Method of Virtual Work for TrussesStevenVukazich

SanJoseStateUniversity

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

Strain Energy of Deformation of a Truss Member

Δ

𝑃Internal strain energy stored through axial deformation of the truss member: 𝑈 = *

+𝑃Δ

𝐿𝐹

∆ 𝛿

𝑃

A = Cross sectional areaE = Modulus of Elasticity

𝑈 =12𝑃Δ =

𝑃+𝐿2𝐴𝐸

Δ =𝑃𝐿𝐴𝐸

External Virtual Work for a Truss Member

𝛿2

𝐿𝐹

𝛿𝑄

A = Cross sectional areaE = Modulus of Elasticity

𝑊2 = 𝑄𝛿4

1. Apply a virtual force Q to the truss member;

2. Apply an additional real force P to the truss member

3. Note that the work done by the virtual force Q only is: 𝑊2 = 𝑄𝛿4

𝛿2

𝑃

𝛿4

𝛿4

𝑄𝑃

Δ𝐿

Virtual work done by the virtual force, Q, acting through the real deformation, 𝛿4

Virtual Strain Energy for a Truss Member𝐹

𝛿𝐹2

𝑑𝑈2 = 𝐹2𝛿674

𝛿672

𝐹4

𝛿674

𝑑𝐿

𝐹2

𝐹4𝑑𝐿

𝛿672

𝛿674

𝐹2𝐹4

𝑈2 = 8 𝐹2𝛿6747

9

𝛿674 =𝐹4𝑑𝐿𝐴𝐸

𝛿2𝛿4

𝑄𝑃

Δ𝐿

𝐿

𝑈2 = 8 𝐹2𝐹4𝑑𝐿𝐴𝐸

7

9

Deformation of small slice of member, dL

Conservation of Energy

𝑊2 = 𝑈2

𝑄𝛿4 = 8 𝐹2𝐹4𝑑𝐿𝐴𝐸

7

9

𝛿2

𝐿

A = Cross sectional areaE = Modulus of Elasticity

𝛿4

𝑄𝑃

Δ𝐿

Principle of Virtual Work for a Prismatic, Elastic Truss member

𝑄𝛿4 = 8 𝐹2𝐹4𝑑𝐿𝐴𝐸

= 𝐹2𝐹4𝐴𝐸

8 𝑑𝐿7

9

7

9

If the internal axial force is constant and A and E are constant:

𝑄𝛿4 = 𝐹2𝐹4𝐿𝐴𝐸

𝛿2

𝐿

A = Cross sectional areaE = Modulus of Elasticity

𝛿4

Q𝑃

Δ𝐿

AB C

P1

E F

Consider a Truss Structure SubjectedTo Joint Loads

P2

P3

D𝐵′

𝛿4

We want to find the deflection of joint B due to the applied loads

Ai = Cross sectional areaEi = Modulus of ElasticityLi = Length of truss membern = Total number of truss members

A

B C

E F

Apply Virtual Force

D

Apply a virtual force in-line with the real displacement 𝛿4

Q

𝑊2 = 𝑄𝛿4

The virtual force will cause an internal axial force to develop in each truss member, FQi

AB C

P1

E F P2

P3

D

Apply the Real Loads to the Truss

The real loads cause an axial deformation of each truss member,Δ𝐿4> =

?@A7ABACA

Ai = Cross sectional areaEi = Modulus of ElasticityLi = Length of truss membern = Total number of truss members

The real loads will cause an internal axial force to develop in each truss member, FPi

AB C

P1

E F P2Ai = Cross sectional areaEi = Modulus of ElasticityLi = Length of truss membern = Total number of truss members

P3

D

Virtual Strain Energy

𝑈2> = 𝐹2> ⋅ ∆𝐿4> = 𝐹2> ?@A7ABACA

Virtual strain energy developed in an individual truss member

𝑈2 =E𝐹2> 𝐹4>𝐿>𝐴>𝐸>

F

>G*

Virtual strain energy for the entire truss

Principle of Virtual Work for Truss Deflections

𝑊2 = 𝑈2A

B C

P1

E F P2

P3

D𝐵′

𝛿4

𝑄𝛿4 =E𝐹2> 𝐹4>𝐿>𝐴>𝐸>

F

>G*

Real Deformation

Virtual Loads

Ai = Cross sectional areaEi = Modulus of ElasticityLi = Length of truss membern = Total number of truss members

Procedure For Virtual Work Deflection Analysis

AB C

P1

E F P2

P3

D𝐵′

𝛿4

We want to find the real deflection of joint B due to the applied loads, 𝛿4

Ai = Cross sectional areaEi = Modulus of ElasticityLi = Length of truss membern = Total number of truss members

Step 1 –Virtual Analysis

A

B C

E F

D

1. Remove all loads from the structure;2. Apply a unit, dimensionless virtual load in-line with

the real displacement, 𝛿4, that we want to find;3. Perform a truss analysis to find all truss member

virtual axial forces, FQi

1Convenient to set Q = 1

Step 2 –Real Analysis

1. Place all of the loads on the structure;2. Perform a truss analysis to find all truss member real

axial forces, FPi

AB C

P1

E F P2

P3

D

AB C

P1

E F

Step 3 – Use the Principle of VirtualWork to Find 𝜹𝑷

P2

P3

D𝐵′

𝛿4

1 ⋅ 𝛿4 =E𝐹2> 𝐹4>𝐿>𝐴>𝐸>

F

>G*

From Step 2 – real analysis

From Step 1 –virtual analysis

Ai = Cross sectional areaEi = Modulus of ElasticityLi = Length of truss membern = Total number of truss members