Mechanical properties and stress-strain models of large diameter high-strength bolts

Abstract

High-strength (HS) bolts (such as grades 8.8, 10.9, 12.9) are extensively used in various types of steel structures. In certain high-altitude or coastal regions, the extremely complex and harsh climatic condition is a great challenge to the long-term serviceability and durability of HS bolts. Based on a steel television tower under construction, grade 8.8 heavy hex large diameter HS bolts were used. In order to obtain their mechanical properties and stress-strain relationships, the extensively used M36 and M48 bolts were selected to be subject to standard tensile tests. The impact of different sampling positions and surface treatment methods (whether or not hot-dip galvanizing) on the mechanical properties was taken into account. The typical mechanical performance, ductility index, bolt pre-tension and tightening torque of the large diameter HS bolts were obtained. The test results indicate that the mechanical properties of HS bolts were slightly influenced by hot-dip galvanizing and sampling positions. Based on the measured engineering stress-strain curves, the true stress-strain curves throughout the entire deformation history were obtained. Constitutive models based on an improved Ramberg-Osgood model for engineering stress-strain relations were proposed and compared with the existing constitutive models. It is recommended to use the high accurate models to simulate the stress-strain relationship in the design and modelling of HS bolts with heavy hex and large diameter.

Keywords

mechanical properties stress-strain model high-strength bolt large diameter hot-galvanizing

Introduction

With the extensive promotion and application of high-performance structural steels in high-rise and large-span steel structures, the performance of various types of steel connections has been improving. In order to extend the applications of prefabricated assembly structures and to quicken the progress of steel modular construction (SMC), large diameter HS bolts are considerably important connector in high-performance steel structures, such as in the fast assemblies of large diameter flange tubular joints and large-sized gusset plate connections. Over recent decades, HS bolted connections are constantly extending the applications of robot-driven intelligent construction in SMiC (Smart Modular-integrated Construction), which profoundly elevates the structural performance and construction efficiency.

A considerable number of scholars have carried out systematic and in-depth experimental investigations, numerical simulations, and theoretical analyses to understand the mechanical properties of HS bolts with regular diameters. The primary behaviors of bolts are tensile behavior (Debnath and Chan, 2021; Lin et al., 2024; Zhang et al., 2021, 2022, 2023a), shear behavior (Qiang et al., 2023; Zhang et al., 2023b, 2023c; Zhu et al., 2023), tensile-shear behavior (Li et al., 2020, 2023), fatigue performance (Milone et al., 2024), corrosion resistance (Chen et al., 2024; Guo et al., 2024; Li et al., 2024; Lian et al., 2023; Nie et al., 2023), impact toughness (Gao et al., 2022), pre-tension (Qiang et al., 2023; Shi et al., 2008; Zhang et al., 2023) and preload losses (Kong et al., 2022; Yang et al., 2023), etc. Besides, the mechanical properties of HS bolts under fire and post fire (Ban et al., 2021; Ketabdari et al., 2019; Kodur et al., 2017; Liu et al., 2022; Mamazizi et al., 2023; Meng et al., 2021; Saglik et al., 2024; Yang et al., 2024a) were studied and relevant formulas of reduction factors and constitutive models were proposed. Since the tensile, shear or tensile-shear failures of HS bolts frequently take place in engineering structures, quite a few scholars have been making tremendous efforts to explore these failure mechanisms. Zhang et al. (2024) conducted experimental and numerical investigations on M20, M24, M30 grade 10.9 HS bolts under monotonic tension. The combined Swift-Voce weighting model was used to characterize the material plasticity properties and the R-T model was adopted to simulate ductile fracture behaviours of HS bolts both at room/elevated temperatures. By choosing an appropriate weight factor k and suitable parameter α, the new proposed model can accurately predict the engineering stress-strain relationships of various grades of HS bolts with yield strength ranging from 600 MPa to 1040 MPa. With respect to the entire deformation history of HS bolt, it is generally recognized that the stress-strain curve are divided into three stages: linear elastic, pre-necking and post-necking stages (Ho et al., 2019, 2022; Zhang et al., 2024). At the first two stages, the engineering stress and strain can be directly converted to the true stress-strain relationship using the integration method (Ho et al., 2019). At the post-necking stage, the power law method and linear law method were proposed by Bridgeman (1952), applicable for converting the engineering stress-strain curve to the true stress-strain relationship. When the bolt coupon tests are not conducted and the engineering stress-strain curves are unavailable, Li et al. (2020) proposed the Ramberg-Osgood equations up to the necking point, and a linear equation for the descending part from necking to fracture was also proposed to construct the full-range engineering stress-strain curves. In addition, the empirical linear law method proposed by Bridgeman (1952) early in 1952, is also used to simulate the post-necking stage of the true stress-strain curves.

In general, the full-size bolt prototype or bolt group in various connections is one of the most concerned research objectives. When the bolt connection is constantly exposed to a harsh and variable condition, effective measures should be taken to protect the bolt connections from corrosion, such as applying hot-dip zinc or corrosion-resistant coatings. Previous publications on the impact of hot-dip galvanizing on the mechanical behaviors of HS bolts are rarely referred. In recent years, a thorough experimental, numerical and theoretical investigation (Firan et al., 2024) was implemented to examine the effect of heat development and transfer through bolt connections between a hot-dip galvanized beam and an intumescent-coated beam. When the HS bolt is used in the high-rise TV tower, the wind effect or earthquake action on the structure becomes an important issue that should be considered. Recent research (Yang et al., 2024b) has revealed that the strain rate significantly affects the mechanical strengths and the strain energy absorption capacities of HS bolts. Under this circumstance, a dynamic constitutive model was proposed for dynamic loading conditions such as wind or earthquake actions.

Because most of the existing design codes or standards specify relevant parameters of bolts with regular diameters, investigations on the mechanical properties, constitutive relationship, and preload design of large size HS bolts with diameter greater than 30 mm become particularly essential. In the design of a 276 m steel TV tower in Yinchuan (see Figure 1, Ningxia Province of China), grade 8.8 bolts with heavy hex and large diameter were used, instead of grade 10.9 or even 12.9 bolts with smaller diameter, which is an optimal balance of shear resistance, construction convenience, and economic saving through large diameter 8.8 grade bolts, while avoiding problems such as gusset plate thickening and increased construction difficulties caused by excessive pursuit of higher strength bolts. The frequent use of large diameter bolts in practical engineering highlights the research significance of the size effect and hot-dip galvanizing effect on the mechanical properties of bolts. In order to provide a solution to the TV tower under construction, a series of comparing experiments were carried out on the mechanical properties of grade 8.8 heavy hex HS bolts with diameter not smaller than 36 mm under hot-dip zinc protection (see Figure 2), so as to determine the mechanical properties, pre-tension value and tightening torque. Based on the measured engineering stress-strain curves, the true stress-strain curves and corresponding constitutive model for HS bolts with large size diameter were obtained. Nonlinear regression analysis and statistical analysis were performed on the model’s parameters to determine the intrinsic relationships, mathematical and physical implications. A deterministic constitutive model and related parameters for characterizing the mechanical properties of each test specimen were obtained. The material property test results and proposed stress-strain constitutive models to be applied in the finite element modelling and numerical analysis are expected to provide a theoretical basis for further analysis of the stress distribution and deformation characteristics of fully bolted connections in the TV tower structure.

Experimental program

Test specimen design

The grade 8.8 heavy hex large diameter HS bolts with pre-tention are extensively used in the 276 m tall steel TV tower. The bolt size includes M36 × 150, M42 × 170, M48 × 180, M52 × 190 (all with hot-dip galvanizing). Obviously, the increased bolt diameter can significantly contribute to the increase of tensile strength and shear resistance of the bolt, due to the expansion of section area. Besides, the pre-tension force and anti-slip capacity of HS bolts are synchronously enhanced. With the increase of bolt diameter, the contact area of the thread increases, which significantly improves the reliability of thread engagement. In addition, the stress concentration factor and the peak contact stress of bolt hole wall also decrease for larger bolt diameter. Remarkably, the rigidity of bolted-connection region using larger diameter bolts is greater than that using smaller ones, on the premise that the same number of bolts are used. It should be noted that large diameter bolts have greater single bolt bearing resistance, making them more suitable for heavy-loaded steel structures such as high-rise structures, large-span bridges and power stations; while ordinary bolts are mostly used for general steel structures/connections. Besides, large diameter bolts require higher pre-tension force normally greater than 400 kN, while for ordinary bolts, the required pre-tension force is basically smaller than 400 kN. Finally, large diameter bolts may use special alloys or heat treatment processes to avoid hardenability issues. It imposes much stricter requirements for thread accuracy and surface treatment (such as galvanizing and Dacromet) to reduce stress concentration and delay fracture risk. Taking these advantages, M36 × 150 and M48 × 180 large diameter bolts with and without hot-dip galvanization were selected to investigate the mechanical properties and stress-strain relationships. The chemical compositions of the tested HS bolts are summarized in Table 1, conforming to GB/T 3077-2015 (2016). Considering that the bolt surface temperature is normally above 420°C during the hot-dip galvanizing process, the microstructure of the hot-galvanized bolts might be changed under high temperature, leading to a substantial transformation of microstructure, which might produce remarkable impact on the fundamental mechanical properties and pre-tensions of HS bolts. Therefore, a series tensile tests of HS bolts (M36 × 150 and M48 × 180, with hot-dip galvanized (MG) and without hot-dip galvanized (M) for comparison, see Figure 3) were carried out. In addition, the material properties of hot-dip galvanized HS bolts at different locations may vary significantly due to the influence of high temperature, thus the test sampling was taken at different positions (Figure 4). The specimen sampling and comparative tests were conducted on the material mechanical properties for three different positions: the standard position (S) according to GB/T 2975-2018 (2018), the bolt rod center (C), and the bolt rod edge (E) (Figure 4). All bolts (see Figure 3) for test specimens were provided by the supplier of Yinchuan TV Tower project. The actual thicknesses of the hot-dip zinc coating are 77–89 μm (for M36 × 150) and 79–96 μm (for M48 × 180), both satisfying the requirement specified in GB/T 5267.3-2008 (2008). All tensile test specimens are standard proportional specimens sampled from the bolt rod. The diameter of the central section of specimen is 10 mm, and the original gauge length is 50 mm (i.e., 5.65

\sqrt{S_{0}}

). The specific dimensions of the test specimen are shown in Figure 5.

Table 1.

Chemical compositions of grade 8.8 heavy hex HS bolts and their assembly (% by weight).

Bolt size	Parts	Value	C	Mn	Si	S	P	Cr
M36 × 150	Bolt rod	Standard	0.37∼0.44	0.50∼0.80	0.17∼0.37	≤0.030	≤0.030	0.80∼1.10
	Bolt rod	Measured	0.39	0.60	0.22	0.005	0.012	0.88
	Nut	Standard	0.42∼0.50	0.50∼0.80	0.17∼0.37	≤0.035	≤0.035	—
	Nut	Measured	0.43	0.58	0.22	0.003	0.013	—
	Washer	Standard	0.42∼0.50	0.50∼0.80	0.17∼0.37	≤0.035	≤0.035	—
	Washer	Measured	0.48	0.66	0.24	0.008	0.013	—
M48 × 180	Bolt rod	Standard	0.37∼0.44	0.50∼0.80	0.17∼0.37	≤0.030	≤0.030	0.80∼1.10
	Bolt rod	Measured	0.39	0.70	0.24	0.005	0.015	0.94
	Nut	Standard	0.42∼0.50	0.50∼0.80	0.17∼0.37	≤0.035	≤0.035	—
	Nut	Measured	0.46	0.58	0.24	0.004	0.019	—
	Washer	Standard	0.42∼0.50	0.50∼0.80	0.17∼0.37	≤0.035	≤0.035	—
	Washer	Measured	0.50	0.59	0.22	0.002	0.015	—

The standard tensile specimens were processed according to GB/T 228.1-2021 (2021) and ISO 6892-1: 2016(E) (2016), and the nominal dimensions and realistic values are shown in Tables 2 and 3. By comparing the values in Tables 1 and 2, it is concluded that the deviations between the actual geometric dimensions and the nominal values are small. Two identical tensile specimens were prepared for each tensile testing, as shown in Table 3.

Table 2.

Geometry of standard tensile specimens for grade 8.8 heavy hex HS bolts. (mm).

Diameter	Original gauge length	Transition arc radius	Gripping diameter	Gripping length	Parallel length	Total length
d ₀	L ₀	r	d ₁	H	L _c	L _t
10	50	8	16	25	60	122.5

Table 3.

Actual dimensions of tensile specimens of grade 8.8 heavy hex HS bolts. (mm).

Specimen	d_0t (mm)	d_0m (mm)	d_0b (mm)	d₀ (mm)	L₀ (mm)	S₀ (mm²)	L_c (mm)	L_t (mm)
M36S-1	9.55	9.64	9.66	9.62	50.64	72.63	60.00	122.5
M36S-2	9.69	9.69	9.68	9.69	50.20	73.70	60.00	122.5
M36C-1	10.01	9.94	10.06	10.00	51.03	78.59	60.00	122.5
M36C-2	9.72	9.80	9.86	9.79	50.57	75.33	60.00	122.5
M36E-1	9.44	9.58	9.69	9.57	50.12	71.93	60.00	122.5
M36E-2	9.67	9.83	9.76	9.75	50.18	74.71	60.00	122.5
MG36S-1	9.71	9.73	9.78	9.74	50.61	74.51	60.00	122.5
MG36S-2	9.85	9.88	9.89	9.87	50.75	76.56	60.00	122.5
MG36C-1	9.51	9.83	9.97	9.77	49.64	74.97	60.00	122.5
MG36C-2	9.51	9.82	9.89	9.74	49.67	74.51	60.00	122.5
MG36E-1	10.00	10.05	9.97	10.01	50.41	78.64	60.00	122.5
MG36E-2	9.83	9.80	9.78	9.80	50.07	75.48	60.00	122.5
M48S-1	9.69	9.72	9.76	9.72	50.02	74.25	60.00	122.5
M48S-2	9.77	9.85	9.92	9.85	50.46	76.15	60.00	122.5
M48C-1	9.90	9.89	9.97	9.92	50.80	77.29	60.00	122.5
M48C-2	9.59	9.61	9.67	9.62	50.53	72.73	60.00	122.5
M48E-1	9.71	9.75	9.79	9.75	50.01	74.66	60.00	122.5
M48E-2	9.76	9.79	9.81	9.79	50.52	75.22	60.00	122.5
MG48S-1	9.75	9.79	9.85	9.80	50.57	75.38	60.00	122.5
MG48S-2	9.67	9.75	9.85	9.76	50.44	74.76	60.00	122.5
MG48C-1	10.14	9.85	10.04	10.01	50.39	78.70	60.00	122.5
MG48C-2	10.14	10.18	10.00	10.11	50.11	80.22	60.00	122.5
MG48E-1	10.01	9.94	10.06	10.00	51.03	78.59	60.00	122.5
MG48E-2	9.66	9.69	9.72	9.69	49.83	73.75	60.00	122.5

Note. M36S: 36 denotes that the nominal diameter of HS bolt is 36 mm, S denotes that the sampling position is standard position of screw; M36C-1: C denotes that the sampling position is central position of screw, −1 denotes the first parallel test specimen; M36E-1: E denotes that the sampling position is edge position of screw; MG36C-1: G denotes that the test specimen is sampled from the hot-dip galvanized bolt; d_0t、d_0m、d_0b denote the diameters of the top, middle, bottom locations of the original gauge range; d₀ denotes the average diameter of the original gauge range, taken as the average value of d_0t, d_0m and d_0b; S₀ denotes the average sectional area of the original gauge range, calculated by $S_{0} = π d_{0}^{2} / 4$ .

Test setup

The tensile test of the mechanical properties of HS bolts was conducted by an electro-hydraulic servo universal testing machine in the Structural Laboratory of Tsinghua University. The loading capacity of the testing machine is 500 kN, and the test setup is shown in Figure 6. Due to the small diameter of the gripping ends of the tensile specimen, a pair of rigid sleeves that matches the M16 threads of the gripping ends was designed, so as to facilitate the testing machine to continuously impose stable tensile force and frictional force to the specimen. During the experiment, the gripping parts of the specimen were screwed into the sleeve before the application of tensile force and frictional force on the sleeve, as shown in Figure 7. The axial deformation and tensile strains during the loading process were synchronously measured by extensometer and strain gauges. The extensometer was clamped at the upper and lower scale lines of the original gauge scope, with measuring range of 20 mm. Becuase the original gauge length calculated by 5.65 $\sqrt{S_{0}}$ according to GB/T 228.1-2021 (2021) and ISO 6892-1: 2016(E) (2016) was taken as 50 mm, the strain range converted from the extensometer’s allowable measuring scope to the calculated original gauge length was 40%. It is believed that the accuracy of the initial linear ascending part measured by strain gauge is generally higher than that by extensometer. In order to correct the deviation of the measurement results by extensometer due to experimental uncertainties and system errors during the initial loading stage, two strain gauges were arranged along the length direction at the center of the original gauge length (both at the same height on opposite sides) to synchronously monitor and record the axial deformation. The arrangement diagram of the strain gauges and extensometer is shown in Figure 8. The real-time outputs of strain gauge, extensometer displacement and axial force were recorded. When the axial deformation over the original gauge length of the specimen approached the extensometer measurement limit, the extensometer was removed and the axial loading continued until the fracture of the specimen.

Standard test method

According to the standard tensile testing methods GB/T 228.1-2021 (2021) and ISO 6892-1: 2016(E) (2016), the mechanical properties of grade 8.8 HS bolts were obtained. To ensure the physical alignment between the specimen and the gripping end, a 5% pre-tension of the standard yield strength was applied to the specimen prior to the formal loading. The minimum elongation of bolt rod after fracture was approximate to 12% without pronounced yield plateau, thus the strain rate control method (Method A) was adopted before removing the extensometer. A recommended strain rate of 0.00025s^-1 was selected for loading, so as to obtain the proof strength at a specified plastic strain (R_p) and the percentage yield point extension (A_e). Regarding the measurements of the tensile strength (R_m), percentage elongation after fracture (A) and percentage reduction area (Z), the estimated strain rate over the parallel length was used, which was achieved by using the crosshead separation rate calculated by multiplying the required strain rate to the parallel length (Method A2 open loop) according to GB/T 228.1-2021 (2021), and a recommended strain rate of 0.0067s⁻¹ wa selected for loading.

