# Simplified Numerical Model Development for Advanced Design of Lightweight-Concrete Encased Cold-Formed Steel Compressed Elements

Authors: Nathalie Eid; Attila László Joó
Journal: Advances in Civil Engineering
 (2022)

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## Abstract

The utilization of cold-formed steel (CFS) elements has become more popular recently, due to their cost efficiency and feasibility in construction work. Nevertheless, since such elements are subjected to a variety of complex stability issues, numerous approaches to strengthening have been examined. Namely, CFS members were braced by using lightweight concrete through a previous experimental program. A simplified finite element model was performed consistently with the previous experimental work, to estimate the ultimate load of encased CFS members subjected to compression. The main aim of this research was to overcome the difficulties of modeling the concrete block by substituting the volume element with springs. In addition, a practical equation was proposed to predict the stiffness of springs, and an imperfection amplitude was set to obtain the design capacity of braced CFS compressed elements.


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## Body

## 1. Introduction

The highly increasing demand for having reliable structures with cost-effective options has led to the use of cold-formed steel (CFS) members in different building types. Nowadays, many novel research projects have been conducted to investigate new configurations of thin-walled steel structures. For example, an experimental project was conducted at the Budapest University of Technology and Economics in Hungary to test CFS elements encased in lightweight concrete [1]. This experimental program was used in this paper to validate the numerical model. In the literature, many projects were also mentioned such as Guan et al. [2], where CFS floor systems were tested as a kind of composite structures by using a specific type of gypsum and a theoretical approach was developed to estimate the displacement of such structures. Erdelyi and Dunai calculated the stiffness, resistance, and ductility of light-gauge composite floor beam [3]. Fiorino et al. [4] tested a two-story sheathing-braced cold-formed steel building, where its seismic behavior was observed.Actually, one of the most important methods of improving the behavior and capacity of CFS elements is the application of concrete for both filling and bracing. For example, Xu et al. [5] tested the cyclic behavior of thin-walled steel tubular columns, where they were filled with ultrahigh-performance steel fiber reinforced concrete (UHPSFRC). The results showed that they showed better performance compared to normal-strength concrete (NSC) and high-strength concrete (HSC) filled columns. Likewise, the flexural [6] and axial [1] behaviors of polystyrene aggregate concrete (PAC) braced CFS specimens were tested. In this case, only the local buckling mode was observed, and the capacity was enhanced.In fact, the main purpose of using PAC is for its high thermal insulation capacity, where many papers examined its properties, especially after high-temperature exposure [7–9].Another kind of strengthening is the use of carbon fiber-reinforced polymer (CFRP); for example, Shen et al. [10] investigated the behavior of partially CFRP-wrapped thin-walled circular normal-strength concrete-filled steel tubular (NCFST) stub compressed columns to increase its axial capacity. Moreover, an improved kind of gypsum was used as a filling material for cold-formed steel framing walls by Wu et al. [11]. Xu et al. [12] have also used a developed type of lightweight concrete with high capacity, as well as straw boards to improve the seismic performance of CFS shear walls.Generally, using PAC for increasing the resistance of CFS elements is considered more advantageous than using FRP for many reasons. Firstly, PAC is much cheaper compared to FRP that is quite expensive, and it has to be glued to the elements too; secondly, PAC has a high insulation capacity that makes it more profitable for CFS structures where insulation is the main concern. Lastly, the density of PAC, which is usually between 200 and 260 Kg/m3, is less than the density of normal concrete (2,400 Kg/m3) by 10 times that makes it significantly lighter, and hence, it is more suitable to use with lightweight structures such as CFS.CFS structures have been numerically analyzed by using different finite element tools, which give accurate results within a short period of time and at a minimal cost. Namely, Shen and Wadee [13] studied thin-walled rectangular hollow section (RHS) struts, where the interaction of buckling modes was analyzed. Additionally, Parenica et al. [14] discussed the simulation of the connection between CFS purlins and the supporting structure. Majdi et al. [15] developed a model for defining the bond-slip behavior in the modeling of galvanized CFS elements embedded in concrete. In addition, the modeling of thin-walled steel built-up members was addressed in many studies, where all details regarding the numerical models were specified [16–20].The purpose of increasing the load-bearing capacity of buildings has led to the development of the mechanical properties of construction materials and the evolution of advanced structural configurations. Thus, conducting the numerical models of such new structural systems consumes a lot of time and energy. Hence, performing simplified models could be a potential solution to analyze effectively the complex structural behavior of many cases. That could be achieved by avoiding the use of solid elements for modeling the filling material, hence keeping the model at the shell element level. Namely, Youssef and Rahman [21] discussed a macromodel to represent the nonlinear performance of reinforced concrete elements under bending. Additionally, many research studies developed finite element models by using springs instead of specific elements [22–25].Consequently, this paper aims to present a simplified numerical model of the experimental tests that were conducted by Hegyi and Dunai [1], where shell and spring elements were used instead of shell and volume elements. This research work has already been started [26], and a summary of the results could be found in Sections 3 and 5.This paper contains a short summary of the experimental results, a description of the numerical model of unbraced specimens, and the simplified model of PAC braced members. Then, the spring stiffness formula development was explained, and finally, the design resistance of each specimen was calculated according to Eurocode and compared to the numerical design values by proposing an appropriate equivalent geometric imperfection for this case.


