2INTRODUCTION: MATERIAL: A shell structure is a thin, curved membrane or slab, usually of reinforced concrete, that functions both as structure and covering, the structure deriving its strength and rigidity from the curved shell forms.Shell structures predominantly resist loads on them by direct compression. That is without bending or flexure.Since most materials are more effective in compression than in bending, shell structures result in lesser thickness than flat structures.MATERIAL:The material most suited to the construction of a shell structure is concrete.
3TYPES:Shell structures are sometimes described as single or double curvature shells.Single curvature shells, curved on one linear axis, are part of cylindrical or cone in the form of barrel vaults and conoid shells.Double curvature shells are either part of a sphere, as a dome, or a hyperboloid of revolution.
4FOLDED PLATES: Folded plates are the simplest of the shell structures. The distinguishing feature of the folded plate is the ease in forming plane surfaces.They are more adaptable to smaller areas than curved surfaces.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.For widths of plate over, say, 12 feet, the thickness of the folded plate must be thicker.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.
5BASIC ELEMENTS:The inclined plates2) edge plates which must be used to stiffen the wide plates3) stiffeners to carry the loads to the supports and to hold the plates in line, and4) columns to support the structure in the air. Are the main 4 elements of folded plates.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.
6THREE 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 size of the frames may be reduced by using a steel tie between the tops of the columns.The depth of the shell should be about 0.10 times the span.The maximum slope of a plate should not be greater than 40 degrees.
7Z SHELL:Each of the units above has one large sloping plate and two edge plates arranged with space between the units for windows.The windows are normally open to the north but most of the light is actually reflected south light.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.
8WALLS 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.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.
9CANOPIES:This folded plate has four segments.A two segment structure is not desirable because it has very little torsional resistance.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 centre valley.
10TAPERED FOLDED PLATES: The height of the shells at the centre 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.
11EDGE 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 vertical columns in the end walls at the crown of the gable takes 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.
12FOLDED PLATE TRUSS:There are horizontal ties across the width only at the ends of the building and the structure acts as an edge supported shell.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 more subtle than the usual shell structure.
13FOLDED 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.
14CYLINDRICAL 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 forms 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.
15ELEMENTS OF BARREL VAULTS: The shell has been allowed to project beyond the edge of the stiffener in order to show the shape of the shell.Stiffeners are required at columns.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.Near the supports the thickness may be greater for containing the larger longitudinal bars.If more than one barrel is placed side by side, the structure is a multiple barrel structure and if more than one span, it is called a multiple span structure.
16MULTIPLE BARRELS - OUTSIDE STIFFENERS: This structure shows a multiple barrel vault with vertical edge beams at the outside edges.The stiffeners have been place above the roof.The advantage of having the stiffeners on top is that there are no interruptions to the space inside the shell so both the inside appearance and the utility are better.The maximum spans for this type shell are again limited by the geometry of the cross section.Assuming the maximum width of barrel to be 50 feet and maximum end slope to be 45 degrees, the rise would be about 14 feet; the maximum span would be in the order of 150 feet.
17UNSTIFFENED EDGES:The thin unstiffened edge of a barrel vault is very dramatic but is subject to considerable deflection if made too thin or used on too long a span.Also, more longitudinal steel is required because the downward deflection of the free edge tends to throw more loads into the reinforcing.An elliptical cross section is superior to a circular section at an unstiffened edge because of the greater curvature of the ellipse, making the shell stiffer at the lower edge.The shell with a complete half circle will be more rigid than one with only a segment of a circle.
18CORRUGATED CURVES:Barrel shells in the form of corrugations may offer structural advantages and may have aesthetic values which make such a roof desirable.Instead of alternative concave and convex circles of the same radius, the curves may be alternate circles of long and short radius.THE LAZY S: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.
19EDGE MEMBERS:The reason for using edge beams is that if a long span is required, say 100 feet or more, the depth of the shell would have to be at least 10 feet, and for a normal column spacing of 20 feet, a full half cylinder having vertical slopes at the edges of the shell would be necessary.Concrete is difficult to place, and, therefore, edge beams should be provided so the radius of the shell is greater and the slope at the top of the edge is not more than 40 degrees.NORTH LIGHT SHELLSThis type of shell structure is used to provide large areas of north light windows for factories requiring excellent natural lighting.The edge member at the bottom forms a drainage trough with the curved shell and materially assists in stiffening the structure.The edges of adjacent shells should be tied together by concrete struts serving as mullions between the window glazing.
20BUTTERFLY 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.Spans for this type of structure must be quire short in comparison to other barrel.
21VAULTS WITHOUT STIFFENERS: Stiffeners on barrel shell vaults are expensive to build and interfere with the interior space if ties are used below the shell.It is possible to eliminate, or to greatly reduce the size of the stiffeners if the thrusts form the shells are taken by the columns.The total force at an interior column is counterbalanced by the thrust of the shells on each side, but the first outside column must take all of the thrust of a shell.The high thrust at the top of the column must be resisted, in turn, by large footings.
