3 - Virtual Work - Advanced Examples

# 3 - Virtual Work - Advanced Examples - Structural Analysis...

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Structural Analysis IV Chapter 3 – Virtual Work: Advanced Examples Dr. C. Caprani 1 Chapter 3 - Virtual Work: Advanced Examples 3.1 Introduction . ........................................................................................................ 2 3.1.1 General . .......................................................................................................... 2 3.2 Ring Beam Examples . ......................................................................................... 3 3.2.1 Example 1 . ..................................................................................................... 3 3.2.2 Example 2 . ..................................................................................................... 8 3.2.3 Example 3 . ................................................................................................... 15 3.2.4 Example 4 . ................................................................................................... 23 3.2.5 Example 5 . ................................................................................................... 32 3.2.6 Review of Examples 1 – 5 . .......................................................................... 53 3.3 Grid Examples . .................................................................................................. 64 3.3.1 Example 1 . ................................................................................................... 64 3.3.2 Example 2 . ................................................................................................... 70 3.3.3 Example 3 . ................................................................................................... 79 Rev. 1

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Structural Analysis IV Chapter 3 – Virtual Work: Advanced Examples Dr. C. Caprani 2 3.1 Introduction 3.1.1 General To further illustrate the virtual work method applied to more complex structures, the following sets of examples are given. The examples build upon each other to illustrate how the analysis of a complex structure can be broken down.
Structural Analysis IV Chapter 3 – Virtual Work: Advanced Examples Dr. C. Caprani 3 3.2 Ring Beam Examples 3.2.1 Example 1 Problem For the quarter-circle beam shown, which has flexural and torsional rigidities of EI and GJ respectively, show that the deflection at A due to the point load, P , at A is: 33 38 44 Ay PR PR EI GJ ππ δ  = ⋅+  

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Structural Analysis IV Chapter 3 – Virtual Work: Advanced Examples Dr. C. Caprani 4 Solution The point load will cause both bending and torsion in the beam member. Therefore both effects must be accounted for in the deflection calculations. Shear effects are ignored. Drawing a plan view of the structure, we can identify the perpendicular distance of the force, P , from the section of consideration, which we locate by the angle θ from the y -axis: The bending moment at C is P times the perpendicular distance AC , called m . The torsion at C is the force times the transverse perpendicular distance CD , called t . Using the triangle ODA , we have:
Structural Analysis IV Chapter 3 – Virtual Work: Advanced Examples Dr. C. Caprani 5 sin sin cos cos m mR R OD OD R R θθ = ∴= = The distance CD , or t , is R OD , thus: ( ) cos 1 cos tRO D RR R θ = = = Thus the bending moment at point C is: ( ) sin M Pm PR = = (1) The torsion at C is: ( ) ( ) 1 cos T Pt PR = = (2) Using virtual work, we have: 0 EI Ay W WW MT F M ds T ds EI GJ δ δδ = = ⋅= + ∫∫ (3)

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Structural Analysis IV Chapter 3 – Virtual Work: Advanced Examples Dr. C. Caprani 6 This equation represents the virtual work done by the application of a virtual force, F δ , in the vertical direction at A , with its internal equilibrium virtual moments and torques, M and T and so is the equilibrium system. The compatible displacements system is that of the actual deformations of the structure, externally at A , and internally by the curvatures and twists, M EI and T GJ . Taking the virtual force, 1 F = , and since it is applied at the same location and direction as the actual force P , we have, from equations (1) and (2): ( ) sin MR δθ θ = (4) ( ) ( ) 1 cos TR = (5) Thus, the virtual work equation, (3), becomes: [ ][ ] ( ) ( ) 22 00 11 1 sin sin 1 cos 1 cos Ay M M ds T T ds EI GJ PR R Rd PR R Rd EI GJ ππ δδ θθ ⋅= + = + −−   ∫∫ (6) In which we have related the curve distance, ds , to the arc distance, ds Rd = , which allows us to integrate round the angle rather than along the curve. Multiplying out:
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3 - Virtual Work - Advanced Examples - Structural Analysis...

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