## 2. Experimental Test Summary of Members Subjected to Compression

All tests [1], which were used in this paper, are detailed in this section. Firstly, C-section members were investigated, in a loading frame (Figure 1), without and with PAC bracing to investigate the effect of PAC mixtures on the load-bearing capacity and failure modes. Consequently, two nomenclatures were used; the first one is for unbraced specimens, which refers to the columns without PAC, and the second one is for braced specimens, which represents the columns with PAC as depicted in Figure 2. The load was applied through load transferring elements of 140 mm length at both ends of elements. The boundary conditions were hinges at both sides restraining bending moment from occurring. The dimensions of the lipped channel section were a height of 90 mm, a width of 41 mm, and a stiffener of 13 mm, and the nominal thickness was equal to 0.9 mm. Additionally, three different element lengths were introduced to investigate different stability phenomena. The lengths of the examined specimens were 300/600/2,000 mm, and the average of the observed thicknesses for each length was 0.866, 0.857, and 0.868 mm, respectively. The yield strength was 372 MPa [1]. For braced specimens, the CFS elements were encased by 400∗200 mm2 PAC block. In particular, two PAC mixtures were used here: type A with Young’s modulus equals 115.36 MPa and X with one equals 270.39 MPa.Figure 1
Loading frame of the specimen tests [1].Figure 2
Cross-section of unbraced (left) and PAC braced (right) specimens [1].The results of unbraced specimens were governed by the height of the element. Namely, the failure modes of the 300 and 600 mm columns were local and distortional buckling (Figures3(a) and 3(b)) [1], while the mode of the 2,000 mm members was flexural torsional buckling (Figure 3(c)). On the other hand, all braced specimens showed the same local failure at the end of the braced area (Figure 4(a)). Figure 4(b) shows the axial load-displacement curves, where all initial axial stiffnesses are close to each other; thus, no remarkable axial stiffening was achieved due to the PAC type. However, it should be mentioned that the end connections have reduced the axial stiffness of the columns. All ultimate loads and increments are detailed in Table 1 [1]. In general, the variation of mechanical properties of PAC does not affect the capacity, and the maximum increment was for the longest elements [1].Figure 3
Failure modes of: (a) 300 mm, (b) 600 mm, and (c) 2,000 mm columns [1].
(a)(b)(c)Figure 4
Failure of encased specimens (a) and axial load-displacement curves (b) [1].
(a)(b)Table 1
Ultimate loads and increments of the first test series [1].
Specimen typeUltimate load [KN]Increment (%)UnbracedBraced – ABraced – XBraced – ABraced – XC90-10-30027.3438.9340.264247C90-10-60027.3437.8037.443837C90-10-200017.7634.4433.579489Secondly, the effect of CFS core stiffness was examined by replacing the lipped channel sections with a bigger one: a height of 140 mm instead of 90 mm, and also, there were sections with a thickness of 0.9 and other sections with a thickness of 1.5 mm [1]. All other test arrangements remained the same as in the first test series. The yield strength was 314 and 305 N/mm2 for 0.9 and 1.5 mm thicknesses, respectively. The Young’s modulus of PAC mixtures, type WM, was varying between 43.30 and 105.43 N/mm2; all the material test results are detailed in [1]. The failure modes were the same as in the first series, except for the 2,000 mm unbraced elements with 0.9 mm thickness, which failed due to distortional buckling [1]. All ultimate loads and increments are detailed in Table 2, where the increments of 1.5 mm thickness elements were marginally smaller [1].Table 2
Ultimate loads and increments of the second test series [1].
Specimen typeUltimate load [KN]Increment (%)UnbracedBraced – WMC140-10-30030.8940.5431C140-10-60031.8840.0826C140-10-200021.3843.52104C140-15-30065.1071.6510C140-15-60064.0879.5624C140-15-200042.4683.2496This section was a summary of the main results of the first and second test groups, which will be used later in this paper. More specific details can be found in [1]. In the next section, a FE model of unbraced specimens will be performed and checked according to the test results, which were specified in this section.


## 3. Numerical Model Development for Unbraced Specimens

A FE model for unbraced specimens was developed [26], using ANSYS software [27]. The test results that were obtained from previous experimental research at the Budapest University of Technology and Economics [1] and described in the previous section were used to validate the model.


### 3.1. Model Description

A shell finite element model was developed by using SHELL181 element type to model the unbraced specimens of the first series as shown in Figure5. The same geometry that was mentioned in Section 2 was used in the model; however, the thickness of each length was taken as the average of the observed thicknesses, which was 0.866, 0.857, and 0.868 mm for 300/600/2,000 mm specimens, respectively. According to EN 1993-1-5 Annex C [28], different material models can be used besides the real one depending on the accuracy and the allowable strain required for the analysis. Hence, the material model was assumed here to an elastic-plastic bilinear, with a nominal plateau slope that was taken as E/1,000. In fact, this material model was chosen for numerical reasons where it is more stable numerically and could capture the post-peak branch of the load-displacement curves for higher displacement values than the elastic-plastic model with linear strain hardening (slope = E/100) or the true stress-strain curve. The elastic modulus (E) was 210 GPa [29]; Poisson’s ratio was 0.3; and the yield strength was 372 MPa. The maximum element size was set to 20 mm. The boundary conditions were hinged support for the first end and roller support in the axial direction for the second one in order to apply the load that was imposed as a displacement load in the longitudinal direction at the center of the roller end. Additionally, the rotation along the longitudinal axis was restrained [26].Figure 5
Applied loads and boundary conditions of the 600 mm member.


### 3.2. Modeling and Suggestion of Equivalent Geometric Imperfections

An imperfection sensitivity study was carried out to determine the equivalent geometric imperfections that also consider the residual stresses for each length. The same ultimate loads and failure modes from the tests were obtained here. The imperfection was applied in ANSYS by performing a buckling analysis; by choosing the eigenvalue buckling option (block Lanczos), the displacement of the obtained modes was scaled by the initial imperfection values using the command “Update Geom.” Namely, the lowest eigenvalue buckling mode was set that was local buckling for the 300 and 600 mm specimens (Figure6(a)) and global buckling for the 2,000 mm specimen (Figure 6(a)); the contour here represents the displacement vector sum. For local buckling, different imperfection amplitudes were implemented ranging from 0.006∗w to 0.03∗w, where “w” is the width of the plate. The value that gave the closest ultimate load to tests was 0.026∗w for the 300 mm column and 0.0118∗w for the 600 mm one. For the global mode, values ranging from L/1,000 to L/35,000 were set, where “L” is the specimen length, and the value that agreed well with test results was L/30,000 [26].Figure 6
Lowest buckling mode of the 600 mm column (a) and global buckling mode of the 2,000 mm column (b).
(a)(b)


### 3.3. Solution Methods, Results, and Discussion

A geometrically and materially nonlinear analysis with imperfections (GMNIA) [28] that assumes a nonlinear material model and takes into account the deformed shape in every step in the analysis was carried out. Actually, it was achieved through carrying out the analysis, which was performed by using Newton–Raphson method with the sparse matrix solver, by setting the large displacement static option (NLGEOM, ON) in ANSYS. The FE models were validated against the previous experimental tests. Namely, Table 3 demonstrates a comparison of the ultimate load values between tests and FE models, Figure 7 shows a comparison of axial load-displacement curves, and Figure 8 illustrates the local failure mechanism. The effect of the loading frame stiffness and the deformation at the end connections of load transferring elements were taken into account, and the displacement values were adjusted. Additionally, the shift in the diagrams at the initial loading stage is because of the small amount of load that was put onto the specimens to adjust and fix them to the right position before starting the measurement during tests. Thus, this effect was taken into account by correcting the load values. The FE models give a quite good agreement with the tests in terms of load-displacement curves and failure modes. The fluctuations in test curves are because of the loading application and measurement tools that have this specific kind of behavior. Here, only the results that were obtained by using the imperfection values specified in the previous subsection and that are in consistent with tests are shown. However, more details about the parametric study of using different element sizes and imperfection amplitudes can be found in [26]. Finally, this section was a proposal for the modeling of unbraced channel columns, and the next section will be about the development of the equation that determines the stiffness of springs, which substitute the lightweight concrete block, and that will be used later in the modeling of filled thin-walled steel members.Table 3
Comparison of test and FEA ultimate loads of unbraced specimens of first test series [26].
Column length [mm]Ultimate load [KN]Fu, testFu, numFu,num/Fu,test30027.3427.591.0160027.3427.431.002,00017.7617.801.00Figure 7
Experimental and numerical axial load-displacement curves of the: (a) 300 mm column, (b) 600 mm column, and (c) 2,000 mm column.
(a)(b)(c)Figure 8
Comparison of local failure mode between test and FEA for 300 and 600 mm specimens.


