Arrangement of reinforcement in variable density timber slab systems for multi-story construction

Abstract

The arrangement of columns and their spacing in multi-story timber construction is restricted to rectangular grids by the production and shipping sizes of floor assemblies. This is particularly true for hollow box floor systems, for which the punctual supports must be placed at the reinforced edges of the hollow boxes. The arrangement of the columns and their spacing is thereby restricted by the production and shipping sizes of the box ceilings to rectangular grids. To overcome these design limits a new wooden box building system is developed that allows for irregular column layouts through a tailored slab interior design. This development allows for the increased applicability of timber floor systems regardless of site shape or architectural design intent. The slab interior design is dependent on occurring forces and fabrication requirements. Three methods for the internal slab layout are developed and compared: a sequential method, a structurally informed agent-based method, and a geometrically informed agent-based method that uses both a sequential and agent-based approach. The structural performance of each method is compared through the analysis of three reinforcement layouts an architectural testing setup.

Keywords

Agent-based modeling integrative design structural analysis computational design timber building system

Introduction

New strategies and systems are required to mitigate the impact of the building industry on the environment in light of climate change. The building industry is responsible for 40% of the global carbon dioxide emissions.¹ Recently, the design and construction of buildings with timber has come to the forefront as one such strategy, as timber is a renewable resource and stores carbon dioxide.² Timber is also a lightweight material and can therefore further reduce the use of concrete beyond what it displaces in the building structure by reducing that needed for foundations.

To reach spans competitive against concrete construction, timber building systems are conventionally supported on grids of beams. These beam grids reduce the architectural design freedom for placing partition walls. This inferior capacity to freely reconfigure their interior minimizes the adaptability of conventional timber buildings for future changes of use, the consequence of which is a shortened building lifespan. Building systems that have flat structural ceilings on the bottom of their slabs, without beams, are less challenged by changing use cases and interior partition layouts.

The two most common means for achieving a flat underside of slab are either a mass timber approach or a box ceiling approach. The hollow cavity inside the box ceiling provides three principal advantages: the reduction of material per built unit volume, the potential to integrate building physics into the slab, and the potential for selective reinforcement. Reinforcement inside a hollow slab system, in the form of linear timber elements, can increase the box ceiling’s spanning capacity much in the way beams below a conventional slab system do.

State of the art

Timber building systems

Timber structures are becoming increasingly common as an approach to reduce the climactic impact of construction. Since the renaissance of timber construction about 20 years ago, timber structures have seen increased height and expanded applications.^3,4 Timber building systems are being applied to building types from single-family houses to mixed-use high-rise buildings. As timber building systems have matured over this time, they have grown to be generally differentiable into three major categories: massive, ribbed, and hollow timber systems.⁵ Any of these three categories can then in turn be fabricated and assembled as either point supported slabs, prefabricated panels, or prefabricated volumetric modules.⁶

The continued development of these system types and fabrication approaches is taking place both in industry and in academia. Some companies have developed their own integrated building system and fabrication solutions, but they are still developing their products.^7,8 In research, the Timber Structures 3.0 (TS3) system⁹ was originally developed based on glued plate-to-plate connections for massive CLT plates. The TS3 system is now being further developed for hollow box systems, in part because hollow systems use less material per built unit area.¹⁰ The change from a solid or ribbed system to a hollow one is not without its drawbacks. Reducing the mass and quantity of material in the building structure effects the acoustic behavior thereof.¹¹ Hollow timber building systems are often filled with additional granular material, such as gravel, to increase their mass and acoustic performance, particularly the reduction of sound transmission through the system.¹² Another drawback is the lack of architectural design flexibility associated with hollow systems. With their mostly open internal areas and reinforced edges, hollow box systems in general allow only for linear or punctual supports at the edges or corners of their prefabricated slab modules.⁵ The restricted placement of columns can also be attributed to the general lack of load transferring plate-to-plate connections in conventional timber buildings.

