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The bending stiffness is the resistance of a member against bending deflection/deformation. It is a function of the Young's modulus E {\displaystyle E} , the second moment of area I {\displaystyle I} of the beam cross-section about the axis of interest, length of the beam and beam boundary condition.
Besides deflection, the beam equation describes forces and moments and can thus be used to describe stresses. For this reason, the Euler–Bernoulli beam equation is widely used in engineering, especially civil and mechanical, to determine the strength (as well as deflection) of beams under bending.
The top and bottom example may be considered structurally equivalent, depending on the effective stiffness of the spring and beam element. A cantilever is a rigid structural element that extends horizontally and is unsupported at one end. Typically it extends from a flat vertical surface such as a wall, to which it must be firmly attached.
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]
Continuum mechanics is valid for a bending beam. The stress at a cross section varies linearly in the direction of bending, and is zero at the centroid of every cross section. The bending moment at a particular cross section varies linearly with the second derivative of the deflected shape at that location. The beam is composed of an isotropic ...
The bending stiffness (EI/L) of a member is represented as the flexural rigidity of the member (product of the modulus of elasticity (E) and the second moment of area (I)) divided by the length (L) of the member. What is needed in the moment distribution method is not the specific values but the ratios of bending stiffnesses between all members.
The proportions of the beam are such that it would fail by bending rather than by crushing, wrinkling or sideways buckling. Cross-sections of the beam remain plane during bending. Deflection of a beam deflected symmetrically and principle of superposition. Compressive and tensile forces develop in the direction of the beam axis under bending loads.
If the beam is bent side to side, it functions as an 'H', where it is less efficient. The most efficient shape for both directions in 2D is a box (a square shell); the most efficient shape for bending in any direction, however, is a cylindrical shell or tube. For unidirectional bending, the Ɪ-beam or wide flange beam is superior. [5]