Optimal Design of Prestressed Concrete Hollow Core Slabs PDF

Advances in Engineering Software 122 (2018) 81–92 Contents lists available at ScienceDirect Advances in Engineering Software...

Advances in Engineering Software 122 (2018) 81–92 Contents lists available at ScienceDirect Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft Research paper Optimal design of prestressed concrete hollow core slabs taking into account T its fire resistance ⁎ V. Albero, H. Saura, A. Hospitaler, J.M. Montalvà, Manuel L. Romero Instituto de Ciencia y Tecnología del Hormigón (ICITECH), Universitat Politècnica de València, Spain A R T I C LE I N FO A B S T R A C T Keywords: Prestressed hollow core slabs are a concrete element widely used as construction floor product, which manu- Structural optimization facturing process has greatly been improved in recent years. Several research studies focused on hollow core slab Fire resistance performance, mainly related to its fire behavior, have provided new limit states to be assessed throughout its life Hollow core slab cycle. Therefore, the hollow core slab design needs to be reviewed to allow for these improvements, a process Prestressed concrete which may involve changes to its geometry. In order to deal with this review, modern computational optimi- zation techniques offer an alternative approach to traditional structural product design procedure, mainly based on the engineer's prior experience. This paper proposes a hollow core slab model (including variables and constraints) to develop heuristic search algorithms, such as simulated annealing, in order to find the most economical slab design including the fire resistant constraint and taking into account all available manufacturing technologies. The optimal designs ob- tained by this process save up to 20% in cross-section area compared with common circular void designs from market, which is taken as a comparison pattern. The results show that traditional designs are deficient when the fire resistant constraint is considered, so that precast manufacturers and machinery designers should use opti- mization techniques to modify their hollow core slab geometry. 1. Introduction Fig. 1a. These tendons may change in position, quantity and cross- section. The concrete casting is continuous and three specific proce- Precast prestressed hollow core concrete slabs are widely used to dures are available in the market: Slipformer, Extruder and Flowformer. make floors in both residential and non-residential buildings and pro- The final geometry of the slab hollows is defined by a specific and in- vide efficient structural behavior for long spans and high loads. terchangeable part of the casting machine called the finishing mold, see Furthermore, the estimated total stock of hollow core floors currently Fig. 1b. This core mold can be replaced to manufacture different cross- installed in Europe is 1.000 million square meters. section designs or slab heights. The slab holes shape, quantity and Since the early seventies, hollow core slabs have been thoroughly position may change using the same machine but different finishing studied in research campaigns, in the USA [2,3] and Europe and mold. Therefore, new proposals of hollow core slab designs may pro- many aspects of the product have been intensively tested, including its duce changes in this manufacturing finishing mold. fire resistance [5–7]. Specific standards have been issued related to its Traditionally, engineers have relied on their experience to design structural behavior, including the PCI Manual for the design of hollow structural products so that the same structural response could be core slabs , FIB Special design considerations for precast prestressed achieved through diverse designs using different resources. hollow core floors and the European product standard EN 1168 The current design of hollow core slabs is a forward design, which. involves three steps. The first one is related to the machinery designer. This type of slab is now widely used and its manufacturing process In this step aspects as manufacturing procedure constraints or the fresh has been greatly improved to include strictly controlled design para- concrete consistency are used to design the finishing mold. Thus, many meters. Specifically, hollow core slabs are manufactured in highly in- hollow core geometry designs appear in the market depending on the dustrialized precast factories. The manufacturing process is to run the manufacturing process. This first step of the current hollow core slab machines on steel beds up to 200 m long, equipped with stressing design does not take into account any aspect of the final behavior of the abutments where tendons are initially positioned and prestressed, see slab in floors. ⁎ Corresponding author. E-mail address: [email protected] (M.L. Romero). https://doi.org/10.1016/j.advengsoft.2018.05.001 Received 13 March 2018; Received in revised form 24 April 2018; Accepted 6 May 2018 0965-9978/ © 2018 Elsevier Ltd. All rights reserved. V. Albero et al. Advances in Engineering Software 122 (2018) 81–92 Fig. 1. Hollow core slab, manufacturing process. (Manufacturer pictures: HORVITEN S.L & HERMO S.L, Spain). In the second step, the hollow core slab manufacturers