Test observation

All test specimens experienced noticeable necking stage on the verge of tensile fracture, as shown in Figure 9(a). After the specimen was fractured to failure, the cross section appeared rough and uneven at the micro-level, but almost perpendicular to the length direction (see Figure 9(b)). Most specimens fractured within the original gauge range, with only a few cases between the parallel and original gauge ranges. The typical failure characteristics are shown in Figure 9(c). The fracture phenomena of all specimens were extremely similar as shown in Figure 10. The fracture locations occurred almost at the same position of the gauge length, which was caused by a unified machine processing deviation, that is, the original diameter at the failure location was unexceptionally slightly smaller than those at other locations.

Experimental results and analyses

Stress-strain curves

The measured engineering stress-strain curves of all test specimens are shown in Figure 11 (denoted with solid lines). All specimens did not show apparent yield plateau, thus the nominal yield strength can be defined as the proof strength R_p0.2 at a specified 0.2% plastic strain. Because all specimens in standard tensile tests are proportional cylindrical coupons, the true stress-strain relationship can be readily obtained by converting the engineering stress-strain curve via the following typical models:

Throughout the entire deformation history (Bridgeman, 1952; Ho et al., 2019, 2022):

ε_{t} = In (1 + ε_{e})

(1)

At the linear elastic and pre-necking stages (Bridgeman, 1952; Ho et al., 2019, 2022):

σ_{t} = σ_{e} (1 + ε_{e}), ε_{t} \leq ε_{n}

(2)

At the post-necking stage (Bridgeman, 1952; Ho et al., 2019):

power model : σ_{t} = σ_{n} {(ε_{t} / ε_{n})}^{ε_{n}}, ε_{t} > ε_{n}

(3)

linear model:

\begin{array}{l} σ_{t} = a + b ε_{t}, ε_{t} > ε_{n} \\ a = σ_{n} (1 - ε_{n}), b = σ_{n} \end{array}

(4)

Where σ_t and ε_t are the true stress and true strain respectively, σ_e and ε_e are the engineering stress and engineering strain respectively, σ_n and ε_n are the true stress and true strain at the onset of necking respectively. For cylindrical coupons, it is generally believed that the onset of necking (ε_n, σ_n) occurs at the ultimate/peak stress point (ε_t,u, σ_t,u) due to Considere’s criterion (Bridgeman, 1952; Ho et al., 2019, 2022; Saboori et al., 2015; Zhang et al., 2024). Therefore, ε_n = ε_t,u, σ_n = σ_t,u, where ε_t,u is the true strain corresponding to the true tensile strength σ_t,u. Besides, at the post-necking stage, the simulation effect by using the power model is almost the same as that by using the linear model, as shown in Figure 12. The true and engineering stress-strain curves throughout the entire deformation history for all test specimens are shown in Figure 11. The deviations of engineering strain/stress from the true strain/stress up till the peak stress are shown in Figure 13. Obviously, through the transformation of equations (1) and (2), the true strain is smaller than the engineering strain, while the true stress is larger than the engineering stress. Besides, Figures 11 and 13 indicate that the proof strength obtained from the engineering stress-strain relationship is almost the same as that from the true stress-strain curve, but the true peak stress (tensile strength) is significantly higher than the engineering value. Furthermore, the true stress-strain curve throughout the entire deformation history did not show descending part, because the necking effect of the cross section was taken into account at the post-necking stage. According to the engineering stress-strain curves, the elongation over the gauge length (50 mm) measured by the extensometer was generally varying from 5% to 15%. When the average strain over the gauge length reached 5%, the axial tensile force almost attained the tensile strength. Therefore, it is concluded that the ultimate strains of all test specimens are around 5%. The primary mechanical properties such as Young’s modulus, proof strength, tensile strength, elongation after fracture and percentage reduction area can be determined from the engineering stress-strain curve.

Young’s modulus

According to Appendix D of GB/T 228.1-2021 (2021), the Young’s modulus of HS bolt specimens can be obtained by regression analysis of the linear segment of the stress-strain curve, as shown in Table 4 and Figure 14. The average Young’s moduli of tested specimens are between 202 GPa and 220 GPa. For MG36E specimens, the average Young’s modulus is lower than that without hot-dip galvanizing, implying that the Young’s modulus at the edge of the HS bolt may be degraded due to the thermal influence of hot-dip galvanizing. The average Young’s moduli of MG48S + MG48C specimens are also lower than their corresponding values of M48 specimens, while MG36S + MG36C specimens present the reverse case, indicating that the hot-dip galvanizing on the Young’s modulus at standard or central positions has little influence. Furthermore, for M36 and M48 specimens, the Young’s moduli sampled at the edge position are significantly higher than those sampled at the central or standard positions. However, for MG36 specimens, the thermal effect significantly weakens the Young’s modulus at the edge position, resulting in lower values than those at the central or standard positions, but MG48 specimens show the opposite effect. In addition, by comparing the test results obtained from specimens sampled at the standard position (see the numbers in brackets in Table 4, for example, 0.6% in bracket is obtained via (204,088–202,893)/202,893 × 100% = 0.6%, −5.3% in bracket is obtained via (202,976–214,323)/214,323 × 100% = −5.3%), it can be seen that except the sampling at the edge of M36 and MG36, the Young’s moduli sampled at different positions have smaller differences from the test results obtained from sampling at the standard position. In summary, the average test results of the Young’s modulus for two parallel specimens vary from 202 GPa to 220 GPa, falling within a reasonable range. For an individual tensile specimen, the minimum and maximum values of Young’s modulus are 199 GPa and 232 GPa, respectively. The deviations are caused by many factors, such as machining errors, different sampling positions, whether or not galvanizing, measurement inaccuracies, systematic errors, etc. Therefore, it is a complex issue that can hardly be determined by a single influencing factor. But reverse analysis can be conducted to find out the potential influencing factors to explain the test results, as elaborated above.

