ADDITIONAL LOAD BEARING CAPACITY OF PRESTRESSED HOLLOW CORE SLABS DUE TO MEMBRANE ACTION
Thomas THIENPONT1, Wouter DE CORTE1, Robby CASPEELE1
1. Department of Structural Engineering and Building Materials, Ghent University, Ghent, Belgium Corresponding author email: [email protected]
Abstract
Due to their efficient design, economic production process and quick installation, prestressed concrete hollow core slabs are frequently used in all kinds of constructions. These prefabricated units are typically installed as single span elements, which are at the joints tied to the neighbouring elements with additional rebars. In a final step, the joints at the edges and between the elements are filled with grout, or a second layer of cast in-situ concrete is added on top of the elements. Although the execution of the joints and the stiffness of the surrounding structure provide a certain level of rigidity, hollow core slabs are typically designed as simply supported single span elements. However, the stiffness of the surrounding structure might facilitate compressive membrane action, which can increase the bearing capacity of the elements. This additional load bearing capacity, which is usually not taken into account, can be beneficial in accidental loading situations. This paper evaluates the additional load bearing capacity of prestressed concrete hollow core slabs due to compressive membrane action using two detailed 3D non-linear finite element models in Abaqus. The influence of the longitudinal restraint forces on the load bearing capacity of a single hollow core element is evaluated and compared to a simply supported configuration. The influence of the element geometry and span to height ratio on the additional load bearing capacity is investigated for both reinforced and prestressed hollow sections.
Keywords: Prestressed hollow core slab, membrane action, finite element analysis.
1. Introduction
When a reinforced concrete (RC) beam or slab is subjected to transverse loading, the cracking of the concrete causes a (upward) migration of the neutral axis (Rankin & Long, 1997). The strains at the tension face become considerably larger in magnitude compared to those on the compression face.
Therefore, the net tensile strain resulting at the slab mid-depth will cause the slab to expand, resulting in an outward displacement of the slab ends (Vecchio & Tang, 1990). Typically this outward expansion will be prevented, to some degree, by the lateral stiffness of the surrounding structure, and as a result, compressive membrane forces will be introduced in the loaded slab. Ockleston (1955), was one of the first to demonstrate that the axial restraint provided by surrounding members could significantly increase the load bearing capacity of concrete elements. Since then, compressive membrane action, and its beneficial effect on the load bearing capacity of RC elements has been thoroughly investigated by several researchers, both experimentally and theoretically (e.g. Park, 1964; Rankin & Long, 1997;
Taylor et al., 2001; Vessali, 2015).
However , with regard to compressive membrane action in concrete hollow core (HC) floor elements, one of the most frequently used construction floor products, there is only a very limited amount of research available in literature. Especially the influence of the longitudinal voids on the formation of compressive arches within the slab is unclear. Therefore, the aim of this paper is to quantify the additional load bearing capacity of prestressed concrete HC slabs due to compressive membrane action using a detailed 3D non-linear finite element model.
2. Numerical model
To evaluate the behaviour of hollow core concrete elements in bending, subjected to excessive vertical loading, two numerical models are developed using the general purpose non-linear finite element analysis package Abaqus (Abaqus, 2016). First a model for RC sections is developed (Model I). This model is validated against the analytical model by Rankin and Long (1997) to evaluate its ability to capture compressive membrane action. The validated numerical model is then used to investigate the influence of the web width on the load bearing capacity of longitudinally restrained RC slab sections. A second model, in which the reinforcing bars are replaced by prestressing strands, is then used to study the effect of longitudinal restraint on prestressed concrete (PC) slab sections and HC slabs (Model II).
2.1. Material properties
In the numerical evaluations, non-linear material models for both concrete and steel are used. A non- linear model for concrete is preferable, since the model aims to correctly capture the concrete behaviour under large deformations, and elastic material models cannot correctly simulate the stiffness loss of concrete elements during and after cracking (Okumus et al., 2012). The concrete properties are based on the concrete damaged plasticity (CDP) model as provided in the Abaqus material library. This model is appropriate for capturing both fundamental types of failure of concrete: crushing and cracking.
The compression behaviour is defined in accordance with Van Meirvenne et al. (2018), and is based on the guidelines of both the fib Model Code 2010 (2012) and AASHTO LRFD Bridge Design Specifications (2012). The stress-strain relation for concrete in uniaxial compression is depicted in Figure 1.a. The tensile behaviour is also modelled based on the fib Model Code 2010 (2012). Herein, in the uncracked state (Figure 2.b), linear elastic behaviour is assumed up until the ultimate tensile stress fctm is reached. After reaching the ultimate tensile strength, a bilinear concrete softening curve is used, which is expressed as a function of the crack width, as depicted in Figure 2.c. The area under the stress- crack opening curve is related to the fracture energy GF, which is calculated according to the fib Model Code 2010.
Figure 1. a) stress-strain model for short-term loading of concrete in uniaxial compression; b) stress-strain model for uncracked concrete in uniaxial tension ; c) stress-crack opening relation for concrete in uniaxial tension
The material model for the reinforcement bars in Model I assumes linear elastic behaviour with Young’s Modulus Es = 210 GPa, and perfect plastic behaviour with an ultimate strain εuk = 5.0%. For the prestressing strands in Model II, a bilinear stress-strain relation is selected. The model assumes linear elastic behaviour up to 0.9 fpk, and a second linear branch up to fpk = 1860 MPa, with an ultimate strain εuk = 3.5%.
The reinforcement steel bars in Model I and the prestressing strands in Model II are modelled using two-dimensional truss elements which are assumed to have perfect bond with the surrounding concrete.
In reality, the condition between the prestressing strands and the concrete is controlled by the adhesion, friction due to the twisted strand configuration and Poisson radial expansion of the strand (Okumus et al., 2012). Exact evaluation of these phenomena requires the use of more complex models, which significantly increases the computational cost, which can be prohibitive in some cases, and leads to convergence issues when modelling large deformations.
2.2. Test setup, mesh and boundary conditions
Sections are modelled considering those described in (Walraven & Merckx; 1983). Due to symmetry, only a single I-shaped section of the hollow core element is modelled, which reduces the computational cost significantly. For the concrete element, C3D8 hexagonal elements (8-node linear brick elements) are selected, whereas the steel is model using 1D beam elements. In order to numerically obtain the ultimate bearing capacity of the slab sections in both restrained and unrestrained conditions, a three point bend test setup is modelled, as depicted in Figure 2. In the unrestrained model, the slab section is simply supported by 25 mm wide support plates, and a concentrated force P is applied at the middle of the span. In the restrained configuration, the outward movement of the concrete element is (partially) prevented by a rigid plate. Rotation of these plates is fully restrained, and the outward movement is controlled using linear springs with varying stiffness K in order to simulate the lateral stiffness of an unknown surrounding structure.
Figure 2. Longitudinally restrained three point bend test, span 4 m. Longitudinal restraint by a linear spring with stiffness K.
3. Model I: Compressive membrane action in reinforced concrete HC sections
3.1. Validation of the solid slab model
In order to evaluate the validity of the RC model, the numerically obtained load bearing capacities for a restrained solid slab slice are compared with the results from analytical calculations based on the model by Rankin & Long (1997). Their model, which has been validated against experimental results, assumes the strength enhancement due to compressive membrane action in restrained slab strips to be the sum of the ultimate bending and arching actions.
Figure 3.a depicts the load displacement curves obtained from the numerical evaluations on a 100 mm × 250 mm RC solid slab section slice, as depicted in first geometry in Figure 4. Herein, the distance between the supports is 4 m and the load P is applied at mid-span. In all cases, after the cracking moment is reached, several cracks appear at the bottom surface, and typically one large crack forms at the bottom surface in the vicinity of the loading point. When the load is increased further, this crack becomes bigger and travels upwards to the compression zone. Finally, the ultimate capacity in compressive membrane action is defined as the point at which the axial restraint force reaches a maximum. Further increase of the load, results in inward movement of the support, a decreasing restraint force and subsequently tensile membrane action is activated. The possible additional load bearing capacity due to tensile membrane action is not considered as it is out of the scope of this paper.
Figure 3. a) Load displacement curves from numerical evaluations on restrained RC slabs; b) Ultimate capacity Pu for a solid RC slab with axial restraint, numerical model vs analytical model Rankin & Long.
0 10 20 30 40 50
0 10 20 30 40 50
Applied loadP[kN]
Displacement at mid-span [mm]
K = 1000 kN/mm K = 64 kN/mm K = 16 kN/mm K = 4 kN/mm
0 10 20 30 40 50
1 10 100 1000
Ultimate capacityPu[kN]
Axial stiffness K[kN/mm]
Rankin & Long Num (full slab)
It should be noted that in models with low axial restraint, the ultimate strain εuk in the reinforcement is reached before the ultimate load bearing capacity due to membrane action is reached. For example, for the case where the axial restrain stiffness K = 4 kN/mm, the 100 mm × 250 mm slab section fails when the strain in the reinforcement reaches its ultimate value εuk.
In Figure 3.b the obtained numerical values are compared with the analytical values of the ultimate load bearing capacity Pu for axial restraint stiffness values from 1 kN/mm up to 1000 kN/mm. From the graphs it can be concluded that a higher degree of restraint (stiffness K) leads to an increased ultimate load bearing capacity Pu. Furthermore, the results from the numerical evaluations show a similar trend to the values obtained from the analytical calculations.
3.2. Influence of voids on the ultimate capacity
Due to the presence of longitudinal voids and the corresponding limited available web width w over which a compression arch can develop, a reduced formation of compressive arching stresses and consequently a reduced bearing capacity Pu is expected. To quantify this reduced capacity Pu, hollow core slab, a set of RC slab sections with rectangular voids are numerically evaluated for a wide range of axial restraint stiffnesses. The geometries of the hollow slab sections are shown in Figure 4. The numerically obtained values for the ultimate load bearing capacities Pu of the tested slab sections are presented in Figure 5.a. From the results it can be concluded that RC slab sections in which only 50% of the width is available for facilitating the compressive arch formation (curve w50), the ultimate load bearing capacity only reduces less than 10% over the entire range of tested axial stiffnesses K. Similarly, when only 25%
of the width remains (curve w25) the ultimate capacity is reduced with less than 40%. The relative performance of the slab sections with rectangular voids is shown in Figure 5.b, based on the dimensionless parameter pu, which represents the relative ultimate capacity compared to a solid slab section (eq. 1).
𝑝 =𝑃 ,
𝑃 , (1)
Herein, Pu,hollow slab is the ultimate bearing capacity for a slab with rectangular voids and Pu,solid slab is the ultimate capacity for a solid slab. The results also show that for all geometries, the ultimate capacity reaches a minimum which corresponds to axial stiffness values between 10 and 100 kN/mm.
Figure 4. Geometries of tested RC slab sections.
Figure 5. a) Ultimate capacity Pu for a solid slab slice and hollow core RC slab sections for varying axial restraint stiffness. b) Relative additional ultimate capacity pu compared to solid RC slab, for varying web width.
0 10 20 30 40 50
1 10 100 1000
Ultimate capacityPu[kN]
Axial restraint stiffness K[kN/mm]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1 10 100 1000
Relative capacitypu[-]
Axial restrain stiffness K[kN/mm]
Full slab w50 w35 w30 w27.5 w25
4. Model II: Compressive membrane action in prestressed hollow core sections
To quantify the ultimate capacity of prestressed slab sections, Model I is adapted to accommodate prestressing strands instead of traditional rebars. The numerical calculation procedure in Model II consists of two steps. First, the prestress is applied to the concrete section, which causes the section to shorten and bend upwards. Subsequently, the load P is applied, and the ultimate capacity is evaluated.
Similar to the previous paragraph, two sections with a web width of 60 and 90 mm are compared to the solid section slice. The evaluated slab section geometries are depicted Figure 6. The shape and dimensions of the second geometry were taken from Walraven & Mercx (1983). The third geometry is a variation on the previous shape in which the web width w is adjusted to 130 mm instead of 60 mm.
Figure 6. Geometries of tested PC slab sections.
