SHELL STRUCTURES Shell structure is a thin, curved membrane of slab, usually of reinforced concrete, that functions both as structure and covering, the structure deriving its strength and rigidity from the curved shell form. It is generally capable of transmitting load in more than two directions to supports. These structures are highly efficient structurally when they are so shaped, proportioned and supported that they transmit the loads without bending or twisting. A shell is defined by its middle surface halfway between it’s inner surface and outer surface. Depending upon the geometry of the middle surface, shells may be classified as: (1) (2) (3) (4)
A Dome A Barrel arch Cone, and Hyperbolic and Parabolic
The strength and rigidity of curved shell structure makes it possible to construct single curved barrel vaults 60 mm thick and double curved hyperbolic paraboloids 40 mm thick in reinforced concrete for spans of 30.0. Single curvature shells, curved on one linear axis, are part of a cylinder or cone in the form of barrel vaults and conoid shells. Double curvature shells are either part of as sphere, as a dome, or a hyperboloid. The term single curvature and double curvature are used to differentiate the comparative rigidity of the two forms and the complexity of the centering necessary to construct the shells forms. Double curvature of a shell adds considerably to its stiffness, resistance to deformation under load and reduction in the need for restrain against deformation. The most straightforward shell construction is the barrel vault, which is part of a cylinder or barrel with the same curvature along its length. The short span barrel vault is used for the width of the arch ribs between which the barrel vaults span. It is cast on similar arch ribs supporting straight timber or metal centering which is comparatively simple and economic to erect and which can, without waste be taken down and use again for similar vaults. A shell structure is many times more expensive than a portal frame structure covering the same floor area because of the considerable labor require to construct the centering on which the shell is cast. The material most suited to the construction of a shell structure is concrete which is a highly a plastic material when first mixed with water that can take up any shape on centering or insert formwork.
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SOME BASIC CONCEPTS FOR SHELL STRUCTURES
The supports for a shell are more important than the shell. Shell structures can usually be understood as a set of beams, arches and catenaries and can be analyzed by that approach. For any shell structure, there will be a simple method of analysis that can be used to check the more precise analysis. Stiffest path concepts are useful in understanding shell structures. Support the edges of shells if they are already supported visually by masonry walls or window walls. Do not throw away all you structural intuition when you design shell structures. For ordinary structures, an adequate preliminary design should be within 10 percent Shell structures can be estimated to within 5 percent because the only usual unknown is the amount of reinforcing. Shell structures can carry relatively large point loads. Shell structures get their strength by shape and not by high strength of materials do not push stresses to their limit. Shell structures are very complex and carry forces by many paths. Shell structures, because of their complexity and unfamiliarity require a large lead time for developing the design.
RULES OF THUMB DESIGN
Long shells: where r/l < .4 Intermediate shells: where .4 < r/l < 2.0 Short shells: where r/l > 2.0 As noted earlier, vertical edge beams would be employed for a long shell while short shells would use horizontal beams.
TYPICAL CONSTRUCTION DETAILS
A minimum of three layers of reinforcement is normally used: top and bottom wire fabric with the main bars in between.
The cover over the fabric should be at least 3/8 in., preferably 1/2 in.
The main bars may be as large as 3/4 in. in diameter and the fabric will probably be about 1/4 in. thick. This requires a minimum of 2 in.
3 in. of concrete will gives ample cover and permit reasonably rapid construction.
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CORRUGATED CURVES
Barrel shells in the form of corrugations may offer structural advantages and may have esthetic values which make such a roof desirable.
Forms are not more difficult to build if the curvatures are not greater than the bending radius of the material used to line the forms.
One structural advantage is that the same area is supplied at the top and bottom of the shells and they are suitable for continuous structures where the maximum area of concrete is required at the bottom of the shell at the support
Instead of alternative concave and convex circles of the same radius, the curves may be alternate circles of long and short radius.
For the mathematically inclined, the shape may be of the pure sine curve.
There are innumerable combinations of curves, or curves and folded plates to serve the particular esthetic or structural function.