Wood’s inherent anisotropy, a fixation on material and fabrication efficiency, and the hegemony of global logistics requirements⁶ all contribute to the mainly grid-based designs for multi-story timber buildings seen today.^13,14 These design limitations further constrain the possible shapes of multi-story timber buildings and reduce the application of timber structures in urban contexts and as building stock extensions. To extend the design space for timber construction a timber building system that allows for multi-directional spans and has load transferring connections is needed. For this, the specific material properties of timber products need to be considered in the system development and connection design. Concepts of multi-directional spanning slabs are presented by Orozco and Krtschil.^15,16 The integration of disciplines leads to a higher complexity in the design process. In these examples, the placement of internal reinforcement is realized through agent-based simulations that consider both the occurring forces in and fabrication requirements of the slab. Because of their capacity to reconcile conflicting priorities, agent-base methods seem well suited to the integrated design of timber buildings.

Agent-based modeling

The application of agent-based modeling (ABM) to architectural applications is common.¹⁷ ABM techniques have previously been used in the design of geometrically complex timber structures.^18,19 Their applicability to multi-story construction was previously tested by Orozco.¹⁵ While the agents in an agent-based simulation can be conceptualized as many things,²⁰ ranging from pedestrians inhabiting a built space²¹ to virtual builders simulating a construction,²² this research chooses to conceptualize the agents as building components. In so doing, the simulation can implement behaviors that allow the building elements to respond to structural and fabrication requirements directly.

Alternative methods for the arrangement of linear elements to responsively cover a space come from graphic fields, either as the application of non-traditional stipple shapes such as bars, or the visualization of vector fields.²³ While the arrangement of linear stipples can be made to respond to environmental factors, such as the color of the original image, they lack the interactivity enabled by agent-based models (ABMs), both between multiple external factors, and also between the model and a designer.²⁴

Different approaches exist for reinforcing architectural slabs made from concrete while aiming to reduce their weight. Some are analogous to the approach taken by a wood hollow box, such as the placement of voids inside a concrete slab.²⁵ Others are more akin to timber beam systems, which have a thin top plate and structurally informed reinforcement below.^26,27 Common to these approaches is topological optimization, which has seen a lot of interest from the automotive and aerospace industries.²⁸ Studies into the application of topological optimization in architecture also focus heavily on issues of fabrication.²⁹ Further research would be necessary to ascertain the applicability of topological optimization to timber building systems such as the one described in this paper, due to both the anisotropic qualities of wood, and the implications of joining discrete aggregated elements instead of subtracting a single plastic material.

Aim

This research aims to benchmark the structural performance of a series of computational design methods for the arrangement of reinforcement in a novel timber slab. In architecture, computational methods allow for the design and rationalization of highly complex systems in the name of performance. Robotic fabrication methods can simplify the implementation of such designs. However, the efficacy of such complexity is not always assured. This research hopes to demonstrate that not only is the addition of informed geometric complexity worthwhile for its increased performance and decreased material use, but also because of the implications it has for the increased applicability of timber within the built environment.

Scope

The arrangement of reinforcement within the slab is tested for its structural performance. The slabs are tested with full mesh loads. The reinforcement of the slabs is achieved through discrete webs and solid reinforcements around the column heads. The webs and solid reinforcements are placed based on the occurring forces in the slab. Three different methods for the placement of field webs are compared: (i) a sequential method (SEQ), (ii) a structurally informed agent-based method (SABM), and (iii) a geometrically informed agent-based method that uses both sequentially arranged webs and webs arranged with ABM (GABM). Each method was used to generate three different layouts, based on target web lengths (l_tw) of 500 mm, 700 mm, and 2000 mm. The methods are compared at each target web length. The quantity of reinforcement inside the slab was kept consistent between methods at each target web length.

The methods are generally applicable to the design of new-build multi-story buildings with box floors, and as such an appropriately analogous testing setup was needed. The design of a testing setup, however, is out of scope. The testing setup described by Orozco,¹⁵ with its exemplary structural and architectural boundary conditions, including a plate segmentation, was chosen for this research. The placing of columns and the segmentation of the slab are also out of scope. Though building system decisions were made with fabricating in mind, the calculation and verification of fabrication considerations, such as material waste due to off-cuts, robotic motion planning efficiency, or complexity of component prefabrication is out of scope. Structural considerations beyond the loading of the slab itself, such as lateral bracing, are out of scope. Building system design considerations, such as connection design, vibration and sound transmission, or the integration of building services are also out of scope.