cast the process as geometrical conditions from production machinery or the concrete slab with the available machinery; however they only can definition of stress limits to avoid undesirable crack during slab pro- change the concrete strength, the prestressed tendons position, quantity duction and transport. and tension. The slab geometry comes from the finishing mold, which is Additionally, due to catastrophic events like the Harbour Edge col- established in advance. The slab manufacturers normally define some lapse in Rotterdam , hollow core slabs have now been greatly im- series of tendon configurations to offer different solutions reaching a proved as regarding its fire resistance. specific range of bending capacity. In 1999 premature collapse of hollow core slabs was observed in Finally, in the third step the floor project designer shall choose the DIFT tests , due to high temperatures, which was not deducible proper configuration which provides enough mechanical capacity to from the standards. This premature collapse was later analyzed by van the floor. Acker and Fontana & Borgogno. These studies identified a strain The forward design described above is clearly uncoupled. The me- compatibility field in the cross-section due to a nonlinear temperature chanical requirements of the floor have no influence in the real hollow field, which caused stress in the webs. The premature collapse of the core slab geometry. Therefore, the current hollow core slab design slab in the fire tests could be explained by this undesirable web stress. procedure is strongly highly inefficient. However, subsequent studies [23,24] concluded that this problem did Alternatively, current computational optimization techniques offer not occur in floors with constrained longitudinal expansion. Advanced another approach to deal with this problem. These techniques can finite element models have now been developed to reproduce this be- manage all constraints along with the hollow core slab life cycle. havior [25,26]. Therefore, a coupled design can be defined to provide optimal and ef- Traditionally, fire resistance requirements are achieved for these ficient solutions. slabs by adding prestressing steel and/or increasing the steel tendon A wide range of structural concrete products, such as frames concrete cover while maintaining the cross-section geometry. Choosing [11,12], bridge piers , precast road bridges , road vaults , between these options is not always obvious. Besides, it has been ob- retaining walls and foundations have been studied using this served that these approaches may provide an over capacity of bending approach. Economic cost has classically been defined as the standard moment resistance at room temperature and therefore, inefficient de- optimization criterion, although other factors like weight, environ- signs. mental or constructability indicators have also been used. This paper therefore proposes a new approach using computational The first study on precast hollow core floor optimization was carried techniques and taking advantage of its controlled manufacturing pro- out by Koskisto and Ellingwood in 1997. The objective function cess. A complete definition of the hollow core slab model is presented, defined was economic cost, including the cost of failure, and focused on including design parameters, variables, and constraints, and a cost- the bending capacity at room temperature as the performance criterion based objective function. This works includes fire resistance as a con- only. This work used only four design variables and concluded that the straint and the models available to assess hollow core slabs at high slab height and reinforcement increment was an efficient way of in- temperatures. All of these constraints are added together taking into creasing the maximum slab span. In contrast, that concrete strength and account all possible interactions. reinforcement eccentricity were found to not have significant effect. Different optimal designs for 60 and 120 min of thermal exposure to Noorzaei et al. studied the optimal design of particular geo- a standard time-temperature fire ISO curve. These standard fire ex- metries of hollow core slabs as circular voids, normally produced by posure times are commonly used for residential and non-residential extrusion (spiroll manufacturing process) in Northern Europe and buildings. The optimal designs obtained are compared to determine the America. This work obtained the optimal depth of this slab type using influence of fire resistance in the optimal cross-section geometry. All bending moment as the principal constraint. the limit states were considered during the complete hollow core slab More recent studies, such as Sgambi et al. , focused on the slab's life cycle, including the fire event. web design defined by five geometric variables, with weight as the objective function. This work only took into account the spalling stress 2. Optimization problem definition constraint to avoid cracking in the web during the manufacturing process. Specifically, this constraint is related to the transient design 2.1. Problem definition situation of prestress transfer. Surprisingly, few studies have dealt with the optimal design of In order to solve the disadvantages of the current uncoupled and hollow core slabs and none has considered fire resistance in the opti- inefficient design procedure of hollow core slabs, described previously, mization process and additional design constraints from manufacturing a new approach is presented. This is based on optimization techniques 82 V. Albero et al. Advances in Engineering Software 122 (2018) 81–92 which allow to find an optimal design subject to all life cycle me- chanical and geometrical constraints. The problem of hollow core slab optimization involves a set of variables x = {xi}, related to cross-section geometry and materials, and an economic optimization to minimize the objective function: min f (x) (1) This optimization is subjected to inequality and equality constraints: gj (x) ≤ 0j = 1,...,n hj (x) = 0j = 1,...,m (2) xi ∈ Di where Di is the discrete variable domain of variable i. Di = (di1,di2,...,di, qi ) i = 1,...,qi. Discrete Variable Design Problem (DVDP) The set of all values (x) satisfying the constraints defines the feasible region A = {x: gj (x) ≤ 0, hj (x) = 0, ∀ j} and any configuration x ∈ A is a feasible solution of the problem. The problem parameter array (p) contains all the remaining data necessary to compute a particular hollow core slab. These parameters are all magnitudes taken as fixed data throughout the optimization problem implementation, and their influence in the optimal design is not taken into account. The aim of this discrete variable nonlinear optimization problem is to find a feasible solution with the lowest value of the objective func- tion. Fig. 2. Void model (10 variables). 2.2. Design variables and solution space The design variables (x) are the magnitudes subject to variation The void curves were modeled as superellipse curves, where a and b during the optimization procedure. Their influence in the optimal de- are the semi-diameters of the curve: sign is studied and the best values are pointed out. x k y k The main objective of this work is to include the influence of fire ⎛ ⎞ +⎛ ⎞ =1 ⎝a⎠ ⎝b⎠ (3) resistance in the hollow core slab optimal design procedure, specifically focused on its geometry, so that the prestressing steel configuration and The superellipse is adapted to concave or convex curves through the void geometry were defined as the main variables. curve degree (k), allowing different configurations, as shown in Fig. 3. Three variable types were defined as follows (see Table 1): In order to complete the slab core definition an additional variable was established (bw) to determine the web thickness. This variable is - Cross-section geometry variables. closely related to the shear capacity of the slab and is mainly con- - Reinforcement geometry variables. strained by the manufacturing technology and maximum aggregate - Mechanical or material variables. dimension of the fresh concrete. In the reinforcement geometry variables, some arrays were defined In the cross-section geometry variables the void is defined by 10 to fix the reinforcement quantity and position. In particular, the yi array variables, see Fig. 2 and Table 1. The void model introduced, which is was defined to indicate the row coordinate from the lower edge and ϕi related to the finishing mold, is able to reproduce any hollow config- defines the tendon diameter in each level. The amount of reinforcement uration (Fig. 3) and can be adapted to any manufacturing process (i.e. is determined through the [n]ij matrix, where the tendon quantity is extruder, slipformer or flowformer). stored. Each matrix column refers to the slab web in which the tendon is placed. Additionally, [n]ij must be defined carefully in order to set Table 1 symmetric prestressing force arrangements around the vertical axis. Model variables. Three mechanical/material variables were defined: concrete Cross-section geometry strength fck, prestressing steel strength fpk and initial prestressing stress h Hollow core slab height c1 Bottom void height σ0. b Hollow core slab width d Bottom void plateau The number of design variables considered in the solution space h0 Concrete topping height a2 Top void width depends on the reinforcement framework and the slab void number. n Number of voids a1 Bottom void width h2 Top flange thickness d2 Top curve center ecc. Working with seven reinforcement rows and five slab voids, the number h1 Bottom flange thickness k1 Bottom curve degree of design variables reaches 48. The set of value combinations for all hw Web height k2 Top curve degree variables may be defined as the solution space, which extends to 1025 c2 Top void height bw Web thickness due to combinatorial explosion; thus it is impossible to identify the (q) Lateral joint geometry global optimum in its entirety. Reinforcement geometry yi Reinforcement position array nij Reinforcement matrix ϕi Reinforcement diameter 2.3. Objective function array Material and mechanics The objective function considered was the economic cost of the fck Concrete strength σp,max Maximum stress applied to the tendon hollow core slab, defined as follows: fp,k Tensile strength f (x) = cc ·Ac (x) + cs As (x) (4) 83 V. Albero et al. Advances in Engineering Software 122 (2018) 81–92 Fig. 3. Void model adaptability. 2.4. Constraints Table 2 Unit prices considered. 