4.1. Influence of axial restraint on solid PC slabs
Similar to the tests for RC slab sections, first the restrained capacity for slab sections without voids is examined. Figure 7.a depicts the load displacement curves obtained from the numerical evaluations on the 260 mm × 300 mm PC solid slab section slice. Also here, a significant increase in load bearing capacity Pu is observed when the stiffness of the axial restraints increases. Moreover, the observed load- displacement behaviour is similar to the numerical evaluations on RC slabs. When the cracking moment is reached, a large crack forms in the vicinity of the load point. Finally the ultimate capacity of the slab sections is characterized by either the tensile reinforcement reaching its ultimate strain for low axial restraint configurations or the maximum compressive membrane force being reached (i.e. the calculations are terminated at that point and are not proceeding in the intermediate and tensile membrane stage.
Figure 7.b depicts the ultimate load bearing capacities from the numerical evaluations, which are again compared with the analytical model by Rankin & Long. From the graph it can be concluded that the theoretical model overestimates the ultimate capacity for low axial stiffness values K, while it underestimates the maximum capacities when very large axial restraint forces are present. This discrepancy for cases with large axial restraint K could be assigned to the positive effect caused by the upward bending of the concrete elements due to prestressing.
Figure 7: a) Load displacement curves from numerical evaluations on restrained PC slabs; b) Ultimate capacity Pu for a 300 mm thick solid PC slab with axial restraint, numerical model vs analytical model.
0 50 100 150 200
0 5 10 15 20 25
Applied loadP[kN]
Deflection at mid-span [mm]
K = 4000 kN/mm K = 1000 kN/mm K = 250 kN/mm K = 64 kN/mm
0 50 100 150 200
1 10 100 1000 10000
Ultimate capacityPu[kN]
Axial Stiffness K[kN/mm]
Rankin & Long Num (full slab)
4.2. Influence of axial restraint on PC hollow core slabs
To study the influence of the web width w on the ultimate bearing capacity Pu of prestressed HC slabs, the results from the calculations on the solid PC slab are compared with the results from two HC slab geometries depicted in Figure 6. The obtained results are similar to those for RC slab sections. As shown in Figure 8, prestressed HC slabs are able to gain significant additional load bearing capacity through compressive membrane action; both the load at which first cracking is observed Pcr and the ultimate capacity Pu increase with increasing axial restraint stiffness. Furthermore, from the tests on the two HC geometries, it can be concluded that a smaller web width corresponds to a slightly lower cracking capacity Pcr and ultimate capacity Pu.
Figure 8. a) Load at first crack Pcr for a solid PC slab and two HC slab sections with axial restraint;
b) Ultimate capacity Pu for a solid PC slab and two HC slab sections with axial restraint.
5. Conclusions
From the results obtained through the numerical modelling of restrained hollow core slab sections, several conclusions can be drawn.
Despite the presence of large voids in the longitudinal direction of hollow core slabs, a significant compressive membrane force can develop in axially restrained hollow core elements. Due to the axial compression both the cracking moment and the ultimate bearing capacity of the single span elements increases significantly.
For solid reinforced concrete slabs, a good correlation was found with the analytical model by Rankin and Long. However, their model slightly overestimates the additional load bearing capacity of prestressed solid slabs compared to the results obtained from the (more refined) numerical calculations, when the stiffness of the surrounding structure is small.
For both reinforced concrete and prestressed concrete hollow core slabs, the additional load bearing capacity reduces with reducing web width over the entire range of restraint stiffnesses. This is attributed to the fact that only the limited width of the webs is available to allow the formation of compressive arches. Nevertheless, the reduction is rather limited and not proportional to the reduction in web width.
Further research is needed to generalize the presented results for geometries with larger spans, as well as hollow core floor elements with a higher effective depth. Furthermore, the possible formation of tensile (catenary) membrane action was not considered in this paper and should be investigated further. Moreover, also the influence of a second phase topping layer on the additional load bearing capacity needs further investigation.
Acknowledgements
Thomas Thienpont is a Research Assistant of the FWO Research Foundation of Flanders. The authors wish to thank the FWO for the financial support on the research project “Performance-based analysis and design for enhancing the safety of prestressed concrete hollow-core slabs in case of fire and unforeseen events”.
0 20 40 60 80
1 10 100 1000 10000
Cracking loadPcr[kN]
Axial Stiffness K[kN/mm]
Full slab w60
w130 0
50 100 150 200
1 10 100 1000 10000
Ultimate capacityPu[kN]
Axial stiffness K[kN/mm]
full slab w60 w130
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