Circular Hollow Section Braces Research Articles

Hot spot stress concentration factor of CFST T/Y joints based on modified equivalent thickness

To extend the calculation method of hot spot stress concentration factors (SCFs) of circular hollow section (CHS) joints in concrete-filled steel tubular (CFST) joints, a calculation method of SCFs of CFST T/Y joints based on the modified equivalent thickness is proposed. Considering the contact relationship between the concrete and tubular wall and the axial force acting on the CFST chord, 36 finite element models of the CFST T/Y joint were established with different geometric parameters, and the stress distribution of the CFST chords and the CHS braces were investigated. According to the SCFs formula of grouted chord joints given by DNV, the modified equivalent thickness of CFST T/Y joints is derived based on the parametric FEA results. The modified calculation method of SCFs of CFST T/Y joints was established based on the modified equivalent thickness and CIDECT formulas. the proposed SCFs formulas for the CFST T/Y joints were verified by FEA and the experiment results with a calculation error of 20%.

Compressive performance of circular hollow section brace with eccentrically installed rib-stiffened splice plates

Hollow section braces are often adopted in steel frame structures because of their superior performance. To connect tubular members with framing elements, field-bolted connections of the slotted-in single splice plate and gusset plate are majorly used because of their ease of construction. However, the eccentricity between the splice and gusset plate axes reduces the compressive strength of the brace and induces local connection failures under seismic excitations. To improve the compressive strength, this paper proposes a new connection configuration for circular hollow section braces in which cruciform rib-stiffened splice plates are eccentrically installed such that the gusset plate axis coincides with the brace axis. The concept of the proposed connection was first described with theoretical considerations. To demonstrate the effectiveness of the proposed connections, compressive loading tests and subsequent cyclic loading tests were conducted for four test specimens. Furthermore, finite element analyses with long-brace models were performed to demonstrate the good performance of the proposed connection for a wide range of slenderness ratios.

Effect of chord length on CHS-RHS T-joints with slender sections

The chord length is an essential geometric property that must be defined in the analysis of isolated joints composed of hollow steel sections, as well as the boundary conditions of the test. The analysis of these parameters’ effect on the behavior of joints has been addressed by other studies, mostly with joints between circular hollow sections with compact or semi-compact cross-sections. Recent research about tubular joints has addressed cases with slender sections, where the design of joints containing these sections is still in development. In this context, this study aimed to evaluate the effect of the chord length in the behavior of T-joints between circular hollow section braces and slender rectangular hollow section chords through experimental tests and a numerical study considering the application of axial compression at the braces. The joint behavior was examined through the load-strain and load-deformation curves and the von Mises stress distribution, which allowed the failure mode's determination and the joint resistance value. Chord face failure was observed in the prototypes. It was concluded that a minimum chord length of 0.6m would be the adequate value for the study of the T-joints containing the geometric properties used in this study, which would be equivalent to a chord length five times higher than the width of the cross-section (5b0).

Behavior of partially concrete-filled X-joints with CHS braces and SHS chord under axial compression

The paper experimentally studies the static behavior of partially concrete-filled X-joints with circular hollow section (CHS) braces and square hollow section (SHS) chord under axial compression. Totally 40 partially concrete-filled CHS-SHS X-joints were designed, including 20 X-joints with concrete-filled CHS braces and hollow SHS chord (CFCHS-SHS) and 20 X-joints with hollow CHS braces and concrete-filled SHS chord (CHS-CFSHS). The impacts of the brace diameter to chord width ratio (β = d/b) and chord width to double chord thickness ratio (γ = b/2 T) on the static behavior of CHS-SHS X-joints were experimentally studied. At the same time, the corresponding finite element analysis (FEA) models were established and validated by comparison with test results, and then an extensive parametric investigation was performed to study the effects of three main influential factors (τ, concrete filling position, and in-filled concrete strength) on the ultimate strength of CHS-SHS X-joints under axial compression. The test results show that the CFCHS-SHS X-joints and CHS-CFSHS X-joints exhibit completely different failure modes under axial compression. All CFCHS-SHS X-joints are failed with chord plasticization, while the local buckling of the brace members is the primary failure mode for the CHS-CFSHS X-joints. The ultimate strength and axial stiffness of CHS-CFSHS X-joints are significantly higher than those of CFCHS-SHS X-joints. However, the ductility of CHS-CFSHS X-joints is slightly superior to that of CFCHS-SHS X-joints. Based on the analyses of test results, the design equations derived from the simplified yield line model were put forward for the estimation of the ultimate strength of CFCHS-SHS X-joints, which was verified to be reasonable and accurate. In addition, the study also revealed that the total cross-section yield strength of the brace could be utilized to accurately predict the ultimate strength of CHS-CFSHS X-joints.

An experimental study on stress concentration factors of stainless steel hybrid tubular K-joints

This paper describes a test program on the stress concentration factors (SCFs) of stainless steel hybrid tubular (SSHT) joints with square hollow section (SHS) braces and circular hollow section (CHS) chord and SSHT joints with CHS braces and SHS chord. The load was applied to the brace for all test specimens. Twenty specimens in total, including ten overlapped and ten gapped tubular K-joints, were tested. Different geometric parameters were used which include brace width-to-chord diameter ratio (β1 = b1/d0), brace-to-chord thickness ratio (τ1 = t1/t0) and chord diameter-to-thickness ratio (2γ1 = d0/t0) for SHS-to-CHS hybrid tubular joints, brace diameter-to-chord width ratio (β2 = d1/b0), brace-to-chord thickness ratio (τ2 = t1/t0) and chord width-to-thickness ratio (2γ2 = b0/t0) for CHS-to-SHS hybrid tubular joints. A number of strain gauges were used at the hot spot locations for various tubular joints. The hot spot stress (HSS) method was used to determine the SCFs of SSHT joints using the quadratic extrapolation method. This was a consequence of the stress distributions around the intersection area of SHS-to-CHS and CHS-to-SHS hybrid tubular joints being nonlinear. Results found the maximum SCFs for hybrid overlapped tubular K-joints with square braces and circular chord to be located at the hot spot location at an angle of 135° between the SHS brace and the CHS chord. This is characterized as line A. The maximum SCFs of hybrid gapped tubular K-joints with square braces and circular chord were located at the hot spot location at an angle of 45° between the SHS tension brace and the CHS chord (characterized as line C) and at an angle of 45° between the SHS compression brace and the CHS chord (characterized as line E). The maximum SCFs of hybrid overlapped tubular K-joints with circular braces and square chord were located at the hot spot location at an angle of 180° to the CHS brace. The maximum SCFs of hybrid gapped tubular K-joints with circular braces and square chord were located generally at the hot spot location at an angle of 0° to the CHS brace. Test results were compared against the design predictions in accordance with the current design guidelines of the International Committee for the Development and Study of Tubular Structures (CIDECT, 2001). Upon comparison, it was found that the design formulas of CIDECT (2001) overestimate the maximum SCFs of SHS-to-CHS hybrid gapped tubular K-joints. For the CHS chord and SHS braces of hybrid overlapped tubular K-joints, the design formulas overestimate the maximum SCFs for most of the specimens. Furthermore, for the SHS chord and CHS braces of hybrid gapped tubular K-joints, the design formulas of CIDECT (2001) overestimate the maximum SCFs. For the SHS chord and CHS braces of hybrid overlapped tubular K-joints, the CIDECT (2001) design formulas were unconservative when determining the maximum SCFs of such joints.

Experimental and numerical study on static behaviour of uniplanar CFDST-CHS T-joints

Experimental and numerical study on static behaviour of uniplanar T-joints between circular concrete-filled double-skin steel tube (CFDST) chord and circular hollow section (CHS) brace, i.e., CFDST-CHS T-joints for short, are presented in this paper. Twelve uniplanar T-joint specimens, including 10 joints with circular CFDST chords and 2 reference ones with CHS chords, were tested under specified boundary conditions. The experimental parameters included hollow ratio of the chord (χ), brace to chord diameter ratio (β) and axial compression ratio of the chord (n). The failure patterns, load versus displacement (strain) curves and bearing capacities of the specimens were collected and analysed. The composite tubular T-joint specimens showed better structural performance than the reference steel ones. For the CFDST-CHS T-joint specimens, the failure patterns change from the local buckling of CHS brace to the compression-bending failure of CFDST chord with the increase of β. With the local buckling failure of CHS brace, the bearing capacity of the composite tubular T-joint specimens increases with the growing of χ and β. However, with the compression-bending failure of CFDST chord, the bearing capacity of the composite tubular T-joint specimens increases with the rising of n. Advanced numerical simulation on the structural performance of uniplanar CFDST-CHS T-joints was implemented using finite element analysis (FEA) method, and the accuracy of the predicted responses was evaluated against the experimental results. Finally, based on the FEA results, the simplified method for the bearing capacity prediction of the composite tubular T-joints was proposed.

Behaviour of high-strength CFDST chord to CHS brace T-joint: Experiment

This paper presents an experimental investigation on the fatigue behaviour of composite T-joints which consisted of circular concrete-filled double-skin steel tubular (CFDST) chord and circular hollow section (CHS) brace, with the high-strength steel used in the chord. The test parameters included the brace to chord diameter ratio, the hollow ratio of the chord and the load range. The brace was subjected to repeated cyclic tension and the development of cracks in test specimens were recorded. The measured fatigue lives were compared with predicted ones using current codes of practice. It is found that the fatigue life of CFDST T-joint was underestimated according to codes for hollow section steel T-joints or joints who have grout between annulus of chord.

Structural performance of cold-formed high strength steel tubular X-Joints under brace axial compression

The experimental investigation presented in this study focused on the static strengths and load-deformation behaviour of cold-formed high strength steel tubular X-joints made up of S900 and S960 steel grades. The X-joints include brace members made up of square, rectangular and circular hollow sections, while chord members were made up of square and rectangular hollow sections. The nominal yield strengths of square and rectangular tubular members were 900 and 960 MPa, while the nominal yield strength of circular tubular members was 900 MPa. Two configurations of tubular X-joints were fabricated, first, where both brace and chord members were made up of square and rectangular hollow sections, and second, where circular hollow section braces were welded to square and rectangular hollow section chord members. The welds were laid using semi-automatic gas metal arc welding process. A total of 34 tests was conducted where an axial compression was applied through the brace members without any chord preload. The ratio of brace-to-chord width (β) ranged from 0.34 to 1, brace-to-chord thickness (τ) from 0.53 to 1.28, chord width-to-thickness (2γ) from 20.2 to 38.9, and chord side wall slenderness (h0/t0) from 12.7 to 39.0. In order to assess the applicability of the existing design provisions, test strengths were compared with the nominal strengths obtained from the Eurocode 3 (EC3) and CIDECT. It is shown that the existing design provisions are not capable of providing accurate and reliable predictions for cold-formed high strength steel tubular X-joints made up of S900 and S960 steel grades.

Static behavior of X-joints with SHS chord and CHS braces

This paper investigates the behavior of X-joints with square hollow section (SHS) chord and circular hollow section (CHS) braces (XSCCB) under brace axial loading. A total of twenty-nine specimens were designed, including twenty-three empty specimens, three specimens filled with concrete only in the square chord and three specimens filled with concrete only in the circular braces. The effect of four influential factors was studied, including brace diameter to chord width ratio (β = d/b), chord width to double wall thickness ratio (γ = b/2 T), brace wall thickness to chord wall thickness ratio (τ = t/T) and partial concrete filling on the behavior of XSCCB under brace axial loading. Initial stiffness and ultimate bearing capacity of empty XSCCB increase as the β increases and the γ decreases. The ductility coefficient of empty XSCCB increases as β increases, while γ has little effect on their ductility coefficient. Also, τ has a minor effect on the ultimate bearing capacity, initial stiffness and ductility coefficient of empty XSCCB. However, for XSCCB filled with concrete only in the chord member, the initial stiffness, ultimate bearing capacity and ductility coefficient increase as τ increases. For the XSCCB filled with concrete only in the braces, τ has little effect on the ultimate bearing capacity, initial stiffness and ductility coefficient. Chinese Code and Eurocode 3 are generally conservative for designing empty XSCCB under brace axial loading. Hence, the corresponding formulas are proposed for calculating the ultimate bearing capacity of empty XSCCB under brace axial loading.

Compression capacities of cold-formed high strength steel tubular T-joints

The paper presents a test programme on cold-formed high strength steel tubular T-joints with fully supported chord members. The T-joints consist of brace members made up of square, rectangular and circular hollow sections, while chord members were made up of square and rectangular hollow sections. The T-joints were fabricated by both brace and chord members made up of square and rectangular hollow sections as well as circular hollow section braces were welded to square and rectangular hollow section chord members. The brace and chord members were welded by semi-automatic gas metal arc welding process. The steel grade of 900 MPa was used for the square, rectangular and circular tubular members, while the steel grade of 960 MPa was used for the square and rectangular tubular members only. A total of 22 tests was conducted where an axial compression was applied through the brace members, while the chord members were resting on a solid flat base. The ratio of brace-to-chord width (β) ranged from 0.33 to 1, brace-to-chord thickness (τ) from 0.53 to 1.27, chord width-to-thickness (2γ) from 20.6 to 38.7, and chord side wall slenderness (h0/t0) from 12.7 to 38.6. The test results were compared with the nominal strengths obtained from the Eurocode 3 (EC3) and CIDECT for the assessment of the existing design rules for tubular T-joints with fully supported chord members. It is found that the existing design rules are not applicable for the design of cold-formed high strength steel tubular T-joints with fully supported chord members.

Tests of cold-formed high strength steel tubular T-joints

The paper presents an experimental investigation on cold-formed high strength steel tubular T-joints. The fabricated T-joints include brace members made up of square, rectangular and circular hollow sections, while chord members were made up of square and rectangular hollow sections. The nominal yield strengths of square and rectangular tubular members were 900 and 960 MPa, while the nominal yield strength of circular tubular members was 900 MPa. Two configurations of tubular joints were fabricated, first, where both brace and chord members were made up of square and rectangular hollow sections, and second, where circular hollow section braces were welded to square and rectangular hollow section chord members. The brace and chord members were joined together by automatic gas metal arc welding process. In total, 24 tubular T-joint tests were conducted where an axial compression was applied through the brace members of the T-joints, while the ends of the chord members were supported on rollers. The ratios of brace-to-chord width (β) ranged from 0.34 to 1, brace-to-chord thickness (τ) ranged from 0.52 to 1.27 and chord width-to-chord thickness (2γ) ranged from 20.63 to 38.55. The resistances and deformations of tubular T-joints were carefully examined. In order to evaluate the applicability of the existing design provisions for high strength steel tubular T-joints, test results were compared with the nominal strengths obtained from the Eurocode 3 (EC3) and CIDECT. It is shown that the existing design provisions are not capable of providing accurate and reliable predictions for the design of cold-formed high strength steel tubular T-joints made up of S900 and S960 steel grades.

Effect of fillet welds on T-joints with thin-walled chords

Tubular joints with thin-walled sections could have a wide range of applications in the construction industry, possibly providing light structures with high structural efficiency. However, current design prescriptions do not recommend their use and more studies are needed to attest their viability. In this context, the effect of the size of fillet welds on cold-formed T-joints consisting of thin-walled rectangular hollow section chords and circular hollow section braces under axial compression loading was analysed. Experimental and numerical analyses were conducted and the results were compared with current design prescriptions. Three experiments were performed, with different weld sizes, in a machine that provided displacement-controlled testing with data acquisition from strain gauges and displacement transducers. Finite-element models were developed to replicate the general experimental setup and the results showed good correlation with the experimental data, indicating that the size of the weld influences joint resistance.

Behaviour of four-legged square CFST latticed members under lateral cyclic loading

Four-leggedconcrete-filled steel tube (CFST) latticed members are extensively used as the piers, arch ribs and host towers of bridges. This paper aims to investigate the seismic behaviour of four-legged square CFST latticed members. A total of eight latticed specimens, i.e., six with square CFST chords and circular hollow section (CHS) braces while another two with square hollow section (SHS) chords and CHS braces, were tested under combined axial compression and lateral cyclic loading. The main experimental parameters included: 1) axial compression ratio, from 0.05 to 0.5; and 2) width-to-thickness ratio of the square chord tubes, taken as 55.6 and 28.6. The experimental results clarify that the four-legged square CFST latticed specimens outperform the reference hollow steel ones in terms of hysteretic behaviour. It has also been revealed that, the well-known inward and outward buckling of chord tubes are observed for the hollow steel latticed specimens, whilst for the CFST latticed specimens, outward tube buckling at the bottom of the chords combined with cracking at the front K-shaped chord-brace connections are found to be the main failure patterns. Bearing capacity and ductility index of CFST latticed specimens increase with the decrease of axial compression ratio and chord tube width-to-thickness ratio, and the initial stiffness of CFST latticed specimens also increase with the decrease of the abovementioned two ratios. Finally, a finite element analysis (FEA) model was developed to investigate the hysteretic performance of four-legged square CFST latticed specimens, the feasibility of which was validated by comparison with the experimental results.

Parametric study and equations of the maximum SCF for concrete filled steel tubular T-joints under in-plane and out-of-plane bending

Due to being increasingly used in infrastructure, studying stress concentration in concrete-filled steel tubular joints is necessary. In this study, the variation of the maximum Stress Concentration Factor (SCFmax) with non-dimensional geometric parameters in steel circular hollow section brace welded to concrete-filled circular hollow section chord (CHS-to-CFCHS) T-joints under in-plane and out-of-plane bending moment has been investigated. A database of the maximum SCFs in CHS-to-CFCHS T-joints under in-plane and out-of-plane bending moment is developed based on three-dimensional (3D) finite element (FE) models. The 3D FE models developed using ABAQUS software, have been verified using experimental results. Graphs showing variation of the maximum SCF, in CHS-to-CFCHS T-joints, with non-dimensional geometric parameters have been produced and compared with those for non-filled empty T-joints. Parametric equations for predicting the maximum SCFs in CHS-to-CFCHS T-joints under in-plane and out-of-plane bending moment have been developed in a multiple nonlinear regression analysis. There is a good agreement between the maximum SCFs predicted by the parametric equations and those determined from the experiments.

EXPERIMENTAL STRESS CONCENTRATION FACTOR IN CONCRETE-FILLED STEEL TUBULAR T-JOINTS

An experimental investigation of stress concentration factor (SCF) in Steel circular hollow section brace welded to concrete-filled circular hollow section chord (CHS-to-CFCHS) T-joints has been performed under axial tension, axial compression, in-plane bending and out-of-plane bending. The distribution of SCF around the welded brace-to-chord intersection on both the brace and chord has been investigated using three CHS-to-CFCHS T-joint specimens. The experimental SCF results have been compared with the predicted SCF in empty T-joints. The relationship between the maximum SCF in relation to parameter β, with fixed other geometrical parameters, has been investigated for the basic load conditions. The experimental maximum SCF under axial tension has been compared with the predicted maximum SCF from parametric equations for CHS-to-CFCHS T-joints previously developed by the authors. The results show that the concrete has a significant effect in reducing the SCF, mostly under axial tension and the parametric equations for predicting SCFs in empty T-joints are not suitable for CHS-to-CFCHS T-joints. The effect of parameter β on the maximum SCF in CHS-to-CFCHS T-joints is significant under axial tension and out-of-plane bending moment.

Formulation of stress concentration factors for concrete-filled steel tubular (CFST) T-joints under axial force in the brace

CFST T-joints consisting of a concrete-filled circular chord and a circular hollow section brace have been used in CFST trussed arch bridges. The stress concentration factors (SCFs) of CFST T-joints have been found to be much lower than those of circular hollow section (CHS) T-joints in the existing researches. At present, no parametric formulae have been proposed for SCFs determination for fatigue design of CFST T-joints. In this study, three-dimensional finite element (FE) models of the existing experiments for CFST T-joints were developed to determine the SCFs distribution at the chord-brace intersection under axial force in the brace. After confirming the validity of the FE models by the comparison of calculated SCFs with existing experimental results, they were provided for the parametric analysis to reveal the influence of four non-dimensional parameters, i.e. diameter ratio (β), diameter to thickness ratio of chord (2γ), thickness ratio (τ) and relative chord length (α), on SCFs of CFST T-joints. In total, 212 FE models with different parameters were analyzed under tensile and compressive axial forces. Based on the results of parametric analysis, a series of parametric formulae to calculate the SCFs was proposed for CFST T-joints referring to those for CHS T-joints. The SCFs determined by the formulae showed good agreements with FE analysis results.

Fatigue behavior and design of welded tubular T-joints with CHS brace and concrete-filled chord

Concrete-filled welded tubular structures have been increasingly used in large-span constructions such as truss bridges where fatigue failure is always a critical issue that should be focused on. This paper deals with fatigue behavior of CHS-CFSHS T-joints, which are made up of circular hollow section (CHS) braces and concrete-filled square hollow section (CFSHS) chords. Experimental results on stress concentration factor (SCF) were briefly summarized. Fatigue tests were conducted under cyclic tensile force in the brace, so that cracking patterns, failure modes and fatigue data of such joints were recorded. The tested CHS-CFSHS T-joints suffered from one of the following three failure modes: (i) brace-90°-side failure, (ii) chord-90°-side failure, (iii) chord-0°-side failure. Based on the fatigue data in terms of hot spot stress range (Sr,hs) versus number of cycles (N), fatigue design Sr,hs-N curves were determined using the deterministic and least-squares methods, respectively. It was found that most of the existing Sr,hs-N curves are unsafe for fatigue design of CHS-CFSHS T-joints, except for the X´ Sr,hs-N curve recommended by American Petroleum Institute [32] and American Welding Society [36]. In addition, the fatigue data of the CHS-CFSHS T-joints were found comparable with CHS-CFCHS T-joints (T-joints with CHS brace and concrete-filled CHS chord), hence a new fatigue design Sr,hs-N curve was proposed for the both types of concrete-filled joints. A comparison between CHS-CFSHS T-joints and empty CHS-SHS T-joints showed that CHS-CFSHS T-joints generally have better fatigue behavior compared to their empty counterparts except for the case of extremely large ratio of brace diameter to chord width (β).

Behaviour of CFDST chord to CHS brace composite K-joints: Experiments

Concrete filled double-skin steel tubular (CFDST) members have a large potential to be used in trusses, bridges and lattice structures. Previous research work has focused on the structural performance of individual CFDST components whilst there is a lack of understanding on the behaviour of typical connections formed by such composite members. This paper intends to fill the knowledge gap in this area. A total of 20 K-joints with circular CFDST chords and circular hollow section (CHS) braces were tested under two typical sets of boundary conditions. Effects of important parameters were investigated based on the test results, including brace to chord diameter ratio (β), chord hollow ratio (χ) and preloads on the chord. Typical failure modes and load-deformation relations were investigated, in-depth study on the interactions between materials and the load-transfer mechanism were conducted, simplified strength-prediction method for this new type of composite joints was also proposed.

Analytical behaviour of CFDST chord to CHS brace composite K-joints

Concrete filled double-skin steel tubular (CFDST) members have been adopted as electrical poles in China and are having increasing utilization in trusses, bridges and lattice structures. Besides previous research work on the structural performance of individual CFDST components and the experimental results on CFDST joints reported by the authors in the companion paper (Hou and Han, 2016), up until now there is little understanding on the detailed analytical behaviour of typical connections formed by such composite members. This paper thus presents a finite element analysis (FEA) on composite K-joints formed by CFDST chords and circular hollow section (CHS) braces. The FEA modelling was validated by the reported test data and then used to perform analysis on multiple types of possible failure modes, full-range load-deformation relations, and material interactions along the cross-sections of such new type of composite joints. Comparative analysis and parametric analysis was also performed, based on which simplified formulae for calculating the strength of CFDST K-joints were presented and validated.

Design of CHS brace-to-H-shaped chord T-joints under axial compression

This paper presents a finite element model of circular hollow section (CHS) brace-to-H-shaped chord T-joints under axial compression, which was verified by the corresponding test results. The parametric study was carried out to reveal the failure modes and plasticity propagation of tubular joints. The effects of CHS brace diameter to H-shaped chord flange width ratio (β), H-shaped chord flange width to thickness ratio (2γ), and CHS brace to H-shaped chord flange thickness ratio (τ) on the ultimate strengths of tubular joints were evaluated. The typical failure modes were obtained from the finite element analysis, which include local buckling failure of CHS brace (LBFB), local buckling failure of flange and web of H-shaped chord (LBFF+LBFW), and local buckling failure of CHS brace and H-shaped chord flange (LBFB+LBFF). The validity range of main geometric parameters was recommended to be β≥0.7, 2γ=20–30, and τ=0.25–0.5. The design equations are proposed by using multiple linear regression for CHS brace-to-H-shaped chord T-joints under axial compression based on the current design rules given in the Eurocode 3. Current Eurocode 3 is dangerous for predicting the ultimate strengths of CHS brace-to-H-shaped chord T-joints under axial compression. It is shown from the comparison that the joint strengths calculated using the proposed design equations agreed well with the finite element analysis results, which means the proposed design equations are verified to be accurate.