2.4.1. Implicit constraints Unit Description Cost (€) Any condition that the problem solution must satisfy in order to become a feasible solution was defined as a problem constraint. kg Steel Y1860 for reinforcement 0.674 m3 Concrete HP-45 281.57 The hollow core slab manufacturing process contains implicit con- straints that were taken into account in the model. Although they are not specifically described here, these include parallel top and bottom The cost function depends on concrete cross-section area (Ac), the edge, equal shape for all slab voids, equal width of slab webs and amount of prestressing steel (As), the concrete (cc) and steel (cs) unit symmetry of reinforcement arrangement around the vertical axis. The price. These unit prices must include the different costs involved in the additional geometrical and structural constraints (not included im- production process, e.g. transport, etc., as provided by de Castilho, et al. plicitly) are shown below.. The unit prices obtained from Spanish hollow core manufacturers are shown in Table 2. 2.4.2. Explicit constraints This cost objective function is the best index to provide information 2.4.2.1. Geometrical constraints. The main explicit geometrical on the resources used in slab production. Other indexes (i.e. environ- constraints are related to cross-section geometrical variables, defining mental indicators) could also be used; however, these are outside the the limits to web and flange thickness. They can be obtained from the scope of this work. product standard EN 1168 or EN 1992-1-1. Some other constraints from EN 1168 on reinforcement position are established to complete the geometrical conditions, which are related to maximum and minimum distance between tendons, which are useful 84 V. Albero et al. Advances in Engineering Software 122 (2018) 81–92 Table 3 the new slab support conditions and the reinforcement prestress losses. Hollow core slab design situations during life cycle. All these stress states are checked through a fiber model developed Transient design situations specifically for the analysis of hollow core slabs in this work, see Fig. 5. SLS during transfer of prestress Bending and shear ULS, deformation and crack control SLS were ULS during transfer of prestress also included in the analysis (persistent design situations, see Table 3). SLS during floor erection Persistent design situation ULS Bending 2.4.2.3. Fire resistance. Mechanical constraint. This work includes an ULS Shear additional structural constraint related to the accidental fire design SLS Deflection SLS Crack control situation, which is one of its important contributions. This constraint of Accidental design situation slab behavior in the fire situation can be used to find the best slab shape ULS Bending in fire for fire resistance. ULS Shear in fire The structural fire design is one of the main topics of the field known as fire safety engineering, which follows the essential require- ments of safety in case of fire. Apart from the load-bearing capacity of to avoid undesirable manufacturing cracks through the appropriate structural members in fire, these requirements also include the limita- reinforcement distribution in the slab. tion of the spread of fire and smoke within the construction and to There are also some geometrical constraints, defined as equality neighboring, the safety egress of occupants and the safety intervention relations, called hard-constraints (h (x) = 0), which are necessary to link of the rescue teams. A fully analytical procedure for hollow core slab variables and parameters. Two void hard-constraints are defined: structural fire design is available to assess its performance in fire. This h1 (x) = 0 h = h1 + c1 + h w + c2 + h2 (5) procedure requires three elements: h2 (x) = 0 2a2 = 2a1 + d (6) 1. Fire model Another was defined to link horizontal void variables with web 2. Thermal model thickness (bw) and void number (n): 3. Mechanical model at elevated temperatures. h3 (x) = 0 b = (n + 1) bw + 2na2 + 2q6 (7) In this work the standard time-temperature curve ISO834 from EN 1363 was adopted as the fire model. This curve reduces the fire 2.4.2.2. Mechanical constraints at room temperature. Apart from event to a thermal boundary condition based on gas-phase temperature previous geometrical constraints, structural constraints were defined and set heat transfer coefficients recommended by EN 1991-1-2. to check all the design situations throughout the slab life cycle (see The second element in this procedure is a thermal model to evaluate Table 3 and Fig. 4). In this approach, the optimal design obtained will the temperatures throughout the cross-section. EN 1168 Annex G be useful for all design situations, taking all the limit states into account provides a specific simplified thermal model for hollow core slabs, (Ultimate Limit States ULS and Serviceability Limit States SLS). For which reduces computation times. instance, ULS and SLS during transient design situations like transfer of The model defines level a50% as the cross-section depth in which presstress are included. Besides, ULS during accidental design situation concrete width equals void width. It assumes a uniaxial temperature like the fire event is also taken into account. The model used to evaluate field, where the temperature of levels under a50% is obtained from EN each limit state is provided by EN 1992-1-1 and EN 1168. 