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### 13.5. Boundary Conditions

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Solution of the plate equation requires that two boundary conditions be satisfied at each edge. These may relate to deflection and slope, or to forces and moments, or to some combination. The principal feature distinguishing the boundary conditions applied to plates from those applied to beams relates to the existence along the plate edge of twisting moment resultants. These moments, as demonstrated next, may be replaced by an equivalent vertical force, which when added to the vertical shearing force produces an effective vertical force.

Consider two successive elements of lengths dy on edge x = a of the plate shown in Fig. 13.4a. On the right element, a twisting moment M_{xy}
dy acts, while the left element is subject to a moment [M_{xy} + (∂M_{xy}/∂y) dy]dy. In Fig. 13.4b, we observe that these twisting moments have been replaced by equivalent force couples that produce only local differences in the distribution of stress on the edge x = a. The stress distribution elsewhere in the plate is unaffected by this substitution. Acting at the left edge of the right element is an upward directed force M_{xy}. Adjacent to this force is a downward directed force M_{xy} + (∂M_{xy}/∂y) dy acting at the right edge of the left element. The difference between these forces (expressed per unit length), ∂M_{xy}/∂y, may be combined with the transverse shearing force Q_{x} to produce an effective transverse edge force per unit length, V_{x}, known as Kirchhoff’s force (Fig. 13.4c):