Development of component stiffness equations for thread-fixed one-side bolt connections to an enclosed rectangular hollow section column under tension

Abstract

The derivation and validation of analytical equations for predicting the tensile initial stiffness of thread-fixed one-side bolts (TOBs), connected to enclosed rectangular hollow section (RHS) columns, is presented in this paper. Two unknown stiffness components are considered: the TOBs connection and the enclosed RHS face. First, the trapezoidal thread of TOB, as an equivalent cantilevered beam subjected to uniformly distributed loads, is analyzed to determine the associated deformations. Based on the findings, the thread-shank serial-parallel stiffness model of TOB connection is proposed. For analysis of the tensile stiffness of the enclosed RHS face due to two bolt forces, the four sidewalls are treated as rotation constraints, thus reducing the problem to a two-dimensional plate analysis. According to the load superposition method, the deflection of the face plate is resolved into three components under various boundary and load conditions. Referring to the plate deflection theory of Timoshenko, the analytical solutions for the three deflections are derived in terms of the variables of bolt spacing, RHS thickness, height to width ratio, etc. Finally, the validity of the above stiffness equations is verified by a series of finite element (FE) models of T-stub substructures. The proposed component stiffness equations are an effective supplement to the component-based method.

Graphical abstract

Keywords

initial stiffness / component based method / thread-fixed one-side bolt / rectangular hollow sections / analytical equation

1 Introduction

In recent years, one-side bolts have been developed as an innovative connection method for tubular members in the field of structural engineering, and they can effectively solve the problem of installing fasteners in closed spaces. The one-side bolts can be generally divided into three categories: 1) those that use a foldable nut or singular-shape head to pass through the bolt hole, such as toggle-anchored bolt, T-head one-side bolt [1] and deformed-washer bolt [2] as shown in Fig.1(a); 2) those that use sleeve deformation when tightening, to form the clamping effect as represented by Blind Bolt [3], Hollo-Bolt [4], and extended Hollo-Bolt [5], as shown in Fig.1(b); 3) those that use internal threads machined in the connected plate to replace the nut, such as flowdrill bolts [6] and thread-fixed one-side bolts (TOBs) [7], as shown in Fig.1(c). The TOB is simple in configuration, applicable to a wide range of steel plate thickness, and can utilize ordinary high-strength bolts. However, a high degree of accuracy is required for the machining of the internal threads.

The strength performance of TOBs has been previously studied. Liu et al. [8] experimentally investigated the tension strength of TOB bolted T-stub and proposed a design method for two new failure modes. Zhu et al. [9] carried out tests to confirm the strengthening effect of a TOB bolted T-stub with a backing plate. Wulan et al. [10] numerically investigated the failure mechanism of TOB bolted T-stubs under tension. Zhang et al. [11,12] presented four yield line patterns of TOB connected T-stubs and made further studies on the yield line patterns of hollow section column with TOB connection. Wang et al. [13] and Liu et al. [14,15] conducted tests on the TOB bolted connection of beam-to-hollow-steel-tube with different strengthening measures. They discussed rotation capacity, strength degradation and stiffness degradation. The above studies showed that with proper design, the TOBs could have the same ability as ordinary high-strength bolts to ensure the strength of beam-column joints.

Regarding the stiffness performance, the special configurations of one-side bolts will result in different stiffness characteristics. In the stiffness calculation of a joint, the component-based method is the most popular due to the advantages of usability and applicability to any common joint in practical design. The method is coded in EC3-1-8 [16] and subsequently extended by works of Huber and Tschemmernegg [17] and Jaspart [18]. Numerous studies, applications and extensions related to the component method can be found in Refs. [19,20]. For the rectangular hollow section (RHS) columns connected by one-side bolts, two factors should be emphasized: the stiffness of the one-side bolt connection and the stiffness of the RHS face. There is currently a lack of full understanding of their behaviors. For a one-side bolt with a nut in Fig.1(a), the stiffness equations for ordinary high-strength bolts specified in EC3 [16] can be applied. The effect of expanding sleeves on stiffness should be considered for the one-side bolts shown in Fig.1(b). A spring component model considering the slippage of sleeves was developed by Pitrakkos [21] for a novel Blind-Bolt. Based on the thin conical shell theory, Wang et al. [4] proposed a theoretical model to predict the stiffness of Hollo-Bolt. However, relatively few studies have been conducted on the stiffness of TOBs of the type shown in Fig.1(c). Generally speaking, the thickness of the connected plate will determine the number of internal threads in a TOB connection, thus affecting the connection stiffness. Sopwith [22] demonstrated that the rounding of the thread roots would contribute little to the stiffness, and the thread could be considered as straight-sided to the full depth. Yamatoto [23] proposed analytical equations of thread deformation in five parts. However, the equations were limited to a single thread and complex in form, with a lack of derivation processes. For the stiffness of column face, some analytical models and equations were developed for an infinitely long RHS column. Based on the plate deflection theory of Timoshenko, Park and Wang [24] derived the deflection equations of the face of an infinitely long column. Thai and Uy [25] concluded numerous finite element (FE) simulation results and proposed an empirical equation for stiffness of the column face. Based on the regression of FE simulation results, equations were proposed to calculate the concrete-filled column face displacement at the bolt location [26,27]. Currently, no literature has been found on the stiffness equations for the face of an enclosed RHS column.

In EC3 [16], a simplified T-stub substructure under tension is always used as the basic object to model the resistance and stiffness behaviors of bolted beam-column joints. A group of bolts are divided into separate rows with an equivalent T-stub, as shown in Fig.2. The overall stiffness of the joint is determined by the individual stiffnesses in parallel. In the TOBs bolted beam-RHS column joints, sometimes stiffener plates are added in RHS at the positions of beam flanges to avoid excessive pressurized deformation of the column face. The two stiffeners (called close-plates) and the column segment between them form the so-called enclosed RHS (Fig.2(a)). Also, a non-through TOBs bolted endplate joint was previously proposed by all the authors of this paper involving the enclosed RHS (Fig.2(b)). Both cases can be simplified to a substructural model of a T-stub connected to an enclosed RHS by TOBs, as shown in Fig.2(c).

The lack of studies on the tensile stiffness of TOB connection and enclosed RHS face limits the potential engineering applications. In this paper, the T-stub substructure model in Fig.2(c) was used to study the above-mentioned stiffness through theoretical derivations and FE verifications. Based on the equivalent treatment of threads, the analytical equations and the thread-shank series-parallel model for the stiffness calculation of TOB connection were proposed and verified. Then, the face of the enclosed RHS was simplified as a two-dimensional (2D) plate with rotation constraints of sidewalls on all four sides. The linear deflection of the plate in tension due to a pair of bolt forces was resolved into three deflections and analytically derived, referring to the plate deflection theory of Timoshenko [28]. Finally, a series of T-stub substructure FE models was established to verify the applicability of the proposed component stiffness equations. This study can be regarded as a development of design guidance for this connection, based on the component method; the proposed equations will widen the use of one-side bolts in engineering applications.

2 Stiffness equations for thread-fixed one-side bolt connection

The clamping effect of the TOB connection is different from that of the ordinary high-strength bolt, due to the different structural details. In the TOB, the bolt nut is replaced by an attached steel plate with threaded hole. The bolt load is transferred turn by turn through the thread pairs of bolts and steel plates, which are usually called external and internal threads respectively. The plate thickness will determine the number of thread pairs, thus affecting the stiffness. Therefore, the derivation of the stiffness equation can be realized based on the force transmission features. It should be noted that the dimensions of threads in this paper were all in accordance with the GB/T 192-2003 [29] and GB/T 196-2003 [30], with no technical differences from ISO standard. The detailed geometric dimensions of the trapezoidal threads are shown in Fig.3 and listed in Tab.1.

2.1 Single thread pair stiffness

The most basic part of force transmission was studied first, namely the stiffness of single turn thread pair (k_thread). The contact forces between the external and internal threads were complicated, and were simplified as concentrated load on the thread section in Refs. [22,23]. However, in practice, the contact forces will be distributed over a band. As shown in Fig.4(a), the vertical contact normal forces (CNORMF), with the amplitude range of 0.8–1.0 of the maximum value of a trapezoidal thread simulated in ABAQUS, are distributed over most of the area. On this basis, the contact forces were simplified to be uniformly distributed. The trapezoidal thread was treated as a cantilever [22,23], and this paper further simplified the trapezoidal thread to a cantilevered beam with uniform cross-sections, based on the bending and shearing stiffness equivalence, as illustrated in Fig.4(b) and Fig.4(c). For the total deformation of a cantilever, the bending and shearing deformations should be predominant. Therefore, the elastic stiffness analyses of a thread in this section were simplified based on the following assumptions: 1) the thread was simplified as an equivalent cantilevered rectangular beam; 2) the thread deformation was mainly composed of bending and shearing deformations; 3) the contact forces between the threads were uniformly distributed. It should be noted that the cantilevered thread and beam analyzed here represented the section in terms of per unit width. The unit-width uniform forces q applied to the beam were the vertical component of the contact forces, the sum of which was equal to the single turn thread pair load F_b1, i.e., q=1\cdot F_{\mathrm{b}1}/\left[\text{π}{\mathit{\text{z}}}_{\mathrm{t}\mathrm{h}}\cdot 0.5\left(d_0+d\right)\right] (unit: N/mm).

2.1.1 Bending deformation δ_b of a single thread

When subjected to the uniform load, the bending deformation Δ_b of the equivalent uniform-cross-section cantilevered beam with moment of inertia I_eq can be calculated by a graphic multiplication method as:

(1)\begin{array}{rl}{\text{Δ}}_{\text{b}}\left(x={\mathit{\text{z}}}_{\text{th}}\right)=& \frac{1}{E{I}_{\text{eq}}}{\int }_0^{{\mathit{\text{z}}}_{\text{th}}}M(x)\overline{M(x)}\text{d}x\\ =& \frac{1}{E{I}_{\text{eq}}}{\int }_0^{{\mathit{\text{z}}}_{\text{th}}}0.5q{x}^2\cdot x\text{d}x=\frac{q{\mathit{\text{z}}}_{\text{th}}^4}{8E{I}_{\text{eq}}}.\end{array}

For a variable-section cantilevered beam, we define the sectional moment of inertia I(x) as a function of x. The equation of bending deformation can be expressed as:

(2)\begin{array}{rl}{\text{Δ}}_{\text{b}}\left(x={\mathit{\text{z}}}_{\text{th}}\right)=& {\int }_0^{{\mathit{\text{z}}}_{\text{th}}}\frac{1}{EI(x)}M(x)\overline{M(x)}\text{d}x\\ =& \frac{1}{E}{\int }_0^{{\mathit{\text{z}}}_{\text{th}}}\frac{0.5q{x}^2\cdot x\text{d}x}{I(x)}=\frac{q}{2E}{\int }_0^{{\mathit{\text{z}}}_{\text{th}}}\frac{x^3}{I(x)}\text{d}x.\end{array}

Merging Eqs. (1) and (2), we can get the expression for I_eq as:

(3)I_{\text{eq}}={\mathit{\text{z}}}_{\text{th}}^4/\left[4{\int }_0^{z_{\text{th}}}\frac{x^3}{I(x)}\text{d}x\right].

Although we can derive a clear analytical equation for I(x), the integral solution of Eq. (3) is extremely hard. To obtain the explicit expressions, the compound Simpson integration method is utilized here. Based on the segmental integration, the calculation complexity for a solution with certain accuracy can be greatly reduced. The basic formula is as follow:

(4)\begin{array}{rl}& F[a,b]\\ \cong & (b-a)\frac{f\left(x_0\right)+4{\displaystyle \sum _{i=1,3,5,...}^{n-1}f\left(x_i\right)+2{\displaystyle \sum _{j=2,4,6,...}^{n-2}f\left(x_j\right)+f\left(x_n\right)}}}{3n},\end{array}

where F[a,b] is the integral value of the original function over a given interval [a,b]; n is the number of segments; f(x_i) is the value of the original function at ith segment point.

In the compound Simpson integration method, theoretically, the more segments there are then the closer the outcome is to the exact solution, but the more complicated the calculation. We take the threads of M16 bolt as an example, and conduct calculations of 4, 6, and 8 equal segments of the trapezoidal thread for I_eq value. The result for 6 segments is −2.94% less than for the 4 segments case, and the result for 8 segments is only −1.45% less than for the 6 segments case. The accuracy improvement with more segments is very limited. Therefore, the case of 8 equal segments is considered sufficient to meet the accuracy requirements of the application. Equation (3) with 8 equal segments (n = 8) is further simplified as:

(5)I_{\mathrm{e}\mathrm{q}}\cong \frac{768}{{\displaystyle \frac{1}{I_1}}+{\displaystyle \frac{4}{I_2}}+{\displaystyle \frac{27}{I_3}}+{\displaystyle \frac{32}{I_4}}+{\displaystyle \frac{125}{I_5}}+{\displaystyle \frac{108}{I_6}}+{\displaystyle \frac{343}{I_7}}+{\displaystyle \frac{128}{I_8}}}.

The moment of inertia of each segment (I₁–I₈) is then calculated according to the dimensional data in Fig.3 and Tab.1, and substituted into Eq. (5) to obtain the unit-width equivalent moment of inertia I_eq (unit: mm⁴) with the only variable p, as expressed in Eq. (6). The equations are different for internal and external threads due to the different dimensions. Note that Eq. (6) apply to all bolts that meet GB/T 192-2003 [29] and GB/T 196-2003 [30], in addition to the M12, M16, and M20 bolts studied here. However, if the thread dimension is different from the GB/T bolts, the calculation can be done again using Eq. (5).

(6)\begin{array}{rl}& I_{\mathrm{e}\mathrm{q}}=0.0299p^3,\text{(for internal thread)}\\ & I_{\mathrm{e}\mathrm{q}}=0.0155p^3.\text{(for external thread)}\end{array}

After the above calculation processes, the trapezoidal thread has been converted to an equivalent uniform-cross-section cantilevered beam with I_eq. Yamatoto [23] concluded that the equivalent contact point for deformation calculation located at a distance c from the thread root (x' = c), as shown in Fig.4(b) and Fig.4(c). Yamatoto [23] proposed that for ISO external thread, c = 0.289p, and for ISO internal thread, c = 0.325p. The bending deformation curve of the equivalent cantilever under uniform load is as follows:

(7){\mathrm{\Delta }}_{\text{b}}\left(x^{\prime }\right)=\frac{q{x}^{\mathrm{\prime }2}}{24E{I}_{\mathrm{e}\mathrm{q}}}\left(x^{\prime 2}-4{\mathit{\text{z}}}_{\mathrm{t}\mathrm{h}}{x}^{\prime }+6{{\mathit{\text{z}}}_{\mathrm{t}\mathrm{h}}}^2\right).

Then the deformation, as the targeted bending deformation δ_b in this subsection, at distance c is calculated using Eq. (8):

(8){\delta }_{\text{b}}={\mathrm{\Delta }}_{\text{b}}\left(x^{\prime }=c\right)=\frac{q{c}^2}{24E{I}_{\mathrm{e}\mathrm{q}}}\left(c^2-4{\mathit{\text{z}}}_{\mathrm{t}\mathrm{h}}c+6{{\mathit{\text{z}}}_{\mathrm{t}\mathrm{h}}}^2\right).

2.1.2 Shearing deformation δ_v of a single thread

When subjected to uniform load, the shearing deformation Δ_v of an equivalent equal-section cantilevered beam with shear area of A_eq can be calculated by a graphic multiplication method as:

(9)\begin{array}{rl}{\text{Δ}}_{\text{v}}\left(x={\mathit{\text{z}}}_{\text{th}}\right)=& \frac{1}{G{A}_{\text{eq}}}{\int }_0^{{\mathit{\text{z}}}_{\text{th}}}F(x)\overline{F(x)}\text{d}x\\ =& \frac{1}{G{A}_{\text{eq}}}{\int }_0^{{\mathit{\text{z}}}_{\text{th}}}qx\cdot 1\text{d}x=\frac{q{\mathit{\text{z}}}_{\text{th}}^2}{2G{A}_{\text{eq}}}.\end{array}

For a variable-section cantilevered beam, we define the sectional shear area A(x) as a function of x. The equation of shearing deformation is:

(10)\begin{array}{rl}{\text{Δ}}_{\text{v}}\left(x={\mathit{\text{z}}}_{\text{th}}\right)=& \frac{1}{GA}{\int }_0^{{\mathit{\text{z}}}_{\text{ih}}}F(x)\overline{F(x)}\text{d}x\\ =& \frac{1}{G}{\int }_0^{{\mathit{\text{z}}}_{\text{th}}}\frac{qx\cdot 1}{A(x)}\text{d}x=\frac{q}{G}{\int }_0^{{\mathit{\text{z}}}_{\text{th}}}\frac{x}{A(x)}\text{d}x.\end{array}

Merging Eqs. (9) and (10), we can get the expression for A_eq as:

(11)A_{\text{eq}}={\mathit{\text{z}}}_{\text{th}}^2/\left[2{\int }_0^{{\mathit{\text{z}}}_{\text{th}}}\frac{x}{A(x)}\text{d}x\right].

The compound Simpson integration method with 8 segments is utilized, as in Subsubsection 2.1.1, and we can get:

(12)A_{\mathrm{e}\mathrm{q}}\cong \frac{48}{{\displaystyle \frac{2}{A_1}}+{\displaystyle \frac{2}{A_2}}+{\displaystyle \frac{6}{A_3}}+{\displaystyle \frac{4}{A_4}}+{\displaystyle \frac{10}{A_5}}+{\displaystyle \frac{6}{A_6}}+{\displaystyle \frac{14}{A_7}}+{\displaystyle \frac{4}{A_8}}}.

The shear area of each segment (A₁–A₈) is calculated with the dimensions shown in Fig.3 and Tab.1, and we can get the unit-width equivalent shear area A_eq (unit: mm²), as expressed in Eq. (13).

(13)\begin{array}{rl}& A_{\mathrm{e}\mathrm{q}}=0.6265p,(\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{d})\\ & A_{\mathrm{e}\mathrm{q}}=0.4875p.(\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{d})\end{array}

The shear deformation curve of the equivalent cantilever under uniform load is as follow:

(14){\mathrm{\Delta }}_{\text{v}}\left(x^{\prime }\right)=\frac{q}{2G{A}_{\mathrm{e}\mathrm{q}}}\left(2{\mathit{\text{z}}}_{\mathrm{t}\mathrm{h}}{x}^{\prime }-x^{\prime 2}\right).

Then the deformation at distance c is calculated using Eq. (15), as the targeted shearing deformation δ_v in this subsection, where G = E/2(1 + μ).

(15){\delta }_{\text{v}}={\mathrm{\Delta }}_{\text{v}}\left(x^{\prime }=c\right)=\frac{q\left(2{\mathit{\text{z}}}_{\mathrm{t}\mathrm{h}}c-c^2\right)}{2G{A}_{\mathrm{e}\mathrm{q}}}.

The analytical equations for the bending and shearing deformations of thread are derived above. The stiffness of a thread pair k_thread considering the deformations of internal and external threads can then be calculated using Eq. (16), where F_b1 is a process variable and we define F_b1 = 1 for simplification. The analytical stiffnesses of single thread pairs of M12, M16, and M20 bolts are listed in Tab.2.

(16)k_{\text{thread}}=\frac{F_{\mathrm{b}1}}{{\delta }_{\text{internal}}+{\delta }_{\text{external}}}=\frac{1}{{\left({\delta }_{\text{b}}+{\delta }_{\text{v}}\right)}_{\text{internal}}+{\left({\delta }_{\text{b}}+{\delta }_{\text{v}}\right)}_{\text{external}}}.

2.2 Analytical stiffness model of thread-fixed one-side bolt connection

In the TOB connection, the bolt load is transferred from the bolt head to the free bolt shank, then to the first thread pair, then to the next thread pair via the inter-thread shank, and so on until the last thread pair. The stiffness of the thread-engaged part of TOB connection can be derived from the above load transfer mode, i.e., each thread pair is connected in parallel by the inter-thread shank in series. The proposed internal thread-external thread-shank series-parallel stiffness model is shown in Fig.5. Based on the stiffness model, the stiffness equation for the threaded connected part of TOB connection k_nThread can be derived from the single thread pair stiffness k_thread and the bolt shank stiffness k_shank as expressed in Eq. (17), which involves the iterative calculation:

(17)k_{n\mathrm{T}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{d}}=\frac{1}{{\displaystyle \frac{1}{k_{\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{k}}}}+{\displaystyle \frac{1}{k_{(n-1)\mathrm{T}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{d}}+k_{\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{d}}}}},

where n is the number of thread pairs, k_shank is the inter-thread shank axial stiffness, k_{\text{shank}}=E{A}_{\text{shank}}/p, where A_shank is the area with the diameter of d₀.

2.3 Verification against ABAQUS modeling

The FE software ABAQUS was used to establish refined models of TOB connection to verify the proposed stiffness equations. The FE modeling methods were first calibrated by the test and simulation data of specimen S20-120-12 (Fig.6(a)) from Ref. [12]. The thick T-stub flange and column wall made the effect of TOB connection on overall stiffness more pronounced. The load–displacement comparison of test and finite element model (FEM) in the initial displacement is shown in Fig.6(b) with good accuracy. The modeling details and complete load–displacement comparison were exhibited in Subsection 4.1. Then, the parametric FE models of TOB connections alone were established. The shank parts, i.e., the internal and external thread parts, were built according to the dimensional data in Fig.3 and Tab.1. The length of the shank was set as np + l₀, where l₀ was the length of the upper shank segment without threads, taken to have a fixed value of 4 mm. The modeling of the bolt head was ignored. The steel plate with bolt hole was represented by a ring part, and the length (i.e., the plate thickness) was np. To avoid the effect of additional deformation on the stiffness simulation due to the excessive thickness of the ring part, the thickness of the ring part was set to 0.3 mm after sensitivity trials, and a three-layer mesh was assigned along the thickness direction. The thread parts were refined meshed with the size of 0.25 mm, while other parts were meshed with a bigger size. The external threads (bolt threads) were tied to the shank surface and the internal threads (plate threads) were tied to the inner surface of ring. It should be noted that when using the Tie command here, the shank surface and the ring surface should be set as the main surfaces, so as to prevent the thread parts from penetrating the surfaces. Since the initial stiffness was analyzed under small deformation, the materials were all in the elastic state. The steel material was set as linear elasticity, E = 200 GPa. The outer surface of ring was fixed. The top surface of the shank was coupled to the center point, which was loaded with a displacement value of 0.1 mm. The diagram of FE modeling is shown in Fig.6(c).

A series of FE models of TOB connections, including M12, M16, and M20 bolts, were built based on the above processes. Each type of bolt was analyzed with plate thicknesses of 3 to 10 threads (i.e., np, n = 3–10), which covered the common range in engineering applications. In total, 3 × 8 = 24 models were built. The representative von-Mises stress distributions are shown in Fig.6(d) where can be seen that the internal and external threads engaged each other to generate contact forces. The closer the threads are to the loading point, the higher the stresses are. The peak stresses are distributed near the root of the first thread pair. The phenomena matched the proposed stiffness model as illustrated in Fig.5, since the first thread pair was the most deformed, and was aligned with the results in other studies [31]. The results of the TOB connection stiffness from Eq. (17) and FE models are listed in Tab.3. Noted that relative error throughout this paper was defined as: (analytical value − simulated value)/simulated value. It can be seen that the theoretical equations can predict the tensile stiffness of TOB connections of M12 to M20 classes with errors within 10%. The errors were partly caused by the simplification of thread force state and deformation mode. Moreover, the shank condition was slightly different from the assumed axial tensile state. Actually, the load was transferred from the thread to the thread-shank interface and finally to the full section of the shank. The M12 bolt had the lowest error level due to its smallest shank diameter (smallest load transfer distance) and the closest condition to the assumed axial tension state. Overall, the prediction errors were within a reasonable range, and the proposed thread-shank series-parallel model reflected the stiffness composition and load transfer mode of TOB connections.

3 Stiffness equation for the face of enclosed rectangular hollow section

3.1 Analytical model of enclosed rectangular hollow section

Based on the component method, the enclosed RHS face subjected to two bolt loads was another important stiffness component. It was proposed that for an infinitely long RHS column, the three-dimensional (3D) column could be simplified to a 2D face plate with left and right sides restrained by rotating springs that represented the effect of the RHS sidewalls [24]. For the enclosed RHS analyzed in this paper, the close-plates were welded on the top and bottom of the RHS column in the joint zone, thus forming four “sidewalls”. The close-plates thickness might not be the same as the column thickness. As well as the left and right sides, the rotation constraints on the up and down sides of the 2D plate needed to be strictly considered, which made the problem more complicated.

As the analytical model of enclosed RHS, the 2D plate representing the column faces were elastically constrained by the equivalent rotating springs at four sides. The stiffness of the left and right springs, as the constraint of the column sidewalls, was denoted as kr₁; the stiffness of the up and down springs, as the constraint of the close-plates, was denoted as kr₂, as shown in Fig.7(a). The derivation of the rotational stiffness coefficients kr₁ and kr₂ was based on a simplified two-beam model in Fig.7(b). Note that the two-beam model was simplified by the symmetry, and the two beams represented the unit-width cross section. The calculated result was the unit-width rotation restraint distributed along the side of the 2D plate. Through the stiffness matrix operation of two vertically connected beams, the rotational stiffness of point C, i.e., the rotational restraint of the sidewall or close-plate on the analyzed plate was obtained. A similar work was done in Ref. [24], but only the case of two beams with equal moment of inertia was considered and there were some mistakes in the matrix parameter identifiers. In the case of enclosed RHS, the thicknesses of RHS and close-plates might not be equal and the work did not apply. The case of two beams with unequal moments of inertia (I_AB and I_BC) was analyzed in this paper. The expression of k_r is shown in Eq. (18), and the detailed derivation is presented in Electronic Supplementary materials.

(18)k_{\mathrm{r}}=\frac{4E{I}_{\mathrm{B}\mathrm{C}}\left(I_{\mathrm{A}\mathrm{B}}d+1.5I_{\mathrm{B}\mathrm{C}}w\right)}{d\left(I_{\mathrm{A}\mathrm{B}}d+2I_{\mathrm{B}\mathrm{C}}w\right)}.

The applicability of theoretically derived Eq. (18) for actual sections was verified as follows. First was the case of beams with equal moments of inertia (i.e., I_AB = I_BC in Fig.7(b)). The I_AB and I_BC terms in brackets in the numerator and denominator of Eq. (18) could be eliminated. Equation (18) was then exactly the same as the equation for the case of beams with equal moments of inertia in Ref. [24]. The equation was based on a simplified two-beam model, which ignored the possible root radius (r) of the corners in cold-formed or hot-rolled RHS columns. The effect of ignoring r was systematically analyzed in Ref. [24]. Through the comparison between the equation and FE models with root radius, it was concluded that the simplified treatment of ignoring r in the derivation of k_r was acceptable for all analyzed sections. The section parameters studied here were basically consistent with those in Ref. [24] and the same conclusion of ignoring r was adopted after verification. Then, the case of two beams with unequal moments of inertia was analyzed. This case only involved the RHS side welded with close-plate, where the root radius was naturally zero. The proposed Eq. (18) was verified with the simplified two-beam model built in OpenSees software in this section. The FE models were built with the same structure of the simplified two-beam model in Fig.7(b) and the two beams adopted elasticBeamColumn elements (E = 200 GPa). A bending moment was applied at point C to determine the rotation and calculate the corresponding rotational stiffness. Before the following parametric analyses, the FE models were pre-validated by the ABAQUS simulated results for the identical structure in Ref. [24]. In parametric analyses, the variables of the FE models were set as RHS width, RHS depth, RHS thickness and close-plate thickness. The results in Tab.4 show that: Eq. (18) could accurately calculate the rotational stiffness of the simplified two-beam model with different moments of inertia, and the prediction error was almost zero. Overall, after the above discussions on beams had equal or unequal moments of inertia, it was concluded that Eq. (18) based on the simplified two-beam model for the derivation of kr was appropriate for all analyzed actual RHS sections.

After the above procedures, the analytical model of an enclosed RHS column subjected to bolt loads was simplified to a 2D plate subjected to two concentrated forces with rotational constraints at four sides. For the analytical solution of the elastic deflection of this kind of hyperstatic plate, the load superposition method was usually adopted in previous studies. In this paper, the deflection of the plate was resolved into the deflections under three kinds of force and boundary conditions: the deflection of a rectangular plate loaded by two concentrated loads with four sides simply supported; the deflection of a rectangular plate loaded by equivalent bending moments on the up and down sides with left and right sides simply supported; the deflection of a rectangular plate loaded by equivalent bending moments on the left and right sides with up and down sides simply supported, as shown in Fig.7(a). The equivalent bending moment here was the reaction moment from the sidewalls or close-plates when the column face was subjected to the concentrated forces. With the elastically retrained boundary conditions as the compatibility condition to be satisfied, the above three deflections could be solved and the sum was the actual deflection w_RHS to be obtained for this analytical model.

3.2 Deflection of four-side simply supported plate subjected to bolt forces (w₁)

Based on the plate theory, Timoshenko [28] proposed the analytical solution of the deflection of a four-side simply supported plate under a single concentrated force P, as presented in Eq. (19). The coordinate origin of this equation is at the corner of the rectangular plate, as shown in Fig.7(a).

(19)w=\frac{4P}{{\text{π}}^{4}abD}\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}\frac{\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{n\text{π}\eta }{b}}\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{n\text{π}y}{b}}}{{\left(m^2{a}^{-2}+n^2{b}^{-2}\right)}^2}\mathrm{s}\mathrm{i}\mathrm{n}\frac{m\text{π}\xi }{a}\mathrm{s}\mathrm{i}\mathrm{n}\frac{m\text{π}x}{a}.

First, the symmetric case is considered where loads are located at the middle length of the plate (b/2). Two concentrate forces are applied at the position of bolt hole centers with the spacing of a_t, i.e., {\xi }_1=\left(a-a_{\mathrm{t}}\right)/2, {\xi }_2=\left(a+a_{\mathrm{t}}\right)/2, n_1=n_2=b/2. The deflection point is the center of the plate, i.e., x=a/2, y=b/2. Considering two P/2 concentrated bolt forces, the two corresponding deflections at the center of the plate are superimposed as w₁:

(20)\begin{array}{rl}{w}_1=& \frac{w\left(\xi ={\xi }_1\right)+w\left(\xi ={\xi }_2\right)}{2}\\ =& \frac{4P}{{\text{π}}^{4}abD}\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}\frac{\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{n\text{π}}{2}}\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{n\text{π}}{2}}\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{m\text{π}}{2}}}{{\left(m^2{a}^{-2}+n^2{b}^{-2}\right)}^2}{\displaystyle \frac{\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{m\text{π}{\xi }_1}{a}}+\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{m\text{π}{\xi }_2}{a}}}{2}}.\end{array}

We find that:

(21)\begin{array}{rl}\frac{\sin {\displaystyle \frac{m\text{π}{\xi }_1}{a}}+\sin {\displaystyle \frac{m\text{π}{\xi }_2}{a}}}{2}=& \frac{\sin {\displaystyle \frac{m\text{π}\left(a-a_t\right)}{2a}}+\sin {\displaystyle \frac{m\text{π}\left(a+a_t\right)}{2a}}}{2}\\ =& \cos \frac{a_{t}m\text{π}}{2a}\sin \frac{m\text{π}}{2}.\end{array}

Then Eq. (20) can be simplified as:

(22)w_1=\frac{4P}{{\text{π}}^{4}abD}\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}\frac{{\mathrm{s}\mathrm{i}\mathrm{n}}^2{\displaystyle \frac{n\text{π}}{2}}{\mathrm{s}\mathrm{i}\mathrm{n}}^2{\displaystyle \frac{m\text{π}}{2}}}{{\left(m^2{a}^{-2}+n^2{b}^{-2}\right)}^2}\mathrm{c}\mathrm{o}\mathrm{s}\frac{a_{\text{t}}m\text{π}}{2a}.

It is observed that both {\sin }^2{\displaystyle \frac{m\text{π}}{2}} and {\sin }^2{\displaystyle \frac{n\text{π}}{2}} are equal to 0 when m and n are even terms and equal to 1 when m and n are odd terms. We denote \lambda =b/a, then Eq. (22) can be further simplified as:

(23)\begin{array}{rl}{w}_1=& \frac{4P}{{\text{π}}^{4}abD}\sum _{m=1,3,5,\dots }^{\mathrm{\infty }}\sum _{n=1,3,5,\dots }^{\mathrm{\infty }}\frac{\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{a_{\text{t}}m\text{π}}{2a}}}{{\left(m^2{a}^{-2}+n^2{b}^{-2}\right)}^2}\\ =& \frac{4P{b}^3}{{\text{π}}^{4}aD}\sum _{m=1,3,5,\dots }^{\mathrm{\infty }}\sum _{n=1,3,5,\dots }^{\mathrm{\infty }}\frac{\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{a_{\text{t}}m\text{π}}{2a}}}{{\left({\lambda }^2{m}^2+n^2\right)}^2}.\end{array}

To quantify the sum of infinite series in Eq. (23), we denote {\displaystyle \sum _{m=1,3,5,\dots }^{\mathrm{\infty }}{\displaystyle \sum _{n=1,3,5,\dots }^{\mathrm{\infty }}{\displaystyle \frac{\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{a_{\text{t}}m\text{π}}{2a}}}{{\left({\lambda }^2{m}^2+n^2\right)}^2}}=S{\displaystyle \frac{\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{a_{\text{t}}\text{π}}{2a}}}{{\left({\lambda }^2+1\right)}^2}}}}. The notation S is defined as follow:

(24)S=\left[\sum _{m=1,3,5,\dots }^{\mathrm{\infty }}\sum _{n=1,3,5,\dots }^{\mathrm{\infty }}\frac{\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{a_{\text{t}}m\text{π}}{2a}}}{{\left({\lambda }^2{m}^2+n^2\right)}^2}\right]/\left[\frac{\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{a_{\text{t}}\text{π}}{2a}}}{{\left({\lambda }^2+1\right)}^2}\right].

Through the calculation of infinite series in MATLAB software, a simple quadratic equation was proposed in this work to fit the exact S values in different λ cases with the coefficient a_t/a as variable. The exact curves of S and the fitted curve are plotted in Fig.8, with the maximum and average error of 4.51% and 2.8%. In this way, the calculation of S was greatly simplified, while maintaining accuracy.

(25)S\approx \left[0.063{\left(a_{\mathrm{t}}/a\right)}^2-0.1479\left(a_{\mathrm{t}}/a\right)+0.5432\right](1+\sqrt{\lambda }).

Equation (23) could be finally simplified to Eq. (26) with notation S defined by Eq. (25), as the deflection of the four-side simply supported plate under two P/2 concentrated forces.

(26)\begin{array}{rl}{w}_1=& \frac{4P{b}^{3}S}{{\text{π}}^{4}aD}\left[\frac{\cos {\displaystyle \frac{a_{\mathrm{t}}\text{π}}{2a}}}{{\left({\lambda }^2+1\right)}^2}\right]=\frac{43.68P{b}^{3}S}{{\text{π}}^{4}aE{t}^3}\left[\frac{\cos {\displaystyle \frac{a_{\mathrm{t}}\text{π}}{2a}}}{{\left({\lambda }^2+1\right)}^2}\right]\\ =& \frac{43.68PS}{{\text{π}}^{4}abE{t}^3}\left[\frac{\cos {\displaystyle \frac{a_{\mathrm{t}}\text{π}}{2a}}}{{\left(a^{-2}+b^{-2}\right)}^2}\right].\end{array}

3.3 Deflection of plate subjected to equivalent bending moment (w₂ and w₃)

The derivation of w₂ and w₃ also starts with the single concentrate force P. Moving the coordinate origin to the center point of the plate, we can get an alternative expression of Eq. (19) as:

(27)\begin{array}{rl}& w=\sum _{m=1,3,5\dots }^{\mathrm{\infty }}\sum _{n=1,3,5,\dots }^{\mathrm{\infty }}{A}_{\text{w1}mn}\mathrm{c}\mathrm{o}\mathrm{s}\frac{m\text{π}x}{a}\mathrm{c}\mathrm{o}\mathrm{s}\frac{n\text{π}y}{b},\\ & A_{\text{w1}mn}=\frac{4P}{{\text{π}}^{4}abD}\frac{\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{n\text{π}\eta }{b}}\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{m\text{π}\xi }{a}}}{{\left(m^2{a}^{-2}+n^2{b}^{-2}\right)}^2}.\end{array}

The boundary and force conditions for w₂ derivation involve a rectangular plate simply supported at the sides x = ±a/2 and bent by moment distributed along the sides y = ±b/2. The deflection under these conditions must satisfy the homogeneous differential equation [28]:

(28)\frac{{\mathrm{\partial }}^4{w}_2}{\mathrm{\partial }{x}^4}+2\frac{{\mathrm{\partial }}^4{w}_2}{\mathrm{\partial }{x}^2\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^4{w}_2}{\mathrm{\partial }{y}^4}=0,

and the following boundary conditions:

(29)w_2=w_{2,xx}=0,\text{for}x=\pm a/2,

(30)w_2=0,\text{for}y=\pm b/2.

The infinite series solution of w₂ is established with each term satisfying Eq. (28):

(31)w_2=\sum _{m=1,3,5,\dots }^{\mathrm{\infty }}{Y}_{2m}\mathrm{c}\mathrm{o}\mathrm{s}\frac{m\text{π}x}{a},

where

(32)\begin{array}{rl}{Y}_{2m}=& A_{2m}\sinh \frac{m\text{π}y}{a}+B_{2m}\cosh \frac{m\text{π}y}{a}\\ & +C_{2m}\frac{m\text{π}y}{a}\sinh \frac{m\text{π}y}{a}+D_{2m}\frac{m\text{π}y}{a}\cosh \frac{m\text{π}y}{a}.\end{array}

Due to the symmetry boundary and load conditions, Y_{2m} must be an even function of y. Thus, we get A_{2m}=B_{2m}=0. We denote {\alpha }_{2m}={\displaystyle \frac{m\text{π}b}{2a}}. And from the boundary conditions in Eqs. (29) and (30), we get B_{2m}\cosh {\alpha }_{2m}=-C_{2m}{\alpha }_{2m}\sinh {\alpha }_{2m}. Then Eq. (31) can be simplified as:

(33)\begin{array}{rl}{w}_2=& \sum _{m=1,3,5,\dots }^{\mathrm{\infty }}{C}_{2m}\left(\frac{m\text{π}y}{a}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\frac{m\text{π}y}{a}-{\alpha }_{2m}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}{\alpha }_{2m}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\frac{m\text{π}y}{a}\right)\\ & \cdot \mathrm{c}\mathrm{o}\mathrm{s}\frac{m\text{π}x}{a}.\end{array}

The deflection w₂ and deflection w₃ have the same boundary and load conditions, with the only difference of coordinate direction. Based on the same processes, the expression for w₃ can be solved as follow, where {\alpha }_{3m}={\displaystyle \frac{m\text{π}a}{2b}}:

(34)\begin{array}{rl}{w}_3=& \sum _{m=1,3,5,\dots }^{\mathrm{\infty }}{C}_{3m}\left(\frac{m\text{π}x}{b}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\frac{m\text{π}x}{b}-{\alpha }_{3m}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}{\alpha }_{3m}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\frac{m\text{π}x}{b}\right)\\ & \cdot \mathrm{c}\mathrm{o}\mathrm{s}\frac{m\text{π}y}{b}.\end{array}

The actual deflection of RHS face w_RHS is the sum of the three deflection components. Due to the elastic rotational constraints (kr₁ and kr₂) around the plate, w_RHS should satisfy as [32]:

(35)\begin{array}{rl}& {\left(\pm \frac{k{r}_1}{D}\frac{\mathrm{\partial }{w}_{\mathrm{R}\mathrm{H}\mathrm{S}}}{\mathrm{\partial }x}+\frac{{\mathrm{\partial }}^2{w}_{\mathrm{R}\mathrm{H}\mathrm{S}}}{\mathrm{\partial }{x}^2}\right)}_{x=\pm a/2,\mathrm{\forall }y}=0\\ \iff & {\left(\frac{k{r}_1}{D}\frac{\mathrm{\partial }{w}_{\mathrm{R}\mathrm{H}\mathrm{S}}}{\mathrm{\partial }x}+\frac{{\mathrm{\partial }}^2{w}_{\mathrm{R}\mathrm{H}\mathrm{S}}}{\mathrm{\partial }{x}^2}\right)}_{x=a/2,\mathrm{\forall }y}=0,\end{array}

(36)\begin{array}{rl}& {\left(\pm \frac{k{r}_2}{D}\frac{\mathrm{\partial }{w}_{\mathrm{R}\mathrm{H}\mathrm{S}}}{\mathrm{\partial }y}+\frac{{\mathrm{\partial }}^2{w}_{\mathrm{R}\mathrm{H}\mathrm{S}}}{\mathrm{\partial }{y}^2}\right)}_{y=\pm b/2,\mathrm{\forall }x}=0\\ \iff & {\left(\frac{k{r}_2}{D}\frac{\mathrm{\partial }{w}_{\mathrm{R}\mathrm{H}\mathrm{S}}}{\mathrm{\partial }y}+\frac{{\mathrm{\partial }}^2{w}_{\mathrm{R}\mathrm{H}\mathrm{S}}}{\mathrm{\partial }{y}^2}\right)}_{y=b/2,\mathrm{\forall }x}=0.\end{array}

The expression w_{\mathrm{R}\mathrm{H}\mathrm{S}}=w_1+w_2+w_3 from Eqs. (27), (33), and (34) is substituted in Eqs. (35) and (36) for the solutions of C_2m and C_3m. However, the equations involve infinite series, which cannot be solved directly to obtain an explicit solution. By applying computational software, we can certainly obtain a value with the specified accuracy, but it is inconvenient for engineering applications. Reference [24] involved a trial for the solution with infinite series that only taking the first term. The same trail is conducted here and the errors will be discussed later, with numerical simulation results. Taking the solution of C₃ (m = 1 and therefore C_3m changes to C₃) as an example, substituting the expressions of three deflections into Eq. (35) with m = n = 1, we get:

(37)\frac{k{r}_1}{D}\left[-\frac{\text{π}}{a}{A}_{w1}\cos \frac{\text{π}y}{b}+C_3\left(\frac{\text{π}}{b}\sinh \frac{\text{π}a}{2b}+\frac{{\text{π}}^{2}a}{2b^2}\cosh \frac{\text{π}a}{2b}-\frac{\text{π}}{b}{\alpha }_3\tanh {\alpha }_3\sinh \frac{\text{π}a}{2b}\right)\cos \frac{\text{π}y}{b}\right]+C_3\frac{2{\text{π}}^2}{b^2}\cosh \frac{\text{π}a}{2b}\cos \frac{\text{π}y}{b}=0,

which is solved as:

(38)C_3=\frac{{\displaystyle \frac{b}{a}}{A}_{w1}}{\sinh {\alpha }_3+{\alpha }_3\cosh {\alpha }_3-{\alpha }_3\tanh {\alpha }_3\sinh {\alpha }_3+2{\displaystyle \frac{\text{π}D}{bk{r}_1}}\cosh {\alpha }_3}.

Similarly, C₂ can be solved as:

(39)C_2=\frac{{\displaystyle \frac{a}{b}}{A}_{w1}}{\sinh {\alpha }_2+{\alpha }_2\cosh {\alpha }_2-{\alpha }_2\tanh {\alpha }_2\sinh {\alpha }_2+2{\displaystyle \frac{\text{π}D}{ak{r}_2}}\cosh {\alpha }_2},

where due to m = n = 1, we get {\alpha }_2={\displaystyle \frac{\text{π}b}{2a}}, {\alpha }_3={\displaystyle \frac{\text{π}a}{2b}}, A_{w1}={\displaystyle \frac{4P}{{\text{π}}^{4}abD}}{\displaystyle \frac{\cos {\displaystyle \frac{\text{π}\eta }{b}}\cos {\displaystyle \frac{\text{π}\xi }{a}}}{{\left(a^{-2}+b^{-2}\right)}^2}}.

Through the above derivations, the analytical equations for the deflections w₂ and w₃ of the rectangular plate subjected to single concentrated force are obtained. The case of two bolt loads is now analyzed as follows. Consistently with Subsection 3.1, the location of the load point is defined at the center of the bolt hole, i.e., {\xi }_1=-a_{\mathrm{t}}/2, {\xi }_2=a_{\mathrm{t}}/2, and {\eta }_1={\eta }_2=0, and the deflection point is at the center of the plate. Superimposing the deflection values under two P/2 bolt loads, it is observed that:

(40)\begin{array}{rl}& \frac{A_{w1}\left({\xi }_1=-{\displaystyle \frac{a_t}{2}}\right)+A_{w1}\left({\xi }_1={\displaystyle \frac{a_t}{2}}\right)}{2}\\ =& \frac{4P}{{\text{π}}^{4}abD}\frac{\cos {\displaystyle \frac{\text{π}\eta }{b}}\left(\cos {\displaystyle \frac{\text{π}{\xi }_1}{a}}+\cos {\displaystyle \frac{\text{π}{\xi }_2}{a}}\right)/2}{{\left(a^{-2}+b^{-2}\right)}^2}=\frac{4P\cos {\displaystyle \frac{\text{π}{a}_t}{2a}}}{{\text{π}}^{4}abD{\left(a^{-2}+b^{-2}\right)}^2}.\end{array}

Then the analytical equations for w₂ and w₃ under a pair of bolt loads can be further simplified from Eqs. (33) and (34) with x = 0 and y = 0 as:

(41)\begin{array}{rl}{w}_2=& -\frac{2P\cos {\displaystyle \frac{\text{π}{a}_{\mathrm{t}}}{2a}}}{{\text{π}}^{3}abD{\left(a^{-2}+b^{-2}\right)}^2}\\ & \cdot {\displaystyle \frac{\tanh {\alpha }_2}{\sinh {\alpha }_2+{\alpha }_2\cosh {\alpha }_2-{\alpha }_2\tanh {\alpha }_2\sinh {\alpha }_2+2{\displaystyle \frac{\text{π}D}{ak{r}_2}}\cosh {\alpha }_2}},\end{array}

(42)\begin{array}{rl}{w}_3=& -\frac{2P\cos {\displaystyle \frac{\text{π}{a}_{\mathrm{t}}}{2a}}}{{\text{π}}^{3}abD{\left(a^{-2}+b^{-2}\right)}^2}\\ & \cdot \frac{\tanh {\alpha }_3}{\sinh {\alpha }_3+{\alpha }_3\cosh {\alpha }_3-{\alpha }_3\tanh {\alpha }_3\sinh {\alpha }_3+2{\displaystyle \frac{\text{π}D}{bk{r}_1}}\cosh {\alpha }_3}.\end{array}

The solved three deflections from Eqs. (26), (41), and (42) are superimposed to obtain the total deflection. Finally, the stiffness of the enclosed RHS face subjected to two P/2 bolt forces can be calculated according to Eq. (43), where P is a process variable and we define P = 1 for simplification.

(43)k_{\mathrm{R}\mathrm{H}\mathrm{S}}=P/w_{\mathrm{R}\mathrm{H}\mathrm{S}}=1/\left(w_1+w_2+w_3\right).

3.4 Verification against ABAQUS modeling

A series of FE models was built in ABAQUS software to verify the proposed analytical equations. The FE modeling methods were firstly calibrated by the test and simulation data of specimen S20-80-6 (Fig.9(a)) from Ref. [12]. The relatively thin column wall made the effect of column face on overall stiffness more pronounced. The load–displacement comparison of test and FEM in the initial displacement is shown in Fig.9(b) with good accuracy. The modeling details and complete load–displacement comparison are reported in Subsection 4.1. The parametric FE models of enclosed RHS columns alone were established. The elastic material was used again, E = 200 GPa. The RHS columns were all taken to be square, with the RHS width, thickness and length as well as the bolt hole spacing as variables. The thickness of close-plates was taken as 16 mm, throughout. The thickness ratio of the column face and close-plate varied with the change of face thickness, which already considered the different thickness ratios. The naming rule of the FE models was R-w_RHS-L_RHS-t_w, where w_RHS indicated the width of the square RHS section, L_RHS and t_w were the length and thickness of RHS. The diameter of the bolt holes was 16 mm, consistent with the size of M16 bolts. The effect of the hole diameter on stiffness is discussed later. The pinned boundary condition was applied to the four edges of the bottom face of the enclosed RHS. Each bolt hole wall was coupled to a reference point at the hole center, and the two reference points were synchronously loaded with small vertical displacements. The sum of the reaction forces of the two reference points divided by the displacement was the objective stiffness. A schematic diagram of the modeling with representative stiffness curves and von-Mises stress distributions is shown in Fig.9(c).

90 enclosed RHS models including 2 RHS section sizes, 3 RHS thicknesses, 3 RHS length and 5 bolt hole spacing ratios were established. Due to the space limitation, part of the stiffness calculation processes by Eq. (43) are listed in Tab.5. The errors of all 90 models are plotted in Fig.10. Noted that the relative error throughout this paper was defined as: (analytical value−simulated value)/simulated value. It can be seen that the errors increased as the values of a_t/a increased, especially for a_t/a > 0.5. The larger errors were due to the stronger shear deflection near the boundary. The Timoshenko theory mainly involves the bending deflection of 2D thickness-free plate and assumes that the shear deflection was negligible. The inconsistency with the actual plate resulted in a certain degree of errors. The deviations of the deflection prediction also changed with the variation of width-thickness ratios. For the three cases of RHS with width of 140 mm and thickness of 20 mm (a/t_w = 5.0), the errors in deflection prediction were all over 20%. And for the cases of RHS with length of 320 or 450 mm and thickness of 12 mm (b/t_w ≥ 26.7), the errors were larger at higher values of a_t/a and exceeded 20% at a_t/a = 0.7. In general, with the limitation of 6.8 ≤ [max(a,b)]/t_w ≤ 24.2 and practical range of a_t/a (0.3–0.7), the analytical equations provided good predictions of initial stiffness for the enclosed RHS face. The maximum absolute error was less than 15% and the average absolute error was only 8.4%.

3.5 Discussion of the effect of bolt diameter

Different bolt hole sizes would change the rigid zone area due to the coupling of hole wall to the reference point, thus affecting the stiffness of RHS face. To quantify the effects, three enclosed RHS configurations with the largest, moderate and smallest width-thickness ratios in Subsection 3.4 were selected and modeled with bolt sizes of M12, M16, and M20. 45 FE models were built, with varying a_t/a. The stiffness results are plotted in Fig.11, which shows that when a/t_w = 100/20 (R-140-140-20) then the influence of hole sizes on the stiffness was large. Taking the M16 case as the baseline, the relative errors were between 7.2% and 9.4% with the average error of 7.9%. While for R-140-140-16 with smaller a/t_w, the relative errors were within 6% with an average value of 5.2%. For R-200-450-12 with the minimum a/t_w, the relative errors were within 4.6% with an average value of 4.0%. The case of R-140-140-20 was not applicable to the proposed equations as discussed in Subsection 3.4. To conclude, within the applicable range, the stiffnesses had slight deviations with different hole sizes. The proposed tensile stiffness equations were applicable to the enclosed RHS faces with M12–M20 bolts.

3.6 Discussions of the asymmetric cases

The above derivation and analysis of the deflection equations were based on the so-called symmetric layout, i.e., whereby the two bolt holes were located in the middle of the RHS length (b_t = 0). Many terms in the deriving process were zero, creating the possibility of a strict solution. However, asymmetric cases also existed in the beam-column joints, where the bolted connections were located off the midline of joint zone length. In the asymmetric cases, the deviation distance was denoted as b_t (η₁ = η₂ = b_t), as shown in Fig.9. After trying the derivation following the similar procedures of symmetric cases, it was found that the absence of symmetry condition made it impossible to derive explicit solutions for deflection equations. To realize the solution in the asymmetric case, it was assumed that the asymmetric equation was a modification of the symmetric equation. When observing the deflection equations Eqs. (27), (38), and (39), it was found that all equations contain A_{\mathrm{w}1} term, and when η₁ = η₂ = b_t the A_w1 term had an additional factor \cos {\displaystyle \frac{\text{π}{b}_{\mathrm{t}}}{b}}. After the fitting based on the FE results, the modification factor was taken as {\cos }^{0.6}{\displaystyle \frac{\text{π}{b}_{\mathrm{t}}}{b}}. This treatment was a combination of rigorous equations and numerical fitting. Then we obtained the expression for the asymmetric case of bolt hole as:

(44)w_{\mathrm{R}\mathrm{H}\mathrm{S},\mathrm{a}\mathrm{s}\mathrm{y}}=w_{\mathrm{R}\mathrm{H}\mathrm{S}}\cdot {\cos }^{0.6}\frac{\text{π}{b}_{\mathrm{t}}}{b}=\left(w_1+w_2+w_3\right)\cdot {\cos }^{0.6}\frac{\text{π}{b}_{\mathrm{t}}}{b}.

It can be seen that Eq. (44) is equivalent to Eq. (43) when b_t = 0. The compatibility is ensured.

36 FE models of enclosed RHS models with asymmetric location of bolt holes were built to demonstrate the applicability of Eq. (44). The models with a_t/a = 0.6 in Subsection 3.4 were selected, and the b_t/b variable were taken as 0.15, 0.25, and 0.35. The relative errors in stiffness between FE models and Eq. (44) are shown in Fig.12. The average absolute error was 9.7%, and the average error was −1.3%, with a standard deviation (SD) of 0.12. The errors of the proposed equation for the asymmetric case were slightly higher than those of the symmetric case. This was due to the fact that the modification factor {\cos }^{0.6}{\displaystyle \frac{\text{π}{b}_t}{b}} was empirically defined and adjusted. And the stiffness was also influenced by many other factors besides b_t/b. The degree of errors and deviations here were of the same magnitude as in similar studies [24,25]. The larger errors occurred in the relatively extreme case of b_t/b = 0.35 and decreased with smaller b_t/b. Therefore, it was considered that Eq. (44) could estimate the tensile stiffness of enclosed RHS face with asymmetric bolt distribution within the practical range of b_t/b.

4 Validation and parametric study of thread-fixed one-side bolt connection to enclosed rectangular hollow section

4.1 Model calibration

Reference [12] conducted tests and numerical studies on the TOB bolted endplate connection to square hollow section (SHS) column under tension. The specimens S20-120-12 and S20-80-6 were selected as the benchmarks of the FE models in this paper. The naming rule of the specimens was “Sd-a_t-t_w”, where d, a_t, and t_w represented the bolt diameter, the bolt distance, and the column wall thickness (in mm), respectively. It should be noted that the SHS columns in the reference were not the enclosed RHS columns. However, there was apparently no difference in the modeling strategies of the two, except two additional close-plates for the latter. The specimens are shown schematically in Fig.13. Two T-stubs were assembled on opposite sides of the SHS column by four TOBs without preload. For the simulation studies here, the 1/4 symmetry models were established. The same material properties according to the coupon tests were defined. After the convergency analysis of the FE mesh, as shown in Fig.13(e), the C3D8R elements were selected with moderately defined mesh density. The modeling of TOB was the same as described in Subsection 2.3. The contact between TOBs and T-stub, T-stub and column face, internal and external threads, etc. were considered, defined as hard contact in the normal direction and finite slip in the tangential direction, with a friction coefficient of 0.3 [12]. Moreover, a sensitivity analysis of the friction coefficient was conducted. The results, in Fig.13(e), indicated insensitivity to the value of friction coefficient, due to the fact that the specimens relied mainly on the tension and bending of the members to provide stiffness and bearing capacity. The symmetric boundary conditions were set. The displacement load was finally applied to the top of the T-stub.

The results of the FE models established in this paper were compared with the experimental and FE results from Ref. [12]. From the von-Mises stress distributions in Fig.13(c) and Fig.13(d), it can be seen that the stress was mainly concentrated in the threads. The load–displacement curves are shown in Fig.14. The established models matched well with the data from the reference, including the initial stiffness and bearing capacity. The reason for the slight difference may be the difference in mesh density. Through the above comparison and analysis, it was concluded that the model strategies of TOB connection to RHS column in this paper were correct and applicable.

4.2 Parametric study settings

Based on the calibrated modeling methods, a parametric modeling study of the TOB connection to enclosed RHS was conducted to validate the proposed component equations in Sections 2 and 3. This paper only involved the stiffness analysis of the TOB connection and the enclosed RHS column face. The T-stub stiffness was not the study object and its stiffness equation has been explicitly specified in EC3 [16]. Therefore, the T-stub was designed to be strong enough so that it would not be the weakest component and thus affect the validation of the total stiffness. The purpose of the FE study was to obtain the initial stiffness in the elastic phase, so the elastic property E = 200 GPa was set for all the steel materials to reduce the computational time. The dimensions of models are schematically shown in Fig.15. In each model, the T-stub was connected to the enclosed RHS by two TOBs without preloads. Based on the symmetry condition, the 1/2 models were built. The dimensions of the TOBs and threads followed the data in Fig.3 and Tab.1. The naming rule of the models was S-w_RHS-L_RHS-t_w, where w_RHS was the height and width of the square RHS, L_RHS and t_w were the length and thickness of RHS. The thicknesses of close-plates, t_c, were all set as 16 mm. The T-stub were all designed with 30 mm-thick flange, 20 mm-thick web, and 100 mm length.

4.3 Validation of the proposed component stiffness equations

In the model of T-stub connected to enclosed RHS column with TOB connection, there were three stiffness components: the T-stub (including T-stub web and flange), the TOB connection (including threaded connected part and bolt elongation) and the enclosed RHS face. The bending stiffness of the T-stub flange has been widely studied and there is no specific investigation of that here. The stiffness of the T-stub flange was calculated using the well-known component equation in Tab.6.11 of EC3-1-8 [16] (Eq. (45)). The T-stub web was calculated as an axially tensioned component. For the stiffness of a pair of TOB connections (Eq. (46)), the equation referred to the bolt elongation stiffness equation in Tab.6.11 of EC3 [16] and calculated in series with the stiffness of the threaded connected part in Eq. (17) proposed in this paper. For the enclosed RHS column face, Eq. (44) proposed in this paper was applied.

(45)k_{\text{Tstub-flange}}=\frac{0.9l_{\text{eff}}{t}_p^3}{m^3},

(46)k_{\mathrm{b}\mathrm{a}\mathrm{r}}=\frac{A_{\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{k}}}{L_b},k_{\mathrm{T}\mathrm{O}\mathrm{B}}=1.6/\left(\frac{1}{k_{\mathrm{t}\mathrm{o}\mathrm{b}}}+\frac{1}{k_{\mathrm{b}\mathrm{a}\mathrm{r}}}\right),

where m is the distance from hole center to T-stub web; L_b is the bolt elongation length [16].

First, 26 models were established for the symmetric cases. All RHS models with [max(a,b)]/t_w in the applicable range in Subsection 3.4 were selected. Tab.6 summarizes part of the analytical results compared with the FE results. The relative errors of all models are shown in Fig.16(a). The results show that the average error in the stiffness prediction of the T-stub substructure models is 9.8% with a maximum error of 15.6% and the SD is 0.04. This degree of error is similar in magnitude to that of the enclosed RHS face in Subsection 3.4. It should be noted that in the compositions of k_TOB, k_bar was always more predominant than k_tob, due to the minimum threaded depth requirement for strength guarantee. The stiffness component calculation for ordinary bolt connection in EC3-1-8 [16] only specified the bolt shank stiffness, and the threaded connection stiffness between shank and nut was considered to be infinite. However, for the TOB connection, the stiffness calculation results for different configurations, in Tab.6, showed that the ignorance of k_tob would lead to larger and unacceptable errors. The value of k_tob should be fully considered in stiffness calculation.

Next, the asymmetric cases were analyzed. In general, the difference due to asymmetry was mainly in the stiffness of the enclosed RHS face. Based on the same modeling procedures, 36 FE models of the asymmetric cases were established. The relative errors of analytical and simulation results are shown in Fig.16(b). The average error is 11.3% with a maximum error of 25.4% and the SD is 0.06. The average error is slightly higher than that of the symmetric cases. Noted that compared with the errors in Subsection 3.5 of enclosed RHS face, the error level here did not expand. In the component-based method, other components with good prediction accuracy would mitigate the effect of the stiffness prediction error from the enclosed RHS face. Referring to the similar studies [24,25] on the component stiffness, there were also cases where the maximum errors were of this magnitude. The predicted results here were considered acceptable. However, follow-up studies are needed to further improve the prediction accuracy in the asymmetric case. The stiffness combinations of k_Bolt and k_Tstub in the T-stub substructure models established here can cover most practical applications. It was concluded from the above analysis that the component stiffness equations proposed in this paper could well predict the initial stiffness of T-stub connected to enclosed RHS column with TOB connection, in both symmetric and asymmetric cases.

5 Conclusions

This paper presents new derivations of analytical equations for the stiffness of enclosed RHS column with TOB connection. In the derivation, the T-stub substructure with a row of two TOBs on the enclosed RHS face was treated as the analytical model. The derivation and verification of two unknown stiffness components, the TOB connection and the enclosed RHS column face, were solved. The derivation process and proposed stiffness equations can be a crucial contribution to the component-based method and facilitate the application of one-side bolts. Main conclusions were summarized as follows.

1) Based on the uniformly distributed simplification of contact force and equivalent treatment, the trapezoidal thread was treated as a cantilevered rectangular beam under uniform force. With the compound Simpson integration method, the equivalent inertia moment and shear area of the thread were theoretically derived as the function of pitch.

2) The bending and shearing deformations were considered to be predominant for a single thread. The bolt shank between the threads was assumed to be axially tensile. With the proposed thread-shank serial-parallel stiffness model, the stiffness of the TOB connection was well predicted.

3) The 3D enclosed RHS column under two bolt loads was simplified into a 2D plate with rotational constraints at four sides. Based on the load superposition method, the deflection of the plate was resolved into three cases and analytically derived from the plate deflection theory of Timoshenko.

4) After calibrating and validating the FE modeling methods through comparisons with test and simulation results in Ref. [12], extensive parametric FE analyses were conducted at component and substructure levels to confirm the validity of the proposed component stiffness equations for practical applications. For TOB connections with M12–M20 and enclosed RHS columns satisfying 6.8 ≤ [max(a, b)]/t_w ≤ 24.2, the analytical equations provided good predictions of initial stiffness.

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