Method of Virtual Work for Beams and Frames

StevenVukazichSanJoseStateUniversity

Work Done by Force/Moment

𝑊 = 𝐹𝛿

𝛿𝐹

Work is done by a force acting through and in-line displacement

𝑊 = 𝑀𝜃

𝑀𝜃

Work is done by a moment acting through and in-line rotation

Recall the General Form of the Principle of Virtual Work

𝑄𝛿( +*𝑅,

�

�

𝛿. = /𝐹,𝑑𝐿(

�

�

Real Deformation

Virtual Loads

External Work due to Support Settlements

Consider a Beam SubjectedTo General Loading

We want to find the deflection at point A and the slope at point B due to the applied loads

PM

w

x

y

𝜃2

Modulus of Elasticity = EMoment of Inertia = I

AB

ab

L

𝛿3

Apply a Virtual Force toMeasure the Deflection at A

Qx

y Modulus of Elasticity = EMoment of Inertia = I

Aa

L

𝛿,𝑑𝑥

𝑑𝜃,

𝑀,𝑀,

For bending, the internal virtual load is MQ

𝑑𝑥

Add the Real Loads and Examine Internal Deformation due to Bending

Q

PM

w

x

yModulus of Elasticity = EMoment of Inertia = I

Aa

L

𝛿,𝑑𝑥

𝑑𝜃( 𝑑𝜃,

𝑀,𝑀( 𝑀(

𝛿3

𝑑5𝑦𝑑𝑥5

=𝑑𝑑𝑥

𝑑𝑦𝑑𝑥

=𝑀(

𝐸𝐼

𝑀,

Can rewrite the moment-curvature relationship

𝑑𝜃(𝑑𝑥

=𝑀(

𝐸𝐼 𝑑𝜃( =𝑀(

𝐸𝐼𝑑𝑥

𝑑𝑥

Modify the General Form of the Principle of Virtual Work for Bending Deformation

𝑄𝛿( = /𝐹,𝑑𝐿(

�

�

Real Deformation

Virtual Loads

𝑑𝜃(

𝑑𝜃,

𝑀,𝑀( 𝑀(

For bending deformation, dLP = d𝜃P and FQ = MQ

𝑄𝛿( = 9 𝑀,𝑑𝜃(:

;

The Principle of Virtual Work expression for bending can be expressed as:

𝑑𝑥𝑀,

Principle of Virtual Work to Measure 𝜹𝑨

𝑄𝛿3 = 9 𝑀,

:

;

𝑀(

𝐸𝐼𝑑𝑥

𝑑𝜃( =𝑀(𝐸𝐼

𝑑𝑥

If the bending stiffness, EI, is constant:

𝑄𝛿3 =1𝐸𝐼9 𝑀,

:

;𝑀(𝑑𝑥

Q

PM

w

x

yModulus of Elasticity = EMoment of Inertia = I

Aa 𝛿, 𝑑𝑥

𝛿3

𝑄𝛿( = 9 𝑀,𝑑𝜃(:

;

To Measure Rotational Deformation, Apply a Virtual Moment

x

y Modulus of Elasticity = EMoment of Inertia = I

B

L

𝜃,

𝑑𝑥𝑑𝜃,

𝑀, 𝑀,

For bending, the internal virtual load is MQ

Q

𝑑𝑥

b

PM

w

x

yModulus of Elasticity = EMoment of Inertia = I

B

L

𝑑𝑥

𝑑𝜃( 𝑑𝜃,

𝑀,𝑀( 𝑀(𝑀,

From the moment-curvature relationship

𝑑𝜃(𝑑𝑥

=𝑀(

𝐸𝐼 𝑑𝜃( =𝑀(

𝐸𝐼𝑑𝑥

𝑑𝑥

Q 𝜃2

b

Add the Real Loads and Examine Internal Deformation due to Bending

𝑑5𝑦𝑑𝑥5

=𝑑𝑑𝑥

𝑑𝑦𝑑𝑥

=𝑀(

𝐸𝐼

𝑄𝜃2 = 9 𝑀,

:

;

𝑀(

𝐸𝐼𝑑𝑥

𝑑𝜃( =𝑀(𝐸𝐼

𝑑𝑥

If the bending stiffness, EI, is constant:

𝑄𝜃2 =1𝐸𝐼9 𝑀,

:

;𝑀(𝑑𝑥

Modulus of Elasticity = EMoment of Inertia = I

PM

w

x

y

B

L

𝑑𝑥

Q 𝜃2

b

Principle of Virtual Work to Measure 𝜽𝑨

𝑄𝛿( = 9 𝑀,𝑑𝜃(:

;

Summary of Procedure for Finding Bending Deformation Using Virtual Work

We want to find the deflection at point A and the slope at point B due to the applied loads

PMw

x

y

𝜃2

Modulus of Elasticity = EMoment of Inertia = I

AB

ab

L

𝛿3

Step 1 – Remove all loads and apply a virtual force (or moment) to measure the deformation at the point of interest

Qx

y

Aa

L

From an equilibrium analysis, find the internal bending moment function for the virtual system: MQ(x)

Convenient to set Q = 1

PMw

x

y

L

Step 2 – Replace all of the loads on the structure and perform the real analysis

From an equilibrium analysis, find the internal bending moment function for the real system: MP(x)

𝑄𝛿3 = 9 𝑀,

:

;

𝑀(

𝐸𝐼𝑑𝑥

If the bending stiffness, EI, is constant:

𝑄𝛿3 =1𝐸𝐼9 𝑀,

:

;𝑀(𝑑𝑥

Table in textbook appendix is provided to help evaluate product integrals of this type

Step 3 – Evaluate the virtual work product integrals and solve for the deformation of interest

Appendix Table.2

Table to Evaluate Virtual Work Product Integrals

9 𝑀,

:

;𝑀(𝑑𝑥

Table is as useful tool to evaluate product integrals of the form: