Symmetrical Structures - Symmetrical Structures The...

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Symmetrical Structures Page 1 of 14 2003 Symmetrical Structures The following structures are symmetrical, with symmetrical loading, and as a result of the symmetry one would only like to analyze one half of the structure. The same structures could have anti-symmetrical loading: In both cases the symmetry of the structure can be used and the analysis can be restricted to a half structure. Sway structures will still stay sway structures. The sway may occur in the symmetrical loading case or in the anti-symmetrical loading case. Slope deflection equations or moment-distribution may be used to obtain the bending moments. To decide how the analysis can be restricted to one half only, it is necessary to investigate the effect of symmetry and anti-symmetry on a member AB.
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Symmetrical Structures Page 2 of 14 2003 Note that in the symmetrical case θ B is equal in size to θ A but has the opposite sign, whereas in the anti-symmetrical case the signs are the same. The moments on either side of the axis will have the opposite sign in the case of symmetry and the same sign in the case of anti-symmetry. This can be seen from the slope-deflection equations: Symmetry: ( ) 2 2 3 AB AB A B AB AB AB EI M FEM L θ θ ψ = + + ( ) 2 2 0 AB AB A A AB AB EI M FEM L θ θ = + ( ) 2 AB AB A AB AB EI M FEM L θ = + Note that with symmetry the fixed end moment at B is the negative of the fixed end moment at A. ( ) 2 2 3 AB BA A B AB BA AB EI M FEM L θ θ ψ = + + ( ) 2 2 0 AB BA A A AB AB EI M FEM L θ θ = ( ) 2 AB BA A AB AB EI M FEM L θ = − Note that in this case the moments have the opposite sign. Anti-symmetry: With anti-symmetry the rotation at B is equal in sign and size to the rotation at A. ( ) 2 2 3 AB AB A B AB AB AB EI M FEM L θ θ ψ = + + ( ) 2 2 0 AB AB A A AB AB EI M FEM L θ θ = + + ( ) 6 AB AB A AB AB EI M FEM L θ = + The fixed end moment at B is equal in size and sign in the anti-symmetrical loading case. ( ) 2 2 3 AB BA A B AB BA AB EI M FEM L θ θ ψ = + + ( ) 2 2 0 AB BA A A BA AB EI M FEM L θ θ = + + ( ) 6 AB BA A BA AB EI M FEM L θ = + Note that in this case the moments have the same sign.
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