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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.
The cantilever method is an approximate method for calculating shear forces and moments developed in beams and columns of a frame or structure due to lateral loads. The applied lateral loads typically include wind loads and earthquake loads, which must be taken into consideration while designing buildings.
Like other structural elements, a cantilever can be formed as a beam, plate, truss, or slab. When subjected to a structural load at its far, unsupported end, the cantilever carries the load to the support where it applies a shear stress and a bending moment. [1] Cantilever construction allows overhanging structures without additional support.
In this case, the equation governing the beam's deflection can be approximated as: = () where the second derivative of its deflected shape with respect to (being the horizontal position along the length of the beam) is interpreted as its curvature, is the Young's modulus, is the area moment of inertia of the cross-section, and is the internal ...
A cantilever Timoshenko beam under a point load at the free end For a cantilever beam , one boundary is clamped while the other is free. Let us use a right handed coordinate system where the x {\displaystyle x} direction is positive towards right and the z {\displaystyle z} direction is positive upward.
Fig. 1: Critical stress vs slenderness ratio for steel, for E = 200 GPa, yield strength = 240 MPa.. Euler's critical load or Euler's buckling load is the compressive load at which a slender column will suddenly bend or buckle.
Non-circular cross-sections always have warping deformations that require numerical methods to allow for the exact calculation of the torsion constant. [2] The torsional stiffness of beams with non-circular cross sections is significantly increased if the warping of the end sections is restrained by, for example, stiff end blocks. [3]
For a 3-point test of a rectangular beam behaving as an isotropic linear material, where w and h are the width and height of the beam, I is the second moment of area of the beam's cross-section, L is the distance between the two outer supports, and d is the deflection due to the load F applied at the middle of the beam, the flexural modulus: [1]