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Sometimes the equilibrium equations – force and moment equilibrium conditions – are insufficient to determine the forces and reactions. Such a situation is described as statically indeterminate. Statically indeterminate situations can often be solved by using information from outside the standard equilibrium equations.
Kinetics deals with states that are not in mechanical equilibrium. According to Newton's second law, an external force leads to a change in velocity (acceleration) of a body. Analogously an external torque means a change in angular velocity resulting in an angular acceleration. The inertia relating to rotation depends not only on the mass of a ...
In order to distinguish between this and the situation when a system under equilibrium is perturbed and becomes unstable, it is preferable to use the phrase partly constrained here. In this case, the two unknowns V A and V C can be determined by resolving the vertical force equation and the moment equation simultaneously. The solution yields ...
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 boundary conditions usually model supports, but they can also model point loads, distributed loads and moments. The support or displacement boundary conditions are used to fix values of displacement ( w {\displaystyle w} ) and rotations ( d w / d x {\displaystyle \mathrm {d} w/\mathrm {d} x} ) on the boundary.
By forming slope deflection equations and applying joint and shear equilibrium conditions, the rotation angles (or the slope angles) are calculated. Substituting them back into the slope deflection equations, member end moments are readily determined. Deformation of member is due to the bending moment.
The boundary conditions that are needed to solve the equilibrium equations of plate theory can be obtained from the boundary terms in the principle of virtual work. For small strains and small rotations, the boundary conditions are
Close to local equilibrium, solution of the Boltzmann equation can be represented by an asymptotic expansion in powers of Knudsen number (the Chapman–Enskog expansion [20]). The first two terms of this expansion give the Euler equations and the Navier–Stokes equations. The higher terms have singularities.