Building Design and Construction Handbook - Merritt - Ventech!

STRUCTURAL THEORY 5.121
FIGURE 5.97 Special provisions made at supports and boundaries of thin shells to
meet requirements of the membrane theory include: (a) a device to ensure a reaction
tangent to the middle surface; (b) stiffened edges, such as the ring girder at the base of
a dome; (c) gradually increased shell thicknesses at a stiffening member; (d) a transition
curve at changes in section; (e) a stiffening edge obtained by thickening the shell; (ƒ)
scalloped edges; (g) a flared support.
transmits normal forces to columns below. The enlarged view of the ring girder in
Fig. 5.97c shows gradual thickening of the shell to reduce the abruptness of the
change in section. The stiffening ring at the lantern in Fig. 5.97d, extending around
the opening at the crown, projects above the middle surface, for compatibility of
strains, and connects through a transition curve with the shell; often, the rim need
merely be thickened when the edge is upturned, and the ring can be omitted. In
Fig. 5.97e, the boundary of the shell is a stiffened edge. In Fig. 5.97f, a scalloped
shell provides gradual tapering for transmitting the loads to the supports, at the
same time providing access to the shell enclosure. And in Fig. 5.97g, a column is
flared widely at the top to support a thin shell at an interior point.
Even when the conditions for geometric compatibility are not satisfactory, the
membrane theory is a useful approximation. Furthermore, it yields a particular
solution to the differential equations of the bending theory.
(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: V. S. Kelkar and R. T. Sewell, ‘‘Fundamentals of the Analysis
and Design of Shell Structures,’’ Prentice-Hall, Englewood Cliffs, N.J.)
5.15.3 Bending Theory for Thin Shells
When equilibrium conditions are not satisfied or incompatible deformations exist
at boundaries, bending and torsion stresses arise in the shell. Sometimes, the design
of the shell and its supports can be modified to reduce or eliminate these stresses
(Art. 5.15.2). When the design cannot eliminate them, provisions must be made for
the shell to resist them.