Methods

System definition

The investigated timber building system in Figure 1 is a hollow box system that combines the strengths of massive and hollow timber building systems.¹⁶ The timber building system consists of a top and bottom slab, both made of 100 mm five-layer CLT. Unlike conventional timber hollow box systems, which generally can only be supported at the assembly edges, this building system can take pointwise supports within the slab plate. More viable locations for potential columns increase the variety of architectural layouts and solutions available to designers. Furthermore, by not supporting panels at their edge, the maximum moment experienced by each panel can be reduced. This innovation is achieved by arranging both planar and linear reinforcement in the space between the top and bottom slabs. Solid, planar elements made of beech LVL,³⁰ called “column crowns,” are placed in the high shear force areas where the column meets the slab. Linear “web” elements are made of reused CLT off-cuts from the shaping of the top and bottom slab. They are therefore 100 mm wide and have the principal direction of the CLT running laterally. Web elements differ in length, are placed robotically in the prefabrication setup, and are glued to top and bottom plate. Some, called “radial webs,” so named for their radial arrangement around the column crowns, allow for a better shear force transmission into the column-to-plate connection and are also glued to the column crowns. The size of the column crowns and the number of radial webs is set according to the occurring shear forces where the column meets the slab. Elsewhere, “field web” elements are placed throughout the slab to transfer shear and moment forces between the top and bottom slabs. The quantity of field webs is determined by the web placement methods. For comparability, the quantity of material inside the slab is kept consistent regardless of the web placement strategy.

Figure 1.

Elements of the timber building system.

The column-to-plate connection design is a glued softwood-hardwood connection based on the work of Tapia.³¹ In addition to the column crown, the column-to-slab connection is reinforced with beech LVL³⁰ on top of and below the slab. The beech LVL pyramid on the top of the top slab is there to strengthen the slab at this high moment area. The one in the bottom slab is there for the transfer of the support forces onto the column below. The plate-to-plate connection is a glued connection that is assumed to be a stiff.

Testing setup

The testing setup (Figure 2) represents the architectural goals of the developed timber building system. Due to its various spans and an irregular column layout, it would not be possible to build this testing setup with a conventional timber building system. Slab openings and cores are not considered in the design as the testing setup is conceived as being a section of a larger multi-story timber building. The spans reach from 3.0 m to 7.9 m. Cantilevers of 2.0 m are considered in the design.

Figure 2.

Architectural testing setup.

Structural analysis

The structural analysis is divided into two parts. The first simulation is based on the slab segmentation using a solid cross-section considering the main fiber direction of each plate. The reinforcement of the slab is defined based on this first simulation. The second simulation is done after the reinforcement placement and represents the hollow box system.

Both simulations use the finite element (FE) software Sofistik.³² The material properties for CLT, shown in 1, are set according to Wallner-Novak³³ as there are as of yet no normative regulations for CLT material properties. The material properties for the beech LVL plates are set according to the product declaration from Pollmeier.³⁰

The supports are pinned punctual supports that are fixed in z-direction at the column positions. To avoid the horizontal movement and rotation of the slab two points of the boundary shape are fixed horizontally. The considered loads are the self-weight of the structure and an additional mesh load of 2 kN/m² (Table 1).

Table 1.

Characteristic strength and stiffness properties.^30,33

Material property	Unit	C24	Beech LVL
$E_{0, g, m e a n}$	(MPa)	11,000	16,800
$E_{90, g, m e a n}$	(MPa)	0.01	470
$G_{0, g, m e a n}$	(MPa)	448.8	760
$G_{r, g, m e a n}$	(MPa)	50	430
$f_{m, g, k}$	(MPa)	24	80
$f_{t, 0, g, k}$	(MPa)	14	60
$f_{t, 90, g, k}$	(MPa)	0.0	1.5
$f_{c, 0, g, k}$	(MPa)	21.0	57.5
$f_{c, 90, g, k}$	(MPa)	0.0	10
$f_{ν, 0 - 90, k}$	(MPa)	4.0	6.9
$f_{ν, 90 - 90, k}$	(MPa)	1.1	3.45
$ρ_{m e a n}$	(kg/m³)	420	800

Initial structural analysis

The initial structural analysis is done for a 200 mm thick CLT cross-section with a five-layer layup (40-40-40-40-40). The global mesh size for the initial simulations is set to 200 mm. The plate-to-plate connection seams are considered in the plate mesh. The main fiber directions of the individual plates are considered in the simulation. The mesh and main fiber direction of the individual plates for the initial structural analysis is shown in Figure 3.

Figure 3.

Initial Analysis Mesh with main fiber directions of the individual plates.

