Introduction
Noise barriers (sound walls) are usually constructed along roadways to mitigate the airborne noise emanating from vehicles. The proposed innovative sound wall (poly-wall) comprises stay-in-place poly-blocks as formwork, rigid polyurethane foam (RPF) as structural cores and polyurea as a coating of the wall surfaces. The poly-blocks are stacked layer by layer on the wall footing and form a wall with cylindrical voids. A fast-curing liquid mixture of RPF is injected into poly-blocks voids to act as the main structural element. As shown in Fig. 1, the RPF cores are reinforced with steel rebars which connect the cores to the wall footing and increase the lateral stiffness of the entire wall system. A polyurea coating is sprayed on both sides of the wall in order to enhance the resistance of poly-blocks surfaces against abrasion, stone impact, weathering, fire development, chemicals and water penetration. Using these advanced materials and technology, the construction of poly-wall induces less obstruction for traffic as construction vehicles are not required over the course of construction. Implementation of modular poly-blocks and fast-setting RPF cores expedite construction of the proposed wall system in comparison with the conventional methods. Furthermore, the unit weight of poly-wall is almost half of a concrete or masonry wall which results in saving in foundation material. Noise-absorbing materials such as polyurethanes reduce the noise reflection more effectively and thereby perform better where the reflection is an issue.
Daee et al. conducted an extensive experimental program including tensile, compression, modulus of elasticity, Poisson’s ratio, flexural, shear, pull-out, creep and cyclic tests on RPF and determined its mechanical properties for the subject application [
1]. In that study, lateral resistance and cyclic behavior of six full-scale poly-walls were obtained and satisfactory performance of the poly-wall system was experimentally established.
This paper is focused on developing a 3D finite element (FE) model of poly-wall employing the commercial FE software, ABAQUS [
2]. The main objectives of this study was to (i) verify and expand the experimental results of the proposed wall system for various geometry and loading conditions (ii) establish a numerical behavior for structural application of RPF material (iii) improve simulation techniques of multi-martial 3D FE models.
For these purposes, the components of the poly-wall, including poly-block, RPF cores and steel rebars were individually modeled and their stress-strain relations were calibrated using the experimental data. All components were assembled and two poly-walls were simulated identical to the full-scale models and their boundary conditions. The lateral behavior of simulated wall was verified using the experimental results. The verified numerical models were then used to conduct an investigation on the behavior of poly-wall under lateral wind loading. Based on the results of finite element analysis (FEA), a simplified approach was proposed for the design of poly-wall.
Verification of finite element models
Steel rebar model
A benchmark FE analysis was carried out to calibrate the behavior of steel rebar model as a structural component of poly-wall. A solid 3D model of 16M rebar with
fy = 400 MPa, grade 400R of steel [
3], and 200 mm length was developed (Fig. 2). The top surface of the rebar was subjected to a tensile displacement and the connection of rebar to the concrete foundation was simulated using a tie constraint. The tie constraint well represents the connection of the rebar and foundation since, in practice, the slip of rebars relative to connection point is prevented by providing a sufficient development length. Therefore, the embedded length of the rebar inside the foundation can be disregarded in the analytical model.
For better discretization of geometry, 6-node linear triangular prism elements (element C3D6 of ABAQUS library) were employed to simulate the inner core of the rebar. The remaining parts of the model were simulated by employing linear hexahedral brick elements (element C3D8R of ABAQUS library). As a result, a uniform mesh across the entire part was produced using these two types of elements. This prevents the formation of an unreal stress concentration across the rebar, particularly along the rebar centerline. The aspect ratio of all elements was kept below 1:3.
All translational and rotational degrees of freedom at the bottom surface of the foundation were restrained to mimic the experimental support conditions. As the failure of concrete foundation was out of scope of this study, it was simulated using a linear concrete model with modulus of elasticity and Poisson’s ratio equal to 25 GPa and 0.2, respectively. The solid line in Fig. 3 displays the typical stress-strain behavior of rebar [
4], which was employed in the FE model.
A static displacement-control analysis was conducted and the stress-strain values were directly obtained from the elements that underwent the highest tensile stress in the vicinity of the connection point of two parts. The numerical results are plotted as a dashed line in Fig. 3, which demonstrates a good agreement with the input material properties, indicating accurate simulation of the rebar behavior.
Poly-block model
Poly-block is made of low-density (50 kg/m
3) closed-cell polyurethane foam. Under uniform pressure, light foams typically deform linearly until the yield point, which is followed by a plateau with small stress variation. The plateau region is irrecoverable and continues up to large strains at which point foam densification initiates. At this stage, cell walls of the foam progressively buckle and collapse under the applied pressure resulting in air escaping, stiffness hardening and ultimate failure of the foam structure [
5].
To determine the compressive behavior and particularly the yield strength of poly-blocks, three compression tests were carried out. In practice, the blocks are coated with polyurea which may slightly provide a circumferential confinement for the blocks. Hence, the blocks were coated with polyurea and then tested to account for possible effects of polyurea on the compressive behavior of poly-block. The load was applied at the rate of 0.5 kN/s through the moving crosshead of the testing machine and was distributed as a uniform pressure using steel plates. Since poly-blocks underwent very large deflections, true strain was calculated rather than engineering strain. The test was stopped at 50% of deflection of the block’s height as it was beyond the engineering strain range. Two poly-blocks before and after compression test are exhibited in Fig. 4.
The dashed lines in Fig. 5 illustrate the compressive stress versus true strain of poly-blocks acquired from the experiments. In all cases, a peak was observed at the yield point, and the average strength within the plastic region was about 0.5 MPa. The solid line in Fig. 5 is a bilinear representation of the average poly-blocks compressive behavior, which was calculated based on the test data. It can be observed that the test results are consistent with the typical behavior of low-density foams as stipulated in the literatures [
5].
The tensile behavior of poly-block was also required for the numerical model. Low-density foams typically demonstrate a linear tensile behavior with a relatively limited plastic region before failure occurs. For the range of 50-60 kg/m
3 density, previous studies have shown that the tensile modulus and strength (
Et,
st) are slightly greater than the compressive modulus and strength (
Ec,
sc) [
6,
7]. However, in the absence of tensile test data, it is common to assume them to be identical. A built-in material model for crushable foams has been developed in ABAQUS assuming a symmetric behavior in tension and compression. To define the evolution of the yield surface in this model, only uniaxial compression test data are required [
2].
A poly-block model was developed considering the same boundary conditions as the actual tests conducted. All degrees of freedom at the bottom surface of the blocks were restrained and a uniform static pressure was applied on the top surface. Linear hexahedral brick elements along with hourglass control and reduced integration points were employed in the FE model and the aspect ratio of the elements were kept under 1:3 (Fig. 6(a)). The obtained compressive test data (the solid line in Fig. 5) was properly input as the stress-strain relation of the block. Elastic properties of the foam were taken as
E = 8.33 MPa and
n = 0.3 [
6] and the plastic region was simulated using the crushable foam model available in ABAQUS. Large-deformation was activated in the software by implementing NLGEOM parameter.
The results of the finite element analysis are demonstrated in Fig. 5 with a square-dotted line. The numerical results are in a good agreement with the average compressive behavior of the blocks across the elastic and plastic regions, indicating satisfactory performance of the numerical model. The undeformed and deformed shapes of the poly-blocks are shown in Fig. 6, which are fairly similar to the test observations.
