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 construc@ng 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 doSed, 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.