1992-1-2 Fig. A.2 for solid slabs. Additionally, a linear interpola- Firstly, the transient design situation during transfer of prestress tion is used between a50% and the unexposed upper edge at 160 °C. must be evaluated. In this situation the concrete still has not achieved A cross-section temperature field obtained following the thermal its full strength and stresses must be kept lower than the limits re- model explained above is shown in Fig. 6 for a hollow core slab exposed commended by the standards. The model for concrete strength devel- to an ISO834 curve for 60 and 120 min (R60-R120). opment is obtained from EN 1992-1-1. Some models obtained from The third element is related to the mechanical model at high tem- the regulations [30,10] are used to avoid spalling, splitting and peratures. Bending and shear models may be found in standards bursting, which may create undesirable cracks during production. Ad- [10,33]. In this work the 500 °C isotherm method from EN 1992-1-2 ditional transient design situations are evaluated for the other con- Annex B was used for assessing bending in fires. This is a simplified struction steps, like transport or floor construction. The stress limits are calculation method applicable to a standard fire exposure based on a checked again, taking into account the higher strength of the concrete, reduction of cross-section size. Concrete at temperatures in excess of Fig. 4. Hollow core slab lifecycle. 85 V. Albero et al. Advances in Engineering Software 122 (2018) 81–92 Fig. 5. Fiber model for hollow core stress analysis. 500 °C is assumed not to contribute to load bearing, while residual account. concrete (temperatures lower than 500 °C) retains its full strength and Therefore, in the present work, in order to verify all the limits states elasticity. On the other hand, the reinforcement strength is reduced due together, they were tied through two conditions: to temperature, according to EN 1992-1-2 Clause 4.2.4.3, as- suming that the prestressing steel temperature matches the surrounding - Pinned-Pinned boundary condition. concrete. - Uniform load distribution. EN 1168 provides a specific model for hollow core slabs for shear behavior at high temperatures: Using these conditions and the slab span and load amplitude, which are established as parameters in Section 4, design values for all bending VRd, c, fi = [Cθ.1 + αk Cθ.2] bw d (8) and shear design situations (MEd,VEd,etc.) can be obtained coordinately. Where Cθ.1 is a coefficient that takes into account high-temperature Fig. 7b shows the safety factor results for a hollow core design following concrete stress, Cθ.2 is related to reinforcement anchorage in fires, this new approach. As all reach values higher than 1.00, all the limit and d is the effective depth of the slab. states are verified, in contrast with results shown in Fig. 7a. It is also shown in Fig. 7b that ULS Bending, SLS Deflection and ULS Bending in 2.4.3. Tie of mechanical constraints fires obtain a safety factor value closest to 1.00, which indicates that the All the structural constraints, defined in 2.4.2, to verify hollow core most important constraints are related to these limit states, therefore slab behavior throughout its lifecycle are tied to ensure that the optimal they have the highest influence in the hollow core slab design. design could be used for any design situation. Therefore, they are not independent and are linked through the load pattern and boundary 3. Applied heuristic search methods conditions established in the project. Traditionally, the optimization of hollow core slabs has focused on Combinatorial optimization problems like the one described herein the verification of the main structural constraint due to bending can require a huge possible combinations, as has been pointed out in [18,19]. However, as explained above, multiple constraints, classified Section 2.2 (e.g. a 1025 solution space). The analysis of the entire so- into limit states, have now been added to the standards. Through the lution space is a difficult task, even with the large computational ca- traditional approach, optimal design could be reached ensuring a safety pacity now available. However, approximation algorithms (i.e. heur- factor for ULS bending higher than 1.00. Nevertheless, it does not take istic methods) are available to explore the most promising space into account other design situations such as ULS shear, SLS deflection or solution areas and can reach an approximate solution in an acceptable accidental fires. Fig. 7a shows the safety factor for all limit states of an time. The simulated annealing algorithm, proposed by Kirkpatrick et al. optimal hollow core design obtained through the methodology de- and Cerny was selected for this analysis. This algorithm is scribed in this work when only bending approach is taken into account. based on randomization techniques and also includes aspects related to It can be seen that while ULS bending shows a safety factor value 1.00, iterative improvement. The simulated annealing application focuses on others such as ULS bending and shear in fire result to unsafe behavior a small perturbation to generate the transition from one configuration with value 0.27 and 0.47 respectively. It shows that alternative ap- to another. This perturbation defines a neighborhood for each config- proaches are needed where all design situations would be taken into uration, consisting of all the configurations that can be reached from 250 R60 R120 200 150 (mm) 100 50 a50% level 0 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 Tª (ºC) Fig. 6. Hollow core slab EN 1168 simplified thermal model. 