The Principle of Virtual Work

1 Duke University Department of Civil and Environmental Engineering CEE 421L. Matrix Structural analysis Fall, 2012. Henri P. Gavin The Principle of Virtual Work Definitions: Virtual work is the work done by a real force acting through a Virtual displace- ment or a Virtual force acting through a real displacement. A Virtual displacement is any displacement consistent with the constraints of the structure, , that satisfy the boundary conditions at the supports. A Virtual force is any system of forces in equilibrium. Example: f (x) and y(x) are real forces and associated displacements.

2 Y (x) is a Virtual displacement consistent with the boundary conditions. 2 CEE 421L. Matrix Structural analysis Duke University Fall 2012. Consider a structure deformed by the effect of n external forces, denoted by the vector {F }. The actual (real) displacements at the same n coordinates are contained in the vector {D}. The stresses and strains at any point in the structure are elements of the vectors { } and { }: { }T = { xx yy zz xy xz yz }. { }T = { xx yy zz xy xz yz }. The total external work done by {F } is 1X n 1. W = Fi Di = {F }T {D}, (1).

3 2 i=1 2. and the total internal work done by {F } is the total strain energy, which can be written compactly as 1Z. U= { }T { }dV. (2). 2 V. Setting W equal to U gives the Principle of real work, 1 1Z. {F }T {D} = { }T { }dV. (3). 2 2 V. CC BY-NC-ND Gavin The Principle of Virtual Work 3. Suppose, that after the structure is subjected to the system of n external forces, {F }, producing internal stresses { }, a system of m Virtual forces {F } are applied, producing additional Virtual deformations {D }, Virtual stresses { }, and Virtual strains { }.

4 The external work done by the application of {F } is 1 m 1. T. F j D j = {F }T {D } + {F }T {D }, X. W = {F } {D } +. 2 j=1 2. and the internal work done by the application of {F } is 1Z Z. U= { } { }dV + { }T { }dV. T. 2 V V. Setting the external work equal to the internal work, 1 T T 1Z Z. {F } {D } + {F } {D } = { } { }dV + { }T { }dV. T. (4). 2 2 V V. If we consider the Virtual system alone, The external work is 21 {F }T {D }, and the internal work is 1R T. 2 V { } { . }dV, or 1 1Z. {F }T {D } = { }T { }dV. (5). 2 2 V. Substituting equation (5) into equation (4) gives Z.

5 {F }T {D } = { }T { }dV. (6). V. CC BY-NC-ND Gavin 4 CEE 421L. Matrix Structural analysis Duke University Fall 2012. Suppose, instead, that before the actual loads {F } and deformations {D} are introduced, the structure was subjected to a system of m Virtual forces, {F }, producing internal stresses { }. The external work done by the application of {F } is now 1 m 1. T. F j Dj = {F }T {D} + {F }T {D}. X. W = {F } {D} +. 2 j=1 2. Note here that the actual deflections {D} are unrelated to the Virtual forces {F }. The internal work done by the application of {F } is now 1Z Z.

6 U= { }T { }dV + { }T { }dV. 2 V V. Setting the external work equal to the internal work, 1 T T 1Z Z. {F } {D} + {F } {D} = { } { }dV + { }T { }dV, T. (7). 2 2 V V. and substituting equation (3) into equation (7) gives Z. {F }T {D} = { }T { }dV. (8). V. Equation (8) is used in the unit load method to find redundant forces or reac- tions, and to find real structural displacements, as will be shown shortly. The left hand side of this equation, {F }T {D},Z is called the external Virtual work, W . The right hand side of this equation, { }T { }dV , is called the internal V.

7 Virtual work, U . Note that equation (8) is valid for both linear and nonlinear elastic structures (why?). CC BY-NC-ND Gavin The Principle of Virtual Work 5. Example: Unit Load Method Find the deflection of a bar under axial tension. The Unit Load Method When the Principle of Virtual work is used to calculate the displacement D , at a coordinate * , the system of external forces, {F } is chosen so as to consist only of a single unit force at coordinate * : equation (8) becomes: Z.. 1 D = { }T { }dV, V. in which { } are the Virtual stresses arising from the single unit force at * , and { } are the real strains due to the actual loading.

8 CC BY-NC-ND Gavin 6 CEE 421L. Matrix Structural analysis Duke University Fall 2012. FORMS OF INTERNAL Virtual WORK. FOR FRAMED STRUCTURES. Virtual Axial Force Consider a rod subjected to a Virtual normal force n, and a real normal force, N: Virtual Stress = { }T = { xx 0 0 0 0 0}. Real strain = { }T = { xx yy zz 0 0 0}. But we only need xx because we are interested in { }T { }. n N. Virtual Stress = xx = Real strain = xx =. A EA. The internal Virtual work due to an axial force is Z Z ZZ nN Z nN. U = { }T { }dV = dA dl = dl. V l A EA2 l EA.

9 For a structure made up entirely of prismatic truss members, M. X nm Nm Lm U = . m=1 Em Am Virtual Bending Moment Consider a beam subjected to pure Virtual and real bending moments about the z-axis, mz and Mz : mz y Mz y Virtual Stress = xx = Real strain = xx = . Iz EIz The internal Virtual work due to a bending moments is Z Z ZZ mz Mz y 2 Z m M. z z U = { }T { }dV = dA dl = dl. V l A EIz2 l EIz Recall that Iz = A y 2 dA when the origin of the coordinate system lies on the RR. neutral axis of the beam. CC BY-NC-ND Gavin The Principle of Virtual Work 7.

10 Virtual Shear Force Consider a beam subjected to a pure Virtual and real shear forces in the y- direction, vy and Vy : vy Q(y) xy Vy Q(y). Virtual Stress = xy = Real strain = xy = = , Iz t(y) G GIz t(y). where Q(y) is called the moment of area. The internal Virtual work due to shear forces is Z Z ZZ v V Q(y)2 Z vy Vy y y U = xy xy dV = dA dl = dl, V l A GI 2 t(y)2 l G(A/ y ). z where A ZZ Q(y)2. y = 2 dA. Iz A t(y)2. Virtual Torsion Consider a circular bar subjected to a Virtual and real torsional moments, t and T: tr Tr Virtual Stress = = Real strain = = = , J G GJ.