Main structural analysis

After the slab interior is defined, the hollow box system is simulated. The top and bottom plates as well as the solid reinforcements and the webs are modeled as shell elements that are connected through translation couplings representing the glued connections between those elements. The global mesh size for the initial simulations is set to 100 mm with a local refinement at the web edges. The local x-direction of the shell elements is oriented along the main fiber direction of the respective plates. The local x-direction of the column crowns equals the direction of the respective top and bottom shell elements. The local x-direction of the web elements is oriented along the web length. The top and bottom plate and the webs are made of 100 mm five-layer CLT (20-20-20-20-20). To represent the column-to-plate connection a local change of the material is applied to the mesh around the column head with a radius of 300 mm. The solid reinforcements are made of 100 mm thick beech LVL with alternating layers of glued plates with a thickness of 20 mm. The full cross-section of the sandwich panel is 300 mm high, with a cavity of 100 mm height. The cross-section is fixed for this comparison but can be adapted to different spans and loading conditions. The supports are located at the bottom slab. The mesh load is applied to the top slab.

Design methods

The placement of the reinforcement happens in two steps: first the reinforcement as part of the column-to-plate connection, including column crowns and radial webs, and second, the remaining field web elements. The column crowns are rounded rectangles of different sizes centered on the point where the column meets the slab. The sharp corners of a rectangle would have caused stress peaks in the top and bottom slabs. A fully curved shape such as an ellipse would completely minimize stress peaks. However, is inefficient to produce from rectangular stock material. Therefore, the corners of the column crowns are filleted to a radius of 300 mm, which is sufficient to mitigate stress peaks, while minimizing column crown pre-processing. Higher forces must be transmitted through the slab into the columns below in the slab’s stiffer main fiber direction. The column crowns are therefore rectangular, with their long axis in the main fiber direction. Three different sizes of column crown were considered, with long edges measuring 1000 mm, 1400 mm, and 2000 mm, and with an aspect ratio of 3:2 (Figure 4). The smallest of the crowns that covered an area with shear force values below a threshold was chosen. The perimeter of the column crown is subdivided evenly into segments approximately two web widths wide, and a 400 mm long web placed oriented directly away from the column at each subdivision point. As shown in Figure 5, radial webs are mitered so that they can be glued to the column crown and allow for a more fluid force transfer between the solid and the hollow parts of the slab. All radial webs are of the same unmitered the same length for ease of fabrication. The placement of these radial webs is fixed for the comparison of all web layouts.

Figure 4.

Column crown dimensions.

Figure 5.

Radial web connections.

The field web placement is applied in the areas beyond the column crowns and radial webs. All webs are placed with a minimum offset of 100 mm from the plate-to-plate connection. For these experiments, the minimum web length was set to 100 mm and the maximum to the length of the plate. Minimum and maximum web length can depend on several factors, including: (A) the fabrication set up for the slab; (B) restrictions from the structural design; (C) or information from other slab-internal design considerations, such as acoustic performance or integrated mechanical, electrical, or plumbing (MEP) services. The relation of webs to each other is restricted by physical, structural, and fabrication constraints. Fabrication constraints dictate that the area around the middle of each web needs to be 170 mm from other webs’ center lines to accommodate the width of adjacent webs and the potential robotic tools used for the web placement. All parts of adjacent web centers must be one web width apart, to avoid physical clashes between web elements during construction. One web width is also a good separation distance to avoid force anomalies in the FE simulation. The maximum distance between webs is set to 1000 mm to avoid local deflections of the slab.

Sequential method (SEQ)

The Sequential Method (SEQ) builds on conventional timber slab design methods. All its linear reinforcement members are as long and continuous as possible and are aligned with the panel’s principal spanning direction. This results in a regular web placement. Webs are placed parallel to each other 300 mm apart if the peak absolute moment value in a slab plate was more than a threshold of 8 kNm/m or 500 mm apart if it was lower. The moment values within a radius of 1.0 m from the column head are not considered. Once spaced, the webs were centered in the slab’s secondary direction, considering the minimum distance from plate edge. At the shorter target field web lengths, the spacing between the webs in their spanning direction was used to adjust for consistent material use across web placement methods.

