5003 Lectures - Faculty of Engineering and Applied Science

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E5003 - Ship Structures I 156 © C.G. Daley To solve the system of twelve by twelve equations we need to identify the twelve unknowns. It is (almost) never the case that we would know twelve deflections and want to know twelve forces. Nor would we know twelve forces and look for the deflections. Typically we know some forces (mostly zero) and some deflections (zero at supports): We should have some combination of unknown loads and deflections that adds up to twelve. If we don't, we can't solve the system. Note that the structure does not know what is an applied force and what is a reaction. All the structure know is whether it is in equilibrium. There are a variety of ways of solving matrix equations like: There are various numerical strategies used in linear algebra that are used to solve such systems.
E5003 - Ship Structures I 157 © C.G. Daley Gaussian elimination is one common method. We can assume that if we have N equations in N unknowns that we can solve it. To solve these in Maple (see 3bar_frame.pdf or 3bar_frame.mw), we would just expand the matrix expression into a set of 12 simultaneous equations; Maple will solve these equations in either numerical or algebraic form, giving expressions for all results in terms of the variable. For example, for this problem, Maple will give; Q1: With the above solution for force and deflections at the nodes (the dofs), how would we find the stresses in each member? A1: To find the stresses we have to return to the individual beams. We use the global stiffness matrix of a single member. For example, for the cross beam in the previous example (beam 2), we find the member forces as follows;
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5003 Lectures - Faculty of Engineering and Applied Science

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