Virtual Displacement Dr. Rakesh K Kapania Aerospace and Ocean Engineering Department Virginia Polytechnic Institute and State University, Blacksburg, VA AOE 5024, Vehicle Structures ©2016 Rakesh K. Kapania, Mitchell Professor, Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061-0203. Virtual Displacement ~v T~ (~v) ~v = Unit Normal u =0 Deformed Structure With Virtual Displacement Superimposed ©Rakesh K. Kapania AOE 5024, Vehicle Structures 2 Virtual Displacement (contd...) I Virtual displacement is an imaginary infinitesimal displacement that is superimposed on the already displaced structure. I Virtual displacement is consistent with the essential or kinematic (displacement) boundary conditions. They vanish where essential boundary conditions are specified. I Applied loads do not change due to the action of these infinitesimal displacements. I Virtual displacements can be expressed as the d operator. ©Rakesh K. Kapania AOE 5024, Vehicle Structures 3 Virtual Work I Work done on a structure by all the forces acting on the structure as the structure is given a virtual displacement. I As a simple example, consider a simple linear spring with spring constant k subjected to a load Fs . The deflection x is given by x = Fs /k. Given a small virtual displacement dx, the work done by the external load, dWe , in moving through the virtual displacement dx is given as: dWe = Fs dx ©Rakesh K. Kapania AOE 5024, Vehicle Structures 4 Virtual Work (contd...) I The virtual work done by the internal spring force, kx, is represented as dWi and can be expressed as: dWi = kxdx Since F = kx, the virtual work done by the external force is equal to the virtual work done by the internal force. ©Rakesh K. Kapania AOE 5024, Vehicle Structures 5 Principle of Virtual Work I At equilibrium: dWe = dWi This is the Principal of Virtual Work, PVW. This principle is equally valid for more general, complex structures; linear or nonlinear response; as well as Conservative or Non-Conservative forces. I The principle of Virtual Work is used both to derive the governing equations for a structure and also for developing approximate methods. ©Rakesh K. Kapania AOE 5024, Vehicle Structures 6 Principle of Virtual Work (contd...) I The internal Virtual Work done can also be thought of as the change in the strain energy, dU, due to a virtual displacement. For example, for the case of spring, dU = Fs dx = dWi . Note: Fs = kx for a linear spring Hence dWe = dU is also a statement of PVW. ©Rakesh K. Kapania AOE 5024, Vehicle Structures 7 Cantilever Beam Example Here, dw(0) = 0 dw 0(0) = 0 ©Rakesh K. Kapania AOE 5024, Vehicle Structures 8 Cantilever Beam Example (contd...) Consider a Cantilever beam, of length L and subjected to a distributed load q(x), a tip moment M in the anti-clockwise direction, and point loads F1 and F2 acting at distances x1 and x2 respectively. An example of valid transverse Virtual displacement dw(x) for this beam is: h px i dw(x) = dA 1 − cos 2L An Note that both dw(0) and dw 0(0) vanish, this is as it should be since at x = 0 both the transverse displacement w(x) and the slope dw/dx are specified to be zero in this case. It is interesting to see what happens at x = L. ©Rakesh K. Kapania AOE 5024, Vehicle Structures 9 Cantilever Beam Example (contd...) Since the end at x = L is free to both deflect and rotate, the assumed virtual displacement and rotation should not be allowed to vanish at the free end. If any of these vanish, then we are not satisfying the consistency requirements for virtual displacements. ©Rakesh K. Kapania AOE 5024, Vehicle Structures 10 Virtual External Work By distributed load, Z L h px i = dA q(x) 1 − cos dx 0 2L h px1 i By F , F , respectively are: F dA 1 − cos , and 1 2 1 2L h px2 i F dA 1 − cos 2 2L And that due to moment M, d h px i MdA 1 − cos j dx 2L x=L ©Rakesh K. Kapania AOE 5024, Vehicle Structures 11 Virtual External Work (contd...) The virtual work done by external forces is then given as: Z L h px i dWe = dA q(x) 1 − cos dx 0 2L h px1 i h px2 i + F dA 1 − cos + F dA 1 − cos 1 2L 2 2L d h px i + M dA 1 − cos dx 2L x=L For a given set of loads, the external virtual work will become dA times some known quantity. ©Rakesh K. Kapania AOE 5024, Vehicle Structures 12 Change in Strain Energy The change in strain energy can be expressed as, " # 