5003 Lectures - Faculty of Engineering and Applied Science

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E5003 - Ship Structures I 116 © C.G. Daley A rotation at #6 require a similar set of forces, so that we can also write; 4EI 2EI k66 = , k36 = L L 6EI −6EI k 26 = 2 , k56 = 2 , k 16 L L = k 46 = 0 We can collect all these terms in the matrix; ⎡ AE ⎢ L ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 K = ⎢ ⎢ − AE ⎢ L ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢⎣ 0 12EI L 6EI L −12EI L 6EI L 3 2 0 3 2 0 6EI 2 L 4EI L 0 − 6EI 2 L 2EI L − AE L 0 0 AE L 0 0 0 −12EI L − 6EI L 0 12EI L − 6EI L 3 2 3 2 ⎤ 0 ⎥ 6EI ⎥ ⎥ 2 L ⎥ 2EI ⎥ L ⎥ ⎥ 0 ⎥ ⎥ − 6EI ⎥ 2 L ⎥ 4EI ⎥ ⎥ L ⎥⎦ Note that the matrix is symmetrical. This means that terms such as k35 (moment at #3 due to displacement at #5) is equal to k35 (force at #5 due to rotation at #3). This may seem quite odd that these two items would be equal. We will examine this in the next Chapter. The standard 3D degrees of freedom for a beam are shown below; The K matrix for a 3D beam is a 12x12 (144 terms). 3D beam = 12 degrees of freedom
E5003 - Ship Structures I 117 © C.G. Daley Topic 11: Problems 11.1. Solve the pinned-pinned beam by using the displacement method as sketched below. The solution for the fixed-fixed beam is the same as above. Then it is necessary to show that MB*+MB**+MB***=0 and MA*+MA**+MA***=0. Note: MA** = ½ MB**, and MB*** = ½ MA***. 11.2. Describe how you would solve the beam shown below by using the displacement method. 11.3. For the simple beam shown below, derive the shear stiffness terms (i.e k15 to k65) 11.4. Solve the beam shown below using the stiffness method. Find the reactions at A and B, and the deflection at B.
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