Types And Forms Of Shell Structures




INTRODUCTION
The purpose of this book is to provide sketches and descriptions of many types of shell structures to aid the architect or engineer in the selection of a structure for a particular use. No claim is made for completeness. Drawings have been used rather than photographs, first because of the limited number of photographs available, and second, because there is always a reluctance on the part of the architect to use something already built because it would seem like copying. Only the structural features are shown in the sketches and details such as windows, gutters, fascia members, etcetera, have been omitted. The criterion has always been to picture the shell after the concrete has been complete but before the brick, stone, windows, or roofing is placed. Most of the nomenclature is standard in the literature but some of it was devised by the writer to fit gaps for which satisfactory terms were not available. In illustrating shell types, many obvious structures have been omitted because they are so similar to the basic types. The criterion for showing examples has in all cases been its usefulness as an architectural or structural unit of construction.



















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CHAPTER I - FOLDED PLATES
• BASIC ELEMENTS
• THREE SEGMENT FOLDED PLATE
• Z SHELL
• WALLS CONTINUOUS WITH SHELL
• CANOPIES
• TAPERED FOLDED PLATES
• EDGE SUPPORTED FOLDED PLATES
• FOLDED PLATE TRUSS
• FOLDED PLATE RIGID FRAME
CHAPTER II - CYLINDRICAL BARREL VAULTS
CHAPTER III - SHORT SHELLS
• BASIC ELEMENTS OF SHORT SHELLS
• PURE ARCH AND SHELL
• MASSIVE ABUTMENTS
• RIGID FRAMES
• CANTILEVER ABUTMENTS
CHAPTER IV - DOMES OF REVOLUTION
• SPHERE SEGMENT - COLUMN SUPPORTS
• HALF SPHERE - VERTICAL WALLS
• TYPICAL PROFILES - SINGLE DOMES
• DOMES - SQUARE IN PLAN
• MULTIPLE DOMES
• TRANSLATION DOMES
CHAPTER V - FOLDED PLATE DOMES
CHAPTER VI - INTERSECTION SHELLS
• INTERSECTION SHELL - SQUARE IN PLAN SHALLOW INTERSECTION
• INTERSECTION - POLYGONAL PLAN, SHALLOW INTERSECTION
• GROINED VAULT - SQUARE IN PLAN
• GROINED VAULT - POLYGONAL PLAN
• INTERSECTION SHELL - CROSS FORM
• INTERSECTION SHELL - FOLDED PLATE
CHAPTER VII - WARPED SURFACES
• CONOID
• HYPERBOLIC PARABOLOIDAL SURFACE
• HYPERBOLIC PARABOLOID - GABLED EDGE MEMBERS
• UMBRELLA SHELLS
• HYPERBOLIC PARABOLOIDAL SADDLE DOME
• HP FLOWER DOME
• STEEP HYPERBOLIC PARABOLID
• TRUMPET SHELL
• THE GROINED VAULT
• NEW FORMS FOR WARPED SURFACES
CHAPTER VIII - COMBINATIONS OF SHELLS
• DOME AND BARREL VAULT
• FOLDED PLATE AND BARREL VAULT
• FOLDED PLATE AND CONE
• BARREL SHELL - DOMED ENDS
• FOLDED PLATE - TAPERED ENDS
• SPACE ACTION OF ARCHES AND VAULTS
• SHELL SLAB
• BARREL AND SLAB
• FOLDED PLATE AND SLA
CHAPTER IX - SHELL ARCHES
• FOLDED PLATE ARCH
• BARREL ARCH












CHAPTER I - FOLDED PLATES

It seems appropriate to start the presentation of examples of shapes and forms for shell structures with the folded plate because it is the simplest of the shell structures.
The distinguishing feature of the folded plate is the ease in forming plane surfaces. Therefore, they are more adaptable to smaller areas than curved surfaces which require multiple use of forms for maximum economy. A folded plate may be formed for about the same cost as a horizontal slab and has much less steel and concrete for the same spans. Folded plates are not adapted to as wide bay spacings as barrel vaults. For widths of plate over, say, 12 feet, the thickness of the folded plate must be thicker than for a barrel vault. Some advantage may be gained by increasing the thickness of the slab just at the valleys so it will act as a haunched beam and as an I section plate girder.
