## 3.1. Model Description

A shell finite element model was developed by using SHELL181 element type to model the unbraced specimens of the first series as shown in Figure5. The same geometry that was mentioned in Section 2 was used in the model; however, the thickness of each length was taken as the average of the observed thicknesses, which was 0.866, 0.857, and 0.868 mm for 300/600/2,000 mm specimens, respectively. According to EN 1993-1-5 Annex C [28], different material models can be used besides the real one depending on the accuracy and the allowable strain required for the analysis. Hence, the material model was assumed here to an elastic-plastic bilinear, with a nominal plateau slope that was taken as E/1,000. In fact, this material model was chosen for numerical reasons where it is more stable numerically and could capture the post-peak branch of the load-displacement curves for higher displacement values than the elastic-plastic model with linear strain hardening (slope = E/100) or the true stress-strain curve. The elastic modulus (E) was 210 GPa [29]; Poisson’s ratio was 0.3; and the yield strength was 372 MPa. The maximum element size was set to 20 mm. The boundary conditions were hinged support for the first end and roller support in the axial direction for the second one in order to apply the load that was imposed as a displacement load in the longitudinal direction at the center of the roller end. Additionally, the rotation along the longitudinal axis was restrained [26].Figure 5
Applied loads and boundary conditions of the 600 mm member.


## 3.2. Modeling and Suggestion of Equivalent Geometric Imperfections

An imperfection sensitivity study was carried out to determine the equivalent geometric imperfections that also consider the residual stresses for each length. The same ultimate loads and failure modes from the tests were obtained here. The imperfection was applied in ANSYS by performing a buckling analysis; by choosing the eigenvalue buckling option (block Lanczos), the displacement of the obtained modes was scaled by the initial imperfection values using the command “Update Geom.” Namely, the lowest eigenvalue buckling mode was set that was local buckling for the 300 and 600 mm specimens (Figure6(a)) and global buckling for the 2,000 mm specimen (Figure 6(a)); the contour here represents the displacement vector sum. For local buckling, different imperfection amplitudes were implemented ranging from 0.006∗w to 0.03∗w, where “w” is the width of the plate. The value that gave the closest ultimate load to tests was 0.026∗w for the 300 mm column and 0.0118∗w for the 600 mm one. For the global mode, values ranging from L/1,000 to L/35,000 were set, where “L” is the specimen length, and the value that agreed well with test results was L/30,000 [26].Figure 6
Lowest buckling mode of the 600 mm column (a) and global buckling mode of the 2,000 mm column (b).
(a)(b)


## 3.3. Solution Methods, Results, and Discussion

A geometrically and materially nonlinear analysis with imperfections (GMNIA) [28] that assumes a nonlinear material model and takes into account the deformed shape in every step in the analysis was carried out. Actually, it was achieved through carrying out the analysis, which was performed by using Newton–Raphson method with the sparse matrix solver, by setting the large displacement static option (NLGEOM, ON) in ANSYS. The FE models were validated against the previous experimental tests. Namely, Table 3 demonstrates a comparison of the ultimate load values between tests and FE models, Figure 7 shows a comparison of axial load-displacement curves, and Figure 8 illustrates the local failure mechanism. The effect of the loading frame stiffness and the deformation at the end connections of load transferring elements were taken into account, and the displacement values were adjusted. Additionally, the shift in the diagrams at the initial loading stage is because of the small amount of load that was put onto the specimens to adjust and fix them to the right position before starting the measurement during tests. Thus, this effect was taken into account by correcting the load values. The FE models give a quite good agreement with the tests in terms of load-displacement curves and failure modes. The fluctuations in test curves are because of the loading application and measurement tools that have this specific kind of behavior. Here, only the results that were obtained by using the imperfection values specified in the previous subsection and that are in consistent with tests are shown. However, more details about the parametric study of using different element sizes and imperfection amplitudes can be found in [26]. Finally, this section was a proposal for the modeling of unbraced channel columns, and the next section will be about the development of the equation that determines the stiffness of springs, which substitute the lightweight concrete block, and that will be used later in the modeling of filled thin-walled steel members.Table 3
Comparison of test and FEA ultimate loads of unbraced specimens of first test series [26].
Column length [mm]Ultimate load [KN]Fu, testFu, numFu,num/Fu,test30027.3427.591.0160027.3427.431.002,00017.7617.801.00Figure 7
Experimental and numerical axial load-displacement curves of the: (a) 300 mm column, (b) 600 mm column, and (c) 2,000 mm column.
(a)(b)(c)Figure 8
Comparison of local failure mode between test and FEA for 300 and 600 mm specimens.


## 4. Spring Stiffness Formula Development for PAC Braced Members

In fact, the modeling of PAC in braced specimens is a tough and time-consuming task, especially for practicing engineers. Thus, the main aim of this paper is to develop a simplified shell finite element model for braced elements that could be a good replacement for the volume model of concrete (Figure9). Therefore, PAC was substituted by springs with specific properties. To predict the stiffness of springs accurately, the analytical solution of critical stresses of steel plate elements resting on an elastic foundation subjected to compression, which was derived by Hegyi [29], was used. Specifically, two plate models were performed to derive the spring stiffness of internal and outstand compressed elements, which are explained in Sections 4.1 and 4.2.Figure 9
Volume and shell model of the braced column.


### 4.1. Internal Compressed Element Model

To model the internal plate, a simply supported plate model was carried out by using the SHELL181 element and elastic material model. All edges were constrained against translation in out-of-plane direction (y-axis). The nodes of the compressed edges were coupled by rigid region means to a master node at the center of each edge. The constraints of the first master node were Ux, Uy, Uz, and Rz, while they were Ux, Uy, and Rz for the second one with a concentrated load of Fz. Additionally, one support in x-direction was set in the middle of the plate to avoid rigid body motion (Figure 10).Figure 10
Applied load and boundary conditions for the internal plate.In order to get a general equation that covers most of the cases, a practical range was set for the following parameters:b/t, Ec, and element size (A) as detailed in Table 4, where b is the plate width, t is the plate thickness, and Ec is the PAC Young’s modulus. The plate thickness was taken as 1 mm in all cases. To ensure that the model could accurately predict the results, the critical stress, which was obtained from the linear buckling analysis, was compared with the theoretical value according to the following equation:(1)σcr,p=4π2Es121−υs2b/t2.Table 4
Parameter range of the numerical investigation.
ParameterRangePlate slenderness (b/t)InternalOutstand50–25020–60Elastic modulus of PAC (Ec)50–200 [MPa]Element size (A)(5∗5)–(20∗20) [mm2]After that, the springs were added along the whole plate to check the strength of the braced plate elements. COMBIN40 element was used to model the springs; it has two nodes and one degree of freedom, which was taken as the translation of out of the plane direction. In addition, constraints in the same direction were applied at the second node of each spring. The behavior of this element type is illustrated in Figure11. It has two spring constants, K1 that represents the elastic part and K2 for the plastic part (par to slide), which was assumed as K1/1,000 consistently with the ratio of the plastic slope that was used in defining the material model of steel, for the same numerical reasons. Other real constants within the element are FSLIDE that is the limiting sliding force, C that presents the damping coefficient, and GAP that refers to the gap size. The damper and the gap were removed from the element by setting C and GAP values to zero. The stiffness of the springs (K1) was set such that the critical stress of the braced plate, obtained from the buckling analysis, was the same as the analytical solution, which was derived by assuming an equivalent Winkler-type foundation stiffness, according to the approximated equation (2). In fact, the mechanical model of the Winkler foundation has been widely used in analyzing the problem of beams or plates resting on an elastic foundation. It is characterized by the vertical stiffness of the Winkler-foundation springs, where a set of unidirectional independent springs is attached along the plate to form the plate-foundation system. Since the calculation method relies only on a one-sided connection, the PAC behavior was assumed as linear elastic half-space without cracking or crushing phenomena, and the separation effect was neglected. Consequently, a large value was set to the sliding force (FSLIDE) of the springs to ensure that the response is linear. Thus, the assumption K2 = K1/1,000 has no effect on the solution since it will not contribute to the analysis. Figure 12 shows the buckling mode of unbraced and braced internal compressed plates.(2)σcr,p=4π2Es121−υs2b/t2−4680MPab/t+2.35Ec+Ec.3025MPa.Figure 11
Behavior of COMBIN40 element [27].Figure 12
Buckling mode of unbraced (a) and braced (b) internal compressed plates (b/t = 50).
(a)(b)The results were represented by a 3D curve where the obtainedK values were drawn as a function of Ec and A as shown in Figure 13(a). Hence, for each b/t, a quadratic equation (3) was derived to determine K based on the numerical data that were obtained from Ansys and by using the incorporated curve fitting tool in MATLAB [30] as shown in Figure 14(a). The results are detailed in Table 5.(3)K=aNmm+bNmm3A+cmmEc+dNmm5A2+e1mmAEc+fmm3NEc2.Figure 13
The 3D curve ofK as a function of Ec and A: (a) internal plate (b/t = 175) and (b) outstand plate (b/t = 50).
(a)(b)Figure 14
The fitted 3D curve ofK as a function of Ec and A: (a) internal plate (b/t = 175) and (b) outstand plate (b/t = 50).
(a)(b)Table 5
Coefficient of equation (3) of internal compressed plates.
b/taN/mmbN/mm3c [mm]dN/mm5e1/mmfmm3/N50−203.25.2311.942−0.02360.026150.001267100−228.73.0282.628−0.006290.019380.0015175−197.33.1472.002−0.00680.021560.003725250−262.93.6682.82−0.008020.021740.001072


