Vukazich Fall 2016 CE 160 Notes - Principle of Virtual Work for Beams and Frames

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1 CE 16 Notes - Principle of Virtual Work for Beams and Frames Recall the generic form of the Principle of Virtual Work to find deformation in structures. 1 δ P = F Q dl P Consider the beam below. Suppose we want to find δ P, the vertical deflection at the tip of the beam and the slope of the tangent to the deformed shape, θ P, at the tip of the beam due to the loads shown. Real system y w P δ P θ P If the properties of beam are: Length = L Moment of inertia = I Modulus of Elasticity = E 1 CE 16 Virtual Work Notes for Beam and Frame Deformations

2 Recall the moment-curvature relationship that describes deformation due to bending: d! y d! = M EI The moment-curvature relationship can be written in terms of the slope of the beam: d! y d! = d d dy d = dθ d = M EI which can be written in differential form: dθ = M the rotation of each cross section along the beam due to the real loads can be written as: dθ! = M! where M P is the bending moment in the beam due to the applied loads. 2 CE 16 Virtual Work Notes for Beam and Frame Deformations

3 Virtual system to measure δ P In order to measure the real vertical displacement at the tip of the beam, apply a unit virtual (dummy) load in-line with the real deformation (vertical) that we want to measure (δ P ). y 1 L The unit virtual load causes an internal moment in the beam, M Q, that varies with. The eternal virtual work done by unit virtual load acting through the real displacement at the tip of the beam is: W! = 1 δ! The figure below illustrates the internal work that the virtual moment, M Q, does acting through the internal rotation of the cross section, dθ P, at each point along the beam: dθ P M Q d 3 CE 16 Virtual Work Notes for Beam and Frame Deformations

4 du! = M! dθ! = M! M! The total virtual strain energy in the truss can be found by adding up all of the strain energy by integrating over the length of the beam.!!! U! = du!! M! = M!!!!! By energy conservation (W Q = U Q ), we have the epression to find beam (and frame) deformation using Virtual Work 1 δ P = M P M Q where: L = Length of the beam; M Q = Moment function due to unit virtual load; M P = Moment function in the real system; I = Moment of inertial of beam; E = Modulus of elasticity of beam. If the bending stiffness is constant over the length of the beam, we can take it outside of the integrand: 1 δ P = 1 EI M Q M P d 4 CE 16 Virtual Work Notes for Beam and Frame Deformations

5 Vukazich Fall 216 The product integral of the two moment functions in parenthesis can be evaluated using Table 4 on the back cover of your tetbook. 5 CE 16 Virtual Work Notes for Beam and Frame Deformations

6 Displacements due to support settlements in beams and frames can also be considered and would add to the eternal virtual work. M P 1 δ P + R Q δ s = M Q We can also use virtual work to find the slope at the tip of the beam. Virtual system to measure θ P In order to measure the real slope at the tip of the beam, apply a unit virtual (dummy) moment at the tip of the beam to do work through the slope (θ P ) at the tip of the beam. y 1 L Note that the unit virtual moment causes an internal moment in the beam, M Q, that varies with. The eternal virtual work done by unit virtual load acting through the real slope at the tip of the beam is: W! = 1 θ! 6 CE 16 Virtual Work Notes for Beam and Frame Deformations

7 The internal work done by the virtual moment, M Q, follows similarly from the previous case and by energy conservation (W Q = U Q ), we can write the epression to find beam (and frame) slopes using Virtual Work: 1 θ P = M P M Q where: L = Length of the beam; M Q = Moment function due to unit virtual moment; M P = Moment function in the real system; I = Moment of inertial of beam; E = Modulus of elasticity of beam. and similarly, if the bending stiffness is constant along the length of the beam: 1 θ P = 1 EI M Q M P d The product integral of the two moment functions in parenthesis can be evaluated using Table 4 on the back cover of your tetbook. As previously discussed, displacements due to support settlements in beams and frames can also be considered and would add to the eternal virtual work. 7 CE 16 Virtual Work Notes for Beam and Frame Deformations