Building Design and Construction Handbook - Merritt - Ventech!

5.122 SECTION FIVE
But even for the simplest types of shells and loading, the stresses are difficult
to compute. In bending theory, a thin shell is statically indeterminate; deformation
conditions must supplement equilibrium conditions in setting up differential equations
for determining the unknown forces and moments. Solution of the resulting
equations may be tedious and time-consuming, if indeed solution if possible.
In practice, therefore, shell design relies heavily on the designer’s experience
and judgment. The designer should consider the type of shell, material of which it
is made, and support and boundary conditions, and then decide whether to apply a
bending theory in full, use an approximate bending theory, or make a rough estimate
of the effects of bending and torsion. (Note that where the effects of a disturbance
are large, these change the normal forces and shears computed by the membrane
theory.) For concrete domes, for example, the usual procedure is to use as support
a deep, thick girder or a heavily reinforced or prestressed tension ring, and the shell
is gradually thickened in the vicinity of this support (Fig. 5.97c).
Circular barrel arches, with ratio of radius to distance between supporting arch
ribs less than 0.25 may be designed as beams with curved cross section. Secondary
stresses, however, must be taken into account. These include stresses due to volume
change of rib and shell, rib shortening, unequal settlement of footings, and temperature
differentials between surfaces.
Bending theory for cylinders and domes is given in W. Flügge, ‘‘Stresses in
Shells,’’ Springer-Verlag, New York; D. P. Billington, ‘‘Thin Shell Concrete Structures,’’
2d ed., and S. Timoshenko and S. Woinowsky-Krieger, ‘‘Theory of Plates
and Shells,’’ McGraw-Hill Book Company, New York; ‘‘Design of Cylindrical Concrete
Shell Roofs,’’ Manual of Practice No. 31, American Society of Civil Engineers.
5.15.4 Stresses in Thin Shells
The results of the membrane and bending theories are expressed in terms of unit
forces and unit moments, acting per unit of length over the thickness of the shell.
To compute the unit stresses from these forces and moments, usual practice is to
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
Nx Mx
ƒx � � z (5.175)
3 t t /12
where z � distance from middle surface
t � shell thickness
M x � unit bending moment about axis parallel to direction of unit normal
force N x
Similarly, shearing stresses produced by central shears and twisting moments may
be calculated from equations of the form
T D
vxy � � z (5.176)
3 t t /12
where D � twisting moment and T � unit shear force along the middle surface.
Normal shearing stresses may be computed on the assumption of a parabolic stress
distribution over the shell thickness: