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

=∫⋅dε 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.

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=−yy1310 −   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

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

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

dU (virtual energy density) Internal work is defined as the function of =δεxx σ displacement.

dU (virtual energy) =∫dUdV de σ =∫ xxdV L =dde σ x ∫o xx LLdd u du = δεEA ε dx =( ) EA ( ) dx ∫∫ooxx dx dx

2) Torsional Member (pure torsion)

δU = δε ⋅= σ dγ ⋅ τ  dθ γβ=rr = ⋅ dx τγ= G =⋅=τβ2 = β Mx ∫∫ rdA G r dA GJ

δU=∫ δγ ⋅ τ dV LA = dγ⋅ τ dAdx ∫∫oo =∫∫dβGr2 β dAdx =∫dβGJ β dx dddθ θ =(xx )GJ ( ) dx ∫ dx dx dθ =dβ ⋅=M dx ( M GJx ) ∫ xxdx

3) flexural member

δU = δε ⋅= σ dε σ  xx dv2d dε =−=−yydφ x dx2

σεxx=E = − Ey φ

δ= δε σ U∫ xx dV =dε σ dA dx ∫∫LA xx =dφEy2 φ dA dx ∫∫A dφ φ 2 = =∫∫EIzz dx ( y dA I ) 22 L dd v  dv  =∫∫22  EIz dx o dx  dx  L =dφM dx ( M= EI φ ) ∫o z zz

6.3.3 External virtual work

δδ=∑+⋅ δ W uii P∫ u b dV  (or du⋅ b dx ) ∫ 

Example 6.2

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 = dεEA ε dx ∫0 xx L duu = 22EA dx ∫0 LL EA = ()duu L 22

ddW= uF22x EA ddUW= ⇒= 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

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

dd u du x dU= vr EA dx u= u ∫ dx dx r L 2 ddu u du =(−+12 )EAr dx ∫ L L dx EA EA = −+dduuuu 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 ddu=−+(1 ) uu v LL12

dd u du dU= vr EA dx ∫ dx dx ddu u uu = (−1 + 2 )EA ( −+ 12 ) dx ∫ L L LL

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 dd u du dU= vr EA dx ∫ dx dx

Integration by part

dd u du dU= vr EA dx ∫ dx dx dd u du vr=a ' = b , a ' b = ab - ab ' dx dx ∫∫ L du d22 u d u dF =EAdd ur- EA u rr dx = 0  = 0 vv∫ 22 dx 0 dx dx dx L dur =EAd 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.

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 dd u du dU= EA dx ∫o dx dx L duu22   x =   EA1(1− ) dx ∫o LL   2 L 3EA = duu 1 22 4L

ddW= uF22 ⋅ x

3EA ddUW= ⇒= F1 u x224L

x uu= L 2 2EA1 for 2 ⇒= Fu x x223L dduu=  2 L

x uu= L 2 0.6817EA for  ⇒= Fu1 π x x22L δδuu= sin2 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.

3) is the condition to obtain the exact solution.

Generally, virtual displacement function is the same as the real displacement Function.

⇒ symmetric matrix

Example 6.5

Rayleigh-Ritz method

ppxx2 u =aasin + sin 12LL

aa12,= generalized displacements

ddWU= ⇒ solve approximate solution 19

6.5 Principle of Virtual Force

Principle of Virtual Force

≡ flexibility method

≡ Force method

6.5.1 Equations of Equilibrium

The fundamental requirement on virtual force systems is that they meet the relevant conditions of equilibrium.

For axial force member,

dF = −b dx x

σ xA= FA/

For torsional member,

dM x = −m dx x

Mr τ = x J

For flexural member,

∑=−+Fy dF yy b dx =0 dF ⇒=y b dx y

∑Mz = dM zy −⋅= F dx 0 dM ⇒= z F dx y dM2 z = b dx2 y

6.5.2 Characteristics of virtual force systems

External Equilibrium Equation. ∑=F 0

Internal Equilibrium Equation.

 dF x = −  bx  dx dM  x = −m  dx x  dF dM  y = bF and z =  dx yydx

Virtual complementary Strain energy

ddU**= ∫ U dv =dσ ⋅ ε dv ∫  = ∫dσE−1 σ dv

δδWU**= gives the conditions of compatibility.

The strains and displacements in a deformable system are compatible and consistent with the constraints if and only if the external complementary virtual work is equal to the internal complementary virtual work for every system of virtual force and stresses that satisfy the conditions of equilibrium.

Even if the real force state does not correspond to a deformational state that exactly satisfies compatibility, δδWU**= can be used to enforce an approximate satisfaction of the conditions of compatibility.

For axial force member,

δU*1= ∫ δσ E− σ dV 1 = dσ σ dV ∫ E A = dσ σ dx ∫ E FFd (σ = xx dσ =) EE 1 = d FF d x ∫ EA xx

For torsional member

δU*1= δσ E− σ dV ∫  1 Mr = dτ τ dV τ = x ∫ GJ 1 = d M⋅ r2 M dA dx ∫∫GJ 2 xx 1 = d MM d x ∫ GJ xx

For flexural member

δU*1= δσ E− σ dV ∫  1 My = dσ σ dV σ = − ∫ EI 1 =d M ⋅ M dx ∫ EI

6.5.4 Construction of Analytical Solutions by virtual force principle

Axial force member

F ddU* =⋅= Fx dx F F ∫ xEA xx2 L = d FF xx22EA * ddW= Fux22 ⋅ L ddWU**= ⇒= U F 22EA x

Flexural member

P Mx= 2 x δδMP= ⋅ 2

M ddU* = M dx ∫ EI 2 2 /2 x =d P ⋅⋅ P dx EI ∫0 4 3 =d PP ⋅ 48EI

By using equilibrium equation with low orders, the solution can be found conveniently.

δδW* = Pv ⋅ P3 δδWU**= ⇒= v 48EI

If we set δ P =1,

The principle of virtual force ≡ unit load method.

Discussion on virtual force system

**** δδδδUUUU① =② =③ = ④

The force-equilibrium cause a relative displacement (deformation).

On the other hand, with different support conditions, the relative displacement could be the same when the force-equilibrium is the same. 26

Discussion on stiffness method vs flexibility method

For principle of virtual displacement ddWU= δW= ∫ δεσ dV dv4 = dvEIv dx (= 0) ∫ dx4 v ⇒ 3rd order equation to satisfy the force-equilibrium

If the cross-section is variable along the length, v function becomes more complicated.

On the other hand, for principle of virtual force,

ddWU**= δU* =∫ δσ ⋅ ε dV M = d M dV ∫ EI

M ⇒ 1st order equation for the determinate system.

As the element properties become more complicate, the flexibility method is easier in the derivation of the flexibility matrix and stiffness matrix.