### 4.2. Outstand Compressed Element Model

To simulate the outstand plate elements, the same model of the previous section was used here. The only difference is that one edge was free. Similar to the previous subsection and in order to validate the model, the critical stress, which was obtained from the linear buckling analysis, was compared to the theoretical value according to the following equation:(4)σcr,p=0.43π2Es121−υs2b/t2.After that, the springs were added along the whole plate, and their stiffnesses were set such that the critical stress of the braced plate was the same as the analytical solution according to the approximated equation (5). Figure 15 shows the buckling mode of unbraced and braced outstand compressed plates.(5)σcr,p=0.43π2Es121−υs2b/t2+1200MPab/t+2.22Ec+Ec.3390MPa−40MPa.Figure 15
Buckling mode of unbraced (a) and braced (b) outstand compressed plates (b/t = 20).
(a)(b)Like the previous subsection, for eachb/t, a quadratic equation (3) was derived to determine K by using MATLAB software, where the curve-fitting tool was used. The results are demonstrated in Figures 13 and 14(b) and Table 6.Table 6
Coefficient of equation (3) of outstand compressed plates.
b/taN/mmbN/mm3c [mm]dN/mm5e1/mmfmm3/N20−190.15.9841.726−0.033030.023990.00143335−231.13.4852.906−0.008140.016830.000450−255.93.8792.891−0.00960.019080.00062560−269.93.7353.024−0.008160.021330.00022


### 4.3. Approximated Equation Suggestion

Finally, a general polynomial with simplified coefficients (6) has been proposed to provide the engineers with a handy equation to predict the stiffness of the foundation for any case within the investigated range of the parameters. Precisely, few suggestions have been tried out, and the one that gave the least error has been chosen as a final practical formula (6) for both internal and outstand compressed plate elements.(6)K=−200Nmm+4Nmm3A+2.5mmEc+0.009−A2Nmm5+AEc1mm+Ec2mm3N.Figures16 and 17 show the relative difference between the exact formula and the simplified one. The error equals (Kaccurate – Ksimplified/Kaccurate) ∗ 100.Figure 16
The error of the simplified formula in the case of the internal plate.Figure 17
The error of the simplified formula in the case of the outstand plate.It could be noticed from the curves in Figures16 and 17 that the relative difference is less than 5% in most of the cases and this percentage in terms of K will not have a considerable effect on the final ultimate load as it is described in the next section. Hence, this simplified equation is practical and could be used effectively. However, for small concrete Young’s modulus along with quite small or large element size, using the exact formulae might be preferable if the target of the problem is defining the critical stress values. At last, this section was a proposal for a practical equation to get a suitable spring stiffness, which is supposed to be equivalent to the effect of PAC on buckling, and the next section will be a parametric study to investigate the effect of this stiffness on the ultimate load. Additionally, the proposed practical range of Ec will be checked to see whether it is located in the effective region, where the increment in Ec leads to an increment in the ultimate load as well, or not.


## 4.1. Internal Compressed Element Model

To model the internal plate, a simply supported plate model was carried out by using the SHELL181 element and elastic material model. All edges were constrained against translation in out-of-plane direction (y-axis). The nodes of the compressed edges were coupled by rigid region means to a master node at the center of each edge. The constraints of the first master node were Ux, Uy, Uz, and Rz, while they were Ux, Uy, and Rz for the second one with a concentrated load of Fz. Additionally, one support in x-direction was set in the middle of the plate to avoid rigid body motion (Figure 10).Figure 10
Applied load and boundary conditions for the internal plate.In order to get a general equation that covers most of the cases, a practical range was set for the following parameters:b/t, Ec, and element size (A) as detailed in Table 4, where b is the plate width, t is the plate thickness, and Ec is the PAC Young’s modulus. The plate thickness was taken as 1 mm in all cases. To ensure that the model could accurately predict the results, the critical stress, which was obtained from the linear buckling analysis, was compared with the theoretical value according to the following equation:(1)σcr,p=4π2Es121−υs2b/t2.Table 4
Parameter range of the numerical investigation.
ParameterRangePlate slenderness (b/t)InternalOutstand50–25020–60Elastic modulus of PAC (Ec)50–200 [MPa]Element size (A)(5∗5)–(20∗20) [mm2]After that, the springs were added along the whole plate to check the strength of the braced plate elements. COMBIN40 element was used to model the springs; it has two nodes and one degree of freedom, which was taken as the translation of out of the plane direction. In addition, constraints in the same direction were applied at the second node of each spring. The behavior of this element type is illustrated in Figure11. It has two spring constants, K1 that represents the elastic part and K2 for the plastic part (par to slide), which was assumed as K1/1,000 consistently with the ratio of the plastic slope that was used in defining the material model of steel, for the same numerical reasons. Other real constants within the element are FSLIDE that is the limiting sliding force, C that presents the damping coefficient, and GAP that refers to the gap size. The damper and the gap were removed from the element by setting C and GAP values to zero. The stiffness of the springs (K1) was set such that the critical stress of the braced plate, obtained from the buckling analysis, was the same as the analytical solution, which was derived by assuming an equivalent Winkler-type foundation stiffness, according to the approximated equation (2). In fact, the mechanical model of the Winkler foundation has been widely used in analyzing the problem of beams or plates resting on an elastic foundation. It is characterized by the vertical stiffness of the Winkler-foundation springs, where a set of unidirectional independent springs is attached along the plate to form the plate-foundation system. Since the calculation method relies only on a one-sided connection, the PAC behavior was assumed as linear elastic half-space without cracking or crushing phenomena, and the separation effect was neglected. Consequently, a large value was set to the sliding force (FSLIDE) of the springs to ensure that the response is linear. Thus, the assumption K2 = K1/1,000 has no effect on the solution since it will not contribute to the analysis. Figure 12 shows the buckling mode of unbraced and braced internal compressed plates.(2)σcr,p=4π2Es121−υs2b/t2−4680MPab/t+2.35Ec+Ec.3025MPa.Figure 11
Behavior of COMBIN40 element [27].Figure 12
Buckling mode of unbraced (a) and braced (b) internal compressed plates (b/t = 50).
(a)(b)The results were represented by a 3D curve where the obtainedK values were drawn as a function of Ec and A as shown in Figure 13(a). Hence, for each b/t, a quadratic equation (3) was derived to determine K based on the numerical data that were obtained from Ansys and by using the incorporated curve fitting tool in MATLAB [30] as shown in Figure 14(a). The results are detailed in Table 5.(3)K=aNmm+bNmm3A+cmmEc+dNmm5A2+e1mmAEc+fmm3NEc2.Figure 13
The 3D curve ofK as a function of Ec and A: (a) internal plate (b/t = 175) and (b) outstand plate (b/t = 50).
(a)(b)Figure 14
The fitted 3D curve ofK as a function of Ec and A: (a) internal plate (b/t = 175) and (b) outstand plate (b/t = 50).
(a)(b)Table 5
Coefficient of equation (3) of internal compressed plates.
b/taN/mmbN/mm3c [mm]dN/mm5e1/mmfmm3/N50−203.25.2311.942−0.02360.026150.001267100−228.73.0282.628−0.006290.019380.0015175−197.33.1472.002−0.00680.021560.003725250−262.93.6682.82−0.008020.021740.001072


## 4.2. Outstand Compressed Element Model

To simulate the outstand plate elements, the same model of the previous section was used here. The only difference is that one edge was free. Similar to the previous subsection and in order to validate the model, the critical stress, which was obtained from the linear buckling analysis, was compared to the theoretical value according to the following equation:(4)σcr,p=0.43π2Es121−υs2b/t2.After that, the springs were added along the whole plate, and their stiffnesses were set such that the critical stress of the braced plate was the same as the analytical solution according to the approximated equation (5). Figure 15 shows the buckling mode of unbraced and braced outstand compressed plates.(5)σcr,p=0.43π2Es121−υs2b/t2+1200MPab/t+2.22Ec+Ec.3390MPa−40MPa.Figure 15
Buckling mode of unbraced (a) and braced (b) outstand compressed plates (b/t = 20).
(a)(b)Like the previous subsection, for eachb/t, a quadratic equation (3) was derived to determine K by using MATLAB software, where the curve-fitting tool was used. The results are demonstrated in Figures 13 and 14(b) and Table 6.Table 6
Coefficient of equation (3) of outstand compressed plates.
b/taN/mmbN/mm3c [mm]dN/mm5e1/mmfmm3/N20−190.15.9841.726−0.033030.023990.00143335−231.13.4852.906−0.008140.016830.000450−255.93.8792.891−0.00960.019080.00062560−269.93.7353.024−0.008160.021330.00022


## 4.3. Approximated Equation Suggestion

Finally, a general polynomial with simplified coefficients (6) has been proposed to provide the engineers with a handy equation to predict the stiffness of the foundation for any case within the investigated range of the parameters. Precisely, few suggestions have been tried out, and the one that gave the least error has been chosen as a final practical formula (6) for both internal and outstand compressed plate elements.(6)K=−200Nmm+4Nmm3A+2.5mmEc+0.009−A2Nmm5+AEc1mm+Ec2mm3N.Figures16 and 17 show the relative difference between the exact formula and the simplified one. The error equals (Kaccurate – Ksimplified/Kaccurate) ∗ 100.Figure 16
The error of the simplified formula in the case of the internal plate.Figure 17
The error of the simplified formula in the case of the outstand plate.It could be noticed from the curves in Figures16 and 17 that the relative difference is less than 5% in most of the cases and this percentage in terms of K will not have a considerable effect on the final ultimate load as it is described in the next section. Hence, this simplified equation is practical and could be used effectively. However, for small concrete Young’s modulus along with quite small or large element size, using the exact formulae might be preferable if the target of the problem is defining the critical stress values. At last, this section was a proposal for a practical equation to get a suitable spring stiffness, which is supposed to be equivalent to the effect of PAC on buckling, and the next section will be a parametric study to investigate the effect of this stiffness on the ultimate load. Additionally, the proposed practical range of Ec will be checked to see whether it is located in the effective region, where the increment in Ec leads to an increment in the ultimate load as well, or not.


## 5. Simplified Numerical Model Development for Braced Specimens

### 5.1. Model Description

The model that was proposed in Section3 was developed to consider the effect of PAC, by putting springs along the concrete block zone (Figure 18) as explained in Section 4. Only the braced specimens of the first test series were modeled. The measured thicknesses were 0.84, 0.843, and 0.843 for 300, 600, and 2,000 mm columns, respectively, with concrete mixture A and 0.842, 0.853, and 0.802 for 300, 600, and 2,000 mm specimens, respectively, with concrete mixture X.Figure 18
Boundary conditions of braced specimens.


### 5.2. Parametric Study of Braced Specimens

To predict in which range the mechanical properties of PAC are effective, a parametric study was carried out by implementing different spring stiffnesses up until 100,000 N/mm. Moreover, different imperfection amplitudes were used to see how it affects the ultimate load and to determine the real imperfection value of filled thin-walled steel structures. The results are depicted in Figures19–21, where the normalized ultimate load was drawn as a function of the spring stiffness. Each curve represents a different imperfection amplitude, and the two horizontal lines refer to the test normalized ultimate load, one for mixture A and the other one for mixture X. In order to have a good evaluation of the results, the horizontal axis (spring stiffness) was defined by using a logarithmic scale. For the 300 mm length columns, the imperfection amplitude that gave the best value was equal to 0.006∗w according to Peköz [31], where w is the width of the plate, which is represented by the blue curve (Figure 19). (It was used for the web only.) While for the 600 mm length columns, the preferable imperfection was 0.013∗w, which is represented by the purple curve (Figure 20). (It was used for the web only.) Finally, the proposed imperfection amplitude for the 2,000 mm length columns was 0.015∗w (it was used for the web and flanges), which is demonstrated by the blue curve (Figure 21).Figure 19
Numerical results for the normalized ultimate load as a function of spring stiffness of the 300 mm column.Figure 20
Numerical results for the normalized ultimate load as a function of spring stiffness of the 600 mm column.Figure 21
Numerical results for the normalized ultimate load as a function of spring stiffness of the 2,000 mm column.


### 5.3. Results and Discussion

It can be seen from the parametric study that after the transition zone, between the enhanced and the original axial load, the most effective increment rate could be achieved at a certain stiffness level that could be called an optimal value. Additionally, it can be noticed that after the transition zone, the rate increment becomes quite small; thus increasing the concrete Young’s modulus more than the optimal value might be useless. In addition, this value varies depending on the specimen length, namely, the longer the column, the smaller the stiffness at which the favorable load raise could be achieved. In general, increasing the stiffness by more than 200 N/mm is considered unprofitable. Also, it can be clearly seen that the longer the specimen, the higher the increment of load-bearing capacity.The global model (Figure9) was conducted by using the SOLID65 element for PAC and by assuming a linear elastic material model for it since the failure during tests occurred without any visible damage to the concrete blocks and consistently with the assumption in Section 4. The elastic modulus of PAC was 115.36 MPa (mixture A) [29], and Poisson’s ratio was 0.2. Indeed, both models showed good agreement with test results and were close enough to each other; thus, the simplified model is acceptable and can be used for other specimens. Figures 22 and 23 show a comparison of test and numerical results obtained from both simplified and global models, for both axial load-displacement curve and failure mechanism of 300 mm braced columns. In addition, Figure 24 demonstrates comparisons for 600 mm (left) and 2,000 mm (right) braced columns for the axial load-displacement curve, where all the results are detailed in Table 7. It could be found from Sections 4 and 5 that the spring properties might influence the critical stress of the element, but it has a quite small effect on the ultimate load, as it will be located in the post-transition zone in most of the practical cases. Thus, the error of the simplified equation (6) is completely negligible, and what makes a difference here is the imperfection amplitude.Figure 22
Experimental and numerical axial load-displacement curves of C90-0.9-300-A braced column.Figure 23
Comparison of local failure between test, FEA with the solid element (a) and FEA with spring element (b) for 300 mm braced column.
(a)(b)Figure 24
Experimental and numerical axial load-displacement curves of the 600 mm (a) and 2,000 mm (b) braced columns.
(a)(b)Table 7
Numerical and test ultimate loads of PAC encased elements [26].
Column length [mm]Ultimate load [KN]Fu, testFu, num atK = 100 [N/mm]Fu,num/Fu,test30038.9339.151.0060037.8037.801.002,00034.4434.641.01To summarize, Section5 reported the results of a comprehensive numerical parametric study on determining the effective range of PAC Young’s modulus in terms of the ultimate load of encased thin-walled columns. The accuracy of the numerical model has been checked by comparing the results with former test outcomes. In Section 6, test-based design capacities will be calculated for all the described tests in Section 2. A numerical model of braced specimens will be performed by using nominal properties, and finally, a design imperfection amplitude will be suggested.


## 5.1. Model Description

The model that was proposed in Section3 was developed to consider the effect of PAC, by putting springs along the concrete block zone (Figure 18) as explained in Section 4. Only the braced specimens of the first test series were modeled. The measured thicknesses were 0.84, 0.843, and 0.843 for 300, 600, and 2,000 mm columns, respectively, with concrete mixture A and 0.842, 0.853, and 0.802 for 300, 600, and 2,000 mm specimens, respectively, with concrete mixture X.Figure 18
Boundary conditions of braced specimens.


## 5.2. Parametric Study of Braced Specimens

To predict in which range the mechanical properties of PAC are effective, a parametric study was carried out by implementing different spring stiffnesses up until 100,000 N/mm. Moreover, different imperfection amplitudes were used to see how it affects the ultimate load and to determine the real imperfection value of filled thin-walled steel structures. The results are depicted in Figures19–21, where the normalized ultimate load was drawn as a function of the spring stiffness. Each curve represents a different imperfection amplitude, and the two horizontal lines refer to the test normalized ultimate load, one for mixture A and the other one for mixture X. In order to have a good evaluation of the results, the horizontal axis (spring stiffness) was defined by using a logarithmic scale. For the 300 mm length columns, the imperfection amplitude that gave the best value was equal to 0.006∗w according to Peköz [31], where w is the width of the plate, which is represented by the blue curve (Figure 19). (It was used for the web only.) While for the 600 mm length columns, the preferable imperfection was 0.013∗w, which is represented by the purple curve (Figure 20). (It was used for the web only.) Finally, the proposed imperfection amplitude for the 2,000 mm length columns was 0.015∗w (it was used for the web and flanges), which is demonstrated by the blue curve (Figure 21).Figure 19
Numerical results for the normalized ultimate load as a function of spring stiffness of the 300 mm column.Figure 20
Numerical results for the normalized ultimate load as a function of spring stiffness of the 600 mm column.Figure 21
Numerical results for the normalized ultimate load as a function of spring stiffness of the 2,000 mm column.


## 5.3. Results and Discussion

It can be seen from the parametric study that after the transition zone, between the enhanced and the original axial load, the most effective increment rate could be achieved at a certain stiffness level that could be called an optimal value. Additionally, it can be noticed that after the transition zone, the rate increment becomes quite small; thus increasing the concrete Young’s modulus more than the optimal value might be useless. In addition, this value varies depending on the specimen length, namely, the longer the column, the smaller the stiffness at which the favorable load raise could be achieved. In general, increasing the stiffness by more than 200 N/mm is considered unprofitable. Also, it can be clearly seen that the longer the specimen, the higher the increment of load-bearing capacity.The global model (Figure9) was conducted by using the SOLID65 element for PAC and by assuming a linear elastic material model for it since the failure during tests occurred without any visible damage to the concrete blocks and consistently with the assumption in Section 4. The elastic modulus of PAC was 115.36 MPa (mixture A) [29], and Poisson’s ratio was 0.2. Indeed, both models showed good agreement with test results and were close enough to each other; thus, the simplified model is acceptable and can be used for other specimens. Figures 22 and 23 show a comparison of test and numerical results obtained from both simplified and global models, for both axial load-displacement curve and failure mechanism of 300 mm braced columns. In addition, Figure 24 demonstrates comparisons for 600 mm (left) and 2,000 mm (right) braced columns for the axial load-displacement curve, where all the results are detailed in Table 7. It could be found from Sections 4 and 5 that the spring properties might influence the critical stress of the element, but it has a quite small effect on the ultimate load, as it will be located in the post-transition zone in most of the practical cases. Thus, the error of the simplified equation (6) is completely negligible, and what makes a difference here is the imperfection amplitude.Figure 22
Experimental and numerical axial load-displacement curves of C90-0.9-300-A braced column.Figure 23
Comparison of local failure between test, FEA with the solid element (a) and FEA with spring element (b) for 300 mm braced column.
(a)(b)Figure 24
Experimental and numerical axial load-displacement curves of the 600 mm (a) and 2,000 mm (b) braced columns.
(a)(b)Table 7
Numerical and test ultimate loads of PAC encased elements [26].
Column length [mm]Ultimate load [KN]Fu, testFu, num atK = 100 [N/mm]Fu,num/Fu,test30038.9339.151.0060037.8037.801.002,00034.4434.641.01To summarize, Section5 reported the results of a comprehensive numerical parametric study on determining the effective range of PAC Young’s modulus in terms of the ultimate load of encased thin-walled columns. The accuracy of the numerical model has been checked by comparing the results with former test outcomes. In Section 6, test-based design capacities will be calculated for all the described tests in Section 2. A numerical model of braced specimens will be performed by using nominal properties, and finally, a design imperfection amplitude will be suggested.


## 6. Design Resistance

### 6.1. Test-Based Design Resistance

In order to predict the necessary design equivalent geometric imperfection value for such kinds of structures, the design capacity was calculated based on the test results by following the rules of Eurocode [32], which are dealing with thin-walled steel structures.The test groups were considered as a single family of tests according to EC 1993-1-3 2006 (A.6.3.2 (1)); based on that, three design values were calculated, one for the tests of braced specimens with PAC mixtures “A” and “X”, one for the tests from the second series with 0.9 mm thickness and WM PAC mixture, and one for the tests from the second series with 1.5 mm thickness and WM PAC mixture. The calculation of the first value is shown in Table 8 as follows (EC 5 1993-1-3 2006 (A.6.2)):Table 8
Test-based design values of specimens with PAC mixtures “A” and “X.”
Measured valuesAdjusted resistancesCharacteristic resistanceRd [KN]tobs [mm]Robs,i [KN]μRRadj,i [KN]Rm [KN]S [KN]RK [KN]0.83736.241.1730.9931.382.6726.5026.500.84339.881.1933.580.84040.661.1834.510.86738.431.2530.640.83037.161.1532.300.83331.831.1627.450.84036.661.1831.110.85734.841.2328.430.86034.721.2428.110.83041.711.1536.250.83738.81.1733.160.84037.491.1831.810.86734.271.2627.300.85340.571.2233.390.81332.531.1029.470.80736.441.0933.50In the first step, the adjusted valuesRadj,i of the experimental data were determined based on the actual measured test results Robs,i and taken into account the difference between the measured properties, such as dimensions and yield strength and their nominal values [32].(7)Radj,i=Robs,iμR,where μR is the adjustment coefficient.(8)μR=fyb,obsfybαtobs,cortcorβ,where α is an exponent that should be obtained as follows: if fyb,obs≤fyb, α = 0; if fyb,obs>fyb, α = 1, where fyb,obs: is the actual measured basic yield strength, fyb, is the nominal basic yield strength, where it was equal to 275MPa according to [29], and β is an exponent that should be taken as follows: if bp/t≤bp/tlim, β = 1; if bp/t>1.5bp/tlim, β = 2, where(9)bp/tlim=0.64Ekσfyb∗fyb/γM1σcom,Ed,where σcom,Ed is the largest compressive stress in the member (σcom,Ed = fyb), kσ is the buckling factor, and bp is the notional flat width.In the second step, the characteristic capacity was calculated according to a statistical basis as follows [32]:(10)Rk=Rm−Ks,where S is the standard deviation, K is a coefficient that should be taken according to the number of tests (here is equal to 2), and Rm is the mean value of the adjusted test results Radj.Finally, the design resistanceRd was derived from the previously determined characteristic value Rk, according to A.6.4 (EN 1993-1-3 2006) using(11)Rd=ηsysRkγm,where γm: is the partial factor for resistance (1 for buildings) and ηsys: is a conversion factor for differences in behavior under test conditions and service conditions. (It could be taken as 1 for this case.)Finally, an imperfection sensitivity study was carried out to define a proper imperfection amplitude for braced columns for design purposes in the next subsection. To achieve that, a numerical model was developed as described in Section5.


### 6.2. Design Resistance from Numerical Model

The same model in part 5 was used here with an element size equal to 15 mm, where all the tests in the first and second groups [1], summarized in Section 2, were modeled by using the same nominal dimensions and material properties, which were mentioned in Section 6.1.Local buckling for both web and flanges was considered (Figure25(a)) by introducing different imperfection amplitudes (α). The values were ranging from a/10 to a/1400, where a is the plate width. Figure 26 demonstrates an example of the imperfection sensitivity study for the C90-0.9-A specimen type. The vertical axis represents the normalized ultimate load calculated at each local imperfection amplitude, and the horizontal axis presents the imperfection scaling factor (f), applied from 10 to 1400, which is the denominator of the imperfection amplitude. The outcomes have shown that the value α = a/66 gives roughly the same axial load as the test-based design resistance and thus provides the necessary local imperfection amplitude as a design value for encased C-profiled CFS members subjected to compression, as detailed in Table 9. However, the results of 1.5 mm members showed that this imperfection value slightly underestimates the design capacity of these elements, thus using a smaller value for plates with a thickness higher than 1 mm might give better results. An example of the failure mode is demonstrated in Figure 25(b). In fact, all the members displayed the same behavior in terms of buckling mode, design axial load-displacement curve, and failure mode. The outcomes confirm the conclusion of Section 5 that the PAC Young’s modulus has a small effect on the design capacity within the proposed practical range and the parameter that has a considerable impact on the design resistance is the imperfection amplitude.Figure 25
Buckling mode of C90-0.9-600-A (a) and failure mode of C140-1.5-600-WM (b).
(a)(b)Figure 26
Imperfection sensitivity study.Table 9
Design values forα = a/66.
Specimen typeTest based design resistance, Fd,test [KN]Numerical design resistance (KN)L = 300 [mm]L = 600 [mm]L = 2,000 [mm]Fd,numFd,num/Fd,testFd,numFd,num/Fd,testFd,numFd,num/Fd,testC90-0.9-A26.5027.221.0327.231.0327.161.02C90-0.9-X26.5027.171.0327.381.0327.461.04C140-0.9-WM27.3327.130.9927.371.0027.341.00C140-1.5-WM63.2459.400.9459.970.9559.870.95


### 6.3. Step-by-Step Design Guide for the Application of the Proposed Simplified Model

In this subsection, practical steps are listed below to be followed by the engineers during the simulation of PAC-encased CFS members to get the design capacity.(1)
Creating a shell model for the core cross-section (CFS) by using nominal dimensions and material properties(2)
Determining the stiffness of springs according to equation (6) by using the elastic modulus of the filling material and the element size of the FEM model(3)
Creating spring elements along the area of the stiffened zone as a replacement for the filling material(4)
Implementing an equivalent imperfection amplitude equalsa/66, a is the plate width(5)
Defining nonlinear material for the steel core and running the GMNI analysis