86 V. Albero et al. Advances in Engineering Software 122 (2018) 81–92 Fig. 7. Safety factor: a) Untied optimization; b) Tied optimization. ( p = min 1; e −ΔE T ) (9) Where p denotes the probabilistic acceptance between (0, 1) in- terval; ΔE is the objective function variation due to the perturbation and T is a control parameter called “temperature”. It is worth noting that this control parameter has no relation with the fire resistance constraint because it has no physical meaning. In the first steps of the application, “temperature” is initiated with a high value providing high probabil- istic acceptance for solutions which do not improve the objective function. The “temperature” is gradually reduced throughout the ex- ecution and the probabilistic acceptance also diminishes. In the final steps only the best solutions are accepted and local optimal solutions are avoided. The simulated annealing implementation requires a cooling schedule to guide the control parameter “temperature” (van Laarhoven P.J.M and Aarts E.H.L ). This cooling schedule should be specifically calibrated for each problem. In the present analysis the initial “temperature” was set to provide an initial probabilistic accep- Fig. 8. Simulated annealing performance. tance of 90% and was reduced as follows, established in steps of 10.000 perturbations (also called the “Markov chain”): the previous one, so that it can be classified as a search algorithm. Ti + 1 = 0.95Ti (10) However, unlike other search techniques, simulated annealing in- corporates a probabilistic acceptance criterion, i.e. a perturbation may The algorithm stops when 50.000 perturbations have been analyzed be accepted even when the feasible solution reached does not improve consecutively without any improvement. An example of the Simulated the objective function. The probabilistic acceptance criterion is defined Annealing performance proposed in this work can be observed in Fig. 8. following the Boltzmann distribution: The computational cost to achieve an optimal solution, through the optimization model described in this work, reaches 6 h using a com- puter core of 2.5 GHz. 87 V. Albero et al. Advances in Engineering Software 122 (2018) 81–92 4. Optimization results of hollow core slabs. In fact, circular voids are very common in the extruder casting procedure. The optimization algorithm described above was applied to the Finally, the algorithm was submitted without any specific void developed hollow core model. The optimal design was focused on a geometry. Two different fire resistant times were defined, in order to 25 cm high and 120 cm of width hollow core slab with 5 voids. This compare their optimal designs. Fig. 10c shows the optimal design ob- study is focused on the hollow core slab geometry, thus the influence of tained at 60 min of fire exposure (R60), which is usual in residential concrete topping (h0 = 0) was neglected. buildings. Fig. 10d shows the optimal design for 120 min of fire re- The boundary and load conditions were established as follows: sistance (R120), which can be used for industrial and commercial floors. - Pinned-Pinned boundary conditions. Significant differences were observed between the optimal designs - Total span: 8 m. obtained (see Fig. 10c and d). While the upper void edge design did not - Dead load: G = 2 kN/m2. Live load: Q = 5 kN/m2. show appreciable differences, the lower side obtained by each run was quite different. The R60 analysis reached the usual concave curvature Additional conditions were defined by the remaining structural in the void lower edge. However, for a higher fire resistance time constraints. For concrete building interiors with moderate humidity or (R120), convex curvature was the most economical geometry to deal external concrete sheltered from rain, the XC3 of the exposure class with all the constraints, which indicates that this curvature change is from EN 1992-1-1 was set for SLS crack control. The following due to the fire constraint. Improved hollow core fire resistance can be deflection limit (wmax ) for SLS deflection design situation was defined obtained by separating the rebars from the lower fire-exposed surface, according to total span (L): although the minimum concrete cover between the lower and void edges should be maintained. When the rebars are moved upwards to- wmax = min { 250L ; 0.01 + 500L } (11) wards the web in order to increase fire resistance, the convex lower void edge appears to be the most economical geometry with the As we focus on the optimal design of void geometry, the variables minimum amount of concrete. related to material properties, lateral joint geometry and reinforcement Table 4 compares