Structurally informed agent-based method (SABM)

The Agent-based Method aims to maximize the effect of structural information on the arrangement of reinforcement within the slab. Not only do higher shear forces cause higher web densities, but the webs are oriented by the principal moment vector at their position in the slab. The Agent-based method consists of two agent-based simulations, one to arrange the agents in the slab, and the other to set their directions and lengths. In both simulations, the linear field webs are conceived of as Cartesian agents. The attributes of these agents are as follows: (a) their position in Cartesian space; (b) their spanning direction, as a unitized 2D vector; (c) their length, in mm; and (d) their center line, as a 2D curve. The same agents are used in both simulations and retain their attributes between simulations.

The agents are divided into separate agent systems and environments, one for each plate of the floor slab. The agents remain in the same, separated agent systems and environments for both simulations. The agent systems for both simulations consist exclusively of web agents. The two-dimensional Cartesian environments have the following attributes. (i) The field web area of the slab plate as a 2D surface. The field web area is the boundary of the plate, offset inwards 100 mm to account for plate-to-plate connections, and with the areas for the column crown and radial webs excluded. (ii) The 2D mesh from the initial analysis approximating the field web area (iii) The 2D principal moment vector at the center of each mesh face. And (iv) The scalar shear value at the center of each mesh face based on the initial analysis, in kN/m.

In the first agent simulation, the designer sets both a number of agents and a web length to instantiate the system with. The agents are divided between the different environments proportionally by environment area (i). They are then instantiated throughout each environment pseudo-randomly and proportionally by shear value (iv). The pseudo-random distribution helps reduce the number of iterations needed for convergence. There is a higher probability that at agent will be placed in an area of greater shear. Agents take their spanning direction (b) from the principal moment direction (iii) of their environment’s closest mesh face (ii) to the agent’s position (a).

The first agent simulation aims to distribute the agents in relation to each other so they may better meet adjacent web spacing criteria. The single behavior agents have in the first simulation is a weighted Lloyd algorithm,³⁴ where the weight of the movement vector in inversely proportional the shear value. Agents in areas of less shear therefore move more than those in areas of greater shear. This behavior is based on the first behavior in the third-level agent simulation described by Orozco.¹⁵ The first simulation runs until the smallest distance between agents crosses a threshold based on the set web length. A diagrammatic representation of the first agent simulation is shown in Figure 6. The web agents were located at the black points after the convergence of the agent simulation, which took 10 iterations. The agents were instantiated at the start of the orange Agent Path.

Figure 6.

Agent-based reinforcement placement.

The second simulation uses the same agents, agent systems, and environments as the first, but uses different agent behaviors and the final agent positions and attributes from the first simulation as its starting positions. The agents for the second simulation instantiate their center line parameter (d) by drawing a linear curve of length (c) in their spanning direction (b), centered at their Cartesian position (a).

The second agent simulation iteratively cleans up the results of the first agent simulation. It does this by giving the agents two behaviors: one for merging webs, and the other for shortening webs. The merging behavior checks for co-linearity within an angle tolerance between agents whose center lines’ (d) closest points are within a distance tolerance. If both these criteria are met, a new agent is instantiated with both behaviors and the following properties: (a) its Cartesian position at the midpoint between the positions of the two co-linear agents; (b) its spanning direction set from the moment vector (iii) of the mesh face (ii) closest to their new position (a); (c) its length set as the distance between the first points on the two co-linear agents’ center lines, up to a set threshold; and (d) its center line derived as before. The two co-linear agents are removed from the agent system. This behavior is based on the third behavior in the third-level agent simulation described by Orozco.¹⁵

The shortening behavior looks for an intersection between the agent’s center line (d) and those of the other agents in its agent system, as well as the environment boundary. When all physical intersections k are found, the curve c that would be intersected furthest from the center, calculated as $c_{trim} = i_{k} at \max {a_{0}, \dots, a_{n}}$ , is trimmed with an additional offset of one web thickness. Here, i_k is the index of the web to be trimmed. n is the total number of intersection events. a_k is the normalized distance from the intersection to the center point of the agent’s center line (d) defined as: $a_{k} = | (0.5 - {\tilde{t}}_{k}) |$ , with ${\tilde{t}}_{k}$ being the normalized distance along the curve at intersection event k. Each iteration of the agent simulation ends once a single agent has successfully executed either of its behaviors. This approach was chosen to avoid agents reacting to webs that are either altered or no long part of the agent system. A diagrammatic representation of the first agent simulation is shown in Figure 7.

Figure 7.