BASIC ELEMENTS


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 should 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.



THREE SEGMENT FOLDED PLATE



This sketch shows a folded plate structure with three segments for each barrel. The end stiffeners are rigid frames rather than deep girders as in the last example. The forces from the reactions of the sloping plates on these rigid frames will be quire large and at an outside column they will not be balanced by thrusts from the adjacent plates. The size of the frames may be reduced by using a steel tie between the tops of the columns which can be concealed in the fenestration.
The dimensions of the plates are dependent on both the width of the barrel and on the span. The depth of the shell should be about 0.10 times the span and the maximum slope of a plate should not be greater than 40 degrees. For example, assume for the above structure that the span is 60 feet and the bay width is 24 feet. The depth of the shell should be about 6 feet and the horizontal width of each plate with a three segment plate should be about 8 feet. The slope of the plates is 6/8, which is about 37 degrees and is satisfactory. The thickness of the plates could be about 3 ½ inches.



Z SHELL


Each of the units above has one large sloping plate and two edge plates arranged with space between the units for windows. This form has been called a Z shell and is similar to the louver used for window ventilation. The architectural effect is very dramatic if the structure can be shown by a cantilever projected out beyond the support. The windows are normally open to the north but most of the light is actually reflected south light. To increase this effect, the roof surface can be painted with aluminum so light from the sun is reflected through the windows to the ceiling and the windows need not be very large. Adjacent units should be tied together by structural window mullions. In constructing the Z shell, movable forms need only be lowered a short vertical distance if construction is started on the right and proceeds to the left.
The Z shell is not an efficient structural shape since it is discontinuous and its effective depth is much less than the actual vertical depth. Therefore, the spans are limited in comparison to the plates having a large number of units side by side.





WALLS CONTINUOUS WITH SHELL


In this structure the walls are of tilt-up concrete construction; concrete is cast flat on the floor and raised into place by cranes. The walls are designed to be continuous with the roof plates. Tilt-up walls usually are joined by poured-in-place columns. In this design, columns are not necessary at the junction of the individual side wall panels because the walls are braced at the top. Only a simple grouted key slot is provided.
The tilt-up panels can serve as their own foundation walls so only a continuous footing pad is used with a notch to receive the tilt-up panel. Dock height interior floors can be constructed by filling the interior of the building up with dirt to the required height. The tilt-up walls can be designed for this lateral load because they are held at the top by the shell and act as vertical beams rather than as cantilever retaining.









CANOPIES


A folded plate structure for a small canopy at the entrance of a building is shown. This folded plate has four segments. A two segment structure is not desirable because it has very little torsional resistance. This instability can be demonstrated by a paper model having the ends of the model glued to vertical pieces of cardboard, acting as stiffening members. If it is absolutely necessary to have a two element system, a torsion member can be placed in the valley which will carry the unbalanced loads.
Stiffeners can often be hidden on the top surface so they are not in evidence and the shell will appear to spring from the vertical column. At the wall of the building there should also be a stiffener hidden in the wall construction. Provision should be made for drainage of the center valley.







TAPERED FOLDED PLATES


Folded plate structures may be built with tapered elements and only one of the many possible combinations is shown here. Another possibility is to place the smaller depths all at one end so that the entire structure forms a circular ring. The height of the shells at the center of the span is the critical dimension for bending strength. Therefore, the structure is not very efficient and not suitable for long spans because of the excess height required for the large ends. Another weak element in this design is the transfer of shear from the small end of the triangular plate to the large end. If a large number of units are used in each span, the transfer of loads may be difficult.
A folded plate may be used for walls as a thin structural element by casting each plate flat on the floor and grouting the joints full of concrete. A wall of this type can be made much thinner than a flat wall.







