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In the Kirchhoff–Love plate theory for plates the governing equations are [1], = and , = In expanded form, + = ; + = and + + = where () is an applied transverse load per unit area, the thickness of the plate is =, the stresses are , and
The plate elastic thickness (usually referred to as effective elastic thickness of the lithosphere). The elastic properties of the plate; The applied load or force; As flexural rigidity of the plate is determined by the Young's modulus, Poisson's ratio and cube of the plate's elastic thickness, it is a governing factor in both (1) and (2).
In physics, Hooke's law is an empirical law which states that the force (F) needed to extend or compress a spring by some distance (x) scales linearly with respect to that distance—that is, F s = kx, where k is a constant factor characteristic of the spring (i.e., its stiffness), and x is small compared to the total possible deformation of the spring.
In continuum mechanics, plate theories are mathematical descriptions of the mechanics of flat plates that draw on the theory of beams. Plates are defined as plane structural elements with a small thickness compared to the planar dimensions. [1] The typical thickness to width ratio of a plate structure is less than 0.1.
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.
in these formulas the following parameters are used: = Stress in outer fibers at midpoint, = load at a given point on the load deflection curve, = Support span, (mm) = Width of test beam, (mm) = Depth or thickness of tested beam, (mm)
Deformation of a thin plate highlighting the displacement, the mid-surface (red) and the normal to the mid-surface (blue) The Kirchhoff–Love theory of plates is a two-dimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments.
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 ...