Agent-based reinforcement definition.

Geometrically informed agent-based method (GABM)

During the development of the SABM, it became apparent that some web agents arranged themselves almost parallel to the panel’s principal direction, sometimes over the entire panel, and sometimes only in a panel’s areas of maximum moment. Furthermore, due to the weighed Lloyd algorithm used in the first agent simulation of SABM, the areas without parallel webs had very evenly and densely distributed agents. The GABM was conceived on the premise that the simplification of the web layout would reduce computational complexity without adversely affecting the performance of the informed reinforcement. The GABM describes the combined use of sequential and agent-based methods depending on the region of the slab. The field web area was divided in two: the transition area and the regular area. Transition areas are defined as those in which the force flow vectors of more than 80% of the mesh faces differ by more than 7.5° from the main fiber direction of the slab plate. Regular areas were everywhere else in the slab, in the span centers where the force flow is mainly parallel to the plate’s principal direction. In the regular areas, those that had previously had almost parallel agent alignment, the Sequential Method is used to arrange the webs along the plate’s principal direction. In the transition areas, where agents had previously been evenly and densely distributed, a simplified agent-based method was used, informed by the sequential webs in the regular areas instead of by structural information stored in the agent environment.

This GABM consists of only a single agent-based simulation based on geometrical rules only. The regular webs influence the placement of the webs in the transition areas as they already consider the occurring forces in the slab. The agent systems and environments are divided in the same manner as in the SABM: one environment and one agent system per slab panel. The GABM Cartesian environment has a single attribute: the transition web area of the slab plate, as a 2D surface. The webs are conceived of as “transition” agents, with only their position in Cartesian space as an attribute. Without the structural information stored in the environment, a second set of simple Cartesian agents was introduced: “boundary agents.” Both transition and boundary agents inhabit the same agent systems.

The total web lengths (l_Σ) from the layouts produced at each target web length by SEQ and SABM were used as the target total length l_Σ for the GABM. After webs were sequentially arrayed in the regular areas, their combined length l_reg was removed from l_Σ. The resulting length was divided by the target web length l_t, to produce the number of transition agents, n_trans. This can be expressed as follows

n_{trans} = \frac{l_{Σ} - l_{reg}}{l_{t}}

n_trans was divided between the different environments proportionally by environment area. That number of transition agents are instantiated pseudo-randomly and evenly within the transition areas. The edges of the agent environments are subdivided evenly into segments approximately 400 mm long, and a boundary agent instantiated at that position.

The GABM agent simulation aims to evenly distribute transition agents within the transition area, maintaining an even distance from other transition agents and from the edges of the environment, as defined by the boundary agents. Boundary agents have no behaviors. Transition agents have a single Move to Centroid behavior, based on that described by Groenewolt.²⁴ It is based on the Voronoi diagram, which uses a set of points to divide a surface into generally uniform areas, regardless of the number of agents present. The agents draw a vector between their current position and centroid of their Voronoi cell. This results in a homogeneous spatial distribution of the agents whereby both local accumulations and large gaps are avoided. Figure 8 illustrates the procedure for creating Voronoi regions and the subsequent movement of the agents by a directional vector into the respective area center point. A separation behavior based on that described by Reynolds³⁵ was also tested, but results proved more consistent across the three l_tw with the Move to Centroid behavior. The GABM agent simulation is run until it converges.

Figure 8.

Simple Cartesian agent system.

The spanning direction of each transition webs line is calculated based on the directions of and distances to its nearest regular and radial webs after the convergence of the agent simulation. It is calculated as follows

v_{trans} = (d_{reg} \cdot v_{radial}) + (d_{radial} \cdot v_{reg})

Here, v_trans is the vector for the spanning direction of the transition web, d_reg and d_radial are the Euclidean distances between the agent position and the closest point on the nearest regular and radial web, respectively, and v_reg and v_radial are the vectors for the spanning directions of the nearest regular and radial webs, respectively. The transition web center lines are drawn sequentially as linear curves of target web length l_tw in direction v_trans, centered at the transition agent position. These center lines are trimmed by the edges of the transition areas. If there are any intersections between these center lines, both intersecting agents are trimmed with an offset of half of one web thickness.

