated 12L 2 x - δδuu= is a special case of Eq. 1 v L 2 12 dδ u du x δU= vr EA dx u= u ∫ dx dx r L 2 δδu u du =(−+12 )EAr dx ∫ L L dx EA EA = −+δδuuuu 122LL 2 δδW= uF11 + δ uF 2 2 EA δδUW= ⇒=− F u 12L EA Fu= 22L If we use for both real displacement and virtual displacement, xx u=−+(1 ) uu r LL12 xx δδu=−+(1 ) uu v LL12 dδ u du δU= vr EA dx ∫ dx dx δδu u uu = (−1 + 2 )EA ( −+ 12 ) dx ∫ L L LL 13 δδW= uF11 + δ uF 2 2 EA δδU= W F= ( uu− ) 1L 12 EA F= () uu− 2L 21 Fu11 EA 1 -1 = Fu22 L -1 1 Different displacement shape for virtual displacement x δδuu= ()2 v L 2 → satisfy displacement boundary condition. π x δδuu= sin v 2L 2 but not satisfy the force-equilibrium. x Real displacement uu= r L 2 dδ u du δU= vr EA dx ∫ dx dx Integration by part dδ u du δU= vr EA dx ∫ dx dx dδ u du vr=a' = b, a' b = ab - ab ' dx dx ∫∫ L du d22 u d u dF =EAδδ ur− EA u rr dx =0 =0 vv∫ 22 dx 0 dx dx dx L dur =EAδ uv dx 0 14 This result indicates that δU is affected by the values at the starting and last points, regardless of the shape of the uv Thus if virtual displacement δuv satisfies the displacement B/C, any form of virtual displacement can be used. (Here, x=0 δ uvv =0 , xL = δδ u = u2 ) x However, the real displacement u (= u ) should satisfy r L 2 du2 r = 0 (when b = 0) , which is the force-equilibrium condition. dx2 x 2 x π x δδuuv = 2 δδuu= sin( ) L v 2 2L 2 π x x uu= sin( ) uur = 2 r 2 L 2L du2 ⇒ cannot get the exact solution, because r ≠ 0 dx2 If the chosen read displacements corresponds to stresses that satisfy identically the conditions of equilibrium, any form of admissible virtual displacement will suffice to produce the exact solution. 15 6.4.2 Approximate solutions and the significance of the chosen virtual displacements Principle of virtual work is applicable to seeking approximate solutions for Frame work Analysis: tapered section, nonlinearity, instability, dynamics All finite element analysis For example, tapered truss element. Equilibrium condition dF = −b dx x dF If b = 0, = 0 x dx x If uu= is used, L 2 du u σε=EE = = E2 x dx L 16 dF x u x But, x =−dσ A(1 ) / dx = d E2 A (1−≠ ) / dx 0 dx x 1122L LL ⇒ violate the force-equilibrium condition Nevertheless, we can use the approximate displacement function x uu= which is exact only for prismatic elements. L 2 L dδ u du δU= EA dx ∫o dx dx L δuu22 x = EA1(1− ) dx ∫o LL 2 L 3EA = δuu 1 22 4L δδW= uF22 ⋅ x 3EA δδUW= ⇒= F1 u x224L x uu= L 2 2EA1 for 2 ⇒= Fu x x223L δδuu= 2 L x uu= L 2 0.6817EA for ⇒= Fu1 π x x22L δδuu= sin2 2L EA Exact solution = 0.721 1 u L 2 17 Discussions 1) Although u is not exact displacement function, δδWU= (energy conservation) force to provide a basis for the calculation of the undetermined parameter u2 2) Although the approximate real displacement cannot satisfy the equilibrium conditions at every locations, the enforcement of the condition δδWU= results in average satisfaction of the equilibrium throughout the structure. 3) The standard procedure for choosing the form of virtual displacement is to adopt the same form as the real displacement for convenience and to make the stiffness matrix symmetric. Requirement of displacement function (real displacement) 1) Displacement Boundary condition should be satisfied. uf= (nodal displacements) = fuu(12 , ) In case of 2-node truss element, there are two nodes. Thus, only two terms can be used when a polynomial equation is used. For example u= a01 + ax uu− x==0, uu, xLuu ==, ⇒= a u a =21 1 2 0 1 1 L 2) Rigid body motion (no strain) and constant strain should be described. 18 u= a01 + ax (O.K) π ua=+ a sin x (Not OK) 012L 3) Force-equilibrium should be satisfied. For axial force member dF d du =⇒=b ()EA b dx xxdx dx 1) is the essential condition to get at least an approximate solution 2) is the convergence condition to get a reasonably accurate solution by increasing the number of elements.