Page 282 - Civil Engineering Formulas

216 CHAPTER NINE

assume normal forces and shears to be uniformly distributed over the shell
thickness and bending stresses to be linearly distributed.
Then, normal stresses can be computed from equations of the form:

N x M x
f x z (9.31)
3
t t /12
where z distance from middle surface
t shell thickness
M unit bending moment about an axis parallel to direction of unit nor-
x
mal force N x
Similarly, shearing stresses produced by central shears T and twisting moments
D may be calculated from equations of the form:

T D
v xy z (9.32)
3
t t /12
Normal shearing stresses may be computed on the assumption of a parabolic
stress distribution over the shell thickness:
V
v xz 3 t 2 z 2 (9.33)
t /6 4
where V unit shear force normal to middle surface.

BEARING PLATES

To resist a beam reaction, the minimum bearing length N in the direction of the
beam span for a bearing plate is determined by equations for prevention of
local web yielding and web crippling. A larger N is generally desirable but is
limited by the available wall thickness.
2
2
When the plate covers the full area of a concrete support, the area, in (mm ),
required by the bearing plate is
R
A 1 (9.34)
0.35 f c
where R beam reaction, kip (kN), f specified compressive strength of the
c
concrete, ksi (MPa). When the plate covers less than the full area of the con-
crete support, then, as determined from Table 9.3
R 2
A 1 (9.35)
0.35 f c A 2
2
2
where A full cross-sectional area of concrete support, in (mm ).
2