EDGE SUPPORTED FOLDED PLATES

The usual upturned edge plate can be eliminated and the roof structure can be made to appear very thin if the edge plate is replaced by a series of columns. The slab between columns must be designed as a beam and it may be convenient to extend the main roof slab as a cantilever canopy. The beam element that carries the load of the roof between columns will then be wider and windows under the slab will have the same function as in the previous examples of folded plates. Note the vertical columns in the end walls at the crown of the gable. These take the reactions of the plates and the horizontal ties may be eliminated. Wind loads are taken by rigid frame action in the columns and stiffeners.







FOLDED PLATE TRUSS


The term "folded plate truss" is intended to indicate the structural action of this structure. There are horizontal ties across the width only at the ends of the building and the structure acts as an edge supported shell as shown in the previous example. The thrusts from the triangular crossed arches are carried lengthwise to the ends. The top chord of the inclined truss is formed by the ridge member. The bottom chords are the ties at the base of the side gables and the diagonals are formed by the sloping valleys at the intersection of the gables and the triangular plates. The top longitudinal compression member may require some additional thickness to form a compression member of sufficient size to carry the compression force.
This is truly a space structure and its structural action is not as obvious and, therefore, the architectural appearance is mote subtle that the usual shell structure.







FOLDED PLATE RIGID FRAME


An arch with straight segments is sometimes called a rigid frame. It is not as efficient as the curved arch because the bending moments are greater. Ties across the plates are required at the knees and at the crown in order to distribute the forces at the ends of each segment.












CHAPTER II - CYLINDRICAL BARREL VAULTS

Barrel vaults are perhaps the most useful of the shell structures because they can span upt o 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 intermediate shells. Short shells are a different structural type and are described in a later chapter.















CHAPTER III - SHORT SHELLS

In the prvious chapter, barrel vaults were described having a length of barrel which is long in comparison to the width. In this chapter, the structure known as the short shell will be described. 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, therefore, the emphasis in this chapter then will be on the arch shape. 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, therefor, must be based on the exploitation of the shape of the arch rather than on the shell itself.










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. The edge beams act like the folded plate structures described in the first chapter.







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 and the slanting member will not be as great a hazard to people's heads. 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.









MASSIVE ABUTMENTS


The abutments to the arch in this structure have been mad in the form of an inverted U rigid frame. If the abutments are made heavy and rigid, then the arch may be lighter so it may be more economical to use the large mass of concrete at the lower elevation to save concrete in the arches. In a monumental structure, such as an auditorium, the side spaces can be used as archways for access to the seating area.
Instead of the U frame, which is subjected to very heavy bending moments, a triangular frame may be used with the apex at the springing of the arch. The structural members of this abutment can be quire thin because they follow the thrust line of the forces better than does the U frame.
An architectural problem of the short shell structure is the proper design of the end walls. On a long span structure there will be large blank areas that require careful architectural treatment to make the structure pleasing.






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 decentering.
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.






CANTILEVER ABUTMENTS


The span of the arch may be reduced and the depth and thickness may be made smaller if the support of the arch is placed at the end of a beam cantilever from the wall of the building. This design provides space under the cantilevers for seating by using area that would otherwise be required for the arch ribs.
The design of this structure requires a balance between the height of the arch and the span so the thrust line will be located in the optimum position.
This structure is most suitable for a large monumental auditorium structure rather than a building where economy is the principle consideration. The large volume of concrete and reinforcing steel in the abutment would not be required of the abutment could follow the thrust line.







CHAPTER IV - DOMES OF REVOLUTION

A dome is a space structure covering a more or less square or circular area. The best known example is the dome of revolution, and it is one of the earliest of the shell structures. Excellent examples are still in existence that were built in Roman times. They are formed by a surface generated by a curve of any form revolving about a vertical line. This surface has double curvature and the resulting structure is much stiffer and stronger than a single curved surface, such as a cylindrical shell. The simples dome of revolution is a portion of a sphere. However, other curves are also satisfactory, such as the ellipse, the parabola, other conic sections, or random curves.
Typical profiles for domes are shown later in the chapter and there are an infinite variety of possible shapes, each suitable for a particular purpose. Parts of domes of revolution, square or polygonal in plan with portions of the shell removed, are also considered in this chapter as domes of revolution. Their structural action is much more complex than the dome circular in plan.