Table 4.

Measured Young’s modulus of standard tensile specimens.

Sampling position	Specimen	Young’s modulus E_b (MPa)	Mean value E_b,ave (MPa)	Specimen	E_b (MPa)	E_b,ave (MPa)
Standard position	M36S-1	206,196	202,893	M48S-1	208,390	210,028
	M36S-2	199,590	202,893	M48S-2	211,666	210,028
	MG36S-1	209,001	214,323	MG48S-1	209,428	208,060
	MG36S-2	219,645	214,323	MG48S-2	206,691	208,060
Central position	M36C-1	199,336	204,088 (0.6%)	M48C-1	219,668	209,501 (−0.3%)
	M36C-2	208,840	204,088 (0.6%)	M48C-2	199,333	209,501 (−0.3%)
	MG36C-1	208,698	210,946 (−1.6%)	MG48C-1	209,548	208,260 (0.1%)
	MG36C-2	213,193	210,946 (−1.6%)	MG48C-2	206,971	208,260 (0.1%)
Edge position	M36E-1	231,782	220,069 (8.5%)	M48E-1	214,835	215,494 (2.6%)
	M36E-2	208,355	220,069 (8.5%)	M48E-2	216,153	215,494 (2.6%)
	MG36E-1	200,066	202,976 (−5.3%)	MG48E-1	212,303	212,532 (2.1%)
	MG36E-2	205,886	202,976 (−5.3%)	MG48E-2	212,761	212,532 (2.1%)

Proof strength

Due to the absence of yield plateau for all experimental stress-strain curves, the proof strength (R_p0.2) can be determined in accordance with the method defined in Appendix J of GB/T 228.1-2021 (2021), so as to provide the nominal yield strength of the grade 8.8 heavy hex HS bolt. The results are summarized in Table 5, and the comparison is shown in Figure 15. All the average proof strengths are between 700 MPa and 860 MPa, unexceptionally larger than 660 MPa specified in GB/T 3098.1-2010 (2011). In addition, the experimental results indicate that the proof strengths of almost all specimens are larger than the values given in the Quality Inspection Report (varying from 712 MPa to 723 MPa), and the average proof strengths of M(G)36S and M(G)48S are 727 MPa and 753 MPa, respectively. The experimental proof strengths of all test specimens basically meet the quality requirements. As shown in Figure 15(b), the proof strengths of M48, MG48, M48S, M(G)36, M(G)48, and all 24 specimens are around 750 MPa. Only the values of M36 and M36S are significantly smaller, while MG36 is significantly larger. For MG36E and MG48E hot-dip galvanized bolts sampled at the edge of the bolt rod, the average proof strengths are larger than those without hot-dip galvanizing, indicating that the proof strength at the edge of the bolt is improved by the influence of hot-dip galvanizing (see Table 5). For all M36 specimens, hot-dip galvanizing significantly increases the proof strengths, and the improvement extent decreases as the sampling position approaches the center of the bolt rod. For M48 bolts sampled at standard or central positions, the opposite case occurs, indicating that hot-dip galvanizing enhances the proof strength at the edge position. The reason and mechanism have not been explored. The test results of the specimens sampled from three different positions show that the smaller proof strengths occur in the specimens sampled at the central position, indicating that the proof strengths obtained from the specimens sampled at the central position are more conservative than those sampled at the standard and edge positions of screws. Generally, hot-dip galvanizing has a significant strengthening effect on the proof strength of M36 specimens, but its impact on the proof strengths of M48 specimens are relatively marginal. The proof strength shows a gradually increasing trend as the sampling location is far away from the center of the bolt rod.

Table 5.

Measured proof strengths of standard tensile specimens.

