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Then
1
2 z
t
2 dz
t / 2
3 p z
2 t
σ z =
z
2 z
t
3
3 p z
4
2
3 +
2 z
t
1
3
=−
(13.46c)
Observe that
τ
xz and
τ
yz vary parabolically over the plate thickness (Fig. 13.8), while
σ
z varies cubically. Furthermore, the z -directed stresses
τ
xz and
τ
yz tend to be very small
relative to the
τ xy stress of Eq. (13.38).
Similar to beam shear stresses, peak plate shear stresses occur at the middle surface where
z
=
0. There
3
2
q x
t
3
2
q y
t
τ
|
=
τ
|
=
(13.47)
xz
max
yz
max
Boundary Conditions
Now that local (differential) governing equations have been established, boundary condi-
tions which the displacements and forces must satisfy will be considered. As in the case of
a beam, the solution to the fourth order plate equation (Eq. 13.41) requires that two bound-
ary conditions be satisfied at each edge of the plate. These conditions can be a combination
of deflection, slope, shear force, and moment. However, unlike the beam, with the plate
there appear to be two moments—a bending moment and a twisting moment. This appar-
ent surplus of moments can be corrected by replacing the twisting moment by equivalent
shear forces. In accordance with St. Venant's principle δ this replacement affects the stress
distribution only in the neighborhood of the boundary. As will be seen later in this chapter,
the problem of an apparent redundant boundary condition disappears when considering
plates from the standpoint of a theory including shear deformation.
The development of an equivalent shear force was explained by Kirchhoff using an energy
relationship and then mechanically by Thomson 2 and Tait 3 in 1883. Statically, the twisting
moment in Fig. 13.5 can be represented by a pair of horizontal forces or equivalently by
a pair of vertical forces (Fig. 13.9a). Consider the two successive elements of length dy on
the x
a boundary as shown in Fig. 13.9b. On one element, the twisting moment m xy dy
(Fig. 13.9b) is replaced by a statically equivalent couple of equal and opposite forces m xy
separated by dy (Fig. 13.9c), and on the next element, the couple is formed by the forces
( m xy
=
+
+
y m xy dy ). The adjoining forces m xy
y m xy dy and m xy have opposite signs so that
their sum is
y m xy dy . Add this force to the shear force q x to obtain the equivalent transverse
δ In 1855 Barre de Saint-Venant enunciated a useful principle that now bears his name. In essence, this principle
can be stated as the redistribution of loading, resulting from a set of forces acting on a small region of the surface
of an elastic body being replaced by a statically equivalent set of forces, causing significant changes in the stress
distribution only in the neighborhood of the loading, while stresses remain essentially the same in those portions
of the body located at large distances from the applied loading. By “large distances” are meant distances great
in comparison with the dimensions of the surface on which the loading is applied. “Statically equivalent” sets of
forces mean that the two distributions of loadings have the same resultant force and moment.
2 William Thomson (1824-1907), the son of an Irish (and later Scottish) professor of engineering, was educated at
home. He was given the title of Baron Kelvin of Largs. He was a prolific scientist and, with Helmholtz of Germany,
is credited with establishing physics as a science at the beginning of the 20th century.
3 Peter Guthrie Tait (1831-1901) was a Scottish physicist and mathematician. In dynamics, Tait backed the use of
quaternions, having promised a dying Hamilton to write an elementary text on the subject. This led to a dispute
over the vector method supported by J.W. Gibbs and Oliver Heaviside. In 1868, he caused further controversy by
writing a pro-British topic Sketch of the History of Thermodynamics.
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