Results and Discussion

Three layouts with different target web lengths l_tw are investigated for the three different methods. The layouts are shown in Figure 9. Here, n_agents is the number of agents that were instantiated at the start of each design method, n_webs is the number of webs at the end of the design method, $l_{\bar{w}}$ is the average length of the resulting web agents, l_Σ is the sum of the length of all webs, $% l_{t_{t Σ}}$ is the percentage of the target total web length The tested layouts used the following target web lengths l_tw: 500 mm, 700 mm, and 2000 mm. For each target web length l_iw, there is also a target total web length l_tΣ. The total web length l_Σ, and hence the amount of material, is kept close to l_tΣ for all methods with a maximum deviation of 2%. The resultant average web length $l_{\bar{w}}$ of each layout is no longer the target web length t_tw, but is within bounds min l_w and max l_w.

Figure 9.

Web layouts for the design methods with different target web lengths (l_tw).

The structural comparison is based on the calculated deformation and utilization. The internal shell forces of the simulated FE models are compared to the resistance for each shell element. The utilization is simulated according to Wallner³³

η_{x} = \frac{m_{x, d}}{m_{R, x, d}} + \frac{n_{x, d}}{n_{R, x, d}} \leq 1

(1)

η_{y} = \frac{m_{y, d}}{m_{R, y, d}} + \frac{n_{y, d}}{n_{R, y, d}} \leq 1

(2)

η_{v} = \sqrt{{(\frac{v_{x, d}}{v_{R, x, d}})}^{2} + {(\frac{v_{y, d}}{v_{R, y, d}})}^{2}} + \frac{m_{x y, d}}{m_{R, x y, d}} + \frac{n_{x y, d}}{n_{R, x y, d}} \leq 1

(3)

The directions are based on the local coordinate system of each element. The x-direction represents the main fiber direction, and the y-direction the secondary direction, perpendicular to the main fiber direction. The utilization and deformation for each simulation is shown in Figure 10, Figure 11 to Figure 12. The total web length (l_Σ) is fixed for each target web length (l_tw).

Figure 10.

Utilization and deformation of the layouts generated by the Sequential (SEQ), Geometrically Informed Agent-Based (GABM) and Structurally Informed Agent-Based Methods (SABM) for a target length of 0.5 m.

Figure 11.

Figure 12.

The webs are more utilized closer to the column head areas. The critical points are at the web ends where all shear forces are transferred into the top and bottom slab. The solid reinforcement has a lower utilization than other slab components because of the relatively higher strength of the beech LVL material. Because the top and bottom plates have an equally high utilization, in Figure 10, Figure 11, Figure 12, only the top plates and interior layouts are shown.

The simulations can be observed to be highly sensitive. Slight geometric changes, and hence slight changes to the mesh, cause high utilization differences. According to the discretization of the webs, punching shear areas occur in the top and bottom plate on all web ends (see Figure 10, Figure 11, Figure 12). These shear forces are the limiting factor for the design of the slab. To understand the punching shear problem at the web ends better, a 3D-simulation and physical tests would be required. This effect is similar for the utilization in x-direction according to local peaks of the bending moment at the web ends.

High utilizations in the y-direction occur at those plate-to-plate connections that have the greatest difference between the main fiber direction of the connected plates. The SEQ does not respect the force flow direction in this area, causing high utilizations perpendicular to the main fiber direction of the top and bottom plates. To prevent the high sensitivity of the simulations, and hence the stress peaks at the web ends, an approach similar to that used for concrete buildings should be evaluated: define a punching shear area around the wall end for the slab.³⁶ In this area the slab nodes are coupled to the wall end to reach a better and more realistic distribution of forces. In a further step this concept could be transferred to the connection between the webs and the top and bottom plates.

In general, the utilization in the x-direction increases if parallel webs end at the same position. A better force transmission can be achieved with overlapping (parallel) webs. Plate-to-plate connections with a high angle $(> 30 °)$ cause higher utilizations in the y-direction, perpendicular to the main fiber direction. The moments can be reduced by adjusting the webs to the force flow, as they are in both GABM and SABM. High shear occurs at the web edges in the punching shear areas. This phenomenon is local and needs further investigation.

The simulation of the deformation is not sensitive to slight changes of the web placement and the singularities at the web ends and is therefore a good indicator for the methods comparison. For all three l_t values tested, SABM layouts produce between 16% and 23% lower deflection than the SEQ layouts, while GABM layouts produce deflection between 28% and 30% lower than SEQ layouts. The reason is similar to that which causes the higher utilizations in y-direction for the SEQ: in both the SABM and GABM layouts, webs are generally aligned to the force flow.

