Method of Virtual Work for Trusses Steven Vukazich San Jose State University Work Done by Force/Moment
Work is done by a force � = �� acting through and in-line displacement � �
� = �� Work is done by a � moment acting through and in-line rotation � Strain Energy of Deformation of a Truss Member � Internal strain energy A = Cross sectional area stored through axial E = Modulus of Elasticity deformation of the truss Δ member: � = �Δ �� Δ = � �� � �
1 � � � = �Δ = 2 2�� ∆ � External Virtual Work for a Truss Member
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: � = �� � � Virtual work done by the virtual force, Q, acting � through the real � deformation, � � = �� �
� A = Cross sectional area � � E = Modulus of Elasticity Virtual Strain Energy for a Truss Member
� � � �� = � � � � �
� � � Deformation of � � � small slice of � member, dL � = � � � �� �
� �� �� � = � �� �� � � = � � �� � Conservation of Energy � � � � � = � �
� �� �� = � � ��
A = Cross sectional area E = Modulus of Elasticity Principle of Virtual Work for a Prismatic, Elastic Truss member
� Q If the internal axial force is Δ� constant and A and E are � constant: �
� �� � � �� = � = � �� �� ��
� � A = Cross sectional area �� = � �� E = Modulus of Elasticity Consider a Truss Structure Subjected To Joint Loads
P1
E F P2 Ai = Cross sectional area Ei = Modulus of Elasticity Li = Length of truss member n = Total number of truss members
A �′ D B C � P3
We want to find the deflection of joint B due to the applied loads Apply Virtual Force
The virtual force E F will cause an internal axial force to develop in each
truss member, FQi A D B C Q
Apply a virtual force in-line with the � = �� real displacement � Apply the Real Loads to the Truss
P1
E F P2 The real loads will cause an internal axial force to develop in each truss member, F A Pi D B C P3
Ai = Cross sectional area The real loads cause an axial Ei = Modulus of Elasticity deformation of each truss L = Length of truss member i n = Total number of truss members member, Δ� = Virtual Strain Energy
P1
E F P2 Ai = Cross sectional area Ei = Modulus of Elasticity Li = Length of truss member n = Total number of truss members
A D B C P3
Virtual strain energy developed in an Virtual strain energy for the entire truss individual truss member � � � = � ⋅ ∆� = � � = � � � Principle of Virtual Work for Truss Deflections
P1 Ai = Cross sectional area E F P2 Ei = Modulus of Elasticity Li = Length of truss member n = Total number of truss members
� = � A �′ D B C � P3 Real Deformation � � �� = � � � Virtual Loads Procedure For Virtual Work Deflection Analysis
P 1 Ai = Cross sectional area E F P2 Ei = Modulus of Elasticity Li = Length of truss member n = Total number of truss members
A �′ D B C � P3
We want to find the real deflection of joint B due to the applied loads, � Step 1 –Virtual Analysis
E F
A D B C Convenient to 1 set Q = 1 1. Remove all loads from the structure; 2. Apply a unit, dimensionless virtual load in-line with the real displacement, � , that we want to find; 3. Perform a truss analysis to find all truss member
virtual axial forces, FQi Step 2 –Real Analysis
P1
E F P2
A D B C P3
1. Place all of the loads on the structure; 2. Perform a truss analysis to find all truss member real
axial forces, FPi Step 3 – Use the Principle of Virtual Work to Find ��
P1 A = Cross sectional area E F P2 i Ei = Modulus of Elasticity Li = Length of truss member n = Total number of truss members
A �′ D B C � P3 From Step 2 – real � � analysis 1 ⋅ � = � � � From Step 1 –virtual analysis