the numerical results of the optimal designs with system were fixed as parameters. this market pattern. A total saving of 24% in terms of cross-section area was achieved in R60 optimal design, while the R120 optimal design - Lateral joint geometry (q) is defined through 7 parameters (see reduced cross-section by 19%. This reduction is related to the economic Fig. 9). saving of the optimal designs and shows that an optimal hollow core slab and considerable savings can be achieved by the proposed model q = {qi} = {40, 35, 137.5, 15, 22.5, 33.5, 12}mm (12) and algorithm. A 5% difference in cross-sectional area was observed when the fire constraint was increased from R60 to R120. - Concrete cylinder strength: fck = 45 MPa. 5. Thermal model uncertainty. Local search improvement - Maximum reinforcing steel strength: fpk = 1860 MPa. - Initial prestressing stress: σ0 = 1324 MPa. One of the main assumptions of this design procedure is related to - The reinforcement arrangement is placed in 7 rows (i = 7). the thermal model that evaluates the temperatures within the hollow - Reinforcement diameter for all rows ϕi = 5 mm (Y 1860 C). core slab cross-section during fires, described in Section 2.4.2c. The thermal model used in the optimal design procedure is a simplistic The optimization results are shown in Fig. 10 and Table 4. proposal provided by EN 1168 Annex G , which assumes a uniaxial Firstly, a hollow core design with circular voids, which worked as a temperature field.The temperature of levels under a50% is obtained market pattern, was computed (see Fig. 10a and b). This specific design from EN 1992-1-2 Fig. A.2 for solid slabs and a linear interpolation was verified using all constraints defined previously, including R60 and between level a50% and the unexposed upper edge at 160 °C is proposed R120 fire exposure, and may be considered as a common market design (see Fig. 6). This simplistic model allows obtaining a temperature field with low computational cost, which permits an efficient algorithm performance. The tied optimization showed that one of the main limit states, which define the obtained optimal design, is ULS Bending in fire (safety factor = 1.00) (see Fig. 7b). Since the thermal model is the most im- portant influence on ULS Bending in fire an analysis was carried out on this assumption. In order to assess the robustness of the solution achieved, the sim- plistic proposal from EN 1168 Annex G was compared with an advanced finite element model (FEM) previously developed by the authors. This finite element model deals with a nonlinear heat transfer problem, where EN 1991-1-2 governing parameters were adopted and EN 1992-1-2 recommended values were used for the concrete and steel temperature dependent thermal properties. This model was previously validated against an experimental campaign and showed a good fitting. Besides, it includes the cavity radiation model for the thermal evolution in holes. The optimal design obtained for R120 (Fig. 10d) was processed by the advanced thermal FE model and a new temperature field was ob- tained (see Figs. 11 and 12). Figs. 11 and 12 show that the simplistic proposal from EN 1168 Annex G provides higher temperatures for the lower edge of the cross- Fig. 9. Lateral joint model (7 parameters). section but underestimates the temperature of the cross-section fibers 88 V. Albero et al. Advances in Engineering Software 122 (2018) 81–92 Fig. 10. Optimal designs. Table 4 Fig. 11 also shows that the 160 °C prediction for the unexposed Optimization results. upper edge from the EN 1168 simplistic thermal model should by re- ID Circular voids Circular voids OPT. R60 OPT. R120 viewed for fire resistance times below 60 min. R60 (pattern) R120 (pattern) Consequently, the thermal model variation causes moderate differ- ences in bending resistance in standard fires. Therefore, the optimiza- Cross-section area 165,941 165,941 125,670 134,639 tion process was resubmitted using the advanced FE thermal model. A (mm2) (−24%) (−19%) This new procedure, with high computational cost, was applied to the Upper flange thickness 36 36 26 26 h2 (mm) previously obtained optimal R120 solution up to reach convergence to a Lower flange thickness 37 37 26 27 sub-optimal design. Specifically, it consist of a local search improve- h1 (mm) ment maintaining the field temperatures from FEM thermal model after Web thickness 36 36 36 36 each local search. The modified cross-section design obtained using the bw(mm) advanced FEM is shown in Fig. 14. This new optimal design maintains lower void convex curvature, but the web thickness of the bottom part is too small due to the tem- perature increment from the thermal model. The reinforcement con- placed 25–55 mm from the lower edge. This position, where the sim- crete cover on the bottom edge was increased and only one tendon was plistic proposal obtains lower temperatures, is where the main bending placed in each position for lower reinforcement eccentricity. Therefore, steel reinforcement is usually placed. Therefore, the simplistic thermal the reinforcement needed to be increased (+16.7%) in order to main- model from EN 1168 Annex G does not provide safer predictions than tain bending resistance. This new optimal design has a smaller concrete the advanced thermal model. Indeed, using the advanced FE thermal cross-section (−2.2%) but greater reinforcement (+16.7%) and the model the bending resistance for the optimal design in fire is reduced by economic cost is approximately the same as the original optimal design up to 20%. Fig. 13 shows that a temperature variation of approximately for R120 (see Table 5). 