Sampling position	Specimen	Proof strength R_p0.2 (MPa)	Mean value R_p0.2,ave (MPa)	Specimen	R_p0.2 (MPa)	R_p0.2,ave (MPa)
Standard position	M36S-1	708.2	708.8	M48S-1	776.4	763.2
	M36S-2	709.4	708.8	M48S-2	750.0	763.2
	MG36S-1	749.4	744.9	MG48S-1	738.2	743.3
	MG36S-2	740.3	744.9	MG48S-2	748.5	743.3
Central position	M36C-1	684.9	702.1 (−0.9%)	M48C-1	757.4	744.0 (−2.5%)
	M36C-2	719.3	702.1 (−0.9%)	M48C-2	730.5	744.0 (−2.5%)
	MG36C-1	761.3	744.2 (−0.1%)	MG48C-1	759.4	726.7 (−2.2%)
	MG36C-2	727.0	744.2 (−0.1%)	MG48C-2	694.0	726.7 (−2.2%)
Edge position	M36E-1	790.6	769.2 (8.5%)	M48E-1	737.7	746.4 (−2.2%)
	M36E-2	747.8	769.2 (8.5%)	M48E-2	755.0	746.4 (−2.2%)
	MG36E-1	925.8	856.2 (14.9%)	MG48E-1	701.6	777.3 (4.6%)
	MG36E-2	786.5	856.2 (14.9%)	MG48E-2	852.9	777.3 (4.6%)

Tensile strength and yield ratio

The tensile strengths and yield ratios for all test specimens are shown in Table 6, and the comparison is shown in Figure 16. The tensile strengths of all specimens are greater than 830 MPa specified in GB/T 3098.1-2010 (2011). The yield ratios of all specimens are between 0.80 and 0.86, satisfying the requirement of GB/T 3098.1-2010 (2011). The average tensile strength of all specimens is 909 MPa, approximate to the values (varying from 930 MPa to 940 MPa) provided by the Quality Inspection Report. As shown in Figure 16(b), the tensile strengths of M48, MG48, M48S, M(G)36, M(G)48 and all 24 specimens are around 910 MPa. Only the values of M36 and M36S are significantly smaller, while MG36 is slightly larger. For MG36 and MG48 specimens sampled at the edge position, the average tensile strengths are greater than those without hot-dip galvanizing, indicating that the tensile strength at the edge position of the HS bolt is slightly improved by hot-dip galvanizing. For all M36 specimens sampled from different positions, hot-dip galvanizing increases the tensile strength at varying extents, and the value taken from the edge position is severely affected by hot-dip galvanizing. For M48 specimens sampled at standard or central positions, the opposite situation occurs, indicating that hot-dip galvanizing increases the tensile strength at the edge position, but weakens the tensile strength at both standard and central positions. It is noted that the smallest tensile strength basically occurs in the specimens sampled at the central position (except M48E). It is concluded that the tensile strength obtained from the central position is more conservative than those obtained from the standard position and edge position. Overall, the hot-dip galvanizing has a strengthening effect on the tensile strengths of M36 specimens sampled at the edge position, but its impact on the tensile strengths of other specimens is negligible. In addition, by comparing the test results obtained from those sampled at the standard position (see the numbers in parentheses in Table 6), it can be seen that except M36 and MG36 specimens sampled at the edge position, the tensile strengths of specimens sampled at different positions slightly differ from the test results obtained from the ones sampled at the standard position, but the difference is insignificant (the deviations are all within 5%). That is, the tensile strength sampled at the central position is the smallest, followed by the standard position and the edge position. It should be noted that the yield ratio of specimen sampled at the edge position is basically enhanced due to the high temperature effect of hot-dip galvanizing. Therefore, it is not recommended to use the specimens sampled at the edge position to obtain the mechanical properties of bolts. Considering that both the yield and tensile strengths of the specimens taken at the central position are the smallest, the most reasonable sampling position for large diameter HS bolt remains at the standard position, which is consistent with GB/T 2975-2018 (2018).

Table 6.

Measured tensile strengths and yield ratios of standard tensile specimens.