Rigid polyurethane foam modeling
Flexural analysis
To simulate the flexural behavior of RPF, a 3D finite element model of the flexural test set-up was developed employing 8-node hexahedral elements (C3D8R). The conducted test set-up was of the “third-point loading” flexural test [
1] which was similarly simulated in FE model. The beam geometry was modeled measuring 150 mm × 150 mm × 530 mm and the contact width of the load and support rollers were modeled equal to 5 mm. The material model used to simulate the behavior of RPF was selected based on the results of the uniaxial tests conducted on RPF specimens [
1] which revealed that the stress-strain relationship in compression is hyperelastic. Also, the uniaxial tests on small RPF specimens demonstrated that the material behavior is almost identical in both compression and tension [
1]. The dashed line in Fig. 7 is the average compressive behavior which was input in the FE model representing the material properties. The strain energy function was simulated using the third-order Ogden model (
N = 3) [
8] available in ABAQUS. The Ogden strain energy function fully matched the input RPF behavior and was also stable throughout the required strain range. The vertical load applied statically in 0.05 increments and the beam deformation was recorded corresponding to the measurement point of the flexural tests.
The dotted curve in Fig. 8 represents the FEA result, which well falls within the measured tests data, confirming the accuracy of the numerical model. Figure 9 demonstrates the FE model of the RPF flexural test depicting the stress contour of the beam under the applied load. The computational maximum compressive and tensile stresses were found to be+ 8.9 MPa and -10.9 MPa, respectively. These values are consistent with modulus of rupture obtained from the experiments [
1].
Uniaxial analysis
A benchmark FE analysis was performed on a cylindrical model of RPF to ensure that the material model is suitable for uniaxial loadings. Hence, a 3D FE model with the same geometric dimensions of the compressive samples (140 mm × 280 mm) was simulated as shown in Fig. 10(a). Material properties, type and aspect ratio of elements were all selected identical to the beam model used in the flexural model. The cylindrical model was subjected to 20 MPa uniform compression on the top surface and the translational degrees of freedom on the opposite side were restrained. Load-controlled static analysis was performed and strain was recorded at each load step.
The deformed shape of the model (illustrated in Fig. 10(b)) indicates a uniform stress throughout the sample excluding the loaded and constrained faces as result of stress concentration. The compressive response of RPF and the results of the FEA are exhibited in Fig. 11. A fairly good match between the two set of results can be observed and the maximum error of the numerical solution due to the approximation involved in Ogden model was found to be less than 6.4%. Since the main focus of this research is on the applicability and design of poly-wall, the fracture of RPF was not considered in the numerical analysis.
Full-scale wall modeling
The lateral resistance of the poly-wall model was experimentally determined for a single configuration and two types of reinforcement [
1]. To further our understanding of the performance of full-scale poly-walls, FE analysis was undertaken to investigate the response of the wall with different configurations under various loading conditions. Hence, a 3D FE model that identical to the tested poly-walls was developed.
Initially, several preliminary analyses were executed in order to address the computational and convergence issues of the numerical solution. The solid element was selected for the modeling of poly-wall in order to capture all the interactions and contacts of the components. The initial models were verified step by step by adding more features to the geometry, material properties and interactions of the elements. The detailed model was then employed for the verification using the experimental results.
The full-scale test walls consisted of five stacked poly-blocks, which were filled with RPF and reinforced with either 2 m × 10 m or 2 m× 15 m rebars in each core. These components were separately simulated according to the geometric dimensions of the full-scale test walls. As depicted in Fig. 12, all parts of the wall were finely meshed using hexahedral brick elements (C3D8R) excluding the inner core of the rebar elements, which were simulated using 6-node triangular prism (C3D6). All elements were employed along with hourglass control option and reduced integration points in order to overcome the convergence issues of the full integration.
The rebars and RPF cores model were meshed using built-in sweep technique in ABAQUS. However, given the detailed geometry of poly-block, it was discretized by implementing the advanced “bottom-up” meshing technique. The average aspect ratio of most of elements was close to unity, with maximum aspect ratio of less than 1:3 at regions where geometry was complex. The properties of different materials including concrete, steel, RPF and poly-block’s crushable foam were defined as described in the previous sections.
Figure 13 displays the assembly sequence of the poly-wall model. The contact areas of the different parts were predefined on the top surface of the foundation. Similar to the benchmark example in Section 2.1, the rebars were placed at their positions and connected to the foundation by means of tie constraints (Fig. 13(a)).
The RPF cores were placed on the specified circular areas and contained the rebars into their cylindrical holes (Fig. 13(b)). The internal surfaces of the holes were tied to the external surfaces of the rebars. The interactions of the RPF cores and the footing were simulated by tangential contacts between the bottom surfaces of each core and the predefined circular regions on the wall footing. The interface was modeled as a “surface to surface” contact and the friction coefficient was assumed equal to 15%. The adhesion of concrete and RPF cores was disregarded since neither it cannot be relied on nor accurately measured in practice.
Five poly-blocks model were stacked up and placed on the specified rectangular area on the wall footing (Fig. 13(c)). In the physical test models, the poly-blocks were thoroughly glued together and also to the foundation. Based on the experimental observations during the loading of the wall, no noticeable disconnection between the poly-blocks occurred until failure of the RPF cores took place. Therefore, the connecting surface of the blocks and the contact surfaces of the bottom course with the wall footing were simulated by implementing tie constraints. Due to slight expansion of the RPF liquid during the curing process, it completely sticks to the internal surfaces of the poly-blocks voids. Since there was no relative displacement between the RPF cores and the poly-blocks; they were tied together. Figure 13(d) demonstrates the completed numerical model after the assembly and mesh generation.
After the assembly stage, the boundary conditions of the FE model corresponding to the experimental test walls were defined. All the translational and rotational degrees of freedom at the bottom surface of the foundation were restrained. As Fig. 14 exhibits, the top block was subjected to a distributed load. For the sake of comparison with the measured experimental response, the lateral displacements of the wall were recorded at the elevation of the linear variable differential transducers (LVDTs) in the experiment (0.90 m above top of the footing). The base bending moment was calculated at each loading interval i.e., lateral load times the distance of measurement point from the top of footing. The analysis was conducted on two poly-walls, one reinforced with 2 m × 10 m and one reinforced with 2 m × 15 m rebars. The static analysis was executed and the applied load was increased by increments of 0.05 of the total lateral load.
Comparison of numerical and experimental results of poly-wall flexural performance
Figure 15 compares the flexural behavior of poly-walls obtained from the numerical and experimental results. The numerical results demonstrate a linear behavior until the rebars yielded and inelastic deformations initiated. The computed response fairly matched the experimental results up to the yield strength of both walls. The yield displacements were also well predicted by the FEA for both cases, however, since the “post-yield” behavior and fracture of materials were not considered in the numerical analysis, the numerical results do not match the experimental results past the yield point. This is considered to be acceptable since the design admissible stress for such walls is considerably lower than the yield stress.
The flexural behavior of the first wall reinforced with 2 m × 10 m rebars obtained from FEA are in good agreement with the test results and form a lower-bound for the experimental lateral resistance. The experimental response of the poly-wall reinforced with 2 m × 15 m exhibited a greater initial stiffness in comparison with the FEA results. This could be attributed to some experimental factors such as tensile strength of polyurea, non-alignment of rebars and the test set-up. Also, this initial flexural stiffness was observed for the applied bending moments below 5 kN.m where non-structural elements could contribute to the load resisting system. This discrepancy was not clearly noticeable in the behavior of the first wall.
Regardless of the higher initial stiffness, it can be seen from Fig. 15 that the slopes of the measured and calculated responses for the second wall are slightly unmatched, particularly for lateral displacements greater than 25 mm. This is due to the fact that the properties of RPF cores may marginally vary along the height of the wall due to slight densification of RPF during the curing time. This causes more expansion in higher elevations and subsequently less stiffness. It must also be noted that since there was a slight variation in the compressive behavior of RPF plotted in Fig. 7, a low-average stress-strain relations was selected for the RPF properties in the FE model. This was done to introduce some level of conservatism in the design of poly-wall and to ensure that the FEA results always form a lower-bound for the lateral resistance. Considering a design moment capacity equal to 65% of the ultimate resisting moment, the flexural behavior of the poly-walls achieved from the numerical models would be on the safe side for design purpose.