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.
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 an a plastic dome on the crown has a rather oriental feeling. Some inspired architect will probably use it in the future by means of openings or windows, or in some fresh and unusual way.
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 points. 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.





TYPICAL PROFILES - SINGLE DOMES

Rather than show each on of these shapes as a separate drawing, it was thought convenient to show only the cross sections.
The ellipse shown in (a) provides a dome with a low rise and a vertical tangent at the support so that a separate thrust ring is not required. This form may have esthetic advantages over the circle but the vertical element requires top forms or placing of the concrete by the shotcrete method.
In (b) a cone is used as a dome. This form does not have as much curvature as the sphere and, therefore, the stresses and deflections may be higher. Also, a heavier thrust ring is necessary for the same height of shell.
The curve in (c) illustrates the principle that a dome of revolution may be formed by the rotation of any curve about a vertical axis. The radius of the curve need not be on that axis.
The last example (d) is a dome formed by a segment of a circle for which the center of the circle is not on the axis of rotation and a dimple is created at the top. A vertical column may be used to support the center of the dome at the low point if the span is very long.













DOMES - 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 quire large. For long spans, it is necessary to place a tie between the knees of the frame. 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.
This shell can be classed as a dome of revolution since the shell is part of a sphere (which is a surface of revolution). However, this surface may also be a translation surface formed by a circle or other curve moving along a line. For a low rise, the translation surface is very little different from a sphere but is much easier to form because all the parallel elements have the same curvature.
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.





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, elipses, 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 recangular dome with the same height of arch on all sides, thus making a rectangular dome feasible.
Most of the load is carried by the side arches with some coming directly to the corners. 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.







CHAPTER V - FOLDED PLATE DOMES

In this category are included all domes made with plane slabs and plates. There are many variations and only a few of them can be shown here. 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.














CHAPTER VI - INTERSECTION SHELLS

The previous shells have all been basic types: the folded plate, the cylindrical barrel shell, the dome of revolution, and the folded plate domes. The next category is made up by 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 each side. 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. On large structures with shallow intersections, massive ribs may be necessary which are very evident and detract from the light appearance.















INTERSECTION SHELL - SQUARE IN PLAN SHALLOW INTERSECTION


This structure is a dome formed by using triangular pieces of a cylindrical shell arranged in the form of a square. The drawing, however, conveys more than the description.
The word "shallow" has been used to indicate that the angle between the components is rather small, especially if the rise of the shell is small. With four sides, however, the ribs formed by the intersection should provide an adequate structural member. It is the best type of dome to cover a square area and maintain a level parapet around the building. The structural action is essentially that of a short shell. Loads are carried by the cross ribs formed by the intersection and by the stiffening element created by the edge beam. The bottom of the shell requires tensile reinforcement as in a short shell.
This structure can be inverted and supported from a central column similar to the "umbrella shell" described in the chapter on warped surfaces.





INTERSECTION - POLYGONAL PLAN, SHALLOW INTERSECTION


This form is suitable for a dome of large span which must be nearly circular in plan. If more than six sides are used, the rib formed by the shell gets rather shallow so a rib is added above the shell surface. Columns are shown in this sketch at the center of each panel rather than at the ribs. This would be suitable only for a small structure since it produces additional bending in the lowest part of the shell. As in other types of domes, a skylight may be placed at the crown of the dome. A thrust ring must be added to take the forces in the ribs. Windows may be placed in the shell except at the lowest points.







GROINED VAULT - SQUARE IN PLAN


The groined vault is an intersection shell composed of four triangular pieces of cylindrical shells, arranged in a cross form so that there are arches on each of the sides. This is one of the most ancient of masonry arch structures and still used for underground water reservoirs of concrete without any reinforcing.
The usual vault is a continuous structure but only a single unit is shown here. The structure obtains its rigidity by the large angle between the shell components at the intersections which creates a very strong rib.
The size of these structures is almost unlimited because the form is inherently very strong and is stiffened by six complete arches. In order to take advantage of the rigs, it is necessary to have the center of the abutments at the center of the effective rib. Otherwise, an additional heavy rib is required which impairs the appearance of the groined vault.





