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 Δ� 3. Note that the work done by �4 the virtual force Q only is: �2 = ��4 �2 � Virtual work done by the virtual force, Q, acting � through the real � deformation, �4 � = �� � 2 4 � A = Cross sectional area �2 �4 E = Modulus of Elasticity Virtual Strain Energy for a Truss Member � � � ��2 = �2�674 Δ� �4 �4 �2 �2 � Deformation of �672 �674 � small slice of �4 7 member, dL �2 = 8 �2�674 �2 �� 9 �674 �4�� �� 7 �674 = �4�� �� �2 � = 8 � �672 2 2 �� 9 �4 Conservation of Energy � � � Δ� �2 = �2 4 �2 7 �4�� ��4 = 8 �2 � 9 �� 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 �4 constant: �2 7 7 �4�� �4 � ��4 = 8 �2 = �2 8 �� 9 �� �� 9 � � 4 A = Cross sectional area ��4 = �2 �� 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 �4 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 �4 2 4 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 ?@A7A n = Total number of truss members member, Δ�4> = BACA 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 F � � ? 7 4> > � = � ⋅ ∆� = � @A A �2 = E �2> 2> 2> 4> 2> B C �>�> A A >G* 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 �2 = �2 A �′ D B C �4 P3 Real Deformation F �4>�> ��4 = E �2> �>�> >G* 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 �4 P3 We want to find the real deflection of joint B due to the applied loads, �4 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, �4, 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 �4 P3 From Step 2 – real F �4>�> analysis 1 ⋅ �4 = E �2> �>�> >G* From Step 1 –virtual analysis.