22EDGE SUPPORTED SHELLS: 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.
23SHORT 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.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.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.
24BASIC ELEMENTS OF SHORT SHELLS: the shell spanning between arches, and2) 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.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.
25MASSIVE ABUTMENTS:The abutments to the arch in this structure have been made in the form of an inverted U rigid frame.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 quite thin because they follow the thrust line of the forces better than does the U frame.
26RIGID 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.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.
27CANTILEVER 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.
28DOMES OF REVOLUTION:A dome is a space structure covering a more or less square or circular area.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 simple 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.
29SPHERE 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.Domes have been built with a thickness of 6 inches for a span of about 300 feet.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.
30DOMES - SQUARE IN PLAN:This structure is a spherical dome with portions sliced off to form a square or rectangle.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.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.
31MULTIPLE 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.This shell can be classed as a dome of revolution since the shell is part of a sphere (which is a surface of revolution).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.
32TRANSLATION DOMES:This structure looks very much like the Square Dome 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.Most of the load is carried by the side arches with some coming directly to the corners.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.
33FOLDED PLATE DOMES: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.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.
34FOLDED PLATE DOME - SQUARE IN PLAN: The simplest arrangement of folded plate elements is the square dome.The spans of the column centres are limited by the span of the triangular slab which would get very thick and heavy if the spans are large.At the outside edges, a member is required, as in the usual folded plate structure.FOLDED PLATE DOME - TAPERED ELEMENTS:This dome makes use of tapered folded plates slanting to the centre 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.
35MULTI-FACET DOME:Domes may be constructed with many planes so they resemble the facets of a diamond.The structural problem in designing these shells is to provide enough angle between the planes so that an actual rib is formed which will be stiff enough to support the plane surface.A dome hexagonal in plan can be made continuous with all the adjacent units if it is necessary to cover a large area.
36INTERSECTION SHELLS:The surfaces that produce the shell appear to meet at an intersection that is why they are called “intersection shells”.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 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 intersection for which the angle is very large is called here a shallow intersection.
37INTERSECTION 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 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.It is the best type of dome to cover a square area and maintain a level parapet around the building.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.
38INTERSECTION - 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.This would be suitable only for a small structure since it produces additional bending in the lowest part of the shell.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.The usual vault is a continuous structure.
39GROINED 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.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.If six sides are used, a continuous series of shells may be constructed and units of this type could alternate with those having a shallow intersection.
40INTERSECTION 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: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, or4) ties may be place between tops of columns.The architectural advantage of this structure is that it appears to float in the air.
41INTERSECTION SHELL - FOLDED PLATE: Almost all the combinations used for curved shells may be used for folded plates, the resulting forms are almost unlimited.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.
42WARPED 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, andThe hyperbolic paraboloid, a name for a particular mathematical surface.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.
43CONOID:A conoidal surface, 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 linesThis structure is suitable for a large entrance 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: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.
44HYPERBOLIC 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 panelsThe 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.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 centre where it is supported by a column.These shells may be diamond shaped in plan rather than rectangular.
45HYPERBOLIC PARABOLOIDAL SADDLE DOME: Dome shaped structures of large span may be made from combinations of hyperbolic paraboloids.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.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 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.
46STEEP HYPERBOLIC PARABOLOID: 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.TRUMPET SHELL:Hyperbolic paraboloids may be formed by using portions of the basic surface.The edges are parabolic arches and all the forming is made with straight lines running from eqi-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.
47THE GROINED VAULT:A vault can be constructed from parts of four trumpet shells.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.
48COMBINATIONS OF SHELLS: Combinations of shells are useful and lend variety to the other shapes and forms.The number of combinations is practically unlimited.The combinations possible are: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, and10) domes of revolution and conoids.
49DOME AND BARREL VAULT: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.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.The ties across the sides of the dome can be eliminated by L shaped walls acting as thrust abutments.
50FOLDED 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.FOLDED PLATE AND CONEThe 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.
51BARREL SHELL - DOMED ENDS: 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.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.
52SPACE 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.
53SHELL SLAB:This structure is a combination of a dome of revolution and a plate.The low rise dome in the centre of an otherwise orthodox flat plate will have shell action at the centre of the pane; and plate action along the column lines.The reduction in concrete should result in smaller column sizes for multi-storey buildings.The thickness of the plate at the columns can be made greater than for the ordinary flat plate structure.
54BARREL 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.FOLDED PLATE AND SLAB:It is intended in this structure to have a centre 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 centre 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.
55SHELL 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.There are types of shells that fit in several categories.The hyperbolic paraboloidal dome is really a shell arch.
56FOLDED 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.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.
57BARREL 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.