GROINED VAULT - POLYGONAL PLAN


This structure is similar to the previous groined vault, square in plan, except that there are five triangular cylindrical elements instead of four. The shells which form the dome all have axes perpendicular to the vertical axis. Again, an excellent structural rib is formed by the intersection.
Arched stiffening ribs are required around the outside of the structure and these ribs exert thrusts at their abutments. Therefore, either a steel tie is required at this level (it can be hidden by the window), or if the area must be open, thrust abutments can be used. There are, of course, many possible variations on the structure sketched here. If six sides are used, a continuous series of shells my be constructed and units of this type could alternate with those having a shallow intersection.



INTERSECTION SHELL - CROSS FORM


Four cylindrical barrels intersect to form a central dome. The structure is supported by four columns at the corners of the intersection so that part of the barrel cantilevers from the central dome. Provision must be made for thrusts from the barrels and the central dome at the column. There are several alternates: 1) the columns may be made very heavy, 2) short lengths of walls in an angle shape may be used at the corners instead of individual columns, 3) diagonal members may be placed in each of the walls, or 4) ties may be place between tops of columns. The latter solution might be unsightly if the interior of the building should be clear.
Shell thicknesses for this structure should correspond to those used for barrel shells. The cantilever span of the barrels should not be made too large to take the bending forces.
The architectural advantage of this structure is that it appears to float in the air. Therefore, windows should be located so that this illusion is preserved.




INTERSECTION SHELL - FOLDED PLATE


A two element folded plate is shown here as an intersection shell and is similar to the previous example. Almost all the combinations used for curved shells may be used for folded plates, the resulting forms are almost unlimited. In the above sketch, a cross form is used. The columns may be place so that there is no column at the corner and the central dome is suspended from four cantilevers. However, it is better to put the column in the corner so that the central intersection may be used as the stiffening element. The resulting thrusts can be carried by diagonal braces in the plane of the outside walls and may be concealed by the wall construction. The same effect is achieved by a solid concrete wall in each corner. Ties around the barrel would be very unsightly in this structure.






CHAPTER VII - WARPED SURFACES

Warped surfaces have a great advantage for shell structures because they may be formed from straight form boards even though they are surfaces of double curvature. There are two types which are most useful: the conoid, which, as its name suggests, is a portion of a cone, and the hyperbolic paraboloid, a name for a particular mathematical surface. This type of shell structure can be built to what appears to be the ultimate in lightness of construction, minimum reinforcing and ease of moving forms.
Stresses in the hyperbolic paraboloid shell are almost entirely membrane (direct tension and compression), and all forces are delivered as shear parallel to the stiffening ribs. The shell thickness in structures built by Candela in Mexico, is on and one-half inches except for slight extra thickness at the intersection of the surfaces. This dimension is based on a cover of one centimeter on each side of two layers of bars and not an any structural requirement for strength. In this country, using No. 3 bars, (3/8 inch diameter), and a cover of ¾ inches, a minimum thickness of 2 ¼ inches is required.

















CONOID


A conoidal surface, as shown in the sketch above, is formed by drawing straight lines between a curve such as a circle and a straight line. It is a ruled surface because it can be formed by straight lines. A cylinder and a cone also are ruled surfaces but a sphere is not in this category. In the above cases, the shell is supported by a wall or a beam at the left and by an arch at the right.
The appearance of the roof of the typical steel bow string truss building can be reproduced in a concrete thin shell construction by using short shells for the middle bays and conoids for the ends.
This structure is suitable for a large entrance canopy. The horizontal line at the rear can be the second floor level, the curved arch the entrance to the canopy.





