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The principle of virtualThe principle of virtualforcesforces

A presentation byA presentation by

Benjamin CzwiklaBenjamin Czwikla

Lukas SchaperLukas Schaper

Adrian BrylkaAdrian Brylka

Felix HegemannFelix Hegemann

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Principle of virtual forces:

If you apply infinitisemal small, virtual forces (stresses) on a field,the external virtual work is equal to the whole inner virtual work

The principle of virtual work is often used for calculation ofdisplacements

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The external work is defined as:The external work is defined as:

q virtual surface forcesb virtual volume forcesu displacement condition

The internal work is defined as:The internal work is defined as:

virtual stresses strains

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Consequentially:Consequentially:

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That means for a simple calculation of a beam structure:That means for a simple calculation of a beam structure:

Internal Work:

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External Work:

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A simple exampleA simple example::

Statically defined, shear fixed beam witha constant distributed load q

That means: N = Q = 0

We are looking for the displacement w at the point s

To get this displacement,we bring up a unity force at point s in the

direction of the displacement we are looking for

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Now we have to calculate seperately the momentdiagrams for both loads and superpose it

The result is the searched displacement w

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Differences betweenDifferences between

the principle of virtual forcesthe principle of virtual forcesandand

the priciple of virtual displacementthe priciple of virtual displacement

Principle of virtual forces Principle of virtual displacement

Simple manual calculation for easysystems equilibrium conditions achievedexactly leads to larger displacements than

real=> soft solution

easier to program for complexsystems kinematic conditions achieved exactly

leads to smaller displacements than

real=> stiff solution

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Complementary workw*:

Force-displacement diagramm

u = virtual displacement

w = virtual work

f = virtual force

Definition of complementary work:

Theoretical involvement in FEMTheoretical involvement in FEM

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For a linear problem, like we assumed before, thecomplementary works are equal to the real works.

That means.:

Equilibrium of Work:

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It is essential:

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From this it follows that:

Force - displacement law:

Both equations leads us to the following flexibility matrix:

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In Consideration of a linear problem:

By using Hookes law:

Like the stiffness matrix, the flexibility matrix is symmetrically:

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A little example:

A system with one degree of freedom which leads to a flexibility matrix withone coefficient:

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Because of c1111 = k1111 = E:

The stress results of the load f in the first degree of freedom:

So we get the flexibility coefficient we are looking for:

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The problem of using the principle of virtual forces in finite elementprograms is, that you can only calculate stresses from external forces, if youhave a statically defined system.

In all other cases you have to manipulate the system to get a primarystructure (statically definded system) by reducing the degree of freedom.

This manipulation is a big problem for an automatical process and its mucheasier to program it with the principle of virtual displacement.

Nevertheless, it is possible using the principle of virtual forces, as you can

see on the example before.

Quintessence

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Thanksfor

yourat

tention!

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