Tutorial Sheet 1 Solutions

Tutorial Sheet 1 Solutions - Imperial College of Science...

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Imperial College of Science, Technology & Medicine Department of Aeronautics Second Year Structural Mechanics and Dynamics II Tutorial Sheet 1 Solutions Q1. Part 1 The unit virtual load R in a pin-jointed structure gives rise to a virtual load i N in member i. Hence the virtual stress in member i is i i i A N = σ The real displacement r arises from the real forces i N in all bars i Hence the real stress in member i is i i i A N = σ and so the real strain is i i i i A E N = ε The integral over the volume can be written as i bars of . no 1 i V i i i i i i i bars of . no 1 i V i V dV A E N A N dV dV r 1 i i = = = ε σ = ε σ = All of these quantities are constant for a bar so that i i V i L A V d i = and the unit load equation for a pin-jointed framework becomes = = bars of . no 1 i i i i i i A E L N N r Q1 Part 2 The unit virtual load R in a beam structure gives rise to a virtual bending moment ( ) z M i in member i. This is not constant over the member; it generally varies with the length, z, along the member. From beam theory, the virtual stress at any distance, z , along the member is ( ) i i i i I y z M = σ . The real displacement, r , arises from the real bending moment, i M , in all of the beams. This gives rise to the bending strains (the curvature) as. ( ) i i i i I E y z M = ε The element of volume is i i i dz dA dV = so that the unit load method gives i i 2 i i i i beams of . no 1 i V i i i i i i i i beams of . no 1 i V
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Tutorial Sheet 1 Solutions - Imperial College of Science...

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