New Methods in Efficient Post-Tensioned Slab Design Using Topology Optimization

Post-tensioned (PT) flat-plate gravity framing systems are highly efficient and reduce embodied carbon when compared to conventional reinforced concrete framing systems. Efficiency is especially apparent in multi-span applications with regular orthogonal support arrangements. Even though PT flat-plate gravity framing systems are less efficient in single-span or irregular support applications, they are being still useful in reducing slab thickness, improving construction efficiency, and reducing seismic mass. A novel approach to determining PT tendon arrangements has been applied to several buildings informed by topology optimization results. Topology optimization is an optimization method which determines optimal load paths in a finite element continuum. Thus, by orienting PT tendons along the optimal load paths suggested by topology optimization, it has been shown that 25% or more of PT quantities can be reduced while maintaining the same mild steel reinforcement. Many of the observed arrangements do not follow traditional uniform/banded arrangements. Also, the deflection performance is significantly more consistent since tendons are resisting load in a manner consistent with the load demands. This can help alleviate common issues with thin flat-plate gravity systems such as irregular floor flatness due to warping incited by PT systems and inconsistent deflection at the exterior wall. This new design method has been applied to three buildings and coordinated with construction teams for efficient application. This presentation will discuss the entire design procedure from initial concepts through complete construction documents as applied to three buildings. This presentation will be of interest to academics and practicing structural engineers. INTRODUCTION Unbonded post-tensioned gravity framing systems are increasingly common in multi-story construction throughout the United States and increasingly in other countries. Buildings with irregular column arrangements are increasingly common given the densification of urban centers, common usage of 3D digital modeling, and advanced construction methods now common to the industry. The concurrent advancement of these building characteristics has resulted in post-tensioning primarily being employed for reductions in slab thickness. However, some designs may not fully realize the material efficiencies once observed in orthogonal column arrangements more current in past decades. A new design methodology which is responsive to irregular support conditions is needed to fully realize the performance and material efficiency potential of post-tensioned gravity framing systems. The proposed innovative design methodology enhances the structural designers understanding of gravity framing through the employment of topology optimization. This optimization method iteratively searches a continuous design space for the stiffest configuration of material given a set of support conditions and static loadings. Previously applied for in-plane investigation of lateral force resisting systems (Sarkisian, 2016), this method is reconsidered for out-of-plane demands common to gravity framing systems. Through these results, new load paths and corresponding tendon layout arrangements are identified. These arrangements avoid the inherent inefficiencies of traditional banded-distributed tendon configurations and more directly address the force demands of the gravity framing system. New geometries are explored for various levels of improvement with attention to both material efficiency and constructability. 2017 SEAOC CONVENTION PROCEEDINGS Figure 1. Common Examples of Irregular Supports in Gravity Framing Systems A series of case studies where this methodology has been applied are presented and associated materials savings identified. Consistently, through various design constraints, a minimum of 25% tendon material savings is realized without any increase in mild reinforcement. Additionally, more consistent deflection results are revealed improving floor flatness results. Each project has been developed with direct input from contractors and construction-related considerations are addressed. TOPOLOGY OPTIMIZATION Topology optimization with density methods is a numerical optimization method that enables the identification of optimal geometries for a variety of structural systems. The numerical process is based on the finite element method and assumes “densities” for each element. The densities are related to the stiffness of the element so that if an element has a significant contribution to the target structural performance (e.g.: the overall stiffness of the structure), it will have a high density. Otherwise the density will be reduced. The target volume of material to be utilized in defining the structure is limited to a prescribed percentage of the original design domain. For applications in architecture, the design domain is taken to be the outer skin or shell of the building or outer envelope of a bridge so that the resulting structural system is expressed in the exterior as an integral part of the architecture itself. Thus, the optimal layout problem in terms of an objective function f can be stated using the design variables, d, and the displacements, u, as follows: