An extensive experimental and analytical investigation of thin-steel hyperbolic paraboloid (hypar) structures was carried out to provide design information. As a result of this work, empirical data is provided regarding the behavior of such structures and computer programs are presented for the analysis of thin steel hypar structures. Hyperbolic paraboloid structures possess a unique combination of structural and architectural properties; some of them are the following: 1) Due to the double curvature of the surface the internal stresses in the deck are generally low and the deflections are small. 2) Since a hypar surface can be generated by straight lines, thin-steel or light-gage panels may be used to form the shell; furthermore such panels are well suited to carry the in-plane shear forces in hypar shells. 3) Basic hypar units can be combined in a large variety of ways to produce attractive roofs (Fig. 1-2, page 212). 4) The dead load to live load ratio is very low in the case of thin-steel shell structures. A hypar unit is a warped surface bounded by straight lines (Fig. 1-1, page 211). The equation of the surface is z = Cxy/AB. According to the simple membrane theory, a uniform load p produces pure shear forces Nxy = ABF/2C. This membrane shear transmits uniform eccentric axial forces to the edge members. The following are the major problems associated with the design of thin-steel hypar structures: 1) The deflections, i stresses, and the stability of hypars depends greatly on the shear rigidity of the thin-steel deck. This property must be evaluated experimentally for each combination of decking, connections to edge members, and seam connections. Furthermore, in the case of hypars the deck is warped and thus the shear rigidity may be different from that of an equivalent flat diaphragm. 2) The deck may buckle due to the shear stresses~ and the buckling load must be evaluated for highly orthotropic shells. 3) The design of thin-steel hypar structures is generally governed by stiffness (deflections or buckling) requirements. The evaluation of the deflections is a very complex matter because it depends on the deck rigidity, the edge member axial and bending stiffnesses, and on the eccentricity of the deck-to-edge member connection. 4) If the curvature (or riseto- span ratio) of a hypar is small, the deflections may be large and a considerable portion of the load is carried by bending rather than by membrane shear. 5) Partial or concentrated loads may cause large local deflections, especially if singlelayer decks are used. The present investigation studied all the above-mentioned factors. The experimental and the analytical studies are summarized briefly in the following paragraphs. The ~~perimental investigatio~ consisted of four types of tests: a) Four medium-scale (12 ft by 12 ft in plan) inverted umbrella tests to study the stresses, deflections, and the deck buckling; b) Test on a small-scale (2 ft by 2 ft) inverted umbrella structure to study scaling effects and the overall i1 buckling of hypars; c) Sixteen flat shear tests to determine the shear rigidity of the decks used in the hypar tests; d) Twelve saddle-s~aped hypar tests (S ft by S ft in plan) with various rise-to-span ratios to evaluate the effect of rise or warping on the shear rigidity and to study other factors such as partial loading and single versus double layered decks. Photos of the various types of tests are shown in Figs. 7.1 to 7.S. The experimental program is described in detail in Chapter VII. Prior to the main test program, several small-scale (2 ft by 2 ft) four-quadrant tests and medium-scale single-quadrant tests were also conducted. These tests were however discontinued because of the severe scaling effects in the case of the small-scale models and the violation of the symmetry conditions in the case of single-quadrant experiments where the neighboring quadrants were missing. Nevertheless, these tests produced useful qualitative information and experience with manufacturing and testing thin-steel hypar structures. The edge members of the umbrella-type specimens were made of tubular members since this afforded easy connection of the warped surface to the straight edges. The decking consisted of single or double layers of standard corrugated panels. One layer was connected to the edge ~enbers with sheet metal screws at various spacings. The seam connections between the panels were also made by means of sheet metal screws. In the case of shells with two layers, the top layer was connected to the bottom layer in a similar manner. iii The medium-scale umbrella models were loaded using air bags under each of the four quadrants. The saddle-shaped hypars were loaded with sand, whereas the small-scale models were loaded through loading pads and suspended weights. The following are the principal conclusions of the experimental part of this investigation: The effective shear stiffness of the cold-formed deck and the rise (or curvature) of the structure are the most important factors influencing the behavior of hypars. For low shear stiffnesses and for small rise-to-span ratios the deflections may be large, the bending stresses tend to increase relative to the membrane stresses, and the possibility of deck buckling increases. As in the case of flat shear diaphragms, the shear stiffness depends strongly on the seam and edge connections. The increase in shear stiffness due to the addition of a second layer of deck was found to be only about 1/3 if the second layer was connected only to the first layer and not directly to the edge members. Similarly, the deflections of a double-layered shell are more than half of those of a corresponding single shell. If the two layers are interconnected with sheet metal screws (on an 8 in. grid in the present saddleshaped hypar tests), the deflections are further reduced by about 10 to 20%, depending on the rise ratio. A particular problem of certain types of hypar structures is the deflection of unsupported outside corners (see Fig. 1-2, page 212). The membrane shear cannot carry the load over such flat corners and thus considerable bending and deflections may develop. The tests showed that the bending stiffness of the iv edge members has a great effect on the corner deflections, in fact, they indicate that the design of the edge members in hypars with flat corners is usually governed by deflection limitations. The measured bending strains in flat saddle shells (riseto- span ratio of 1.8), was much greater than the bending in hypars with greater curvature (rise ratio of 