ANALYSIS AND DESIGN OF STRUCTURES IV
NAVEEN KUMAR S
SELL STRUCTURE & FOLDED PLATES ?
Thin shell is defined as a shell with a thickness which is small compared to its other
dimensions and in which deformations are not large compared to thickness.
A primary difference between a shell structure and a plate structure is that, in the
unstressed state, the shell structure has curvature as opposed to plates structures
which are flat. Membrane action in a shell is primarily caused by in-plane forces (plane
stress), though there may be secondary forces resulting from flexural deformations
Domes of Revolution
Folded Plate Domes
CYLINDRICAL BARREL VAULTS
• Barrel vaults are perhaps the most useful of the shell structures because they can span upto
150 feet with a minimum of material.
• They are very efficient structures because the use the arch form to reduce stresses and
thicknesses in the transverse direction.
• Barrel vaults are essentially deep concrete beams with very thin web members and may be
designed as such by the ordinary methods of reinforced concrete. The curve of the cross
section of the barrel is usually a circle. However, any other form maybe used, such as the
ellipse, a parabola, or a funicular curve which fits the thrust line of the applied load.
• Each curve has its particular structural and esthetic qualities.
• A number of terms have been developed to describe cylindrical shells. If the span is large in
comparison to the width, the form is called a long shell. If the length is short, it is called a
short shell. An arbitrary ratio for long shells is a span/radius ratio of 5.
• A short shell has a span/radius ration less than 1 and shells between these limits are called
ELEMENTS OF BARREL VAULTS
Stiffeners are required at columns. They do not necessarily have to be complete
diaphragms, as shown here, but may be arches with a horizontal tie. In contrast to folded plates
where the thickness is based on the design of a slab element, the thickness of the barrel shell is
usually based on the minimum thickness required for covering the steel for fireproofing, plus the
space required for three layers of bars, plus some space for tolerance. If these bars are all half
inch rounds, a practical minimum would be 3 1/4 inches. Near the supports the thickness may be
greater for containing the larger longitudinal bars.
MULTIPLE BARRELS - OUTSIDE STIFFENERS
THE LAZY S EDGE MEMBERS
EDGE SUPPORTED SHELLS NORTH LIGHT SHELLS
barrel vaults were described having a length of barrel which is long in
comparison to the width. the short shell structure is a cylindrical shell
having a large radius in comparison to the length. The two types of shells
have uses which are altogether different and the architectural and
engineering problems require a different approach. There are, of course,
borderline cases where it is difficult to distinguish between the long an
BASIC ELEMENTS OF SHORT SHELLS
1) the shell spanning between arches, and 2) the arch structure. In
this structure, the edge beams are provided at the lowest point of
the shell and the arch is placed on top of the shell so that forms
may be moved through the barrel. In small structures, the edge
beam can be omitted if the shell is thickened. The curve of the shell
is determined by the proper shape of the arch and may be a circle
for small structures or may conform to the thrust line of the arch for
long span structures. :
PURE ARCH AND SHELL
Short shells may be used with concrete rigid frames as the principle structural
The rigid frame without a horizontal tie at the low point of the shell is suitable only for
short spans because of the massive proportions required for the knees. It is not
necessary to have the spans of all the rigid frames equal, and the bending moments in
the frames may be reduced if shorter side spans are used.
Skylights may be used in a short shell and they may be continuous transversely if
they are placed in every other span so the shell on each side of the skylight cantilevers
out from the adjacent span. Rigid frames are usually built with tie rods connecting the
base of the columns, especially if soil conditions will not permit lateral loads on the soil
The primary response of a dome to loading is development of membrane compressive
stresses along the meridians, by analogy to the arch.
The dome also develops compressive or tensile membrane stresses along lines of
latitude. These are known as ‘hoop stresses’ and are tensile at the base and
compressive higher up in the dome.
SPHERE SEGMENT - COLUMN SUPPORTS
• If a dome is built as less than a half sphere, a tension ring of steel bars, plates, or
wires is required at the base to carry the thrusts of the shell.
• In this case, the ring has been made big enough so that it assists in distributing the
reaction of the columns into the dome.
• The direct stresses in the shell are mostly compressive in this structure and are so
small that the stress calculations are hardly necessary. There are bending stresses
in the shell wall due to restraint of the thrust ring and to change in temperature. T
• herefore, the thickness of the shell is increased in the vicinity of the thrust ring.
Otherwise, the shell thickness is a minimum and may be 2 1/2 to 3 inches for spans
up to 150 ft.
combining portions of the previous types arranged to form more stable combinations than
the individual elements alone. The most appropriate name is "intersection shell" because the
surfaces that produce the shell appear to meet at an intersection. Any of the basic types may
be used in this manner but the barrel shell is the most familiar and useful. The structural
efficiency of the intersection shell depends on the angle of the intersection of the surfaces. If
the angle is small (called here for descriptive purposes, sharp), then a natural rib is formed by
the adjacent elements of the basic shells which is much stiffer than the adjacent shells on
An itersection for which the angle is very large is called here a shallow intersection. An
intersection of 90 degrees is the optimum value because it gives a stiff rib.
Folded plates are assemblies of flat plates rigidly connected together along their
edges.in such a way so as to make the structural system capable of carrying
loads without .the need for additional supporting beams along mutual edges.
Types of folded plates :
1- Prismatic : if they consist of rectangular plates.
2- Pyramidal : when non-rectangular plates are used.
3- Prismoidal, triangular or trapezoidal.
Perpendicular to the main span,
the shell acts as short span
plates in transverse bending
In the main span direction, the
shell develops membrane tension at
the top and compression at the
bottom, in analogy to a beam in
• Folded plates with simple joints.
• Folded plates with multiple joints.
• Folded plates with opened cross
• Folded plates with closed cross
Folded Plate Behaviors :
Each plate is assumed to act as a
beam in its own plane, this
assumption is justified when the ratio
of the span "length" of the plate to its
height "width" is large enough. But
when this ratio is small, the plate
behaves as a deep beam.
Assumptions For the analysis of Folded Plates :
1- Material is homogenous, elastic, isotropic, Hook's Law is valid, thickness of
plate is small when compared to plate dimensions.
2- Problem will be treated as one-dimension if plate is assumed to behave in
beam action, but in two dimensions if based on the theory of elasticity.
3- Joints are assumed to be rigid enough.
The main problem of folded plates is the analysis, not the design.
When can ridge displacements be neglected ?
Ridge displacements be neglected if there is no ridge displacement such as:
- closed systems.
- If the folded plate system is under symmetrical loading.
- In folded plate systems with interior symmetrical planes, subjected to
symmetrical loads. But in other cases ridge displacements may take place, yet
they are such a small order of magnitude that their effects can be neglected, as
folded plate system provided with adequate number of intermediate diaphragms.
Folded Plates With Simple Joint
When the folded plate is that with simple joint , which mean that no more than 2
elements are connected to the joint. But when more than 2 elements are
connected to the joint, it can be named as multiple joint. The width of any plate
should not be larger than 0.25 its length to be considered to act as beam.
Actions of Folded plate due to loads :
1- Slab action : loads are transmitted to ridges by the bending of plates normal to
2- Beam action : Loads are transmitted through plates in their planes to
TAPERED FOLDED PLATES
The principle components in a folded plate structure are illustrated in the sketch above. They consist of, 1) the inclined
plates, 2) edge plates which must be used to stiffen the wide plates, 3) stiffeners to carry the loads to the supports and
to hold the plates in line, and 4) columns to support the structure in the air. A strip across a folded plate is called a slab
element because the plate is designed as a slab in that direction. The span of the structure is the greater distance
between columns and the bay width is the distance between similar structural units. The structure above is a two
segment folded plate. If several units were placed side by side, the edge plates sould be omitted except for the first and
last plate. If the edge plate is not omitted on inside edges, the form should be called a two segment folded plate with a
common edge plate.The structure above may have a simple span, as shown, or multiple spans of varying length, or the
folded plate may cantilever from the supports without a stiffener at the end.
FOLDED PLATE RIGID FRAME
REINFORCEMENT DETAIL OF FLAT SLAB ?
The spacing of bars in a flat slab, shall not
exceed 2 times the slab thickness.
AREA OF REINFORCEMENT :
When the drop panels are used, the
thickness of drop panel for determining area
shall be the lesser of the following:
(a) Thickness of drop, and
(b) Thickness of slab plus one quarter the
distance between edge of drop and edge of
capital. The minimum percentage of the
reinforcement is same as that in solid slab
i.e., 0.12 percent if HYSD bars used and
0.15 percent, if mild steel is used.
MINIMUM LENGTH OF REINFORCEMENT
:At least 50 percent of bottom bars should be
from support to support. The rest may be
bent up. The minimum length of different
reinforcement in flat slabs should