## 6.1. Test-Based Design Resistance

In order to predict the necessary design equivalent geometric imperfection value for such kinds of structures, the design capacity was calculated based on the test results by following the rules of Eurocode [32], which are dealing with thin-walled steel structures.The test groups were considered as a single family of tests according to EC 1993-1-3 2006 (A.6.3.2 (1)); based on that, three design values were calculated, one for the tests of braced specimens with PAC mixtures “A” and “X”, one for the tests from the second series with 0.9 mm thickness and WM PAC mixture, and one for the tests from the second series with 1.5 mm thickness and WM PAC mixture. The calculation of the first value is shown in Table 8 as follows (EC 5 1993-1-3 2006 (A.6.2)):Table 8
Test-based design values of specimens with PAC mixtures “A” and “X.”
Measured valuesAdjusted resistancesCharacteristic resistanceRd [KN]tobs [mm]Robs,i [KN]μRRadj,i [KN]Rm [KN]S [KN]RK [KN]0.83736.241.1730.9931.382.6726.5026.500.84339.881.1933.580.84040.661.1834.510.86738.431.2530.640.83037.161.1532.300.83331.831.1627.450.84036.661.1831.110.85734.841.2328.430.86034.721.2428.110.83041.711.1536.250.83738.81.1733.160.84037.491.1831.810.86734.271.2627.300.85340.571.2233.390.81332.531.1029.470.80736.441.0933.50In the first step, the adjusted valuesRadj,i of the experimental data were determined based on the actual measured test results Robs,i and taken into account the difference between the measured properties, such as dimensions and yield strength and their nominal values [32].(7)Radj,i=Robs,iμR,where μR is the adjustment coefficient.(8)μR=fyb,obsfybαtobs,cortcorβ,where α is an exponent that should be obtained as follows: if fyb,obs≤fyb, α = 0; if fyb,obs>fyb, α = 1, where fyb,obs: is the actual measured basic yield strength, fyb, is the nominal basic yield strength, where it was equal to 275MPa according to [29], and β is an exponent that should be taken as follows: if bp/t≤bp/tlim, β = 1; if bp/t>1.5bp/tlim, β = 2, where(9)bp/tlim=0.64Ekσfyb∗fyb/γM1σcom,Ed,where σcom,Ed is the largest compressive stress in the member (σcom,Ed = fyb), kσ is the buckling factor, and bp is the notional flat width.In the second step, the characteristic capacity was calculated according to a statistical basis as follows [32]:(10)Rk=Rm−Ks,where S is the standard deviation, K is a coefficient that should be taken according to the number of tests (here is equal to 2), and Rm is the mean value of the adjusted test results Radj.Finally, the design resistanceRd was derived from the previously determined characteristic value Rk, according to A.6.4 (EN 1993-1-3 2006) using(11)Rd=ηsysRkγm,where γm: is the partial factor for resistance (1 for buildings) and ηsys: is a conversion factor for differences in behavior under test conditions and service conditions. (It could be taken as 1 for this case.)Finally, an imperfection sensitivity study was carried out to define a proper imperfection amplitude for braced columns for design purposes in the next subsection. To achieve that, a numerical model was developed as described in Section5.


## 6.2. Design Resistance from Numerical Model

The same model in part 5 was used here with an element size equal to 15 mm, where all the tests in the first and second groups [1], summarized in Section 2, were modeled by using the same nominal dimensions and material properties, which were mentioned in Section 6.1.Local buckling for both web and flanges was considered (Figure25(a)) by introducing different imperfection amplitudes (α). The values were ranging from a/10 to a/1400, where a is the plate width. Figure 26 demonstrates an example of the imperfection sensitivity study for the C90-0.9-A specimen type. The vertical axis represents the normalized ultimate load calculated at each local imperfection amplitude, and the horizontal axis presents the imperfection scaling factor (f), applied from 10 to 1400, which is the denominator of the imperfection amplitude. The outcomes have shown that the value α = a/66 gives roughly the same axial load as the test-based design resistance and thus provides the necessary local imperfection amplitude as a design value for encased C-profiled CFS members subjected to compression, as detailed in Table 9. However, the results of 1.5 mm members showed that this imperfection value slightly underestimates the design capacity of these elements, thus using a smaller value for plates with a thickness higher than 1 mm might give better results. An example of the failure mode is demonstrated in Figure 25(b). In fact, all the members displayed the same behavior in terms of buckling mode, design axial load-displacement curve, and failure mode. The outcomes confirm the conclusion of Section 5 that the PAC Young’s modulus has a small effect on the design capacity within the proposed practical range and the parameter that has a considerable impact on the design resistance is the imperfection amplitude.Figure 25
Buckling mode of C90-0.9-600-A (a) and failure mode of C140-1.5-600-WM (b).
(a)(b)Figure 26
Imperfection sensitivity study.Table 9
Design values forα = a/66.
Specimen typeTest based design resistance, Fd,test [KN]Numerical design resistance (KN)L = 300 [mm]L = 600 [mm]L = 2,000 [mm]Fd,numFd,num/Fd,testFd,numFd,num/Fd,testFd,numFd,num/Fd,testC90-0.9-A26.5027.221.0327.231.0327.161.02C90-0.9-X26.5027.171.0327.381.0327.461.04C140-0.9-WM27.3327.130.9927.371.0027.341.00C140-1.5-WM63.2459.400.9459.970.9559.870.95


## 6.3. Step-by-Step Design Guide for the Application of the Proposed Simplified Model

In this subsection, practical steps are listed below to be followed by the engineers during the simulation of PAC-encased CFS members to get the design capacity.(1)
Creating a shell model for the core cross-section (CFS) by using nominal dimensions and material properties(2)
Determining the stiffness of springs according to equation (6) by using the elastic modulus of the filling material and the element size of the FEM model(3)
Creating spring elements along the area of the stiffened zone as a replacement for the filling material(4)
Implementing an equivalent imperfection amplitude equalsa/66, a is the plate width(5)
Defining nonlinear material for the steel core and running the GMNI analysis


## 7. Conclusions

In this paper, a numerical study was presented in which a guideline to model encased cold-formed steel elements subjected to compression was proposed. In order to get the design resistance, the stiffness of springs that replace the impact of PAC on buckling was determined and an equivalent imperfection amplitude was set to be used in this simplified model of PAC-filled CFS compressed elements. All the related tests [1] were compared with the results that were in accordance with each other. This simplified model was developed to get principally the peak resistance and the corresponding failure mechanism, which is a local mode out of the braced zone. Particularly, several consequences could be outlined as follows:(i)
Equivalent geometric imperfection amplitudes of unbraced specimens have been suggested by using real material properties and measured dimensions, where the numerical model was developed and verified by tests.(ii)
Based on the experimentally verified numerical model, a simplified FE model with shell and spring elements of braced specimens has been developed to estimate mainly the ultimate axial capacity of lightweight concrete braced CFS elements. This simplified model was developed to tackle the awkwardness of modeling the concrete block as solid elements and the interaction surfaces with CFS components.(iii)
The effective practical range of PAC Young’s modulus was determined through a parametric study.(iv)
A simplified formula to define the solid-replacement spring stiffness was proposed based on the plate slenderness, PAC Young’s modulus, and the element size (equation (6)).(v)
An equivalent geometric imperfection amplitude was suggested to anticipate the design resistance of braced CFS structures where the value was equal toa/66, a is the plate width.(vi)
The imperfection amplitude inside the braced part has a small effect on the design capacity because the local buckling happened at the border of the braced area; thus, the connection part of the structure will be always the part that is prone to buckle and where the equivalent geometric imperfection value should be determined.(vii)
The imperfection amplitude outside the braced part has more effect on the design capacity compared with the impact of springs stiffness; thus, it might be concluded that the imperfection value plays the main role in determining the design capacity of braced columns and the derived formula (6) is quite accurate in the practical range.(viii)
Further research study should be performed to extend the modeling of the simplified shell-spring finite element model of PAC-filled CFS members. Namely, the modeling of connections and wall panels of this type of structure will be the next step of this research.



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*Source: 1207885-2022-05-20.xml*