THE LAZY SHELLS
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The name applied to this shell is derived from the resemblance of the shape to the term for a cattle brand, called Lazy S. The clerestory provides for natural light .The combination of concave and convex shapes increases the effective depth over the depth furnished by a single barrel. The edges are unstiffened but if they have considerable curvature, the stresses can be satisfactory. However, spans should be kept fairly short to reduce both deflections and stresses. The design of graceful stiffening members to support these shells is difficult because the shell does not conform to the shape of either a beam or an arch and is unfamiliar to the eye. However, since there is plenty of light coming form the clerestory, the stiffeners can be made like walls and the windows may be omitted.
NORTH LIGHT SHELLS
This type of shell structure is used to provide large areas of north light windows for factories requiring excellent natural lighting. The windows may be slanting as shown here, or may be vertical. The edge member at the bottom forms a drainage trough with the curved shell and materially assists in stiffening the structure. The effective depth of the shell is not the vertical distance between the two ends but is more nearly represented b the depth if the shell is laid flat with the ends of the circle on the same horizontal line. Therefore, the spans for a north light shell must be rather small in comparison to the vertical depth of construction. The edges of adjacent shells should be tied together by concrete struts serving as mullions between the window glazing.
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BUTTERFLY SHELLS
Cylindrical shell vaults can be constructed with partial segments of arches, arranged in the form of a Y and called a butterfly roof. This shape is often used for canopies for buildings with skylights and for railroad station platform covering. At the ends of this building, a complete stiffening truss is used to tie the shells together. Horizontal struts are sometimes provided to tie the tops of the shells together at frequent intervals. At supports, the skylight is omitted and may be used to increase the stiffness and strength. Spans for this type of structure must be quire short in comparison to other barrel shells because the effective depth is, in effect, the minimum depth measured on a slant of one of the individual segments. This depth can be increased by adding a longitudinal stiffening beam at the top of the shell. A transverse stiffening girder is required at each column and may be placed on top of the shell.
EDGE SUPPORTED SHELLS
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The stresses and deflections in single barrel vaults (or end bays of multiple vaults) may be reduced by using columns or walls to support the edges. This makes it possible to design a single barrel shell for a large auditorium or gymnasium without using intermediate stiffeners. Most of the load is carried to the end stiffeners and columns. The intermediate columns merely act as a vertical support and do not carry lateral load. If there are continuous windows at the sides, the columns may be thin steel pipe columns which appear to be window mullions. It is a basic concept of shell structures that edges, wherever possible, should be supported rather than leave an expansion space between the top of the wall or windows and the shell. It may be argued that it is not really necessary to fireproof these columns since the structure is capable of supporting itself in the event of a fire which will melt the steel columns. The spans and widths of this type of structure may be increased by using ribbed slabs or waffle slabs.
SHORT SHELLS
This 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 short shell.
In structures making use of the short shell, the principle structural element is the stiffener, usually a reinforced concrete arch, although steel arches or trusses have been used. The short shell serves only a minor role. Many structures built with short shells, such a large hangars and auditoriums, could have been built with little more dead load by using a ribbed slab or other lightweight concrete framing system rather than the shell. The architecture of short shells, therefore, must be based on the exploitation of the shape of the arch rather than on the shell itself
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BASIC ELEMENTS OF SHORT SHELLS
This sketch illustrates some of the principle parts of a short shell structure: 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. The minimum shell thickness should be at the top in the center of the span. At the arch, the shell thickness is increased slightly for local stresses. The thickness increases toward the springing line of the arch and if not supported by an edge beam, the thickness here should be based on the thickness for a slab spanning the same distance.
PURE ARCH AND SHELL
The classic simplicity of this structure may be used with startling effect. There are only two structural elements and these are clearly expressed so that their function is evident. Obviously, if the shells are obscured by the walls necessary to enclose this space, much of the effect is lost. However, window walls would be in keeping with the spirit of the design and can be made to follow the curve of the arch. If this structure is to be used as a canopy, the obvious curve of the arch is a ellipse because the arches can spring almost vertically from the ground. The curve requiring the least material would be the thrust line, or funicular curve, for the loads on the structure. This form would have considerable curvature at the top but would be practically straight from the edge of the shell to the ground. The larger the arch span, the greater the saving of concrete and reinforcing by the use of a funicular curve.
