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Simply supported beam with a single eccentric concentrated load. An illustration of the Macaulay method considers a simply supported beam with a single eccentric concentrated load as shown in the adjacent figure. The first step is to find . The reactions at the supports A and C are determined from the balance of forces and moments as
Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) [1] is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It covers the case corresponding to small deflections of a beam that is subjected to lateral ...
Otherwise methods such as virtual work, direct integration, Castigliano's method, Macaulay's method or the direct stiffness method are used. The deflection of beam elements is usually calculated on the basis of the Euler–Bernoulli beam equation while that of a plate or shell element is calculated using plate or shell theory.
Direct integration is a structural analysis method for measuring internal shear, internal moment, rotation, and deflection of a beam. Positive directions for forces acting on an element. For a beam with an applied weight w ( x ) {\displaystyle w(x)} , taking downward to be positive, the internal shear force is given by taking the negative ...
Beam – A measure of the width of the ship. There are two types: Beam, Overall (BOA), commonly referred to simply as Beam – The overall width of the ship measured at the widest point of the nominal waterline. Beam on Centerline (BOC) – Used for multihull vessels. The BOC for vessels is measured as follows: For a catamaran: the ...
After solving the differential equation for the normal forces in the cover sheets for a single span beam under a given load, the energy method can be used to expand the approach for the calculation of multi-span beams. Sandwich continuous beam with flexible cover sheets can also be laid on top of each other when using this technique.
The attempts to provide precise expressions were made by many scientists, including Stephen Timoshenko, [12] Raymond D. Mindlin, [13] G. R. Cowper, [14] N. G. Stephen, [15] J. R. Hutchinson [16] etc. (see also the derivation of the Timoshenko beam theory as a refined beam theory based on the variational-asymptotic method in the book by Khanh C ...
Releasing the vertical reaction for A allows the beam to rotate to Δ. Likewise for part (c). Δ is typically taken as positive upwards. Part (d) of the figure shows the influence line for shear at point B. Using the beam sign convention and cutting the beam at B, we can deduce the figure shown.