Building Design and Construction Handbook - Merritt - Ventech!

5.120 SECTION FIVE
shear stresses that are necessary to satisfy boundary conditions (bending theory,
Art. 5.15.3).
Ribbed Shells. For long-span construction, thin shells often are stiffened at intervals
by ribs. Usually, the construction is such that the shells transmit some of the
load imposed on them to the ribs, which then perform structurally as more than
just stiffeners. Stress and strain distributions in shells and ribs consequently are
complicated by the interaction between shells and ribs. The shells restrain the ribs,
and the ribs restrain the shells. Hence, ribbed shells usually are analyzed by approximate
methods based on reasonable assumptions.
For example, for a cylindrical shell with circumferential ribs, the ribs act like
arches. For an approximate analysis, the ribbed shell therefore may be assumed to
be composed of a set of arched ribs with the thin shell between the ribs acting in
the circumferential direction as flanges of the arches. In the longitudinal direction,
it may be assumed that the shell transfers load to the ribs in flexure. Designers may
adjust the results of a computation based on such assumptions to correct for a
variety of conditions, such as the effects of free edges of the shell, long distances
between ribs, relative flexibility of ribs and shell, and characteristics of the structural
materials.
5.15.2 Membrane Theory for Thin Shells
Thin shells usually are designed so that normal shears, bending moments, and
torsion are very small, except in relatively small portions of the shells. In the
membrane theory, these stresses are ignored.
Despite the neglected stresses, the remaining stresses ae in equilibrium, except
possibly at boundaries, supports, and discontinuities. At any interior point, the number
of equilibrium conditions equals the number of unknowns. Thus, in the
membrane theory, a thin shell is statically determinate.
The membrane theory does not hold for concentrated loads normal to the middle
surface, except possibly at a peak or valley. The theory does not apply where
boundary conditions are incompatible with equilibrium. And it is in exact where
there is geometric incompatibility at the boundaries. The last is a common condition,
but the error is very small if the shell is not very flat. Usually, disturbances
of membrane equilibrium due to incompatibility with deformations at boundaries,
supports, or discontinuities are appreciable only in a narrow region about each
source of disturbance. Much larger disturbances result from incompatibility with
equilibrium conditions.
To secure the high structural efficiency of a thin shell, select a shape, proportions,
and supports for the specific design conditions that come as close as possible to
satisfying the membrane theory. Keep the thickness constant; if it must change, use
a gradual taper. Avoid concentrated and abruptly changing loads. Change curvature
gradually. Keep discontinuities to a minimum. Provide reactions that are tangent to
the middle surface. At boundaries, ensure, to the extent possible, compatibility of
shell deformations with deformations of adjoining members, or at least keep restraints
to a minimum. Make certain that reactions along boundaries are equal in
magnitude and direction to the shell forces there.
Means usually adopted to satisfy these requirements at boundaries and supports
are illustrated in Fig. 5.97. In Fig. 5.97a, the slope of the support and provision for
movement normal to the middle surface ensure a reaction tangent to the middle
surface. In Fig. 5.97b, a stiff rib, or ring girder, resists unbalanced shears and