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 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.

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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 {}: {σ}T = {σxx σyy σzz τxy τxz τyz } {}T = {xx yy zz γxy γxz γyz }. The total external work done by {F } is W =

n 1X 1 Fi Di = {F }T {D}, 2 i=1 2

(1)

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

(3)

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The Principle of Virtual Work

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 ¯ virtual stresses {¯ applied, producing additional virtual deformations {D}, σ }, and virtual strains {¯}.

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

¯ and the internal work is The external work is 21 {F¯ }T {D}, 1Z 1 ¯ T ¯ {F } {D} = {¯ σ }T {¯}dV. V 2 2 Substituting equation (5) into equation (4) gives ¯ = {F }T {D}

Z V

{σ}T {¯}dV.

1R σ }T {¯}dV, 2 V {¯

(4)

or (5)

(6)

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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 {¯ σ }.

The external work done by the application of {F } is now m X 1 1 T F¯j Dj = {F }T {D} + {F¯ }T {D}. W = {F } {D} + 2 2 j=1

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 U=

Z 1Z {σ}T {}dV + {¯ σ }T {}dV. V V 2

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

(7)

and substituting equation (3) into equation (7) gives {F¯ }T {D} =

Z V

{¯ σ }T {}dV.

(8)

Equation (8) is used in the unit load method to find redundant forces or reactions, 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, ¯ . The right hand side of this equation, {¯ W 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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The Principle of Virtual Work

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: ∗

1·D =

Z V

{¯ σ }T {}dV,

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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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 = {¯ σ }T = {¯ σxx 0 0 0 0 0} Real Strain = {}T = {xx yy zz 0 0 0} But we only need xx because we are interested in {¯ σ }T {}. N n Real Strain = xx = A 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, Virtual Stress = σ ¯xx =

U¯ =

M X

nm Nm Lm . 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 Real Strain = xx = − Iz EIz The internal virtual work due to a bending moments is Virtual Stress = σ ¯xx = −

U¯ =

Z V

{¯ σ }T {}dV =

Z ZZ l

Z m M mz Mz y 2 z z dA dl = dl. A l EIz EIz2

Recall that Iz = A y 2 dA when the origin of the coordinate system lies on the neutral axis of the beam. RR

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The Principle of Virtual Work

Virtual Shear Force Consider a beam subjected to a pure virtual and real shear forces in the ydirection, vy and Vy :

Virtual Stress = τ¯xy =

vy Q(y) Iz t(y)

Real Strain = γxy =

τxy Vy Q(y) = , G GIz t(y)

where Q(y) is called the moment of area. The internal virtual work due to shear forces is Z ZZ v V Q(y)2 Z Z vy Vy y y ¯ dA dl = U = τ¯xy γxy dV = dl, l A GI 2 t(y)2 l G(A/αy ) V z where A ZZ Q(y)2 αy = 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 τ Tr Real Strain = γ = = , J G GJ The internal virtual work due to torsional moments is Z Z ZZ tT r 2 Z tT ¯ U = τ¯ γ dV = dA dl = dl, V l A GJ 2 l GJ RR (J = A r2 dA) Virtual Stress = τ¯ =

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CEE 421L. Matrix Structural Analysis – Duke University – Fall 2012

Total Internal Virtual Work As a review of the material above, consider general three-dimensional superimposed real and virtual forces

The total virtual strain energy due to these combined effects is U¯ =

Z m M Z m M nN z z y y dl + dl + dl + l EIz l EIy l EA Z Z Z tT vy Vy vz Vz dl + dl + dl l G(A/αy ) l G(A/αz ) l GJ Z

where 

2



2

A ZZ  Qy (y)  dA αy = 2 Iz A tz (y) A ZZ  Qz (z)  αz = 2 dA Iy A ty (z)

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