The different methods perform similarly throughout the tested target web lengths. The highest changes occur for the maximum shear utilization of the GABM. The authors believe that the SABM produces its best results with the shortest target webs length because shorter webs can better align themselves with the local principal moment direction. This simulation also uses the least material. The GABM performs best for the longest webs and hence uses the highest amount of material despite it having the lowest number of webs. In general, layouts with a target web length of 700 mm lead to the worst performance for all the methods. Though the authors originally believed that longer webs would improve system performance, this does not prove to be the case. Instead, it is the negotiation between the web length and the adaptability to the force flow in the slab that produces the best results.

In summary, the two agent-based methods, the SABM and the GABM, perform better than the SEQ. The added complexity in the design method leads to less deformation and less utilization in the main and secondary fiber directions. Integrating more structural design principles in the methods causes better structural performance. The influence on fabrication of the more geometrically complex layouts remains to be investigated. Furthermore, for general statements about the performance of one specific set of behaviors over another, it is necessary to investigate further case studies.

Conclusions and outlook

Three different methods for the placement of the reinforcement in a novel hollow box system have been investigated: a sequential method, a geometrically informed agent-based method, and a structurally informed agent-based method.

Sequential methods are generally simpler and faster to setup, and at least in theory allow for greater designer influence than ABMs because the latter rely on emergence. However, though not evaluated in this study, ABMs can also allow for real time designer interaction, as demonstrated by Groenewolt.²⁴ A similar user-influenced ABM could provide the benefits of emergence and user control, and therefore improve structural performance. That the geometrically informed agent-based method (GABM), itself a combination of a sequential and an agent-based method, led to the lowest deflection for the chosen case study supports this assumption. This decreased deflection came because the GABM negotiated the transition zones between column crowns and field webs better than either of the other two methods. The informed placement of webs in the transition zone allowed for a better force transfer in the slab. A user manually influencing the results of a simulation like the SABM to do the same may be able to produce similarly desirable results.

The addition and comparison of additional agent behaviors may yet result in even more performative reinforcement layouts. Such additional behaviors should be developed and tested. Simple ABMs such as the ones described in this paper could also be improved through the application of optimization methods to their input parameters. As a first study, more granular changes to target web numbers, target web length, and even the size and shape of the column crown could be optimized. As a further study, the structural simulation results of each iteration of the agent system could be fed back into the following iteration to drive the web positions. As a final study, the results of the Main Structural Analysis could be fed back into the web placement algorithm. As a result of these studies, the parameters of the web agents could be more finely controlled, and their movement could be influenced by previous results of the agent system as well as by the subdivision of the slab.

Further studies could also compare structurally informed agent-based methods to other established methods for slab optimization, including topological optimization. Agent-based methods allow for the integration of disparate requirements into the system and the negotiation of conflicting design goals. Future studies could investigate behaviors that respond to additional architectural considerations, including building physics or MEP. Different material buildups for the slab, perhaps with additional LVL, hardwood, or hardwood LVL may be able to address certain shortcoming of the timber building system.

Methods such as those described in this paper could be used for the design of any timber box floor. However, they are more applicable to new-build multi-story buildings because the building system they were developed for rests on the columns below. Further development of the building system for compatibility with existing building systems would increase its applicability for refurbishment. Research into the refurbishment of large-scale existing buildings is important because it a proven way of extending a building’s life span, one of the goals of this research.

The reinforcement arrangements produced by the methods tested differed in how integrated structural information was and how geometrically complex they were. Increased structural integration in the design can allow for material reduction but may lead to increased fabrication complexity for a given building design. An analysis that incorporates fabrication benchmarks, such as production time, would better identify the applicability of informed reinforcement methods for real world construction. Further investigations would be better able to gauge the applicability of reinforcement arrangement methods such as these by comparing them across multiple floor plans, especially those of existing timber and concrete buildings. Regardless, that a Testing Setup such as this could be made structurally viable using a timber building system, and improved through the informed placement of reinforcement, already shows the potential for the increased application of timber in the built environment.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Deutsche Forschungsgemeinschaft (EXC 2120/01 -- 390831618).

ORCID iDs

Luis Orozco

Anna Krtschi

Lior Skoury

Jan Knippers

Achim Menges

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