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RIGID FRAMES
Short shells may be used with concrete rigid frames as the principle structural element. 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. The ribs are shown in this sketch and are placed below the shell. To save the cost in the forming, it may be better to place the ribs above the shell so they may be moved with very little decent ring. 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 material.
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. Therefore, 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.
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Due to the double curvature of domes, buckling is seldom a factor in the design. Domes have been built with a thickness of 6 inches for a span of about 300 feet. With long spans, however, walking on the roof is like walking on a giant balloon because of the spring action of the shell.
HALF SPHERE - VERTICAL WALLS
A half sphere for a dome of revolution does not require a thrust ring at the base so it can be placed on vertical walls and made continuous with the walls. This design is used for tanks because the roof becomes a part of the tank. The vertical portion of the sphere is not difficult to construct if pneumatically applied shotcrete or a similar process is used. The structure shown above with arched openings and a plastic dome on the crown has a rather oriental feeling. One of the most serious problems in the architecture of domes is acoustics. The reflections of sound tend to come to a focus a single point. In a domed ceiling, the sound may reverberate as many as twenty times unless there is acoustical treatment or unless there is equipment or broken surfaces to break up the sound. This problem should always be taken into account in the design of domed structures.
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DOMES Dome is a common structural element of architecture that resembles the hollow upper half of a sphere. Domes do not have to be perfectly spherical in cross-section, however; a section through a dome may be an ellipse. CHARACTERISTICS:
A dome can be considered as an arch which has been rotated around its vertical axis. As such, domes have a great deal of structural strength.
A small dome can be constructed of ordinary masonry, held together by friction and compressive forces.
Larger domes have all been built as double domes, with inner and outer shells.
A dome can sit directly on a circular base, however, this is not possible if the base is square.
The concave triangular or trapezoidal sections of vaulting that provide the transition between a dome and the square base on which it is set and transfer the weight of the dome are called pendentives.
Domes are semi-spherical or semi-elliptical in shape.
They are used as roof structures. Constructed of stone or bricks or concretes.
They are supported on circular or regular polygon shaped walls.
Dome structures have within certain height and diameters vary small thickness.
Dome structures are generally used in monumental works were roof are to be build on building circular or hexagonal in plan.
The domes can be either (1) Smooth shaped domes (2) ribbed domes.
Smooth shell domes can have either shell of uniform thickness or with shell of uniformly varying thickness.
A dome can be constructed with or without lantern.
Space frame dome exceptionally light structures, which permit the spanning of large distances with relative reduction materials.
The dome surface can be subdivided into a number of triangles or other regular polygons the sides of which are hinge bars.
Any dome shell roof will tend to flatten due to the loading and this tendency must be resisted by stiffening beams or similar to all the cut edges.
As a general grid domes which rise in access of 1- 6 of their diameter required a ring beam.
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Timber domes like their steel counter parts are usually constructed in a single layer grid system and covered with a suitable thin skin membrane.
FOLDED PLATE DOMES:
A curve can be made with small segments of straight lines. In the same way a sphere can be made with small triangular planes. In this category are included all domes made with plane slabs and plates. Domes may be constructed with small angles between the plates or with large angles between plates and the structural action may be considerably different for each type. The obvious advantage of the folded plate dome is that the surfaces are easier to form because they are flat. On the other hand, for slab spans over 16 ft, the shell wall is thicker than a curved surface because bending must be considered. The acoustical properties of a structure with plane surfaces are much better since the sound rays do not come to focus. This characteristic may be enough to make the folded plate dome superior to the curved dome for use in an auditorium. The structural design of folded plate domes follows that of folded plate barrels. Slab elements are designed first and loads are carried to the fold lines. These forces are then carried by direct compressive stresses by the fold lines acting as struts in a space structure.
FOLDED PLATE DOME - SQUARE IN PLAN
This structure is a spherical dome with portions sliced off to form a square or rectangle. Most areas to be covered are rectangular so a circular dome is not always a good solution to the planning requirements. This dome is supported by four rigid frames and would only be suitable for small spans because the frames would get quite large. For long spans, it is necessary to place a tie between the knees of the frame.