Stress contour of poly-wall connection
The stress contour at the interface of the poly-wall and the foundation is illustrated in Fig. 16. As expected, the RPF cores have “toe” and “heel” regions representing the compressive and tensile areas. The maximum compressive stress of the core occurred at the toe region, where it contributes to the load bearing mechanism by transferring the compressive forces to the foundation. A close examination of Fig. 16 reveals that both tensile and compressive rebars play a major role in the anchorage system due to a significant difference between the stiffness of steel and RPF. It can be also observed that the tensile and compressive stresses in the rebars exceed the yield strength.
The stresses in the poly-blocks were minimal due to their negligible stiffness. The maximum tensile stress in the RPF elements was equal to 10MPa and occurred in the vicinity of the tensile rebar immediately above the foundation. This value well matched with the modulus of rupture obtained from the flexural tests [
1]. Also, this maximum stress occurred at the same location where the RPF cores cracked and failed in the full-scale experiments according to the test observations.
Discussion on finite element analysis
Several attempts were made to improve the accuracy and efficiency of the analyses because the computation time for analysis of a finely meshed poly-wall with all geometrical features using load-controlled analysis exceeded 70 h. To optimize the numerical solution, the results of displacement-controlled and load-controlled analyses were compared and the former was found to be superior in terms of convergence rate in the nonlinear domain.
To ensure that the mesh density has no impact on the accuracy and solution convergence, sensitivity analysis was conducted considering different element size of poly-wall components. It was concluded that finer mesh of the rebars and to some extent of the RPF cores, resulted in higher accuracy. On the other hand, no effect on the results was observed by refining the poly-blocks mesh. The results were also found insensitive to the friction percentage of the RPF cores and footing.
The effect of the number of tie constraints on the computations time was also evaluated. In this case, the poly-blocks, RPF cores and the rebars were assembled as a single part with different properties and their interconnection tie constraints were removed. However, their interfaces with the foundation were kept unchanged (i.e., tie constraints of rebars and contact surfaces of RPF cores). These changes significantly reduced the computation time to less than 11 h per run and no noticeable change in the accuracy of the results was observed. The only disadvantage of this method is that discretization of a single part with a sophisticated geometry is more complicated and time-consuming than several independent parts.
It was also found that the effect of poly-blocks stiffness on the performance of the poly-wall model is trivial. Therefore, the details of poly-block geometry were reduced by removing its edges and grooves. Thus, the detailed model of poly-block shown in Fig. 12(b) was replaced with a simplified poly-block model similar to that shown in Fig. 6(a). This modification reduced the solution time to 3 h per run with no impact on precision of the results.
Wind load on poly-wall
The structural performance of poly-wall was investigated for the application as a sound wall. Design of sound walls is mostly dominated by wind load. Hence, in order to evaluate the response of the poly-wall to lateral wind pressure, the two FE models of the poly-wall that were verified using the experimental results were employed in the analysis and were subjected to a uniform wind pressure on their windward surface. Since, the total height of a sound wall is less than 4 m in practice, the wind pressure was assumed constant along the wall height. Similar to previous analyses, the load was increased gradually by increments of 0.05 of the total load.
Figure 17 compares the behavior of poly-walls under uniform lateral load with that due to semi-concentrated loads for two types of reinforcement. The plots in Fig. 17 show the resultant lateral load versus the displacement at the wall top. Since, the resultant of the uniform load acts at the mid-height of the wall, the base bending moment produced by this load distribution is almost half of that produced by the concentrated lateral load applied at the top block. Given that response is dominated by bending deformations, the lateral yield load in the case of uniform loading is twice the yield load for concentrated loading at the wall top.
The walls deflection resulting from both types of loadings were also obtained from “linear elastic solution.” The poly-wall was simplified with a cantilever beam as shown in Fig. 18. Assuming identical flexural stiffness and equal total load, the ratio of end displacements of beams 1 and 2 is obtained as D2/ D1 = 2.16.
The same ratio was calculated for the poly-walls using the numerical results shown in Fig. 17. For the poly-walls reinforced with 2 m × 10 m and 2 m × 15 m, the deflection ratios were 2.24 and 1.96, respectively, i.e., difference of 3.7% and 9.2% relative to the equivalent beam results.
This comparison demonstrates the possibility of employing an equivalent flexural stiffness (EI) for the design of poly-walls with different load distributions and heights. It is believed that the equivalent cantilever beam can reasonably estimate the lateral displacement of poly-walls. On the other hand, the maximum factored moment induced by wind loading can be calculated according to the building codes and then compared with the flexural yield capacity of the anchoring system of the wall (i.e., RPF cores with reinforcing bars), which was experimentally established. Hence, this equivalent approach may be employed as a replacement of the rigorous FEA for the preliminary design of poly-walls under different lateral load profiles. This conclusion would not have been drawn without an accurate supportive FEA as the interaction of the elements in load bearing system of the wall is fairly complex.
Effect of external vertical load on the lateral resistance of poly-wall
Poly-wall can be used as a load bearing wall system and carry vertical load in addition to lateral load. Sound barriers sometimes have attachments, which may apply external gravity loads on the wall. Also, poly-wall can be employed as perimeter walls of buildings which carry vertical load in addition to wind load. The effect of combined loading was investigated herein assuming that the vertical load is applied on the RPF cores of the wall to be transferred to the foundation.
A parametric study was conducted to determine the effect of vertical load on the lateral resistance of poly-wall reinforced with 2 m × 15 m rebars in each core. For these analyses, the total vertical (dead) loads were uniformly distributed on the RPF cores at the wall top and a uniform lateral wind load was then gradually applied to the surface of the wall. The analyses were performed for 0.0, 0.5, 1.0, 1.5 and 2 MPa vertical pressures on each core and the total lateral resistance of the poly-wall was obtained in each case.
The numerical analyses results indicated that up to 0.5 MPa vertical pressure, lateral yield resistance remained almost unchanged. However, the yield resistance decreased proportional to the increase of vertical loads beyond 0.5 MPa. This is due to the fact that the rebars, which are the main load-bearing components of the wall are preloaded before the application of lateral load. This results in a reduction of their lateral load bearing capacity. Table 1 shows the percentage of the reduction in lateral resistance of the wall which is less than 10.7% up to 2 MPa vertical pressure.
Conclusions
A comprehensive 3D finite element model was developed to investigate the structural performance of the proposed sound wall (poly-wall) under lateral loading. For this purpose, 3D FE models of the wall components were calibrated using the test data. Numerical models of the entire poly-wall was then developed and verified using the experimental results. Finally, the lateral performance of the poly-wall model was assessed under wind loading and a simplified equivalent model was proposed for the design applications. The effect of vertical load on lateral strength of poly-wall was discussed for different levels of pressure. According to the results of this study, the following conclusions can be drawn:
1) A fairly good agreement was achieved between the numerical predictions and the experimental observations of the poly-wall components including steel rebar, poly-block and RPF cores indicating acceptable accuracy of FE modeling.
2) The Ogden hyperelastic model was capable to accurately simulate the behavior of RPF under uniaxial and flexural loading.
3) The calculated responses of the full-scale poly-walls using the numerical model were in reasonable agreement with the measured responses during the experiments. The numerical lateral capacity formed a lower bound for the experimental results within the desired range of the design requirement.
4) Fine mesh is more crucial for high-stiffness components in a multi-material FE model. In this study, the numerical accuracy was more dependent on steel rebars mesh size than other components.
5) Removing the number interfaces and unessential geometry features of a FE model, significantly reduce the computation time. Discretizing of the entire model rather than individual component, although more rigorous, improves the efficiency of the finite element analysis, where applicable.
6) An equivalent model or approximate methods can be employed to determine structural performance of a composite structures with reasonable accuracy for design purposes.
To the authors’ knowledge, there are very limited published reports in connection to the application of rigid polyurethane foam as a construction material as well as their 3D finite element modeling. It is believed that this study provides a basis for further investigations toward the applicability of polyurethanes for the construction of main load-bearing members of structures.
Higher Education Press and Springer-Verlag Berlin Heidelberg
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