Following these successful demonstrations, it quickly became a cornerstone of engineering and an enabler of the Second Industrial Revolution.Īdditional mathematical models have been developed such as plate theorybut the simplicity of beam theory makes it an important tool in the sciences, especially structural and mechanical engineering. It was first enunciated circa but was not applied on a large scale until the development of the Eiffel Tower and the Ferris wheel in the late 19th century. It is thus a special case of Timoshenko beam theory.
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It covers the case for small deflections of a beam that are subjected to lateral loads only. Consider a cantilever beam uniform section with a single concentrated load at the end.įrom the equilibrium balance.Euler-Bernoulli beam theory also known as engineer's beam theory or classical beam theory is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. Singularity functions can be used for determining the values when the loading a not simple ref Singularity Functions. The properties of the supports or fixings may be used to determine the constants. The deflection between limits is obtained by further integration. From this simple approximation the following relationships are derived. Values of Z are provided in the tables showing the properties of standard steel sections. It is clear from above that a simple beam subject to bending generates a maximum stress at the surface furthest away from the neutral axis.įor sections symmetrical about Z-Z the maximum compressive and tensile stress is equal. From the above the following important simple beam bending relationship results.
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Therefore the couple of the force resulting from the stress on each area when totalled over the whole area will equal the applied moment.
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This can only be zero if the axis passes through the centre of gravity centroid of the section. Assuming E is the same for compression and tension the relationship is the same.Īs the beam is in static equilibrium and is only subject to moments no vertical shear forces the forces across the section AB are entirely longitudinal and the total compressive forces must balance the total tensile forces. The compressive stress is also directly related to the distance below the neutral axis. Therefore, for the illustrated example, the tensile stress is directly related to the distance above the neutral axis. For beams subjected to several loads of different types the resulting shear force, bending moment, slope and deflection can be found at any location by summing the effects due to each load acting separately to the other loads.Ī straight bar of homogeneous material is subject to only a moment at one end and an equal and opposite moment at the other end After bending the sections will be at A'B' and C'D' and are no longer parallel. The superposition principle is one of the most important tools for solving beam loading problems allowing simplification of very complicated design problems. This theory relates to beam flexure resulting from couples applied to the beam without consideration of the shearing forces.
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Newton's Law of Viscosity.Simple Bending Area Moment Method Composite Beams The stress, strain, dimension, curvature, elasticity, are all related, under certain assumption, by the theory of simple bending.