HYPERBOLIC PARABOLOIDAL SURFACE


The hyperbolic paraboloid surface is so useful for shell structures that it is important to describe the method of constructing the surface. It is formed in the following manner: 1) Lines OA and OB are level and at right angles to each other, 2) Lines AC and C are also level and are shown above and dotted, 3) Point C* is directly below point C, 4) Mark off equal intervals on line OB and divide line AC* into the same number of increments (but of slightly greater length). Connect intervals on line OB with those on line AC* with straight lines, 5) Repeat for OA and BC*, 6) The surface formed by this network is a hyperbolic paraboloidal surface.
In practice, lines OA and OB may not be level and at right angles to each other and point C* may be above C. Also, only part of the surface may be used, so that the boundaries are not along OA, OB, BX* and AC*.
Note that a diagonal from A to B has a sag which is in the shape of a tension catenary. The diagonal OC* (not drawn above) has a corresponding arch shape.

HYPERBOLIC PARABOLOID - GABLED EDGE MEMBERS


Four rectangular units of the surface are used with this structure and are supported by gabled rigid frames at the outside edges. The ridges at the top, formed by the intersection of the surfaces, are also edge members of the individual panels. These ribs may require additional area which may be either on the top of the shell or may be placed below by constructing the form with a drop.
The stresses in this shell, if the rise of the shell is low in comparison to the span, are direct tension across the diagonals which sag, and direct compression across diagonals which are arched. The shell delivers forces to the ribs that are parallel to the rib.
A tie is shown connecting the knees of the rigid frame. The thrusts are quite high on the edge members. The member sizes must be quite large if the tie is omitted. The open space in the gables may be used for windows. The structure may be built continuously with units side by side to cover a large area.



UMBRELLA SHELLS


Four of the rectangular hyperbolic paraboloidal surfaces may be arranged so that the outer edge of the shell is level and the low point is at the center where it is supported by a column. It is necessary to provide drainage for rain at the low point through a pipe in the column. A row of these units may be placed side by side and tilted so that a clerestory is formed between the rows, or a skylight may be provided by leaving a space between each unit. Individual glass blocks are sometimes placed in the shell to provide lighting. These shells may be diamond shaped in plan rather than rectangular.








HYPERBOLIC PARABOLOIDAL SADDLE DOME


Dome shaped structures of large span may be made from combinations of hyperbolic paraboloids, as sketched above. They may be square, rectangular, or diamond shaped. The shell depends for its strength on one of the corners being raised relative to the others. Therefore, this shape produces an enclosure with large tapered windows on the side. The thrusts in the edge members become very large and these members should be terminated at the ground in a thrust abutment, or a steel tie should be provided between corners.
In addition, another support is necessary on one of the ribs, preferably at one of the corners. Window mullions, if they are at the rib, should be made structural columns to prevent relative movement between the rib and the window.










HP FLOWER DOME

This structure is called a flower dome as being the best description of the appearance. It consists of four of the hyperbolic paraboloid dome units as described on the previous page with the highest point at the middle. However, the lower outside edges and stiffeners have been trimmed in a circle so the units resemble petals of a flower, and the structure is circular in plan.
There are many combinations of these types of hyperbolic paraboloidal shapes possible and the architectural combinations are almost infinite. The most important thing as far as the structure is concerned is that the be a system of valid structures that carry the loads. In some cases the shell must take bending stresses and the membrane stress theory is no longer valid. However, the bending stresses may be no larger than in a barrel shell structure.



















STEEP HYPERBOLIC PARABOLID


This type of warped surface shell utilizes steeply pitched surfaces. Each panel is arranged so that one corner is out of the plane of the other three corners. Ribs are formed between adjacent units and additional ribs are required at the outer edges of the structure. A church using hyperbolic paraboloids was built by Felix Candela in Mexico City, and some of the surfaces are nearly vertical. No top forms were used for the concrete. A network of reinforcing held the concrete to the form. In cases where four panels came in at a low point, columns were used to support the structure.









TRUMPET SHELL


Hyperbolic paraboloids may be formed by using portions of the basic surface that was illustrated previously. In the above shell, the edges are parabolic arches and all the forming is made with straight lines running from equa-distant points on parabolas. The arch rib at the ends must be of sufficient strength to carry the principle loads.
On account of its double curvature, this shell may be made much less thick than the equivalent short shell. The stresses are almost entirely tension or compression.
Another variant is the trumpet intersection shell where crossed vaults are used rather than a single vault. This structure is shown on the next page.








