The Principle of Virtual Work

Duke University Department of Civil and Environmental Engineering CEE 421L. Matrix Structural Analysis Fall, 2012 Henri P. Gavin

Deﬁnitions:

Virtual work is the work done by a real force acting through a virtual displacement or a virtual force acting through a real displacement.

A virtual displacement is any displacement consistent with the constraints of the structure, i.e., 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. 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 {σ} and {}:

T {σ} = {σxx σyy σzz τxy τxz τyz} T {} = {xx yy zz γxy γxz γyz}.

The total external work done by {F } is

n 1 X 1 T W = Fi Di = {F } {D}, (1) 2 i=1 2 and the total internal work done by {F } is the total strain energy, which can be written compactly as 1 Z U = {σ}T {}dV. (2) 2 V Setting W equal to U gives the principle of real work,

1 1 Z {F }T {D} = {σ}T {}dV. (3) 2 2 V

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Suppose, that after the structure is subjected to the system of n external forces, {F }, producing internal stresses {σ}, a system of m virtual forces {F¯} are applied, producing additional virtual deformations {D¯}, virtual stresses {σ¯}, and virtual strains {¯}.

The external work done by the application of {F¯} is m 1 T X 1 T T W = {F¯} {D¯} + F¯jD¯ j = {F¯} {D¯} + {F } {D¯}, 2 j=1 2 and the internal work done by the application of {F¯} is 1 Z Z U = {σ¯}T {¯}dV + {σ}T {¯}dV. 2 V V Setting the external work equal to the internal work, 1 1 Z Z {F¯}T {D¯} + {F }T {D¯} = {σ¯}T {¯}dV + {σ}T {¯}dV. (4) 2 2 V V If we consider the virtual system alone,

1 ¯ T ¯ 1 R T The external work is 2{F } {D}, and the internal work is 2 V {σ¯} {¯}dV, or 1 1 Z {F¯}T {D¯} = {σ¯}T {¯}dV. (5) 2 2 V Substituting equation (5) into equation (4) gives Z {F }T {D¯} = {σ}T {¯}dV. (6) V

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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 {σ¯}.

The external work done by the application of {F } is now

m 1 T X 1 T T W = {F } {D} + F¯jDj = {F } {D} + {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

1 Z Z U = {σ}T {}dV + {σ¯}T {}dV. 2 V V Setting the external work equal to the internal work,

1 1 Z Z {F }T {D} + {F¯}T {D} = {σ}T {}dV + {σ¯}T {}dV, (7) 2 2 V V and substituting equation (3) into equation (7) gives

Z {F¯}T {D} = {σ¯}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}, is called the external virtual work, Z W¯ . The right hand side of this equation, {σ¯}T {}dV , is called the internal V virtual work, U¯. Note that equation (8) is valid for both linear and nonlinear elastic structures (why?).

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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∗ = {σ¯}T {}dV, V in which {σ¯} are the virtual stresses arising from the single unit force at “*”, and {} are the real strains due to the actual loading.

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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:

T Virtual Stress = {σ¯} = {σ¯xx 0 0 0 0 0} T Real Strain = {} = {xx yy zz 0 0 0}

T But we only need xx because we are interested in {σ¯} {}. n N Virtual Stress =σ ¯ = Real Strain = = xx A xx EA The internal virtual work due to an axial force is Z Z ZZ nN Z nN U¯ = {σ¯}T {}dV = dA dl = dl. V l A EA2 l EA For a structure made up entirely of prismatic truss members, M n N L U¯ = X m m m . m=1 EmAm

Virtual Bending Moment

Consider a beam subjected to pure virtual and real bending moments about the z-axis, mz and Mz:

mzy Mzy Virtual Stress =σ ¯xx = − Real Strain = xx = − Iz EIz The internal virtual work due to a bending moments is

Z Z ZZ m M y2 Z m M U¯ = {σ¯}T {}dV = z z dA dl = z z dl. V l A 2 l EIz EIz RR 2 Recall that Iz = A y dA when the origin of the coordinate system lies on the neutral axis of the beam.

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Virtual Shear Force

Consider a beam subjected to a pure virtual and real shear forces in the y- direction, vy and Vy:

vyQ(y) τxy VyQ(y) Virtual Stress =τ ¯xy = Real Strain = γxy = = , Izt(y) G GIzt(y) where Q(y) is called the moment of area. The internal virtual work due to shear forces is 2 Z Z ZZ vyVyQ(y) Z vyVy U¯ = τ¯xyγxydV = dA dl = dl, V l A 2 2 l GIz t(y) G(A/αy) where A ZZ Q(y)2 αy = dA. 2 A 2 Iz t(y)

Virtual Torsion

Consider a circular bar subjected to a virtual and real torsional moments, t and T :

tr τ T r Virtual Stress =τ ¯ = Real Strain = γ = = , J G GJ The internal virtual work due to torsional moments is Z Z ZZ tT r2 Z tT U¯ = τ¯ γ dV = dA dl = dl, V l A GJ 2 l GJ RR 2 (J = A r dA)

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Total Internal Virtual Work

As a review of the material above, consider general three-dimensional super- imposed real and virtual forces

The total virtual strain energy due to these combined eﬀects is

Z nN Z m M Z m M U¯ = dl + z z dl + y y dl + l EA l EIz l EIy Z v V Z v V Z tT y y dl + z z dl + dl l G(A/αy) l G(A/αz) l GJ where  2 A ZZ Qy(y) αy =   dA 2 A Iz tz(y)  2 A ZZ Qz(z) αz =   dA 2 A Iy ty(z)

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