• Structural Engineering and Mechanics
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• v.5 no.5
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• pp.507-521
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• 1997

The ultimate behavior of a reinforced concrete hyperbolic paraboloid saddle shell under uniformly distributed vertical load is investigated using an inelastic, large displacement finite-element program originally developed at North Carolina State University. Unlike with the author's previous study which shows that the saddle shell possesses a tremendous capacity to redistribute the stresses, introducing tension stiffening in the model the cracks developed are no longer through cracks and formed as primarily bending cracks. Even though with small tension stiffening effect, the behavior of the shell is changed markedly from the one without tension stiffening effect. The load-deflection curves are straight and the slope of the curves is quite steep and remains unchanged with varying the tension stiffening parameters. The failure of the shell took place quite suddenly in a cantilever mode initiated by a formation of yield lines in a direction parallel to the support-to-support diagonal. The higher the tension stiffening parameters the higher is the ultimate load. The present study shows that the ultimate behavior of the shell primarily depends on the concrete tensile characteristics, such as tensile strength (before cracking) and the effective tension stiffening (after cracking). As the concrete characteristics would vary over the life of the shell, a degree of uncertainty is involved in deciding a specified ultimate strength of the saddle shell studied. By the present study, however, the overload factors based on ACI 318-95 are larger than unity for all the cases studied except that the tension stiffening parameter is weak by 3 with and without the large displacement effect, which shows that the Lin-Scordelis saddle shell studied here is at least safe.

• Structural Engineering and Mechanics
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• v.14 no.2
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• pp.171-190
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• 2002

Two cases of design are performed for the hyperbolic paraboloid saddle shell (Lin-Scordelis saddle shell) and the hyperbolic cooling tower (Grand Gulf cooling tower) to check the design strength against a consistent design load, therefore to verify the adequacy of the design algorithm. An iterative numerical computational algorithm is developed for combined membrane and flexural forces, which is based on equilibrium consideration for the limit state of reinforcement and cracked concrete. The design algorithm is implemented in a finite element analysis computer program developed by Mahmoud and Gupta. The amount of reinforcement is then determined at the center of each element by an elastic finite element analysis with the design ultimate load. Based on ultimate nonlinear analyses performed with designed saddle shell, the analytically calculated ultimate load exceeded the design ultimate load from 7% to 34% for analyses with various magnitude of tension stiffening. For the cooling tower problem the calculated ultimate load exceeded the design ultimate load from 26% to 63% with similar types of analyses. Since the effective tension stiffening would vary over the life of the shells due to environmental factors, a degree of uncertainty seems inevitable in calculating the actual failure load by means of numerical analysis. Even though the ultimate loads are strongly dependent on the tensile properties of concrete, the calculated ultimate loads are higher than the design ultimate loads for both design cases. For the cases designed, the design algorithm gives a lower bound on the design ultimate load with respect to the lower bound theorem. This shows the adequacy of the design algorithm developed, at least for the shells studied. The presented design algorithm for the combined membrane and flexural forces can be evolved as a general design method for reinforced concrete plates and shells through further studies involving the performance of multiple designs and the analyses of differing shell configurations.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.14 no.4
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• pp.807-814
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• 1994

One case of pointwise limit design is performed for a hyperbolic paraboloid saddle shell(originally used by the Lin-Scordelis) to check the design strength against a consistent design loads, therefore, to verify the adequacy of current design practice for reinforced concrete shells. The design method which was based on stresses from membrane analysis in conjunction with pointwise limit state design equations shows a good performance, which means that the design method gives a lower bound on the ultimate load. This shows the adequacy of the current practice at least for this saddle shell case studied. To generalize the conclusion many more designs-analyses are performed with different shell configurations.

• Proceedings of the Korea Concrete Institute Conference
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• 1998.10a
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• pp.405-411
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• 1998

An iterative numerical computational algorithm is presented to design a plate of shell element subjected to membrane and flexural forces. Based on equilibrium consideration, equations for capacities of top and bottom reinforcements in two orthogonal directions have been derived. The amount of reinforcement is determined locally, i. e., for each sampling point, from the equilibrium between applied and internal forces. One case of design is performed for a hyperbolic paraboloid saddle shell (originally used by Lin and Scordelis) to check the design strength against a consistent design load, therefore, to verify the adequacy of design practice for reinforced concrete shells. Based on nonlinear analyses performed, the analytically calculated ultimate load exceeded the design ultimate load from 14-43% for an analysis with relatively low to high tension stiffening, ${\gamma}$ =5~20 cases. For these cases, the design method gives a lower bound on the ultimate load with respect to Lower bound theorem. This shows the adequacy of the current practice at least for this saddle shell case studied. To generalize the conclusion many more designs-analyses are performed with different shell configurations.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.13 no.2
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• pp.11-20
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• 1993

Nonlinear inelastic behavior of an HP saddle shell has been examined by a finite element computer program developed on a Cray Y-MP. The mesh convergence is studied using three progressively refined finite element mesh models, $16{\times}16$, $32{\times}32$ and $64{\times}64$, for the elastic and inelastic analyses. It is shown that the $32{\times}32$ mesh model gives a solution that is very close to that given by the $64{\times}64$ mesh model, thus, showing a convergence. The inelastic analysis shows that the shell has a tremendous capacity to redistribute the stresses. At the ultimate, the concrete cracks and the reinforcement yieldings are spread out all over the shell, indicating that the stress distribution in the shell is approaching that given by the classical membrane theory. The present computer program provides a very useful tool for evaluating the nonlinear ultimate behavior of concrete shells during the design process.