THE GROINED VAULT


A vault can be constructed from parts of four trumpet shells, as shown in the sketch. It may be built without ribs because the curvature of the edges makes the shell sufficiently stiffer and the intersection of the surfaces creates two rigid crossed arches which carry the loads to the supports. Again, this structure is formed with straight lines even though there is considerable curvature to the final surface.
A very dramatic effect can be obtained by continuing the shell beyond the edges shown in the sketch.








NEW FORMS FOR WARPED SURFACES

An infinite variety of forms and structures can be produced using warped surfaces. There are two useful approaches:
1) Instead of a rectangular or diamond grid, use a grid for which the lines are not parallel. The effective depth of some portions of the shell can be increased so that the shell is considerably stronger.
2) Make small models of surfaces, cut them apart and put them back together in new forms which obtain the desired esthetic and functional result and at the same time are satisfactory from the structural point of view. If the form has valid structural elements such as beams and arches which are properly supported, then the structure will be satisfactory.



















CHAPTER VIII - COMBINATIONS OF SHELLS

Combinations of shells are useful and lend variety to the other shapes and forms. The number of combinations is practically unlimited so that only a few may be shown here. The first step will be to list all of the combinations previously described and indicate which types can be combined. The basic types of shells were previously classified as: 1) folded plates and domes, 2) barrel shells, 3) short shells, 4) domes of revolution, and 5) warped surfaces. The intersection shell has been omitted since it is really a combination. Warped surfaces do not combine very well with other types, particularly the folded plate so they will not be considered in combinations.
The combinations possible from the above list are: 1) barrel shells and folded plates, 2) barrel shells and short shells, 3) barrel shells and domes of revolution, 4) barrel shells and conoids, 5) folded plates and short shells, 6) folded plates and domes of revolution, 7) folded plates and conoids, 8) short shells and domes of revolution, 9) short shells and conoids, and 10) domes of revolution and conoids.

















DOME AND BARREL VAULT


In the structure shown above the side of the square dome suggests the shape of a barrel vault. These are really independent structures since the structural elements are all formed before the attachment has been made and could be cut apart without destroying the structure.
Because of the form of the plan, this structure suggests a church or other auditorium with a large central area and adjacent seating.
Vaults can be attached to any of the four sides to produce a T shape or a cross shaped building and the wings may be of various lengths to suit the seating arrangement. The ties across the sides of the dome can be eliminated by L shaped walls acting as thrust abutments.






FOLDED PLATE AND BARREL VAULT


In this structure, a folded plate structure is combined with a barrel vault. For the same width of element, the transverse bending moments in the folded plates are usually larger than the barrel vault so it is important to keep the width of the plates so the slab will not be thick. The form is not especially suitable for long spans since the structural efficiency of the folded plate is not very great. However, it does provide a chance to develop an unusual form.
A very interesting combination is the folded plate Z shell with the north light shell by making the upper end of the Z a smooth curve and the lower end a folded plate.







FOLDED PLATE AND CONE


The cone on the ends of the folded plate finishes off an otherwise angular structure with a curved façade and might have some architectural applications where a bay window effect is required. The cone can hardly be said to have enough stiffness to replace the rib that is otherwise required at the columns. This cone acts like a curved folded plate rather than as a dome.
A similar structure is a dome of revolution used on the end of a curved barrel shell to provide a circular rotunda.










BARREL SHELL - DOMED ENDS


It was a little difficult to say exactly where the above type should be placed in this classification of shells. Finally it was included in this chapter on the theory that it was really a combination of types. The folded plate with tapered ends was included for the same reason.
Tapered ends have the advantage that the eave line can be kept the same all around the building. Less material is required because the stiffener is in domed ends. The amount of formwork is probably less but the forms cannot be moved lengthwise of the shell and must be lowered the full depth of the barrel.
A plane surface can be used instead of the curve at the end. The thickness would be greater because the bending moments are larger in the plate than in the domed shell. However, it would probably not be thicker than the vertical wall type stiffener often used at the ends of barrel shells.