100 °C in the steel reinforcement considerably reduces tendon strength. 89 V. Albero et al. Advances in Engineering Software 122 (2018) 81–92 unit prices did not provide any change in the results obtained. The variation of the material price ratio (steel/concrete) was studied (see Table 6). The initial price ratio was 18.8 (price of steel over concrete) and was modified to 16.9 (−10%) and 20.7 (+10%). When the algo- rithm was resubmitted the optimal design did not change significantly. Indeed, the cross-section area underwent a variation of less than 1.3% and the main variables of flange and web thickness remained constant. It was therefore seen that the optimal design reached is sufficiently robust, even when material price ratio uncertainty is included. 6.2. Concrete strength The analysis of concrete strength variation (see Table 6) shows that the optimal designs obtained from the algorithm (using different con- crete grades) did not cause significant changes and the cross-section area variation was less than 1.63%. It can therefore be concluded that the optimal design did not un- dergo significant changes when concrete strength was modified. Fig. 11. Temperature comparison for optimal design R120. 6.3. Prestressing steel system 6. Sensitivity analysis One of the parameters fixed in the optimization carried out was related to the configuration of the reinforcement; following the After obtaining the optimal designs, a sensitivity analysis was car- common practice in Spain only wires (Y 1860 C ϕ5) were used for re- ried out to check their robustness by the uncertainty of the input as- inforcement. However, 3/8′′ strands (Y 1860 S7) may also be used sumptions of material unit prices, concrete strength and reinforcement when higher bending moments are required. In order to analyze the system. influence of the reinforcement system on slab geometry, the optimiza- tion process was resubmitted with 3/8′′ strands for the lower re- 6.1. Uncertainty of material prices inforcement. The results obtained showed that this reinforcement re- duced the concrete cross-section of the hollow core slab by only 1.5% The material prices are introduced in the objective function, which (see Table 6). was defined as a linear function. Consequently, linear variations in the Fig. 12. Temperature field comparison for optimal design R120. 90 V. Albero et al. Advances in Engineering Software 122 (2018) 81–92 Fig. 13. Tendon strength reduction due to thermal model variation. 7. Conclusions Table 5 Thermal model analysis. The results obtained show that an optimal design of precast hollow Thermal model analysis core slab could be achieved by the proposed model and heuristic al- gorithm. The optimal design obtained approximately 20% savings in Thermal Optimal design Ab (mm2) h2 (mm) h1 (mm) bw (mm) cross-sectional area over common market circular void hollow core model cross-section Ab Variation (mm2) slab. With regard to the main objective of this work, the fire resistant EN 1168 134,639 – 26 27 36 constraint was shown to have a strong influence on the optimal hollow Annex G core slab design, as significant differences were obtained for R60 and Advanced FE 131,612 −2.2% 26 28 36 model R120 fire resistance times. The slab void with lower edge convex Fig. 14. Optimal design R120 using FE thermal model. 91 V. Albero et al. Advances in Engineering Software 122 (2018) 81–92 Table 6 Concrete 2000:768–79. Sensitivity analysis. Jansze W, Van Acker A, Della Bella B, Klein-Holte R, Linström G, Py J-P et al. Structural behavior of prestressed concrete hollow core floors exposed to fire. 2014. Price ratio analysis p. 226. PCI. Manual for the Design of Hollow Core Slabs. Chicago: PCI, Precast Prestressed Price ratio (Cs Optimal design Ab (mm2) h2 (mm) h1 (mm) bw (mm) Concrete Institute; 1998. / Cc) cross-section Ab Variation CEB-FIB. Special design considerations for precast prestressed hollow core floors: (mm2) FIB Comission 6 - Prefabrication; 2000. EN-1168:2005+A3:2011. Precast concrete products. Hollow core slabs. Brussels, 18.8 (initial) 134,639 – 26 27 36 Belgium: CEN; 2011. Payá I, Yepes V, González-Vidosa F, Hospitaler A. Multiobjective optimization of 20.7 (+10%) 132,889 −1.3% 26 27 36 reinforced concrete building frames by simulated annealing. Comput. Aided Civ. 16.9 (−10%) 133,562 −0.8% 26 27 36 Infrastruct. Eng. 2008;23:575–90. Esfandiari MJ, Urgessa GS, Sheikholarefin S, Manshadi SHD. Optimum design of 3D Concrete strength analysis reinforced concrete frames using DMPSO algorithm. Adv. Eng. Software 2018;115:149–60. Concrete Optimal design Ab (mm2) h2 (mm) h1 (mm) bw (mm) Martínez FJ, González-Vidosa F, Hospitaler A, Yepes V. Multi-objective optimiza- strength fck cross-section Ab Variation tion design of bridge piers with hybrid heuristic algorithms. J. Zhejiang Univ. (mm2) 2012;13:420–32. 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