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:

(9.31)

where zdistance from middle surface
tshell thickness
Mxunit bending moment about an axis parallel to direction of unit nor-
mal force Nx

Similarly, shearing stresses produced by central shears Tand twisting moments
Dmay be calculated from equations of the form:

(9.32)

Normal shearing stresses may be computed on the assumption of a parabolic
stress distribution over the shell thickness:

(9.33)

whereVunit shear force normal to middle surface.

BEARING PLATES

To resist a beam reaction, the minimum bearing length Nin 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 Nis generally desirable but is
limited by the available wall thickness.
When the plate covers the full area of a concrete support, the area, in^2 (mm^2 ),
required by the bearing plate is

(9.34)

whereRbeam reaction, kip (kN), fcspecified compressive strength of the
concrete, ksi (MPa). When the plate covers less than the full area of the con-
crete support, then, as determined from Table 9.3

(9.35)

whereA 2 full cross-sectional area of concrete support, in^2 (mm^2 ).

A 1 

R

0.35fcA 2 

2

A 1 

R

0.35fc

vxz

V

t^3 /6

t^2

4

z^2 

vxy

T

t



D

t^3 /12

z

fx

Nx

t



Mx

t^3 /12

z