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The curve () describes the deflection of the beam in the direction at some position (recall that the beam is modeled as a one-dimensional object). is a distributed load, in other words a force per unit length (analogous to pressure being a force per area); it may be a function of , , or other variables.
The starting point is the relation from Euler-Bernoulli beam theory = Where is the deflection and is the bending moment. This equation [7] is simpler than the fourth-order beam equation and can be integrated twice to find if the value of as a function of is known.
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 ...
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. It is given by the formula: [1] = where
The equations below assume a beam with a circular cross-section at all values of z; this can be seen by noting that a single transverse dimension, r, appears.Beams with elliptical cross-sections, or with waists at different positions in z for the two transverse dimensions (astigmatic beams) can also be described as Gaussian beams, but with distinct values of w 0 and of the z = 0 location for ...
The attenuation coefficient of a volume, denoted μ, is defined as [6] =, where Φ e is the radiant flux;; z is the path length of the beam.; Note that for an attenuation coefficient which does not vary with z, this equation is solved along a line from =0 to as:
For the use in differential geometry, see Cesàro equation. For the radius of curvature of the Earth (approximated by an oblate ellipsoid); see also: arc measurement; Radius of curvature is also used in a three part equation for bending of beams. Radius of curvature (optics) Thin films technologies; Printed electronics; Minimum railway curve ...
The resulting equation is of fourth order but, unlike Euler–Bernoulli beam theory, there is also a second-order partial derivative present. Physically, taking into account the added mechanisms of deformation effectively lowers the stiffness of the beam, while the result is a larger deflection under a static load and lower predicted ...