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These ties can be made a part of the window mullions if it is desirable to conceal them. Stresses in the shell are direct compression (membrane) stresses except across the corner where there are direct tensile forces due to the outward spread of the forces. The arches, or rigid frames, pick up the shell forces by shears parallel to the arches which are zero at the top and maximum at the bottom. There is no component of force in the shell perpendicular to the arches.
MULTIPLE DOMES
In this example the dome is rectangular and is continuous with the adjacent domes. The edges of the dome are supported by tied arches or bowstring trusses. If windows are needed in these arches, the mullions may be made to serve as vertical hangers for the bottom chords of the arch. In constructing this shell, each one of the dome elements is an independent structural unit so the forms may be moved without shoring all or part of the dome already cast. The shell thickness of this type of dome does not need to be greater than a circular dome except at the triangular corners. Membrane action ceases to exist and the corner should be designed as a slab.
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FOLDED PLATE DOME - TAPERED ELEMENTS
This dome makes use of tapered folded plates slanting to the center in the form of a tent. It can be built so that each of the triangular elements is self supporting during construction except for possibly a single shore at the crown. The forms, therefore, can be re-used many times in contrast to the usual dome structure. To obtain natural light, the top may be cut off and a ring inserted with a skylight. The arch thrusts are taken through this ring and the difficultly forming of the narrow plates at the crown is avoided. The acoustics of this dome should be better than the domes of revolution because the reflections of sound do not focus at a point. The structure shown here is a circular thrust ring. If the structure is large, there would be very high bending stresses due to the curvature, and the ring would be very large.
TRANSLATION DOMES
This structure looks very much like the Square Dome shown previously except the shape is generated by an entirely different method. A translation shell is generated by a vertical curve sliding along another vertical curve. The curves can be circles, ellipses, or parabolas. Therefore the vertical sections are all identical as opposed to a circular dome in which all vertical sections vary in height. This is a big advantage in construction of the formwork. This method can provide a rectangular dome with the same height of arch on all sides, making a rectangular dome feasible. Most of the load is carried by the side arches with some coming directly to the corners.
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The sketch shows a tie at the springing of the arches, but usually this will be covered by the walls or window mullions. Such shells are suitable for quite long spans with some interior lighting furnished by skylights in the shell. Barrel shells, folded plates, and shell arches are all special cases of translation shells.
GEODESIC DOMES
In this the triangles and the pentagon are used to obtain a subdivision in terms of bars of equal lengths. The connections between bars are delicate elements and weigh heavily in construction costs. These structures are becoming popular for industrial buildings hangers and other large buildings because they provide uninterrupted space without columns. Thin shell structures result in reduction of dead weight and in turn the economy. Generally, concrete is a very good material to take compressive force if used for the construction of a shell whose major portion is subjected to the compression the centering used for the construction of the shells structure can be reused repeatedly. This process results in economy.These structures are required very low maintenance costs. The amount steel used is also lesser.
HYPERBOLIC PARABOLOIDS
These are obtained by sliding a vertical parabola with upward curvature on another parabola with downward curvature in a plane at right angle to the plane the first.
Here directions, up in one and down in the other.
This surface generally called a saddle surface.
There are different ways in which saddle surfaces can be supported.
The surfaces are generally design with small rises so as to produce fairly flat roofs.
If cut by planes parallel to the two parabolas, the edges will be parabolic and the supporting structure must be parabolic.
To obtain a more practical shaped than the true saddle the usual shaped hyperbolic paraboloid which is formed by rising of or lowering one or more corners of a square.
By virtue of its shape this form of shell roof has a greater resistant to buckling than dome shapes.
Hyperbolic paraboloid shells can be used singly or conjunction with one another to cover any particular plan shape or size.
If the rise is small the result will be the hyperbolic paraboloid of low curvature acting structurally like a plate which will have to be relatively thick to provide the necessary resistance to deflection.
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To obtain full advantage of the inbuilt strength of the shape the rise to diagonal span ratio should not be less than 1:15 indeed the higher the rise the greater will be the strength and the shell can be thinner.
By adopting a suitable rise to span ratio it possible to construct concrete shells with diagonal spans of 35 mts. with a shell thickness of only 50mm.
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