FOLDED PLATE - TAPERED ENDS


This structure is a combination of a folded plate and a folded plate dome. The taper acts as a stiffener and transfers the thrust of the inclined plates to the columns. It has many of the same advantages and disadvantages of the barrel shell with dome ends shown on the previous page.
The outer form of this structure is similar to the hipped roof used for house construction and it is conceivable that folded plates might be used for an inexpensive structure for mass produced houses.










SPACE ACTION OF ARCHES AND VAULTS


This structure consists of two intersection shell domes made from parts of cones. The edges of the domes are supported by ribs acting as arches. The thrusts from these arches are transferred to horizontal girders which, in turn, carry the load to horizontal ties at each end of the building. The horizontal girders also serve as slabs over the side aisles. Two rows of columns are required to support each of these slabs.
This type of space structure is very useful for church buildings because the arches can spring from a higher level and cross ties are not needed except at the ends of the building.









SHELL SLAB


This structure is a combination of a dome of revolution and a plate. The low rise dome in the center of an otherwise orthodox flat plate will have shell action at the center of the pane; and plate action along the column lines. This should result in some savings in both steel and concrete. The flat plate elements should be prestressed so that the thrust of the dome can be fully utilized. The reduction in concrete should result in smaller column sizes for multistory buildings. The thickness of the plate at the columns can be made greater than for the ordinary flat plate structure.












BARREL AND SLAB


This shell is a combination of a barrel shell and a slab. The slab must be of sufficient thickness to carry the required bending moment and shear at the end of the span. The small columns shown on the outside edges are for support of the edges only; otherwise, there are no inside columns for this structure. The barrels also must be of sufficient depth at the center of the span for the required bending moment. Since this is half of the depth at the ends, it is evident that the spans possible with this construction are not very great. This structure would be very suitable when high windows were required to the north and a cantilever slab for sun protection to the south. Then the structural characteristics would be fully utilized.










FOLDED PLATE AND SLA


It is intended in this structure to have a center row of columns so that the greatest depth is at the point of maximum moment and the shear and bending moment at the outside edges are low. A stiffener will be required at the center row of columns. However, the slab acts as its own stiffener. The slab thickness might be somewhat thicker than the shell but as soon as the total depth of the shell became sufficient to resist the stresses, the thickness could be reduced.












CHAPTER IX - SHELL ARCHES

Folded plates and cylindrical barrel shells are essentially beams. The same cross sectional shapes can be used for arches and a new set of forms, having different structural properties, is obtained. It was thought worthwhile to illustrate them separately in this chapter rather than include them in the basic shapes. Hyperbolic paraboloidal surfaces can also be used to form these arches, the virtue being that they can be formed by straight lines.
Shell arches are somewhat in the same category as short shells in that the shell action is subservient to the arch action. All the thicknesses can be made quite small of an arch is used because the stresses will be principally compression. The curve of the arch has to be generally a funicular form, that is, it should fit the thrust line of the applied loads. Shells are not very efficient structures if the bending moments are high, as in the folded plate rigid frame.
There are types of shells that fit in several categories. The hyperbolic paraboloidal dome is really a shell arch.

















FOLDED PLATE ARCH


This structure is suitable for quite long spans and forms for the concrete can be used many times because each unit can be made self-supporting.
All of the different section shapes of folded plates are possible with this type of structure. The Z shape can be used to provide north light.
As in the folded plate shapes, an edge plate is required for the outside member. Placing of concrete on the steep slope at the springing of the arches may be a problem unless blown-on concrete is used or the lower portion of the shell may be precast on the ground and lifted into place.











BARREL ARCH


This shape is similar to the folded plate shell arch except that cross sectional elements are curved instead of being made with plane surfaces. The surface is more difficult to form but the widths of the individual elements may be made greater than for the folded plate shape. Arches of very long span are possible because the bending moments in an arch are much less than in a beam of comparable span. Any number of different shapes may be used, such as the corrugated shape or the north ligh Lazy S shape. Lighting may be obtained by using skylights. The shearing forces are not very large in an arch so larger holes may be used than for a barrel shell.

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