1 Z L d2w 2 dU = d EI 2 dx 2 0 dx 1 Z L d2w d2w = EI .2 2 d 2 dx 2 0 dx dx Assuming the deflection profile to be given as: h px i w(x) = A 1 − cos 2L dU becomes: p 4 Z L px dU = AdA EI cos2 dx 2L 0 2L ©Rakesh K. Kapania AOE 5024, Vehicle Structures 13 Change in Strain Energy (contd...) The variation in the strain energy dU, after performing the integral, depends upon product of A, dA and some known quantity. By equating dU = dWe , from the previous slide, we can obtain A; leading to a one-term approximate solution of the problem. ©Rakesh K. Kapania AOE 5024, Vehicle Structures 14 Example For x1 = L/3, x2 = 2L/3, F1 = F2 = F , and, M = 2FL dWe becomes Z L h px i h p i = dA q(x) 1 − cos dx + F dA 1 − cos 0 2L 6 h p i d h px i +F dA 1 − cos + 2FLdA 1 − cos 3 dx 2L x=L " p # Z L h px i 3 3 dWe = dA q(x) 1 − cos dx + F dA p + − 0 2L 2 2 ©Rakesh K. Kapania AOE 5024, Vehicle Structures 15 Example (contd...) Similarly after substituting values for x1, x2, F1, F2, and, M , dU becomes: p 4 L dU = A.dA.EI 2L 2 By equation dWe = dU, dA cancels out and A becomes, " p # L h px i 3 3 R q(x) 1 − cos dx + F p + − 0 2L 2 2 A = p 4 L EI 2L 2 ©Rakesh K. Kapania AOE 5024, Vehicle Structures 16 Example (contd...) F x Assuming: q(x) = . , L L " p # 3 3 8 − 4p + p2 p + − + 2 2 2p2 A = F p 4 L EI 2L 2 FL3 A = 1.3285 EI This is a one-term approximation to the deflection of the beam. As will be seen subsequently, the accuracy of the solution can be increased by adding more terms to the solution and applying the PVW as many times there are number of unknown coefficients. ©Rakesh K. Kapania AOE 5024, Vehicle Structures 17 Principle of Virtual Work, General Case For general 3-D case, subjected to body forces Bi and surface forces (~n) Ti , where ~n represents the unit normal on the surface area S, ©Rakesh K. Kapania AOE 5024, Vehicle Structures 18 Principle of Virtual Work, General Case (contd...) The external virtual work, due to virtual displacement dui , can be written as: ZZZ ZZ ZZ (~n) (~n) dWe = Bi dui dV + Ti dui dS + Ti dui dS V S1 S2 Since dui = 0 on S2 ZZZ ZZ (~n) dWe = Bi dui dV + Ti dui dS V S1 The first term represents the virtual work done by the body forces and requires an integral over the volume V ; the second term represents the virtual work done by surface traction and requires an area integral over the surface S. ©Rakesh K. Kapania AOE 5024, Vehicle Structures 19 Principle of Virtual Work, General Case (contd...) Recall: ai bi = a1b1 + a2b2 + a3b3. If S = S1 [ S2 such that u is prescribed on S2, then du vanishes on S2 and the second integral is then performed over only S1, the surface area over (~n) which traction Ti is specified. (~n) Using Cauchy's relation, Ti = tij nj , the external virtual work becomes: ZZZ ZZ dWe = Bi dui dV + tij nj dui dS V S ©Rakesh K. Kapania AOE 5024, Vehicle Structures 20 Principle of Virtual Work, General Case (contd...) The surface integral in the above equation can be converted into a volume integral using Gauss Divergence Theorem. ZZZ ZZZ dWe = Bi dui dV + (tij dui ),j dV V V ZZZ = [(tij,j + Bi ) dui + tij dui,j ] dV V Recall: tij,j + Bi = 0 from equilibrium equations. ZZZ dWe = tij dui,j dV V ©Rakesh K. Kapania AOE 5024, Vehicle Structures 21 Principle of Virtual Work, General Case (contd...) One can write: 1 1 u = [u + u ] + [u − u ] i,j 2 i,j j,i 2 i,j j,i ui,j = eij + wij This gives: ZZZ dWe = tij d (eij + wij ) dV V ZZZ = tij deij dV Recall : tij wij = 0 V dWe = dU = dWi (Internal Virtual Work) ©Rakesh K. Kapania AOE 5024, Vehicle Structures 22 Principle of Virtual Work, General Case (contd...) If a structure is in equilibrium and remains in equilibrium while it is subjected to a virtual distortion, the external virtual work, dWe , done by external forces acting on the structure is equal to the internal virtual work dWi done by the internal stresses. Conversely; If dWe = dWi for an arbitrary virtual distortion then the body is in equilibrium. ©Rakesh K. Kapania AOE 5024, Vehicle Structures 23 Principle of Virtual Work, General Case (contd...) ©Rakesh K. Kapania AOE 5024, Vehicle Structures 24 Principle of Stationary Potential Energy The internal virtual work done can be written as; ZZZ dWi = tij deij dV V ZZZ ¶u = deij dV V ¶eij = dU Here u is the strain energy density and U is the strain energy of the complete structure.