1/3). The membrane theory is insufficient for the design of flat hypar structures. However, the design of the connections (seam or edge) may be based on the shear forces obtained from the simple membrane theory. Several single and double layered saddle-shaped models were tested under partial loading. Since such loads must be carried mainly by bending of beam strips along the deformations of single decks, relatively large deflections were noted. The deflections under the 8 in. by 8 in. loaded area were about three times greater in the single decks than on the doublelayered structures. Since the effective shear rigidity of the deck is of paramount importance, the effect of curvature (warping) on it is an important question. The effective shear rigidity of various deck, edge member, and connection configurations are determined by tests on flat diaphragms. The comparison of the measured deflections for saddle hypars with various rise-span ratios and the evaluation of the effective shear rigidities backwards from the measured deflections indicated that the shear rigidity is reduced by about 20% due to the warping effect. The buckling of the deck is one of the design factors. For small rise-span ratios and for low deck shear rigidities the deck may buckle. As an example, a 12 ft by 12 ft model v having a single layer 24 gage corrugated sheet deck buckled at a uniform load of 70 psf (see Fig. 6.14). This model had relatively stiff edge members (3 in. dia. tubular sections). The corner deflections remained linear with increasing load beyond the buckling load. The buckling load of double-layered structures is much larger than that for single deck shells. A model, similar to the above but with two layers of 28 gage standard corrugated decks, did not buckle up to a load of 145 psf, when the test was discontinued. The major part of the ~~lytical investigation consisted of two finite element approaches for the calculation of deflections, stresses, and instability. In addition, two simple methods were developed for estimating the deck buckling load and the buckling of the compression edge members, which would suffice in preliminary designs. Two types of finite elements were used: curved shallow shell elements and flat elements. The details of the analysis are described in Chapter III. Both approaches were verified by comparisons with existing experimental and analytical results. The stiffness of the eccentric edge members were properly accounted for in the mathematical representation of the structure. The connection of the decks to the edge members may allow rotation about the axis of the edge members and movement normal to the edges dUe to slip at the connections. These possibilities were also considered in the analysis. The instability of the decks was studied with the help of the incremental stiffness matrix approach. The effective stiffvi ness of the system is reduced due to the in-plane forces in the deck. The in-plane forces depend on the deflections of the shell and to obtain the buckling load, the eigenvalues of a large order system need to be evaluated. In the present study the load increment at ion method was used instead. The effect of the in-plane forces was evaluated iteratively at successive load increwents. The buckling load is obtained from the nonlinear load-deflection curve, (Fig. 6-6, Page 276). The comparison of the results of the flat element and the curved element approaches reveals that both give good results for shells supported around the perimeter. However, the flat element method gave better results in the neighborhood of unsupported flat corners. The analysis of the structures tested in this and in other studies confirmed the conclusions of the experimental part of the investigation. The stresses in most types of hypars are low and the design is usually controlled by deflection or buckling linitations. The relative stiffness of the deck and the edge members is an important factor. For stiff edge members the deck tends to bend between opposite edges, whereas in the case of flexible cantilevered edge members the shell partially supports the edge members. Analysis of a structure including the weight of the edge members indicated that this effect may have to be considered in the design of hypar structures. The analysis of buckling of hypar decks showed that the buckling load of double-layered shells is three to four times vii greater than that of single decks. The predicted buckling loads compared well with experimental or previous analytical evidence. The buckling load does not depend much on pre-buckling deflections, however it depends on the axial stiffness of the edge members. The finite element analysis was also used to calculate the deflection of an unsymmetrically loaded inverted umbrella structure. The results, which compared well with experimental data, showed that these deflections are about four times greater than those due to symmetric loading. This increase of deflections obviously depends on the type of structure; in this case much of the flexibility was due to the bending of the central column of the umbrella structure. Since the instability a~alysis of hypars by the finite element method involves considerable amount of computer capacity and expense, approximate methods were developed for the calculation of buckling loads. The buckling of the compression edge members was studied by isolating them from the structure. The instability of columns loaded by tangential axial forces that remain parallel to the member during deflection was evaluated. The results are tabulated in Fig. 6-13, page 284. The buckling of hypar decks was also investigated by the energy method (Section VI-7. The resulting equation has to be minimized to get the critical load; this can easily be done with the help of a computer. This approach is much simpler than the finite element instability analysis and is preferable in preliminary designs. viii A few buckling analyses of cold-formed hypar shells showed that the critical load for double-layers is about three to four times greater than a shell with a single deck. The finite element analysis computer program will be made available to designers by the A~erican Iron and Steel Institute.
Civil, Architectural and Environmental Engineering
American Iron and Steel Institute
Wei-Wen Yu Center for Cold-Formed Steel Structures
01 Sep 1971
Technical Report Number
Report No. 338
Gergely, Peter; Banavalkar, P. V.; and Parker, J. E., "The analysis and behavior of thin-steel hyperbolic paraboloid shells" (1971). CCFSS Library (1939 - present). 19.