Sampling position	Specimen	Tensile strength R_m (MPa)	Mean value R_m,ave (MPa)	Yield ratio	Specimen	R_m (MPa)	R_m,ave (MPa)	Yield ratio
Standard position	M36S-1	869.1	870.2	0.81	M48S-1	926.6	919.7	0.84
	M36S-2	871.3	870.2	0.81	M48S-2	912.7	919.7	0.82
	MG36S-1	904.0	898.7	0.83	MG48S-1	892.9	900.0	0.83
	MG36S-2	893.3	898.7	0.83	MG48S-2	907.2	900.0	0.83
Central position	M36C-1	842.8	851.3 (−2.2%)	0.81	M48C-1	922.5	918.4 (−0.1%)	0.82
	M36C-2	859.8	851.3 (−2.2%)	0.84	M48C-2	914.4	918.4 (−0.1%)	0.80
	MG36C-1	902.6	886.3 (−1.4%)	0.84	MG48C-1	918.0	888.2 (−1.3%)	0.83
	MG36C-2	870.1	886.3 (−1.4%)	0.84	MG48C-2	858.4	888.2 (−1.3%)	0.81
Edge position	M36E-1	940.8	919.1 (5.6%)	0.84	M48E-1	905.7	911.8 (−0.9%)	0.81
	M36E-2	897.3	919.1 (5.6%)	0.83	M48E-2	917.9	911.8 (−0.9%)	0.82
	MG36E-1	1079.1	1001.6 (11.5%)	0.86	MG48E-1	874.4	939.5 (4.4%)	0.80
	MG36E-2	924.1	1001.6 (11.5%)	0.85	MG48E-2	1004.6	939.5 (4.4%)	0.85

Post fracture properties

The post fracture parameters of all test specimens are shown in Table 7, including the post fracture length, post fracture diameter, post fracture cross-sectional area, percentage elongation and percentage reduction area. It can be seen that the percentage elongations after fracture for all test specimens are not less than 12% as required by GB/T 3098.1-2010 (2011), and the percentage reduction area is generally higher than the minimum requirement of 52% in GB/T 3098.1-2010 (2011) (M48E-1 is slightly lower). Different bolt sizes, sampling positions, and surface treatment methods (whether or not hot-dip galvanizing) have a certain impact on the post fracture percentage elongation and percentage reduction area.

Table 7.

Measured post fracture percentage elongation and area reduction of standard tensile specimens.

Sampling position	Specimen	Post-fracture length L₁ (mm)	Post-fracture diameter d₁ (mm)	Post-fracture cross-section area S₁ (mm²)	Percentage elongation A (%)	Mean value A_ave (%)	Area reduction Z (%)	Mean value Z_ave (%)
Standard position	M36S-1	58.23	5.93	27.62	15.0	14.9	62.0	60.0
	M36S-2	57.64	6.27	30.88	14.8	14.9	58.1	60.0
	MG36S-1	57.38	6.18	30.00	13.4	14.4	59.7	61.8
	MG36S-2	58.54	5.93	27.62	15.4	14.4	63.9	61.8
Central position	M36C-1	60.71	6.18	30.00	19.0	16.3	61.8	61.1
	M36C-2	57.49	6.17	29.90	13.7	16.3	60.3	61.1
	MG36C-1	55.60	5.95	27.81	12.0	12.2	62.9	62.6
	MG36C-2	55.87	5.98	28.09	12.5	12.2	62.3	62.6
Edge position	M36E-1	57.08	6.30	31.17	13.9	15.0	56.7	58.5
	M36E-2	58.23	6.15	29.71	16.0	15.0	60.2	58.5
	MG36E-1	58.95	5.92	27.53	16.9	19.5	65.0	62.6
	MG36E-2	61.13	6.19	30.09	22.1	19.5	60.1	62.6
Standard position	M48S-1	56.86	6.26	30.78	13.7	13.9	58.6	59.2
	M48S-2	57.55	6.24	30.58	14.1	13.9	59.8	59.2
	MG48S-1	57.41	6.13	29.51	13.5	14.0	60.9	57.8
	MG48S-2	57.71	6.57	33.90	14.4	14.0	54.7	57.8
Central position	M48C-1	58.21	6.65	34.73	14.6	14.7	55.1	54.3
	M48C-2	58.00	6.56	33.80	14.8	14.7	53.5	54.3
	MG48C-1	56.64	6.79	36.21	12.4	13.6	54.0	57.4
	MG48C-2	57.53	6.33	31.47	14.8	13.6	60.8	57.4
Edge position	M48E-1	58.40	7.06	39.15	16.8	15.2	47.6	51.9
	M48E-2	57.41	6.48	32.98	13.6	15.2	56.2	51.9
	MG48E-1	60.71	6.35	31.67	19.0	17.1	59.7	59.6
	MG48E-2	57.41	6.17	29.90	15.2	17.1	59.5	59.6

Bolt pre-tension and tightening torque

The pre-tension of grade 8.8 heavy hex HS bolts with large diameters of 36 mm and 48 mm are not specified in the current standards GB 50017-2017 (2018) and JGJ 82-2011 (2011), thus it is essential to determine their pre-tensions and torques based on the test results. According to the Chinese standard GB 50017-2017 (2018), the design value P of the pre-tension force for HS bolts is based on the tensile strength of the bolts, reduction factors and net cross-section of the threaded bolt rods according to GB 50017-2017 (2018). The pre-tension value of HS bolts is given by the following expression:

P = \frac{α_{1} α_{2} α_{3}}{β_{1}} f_{u} A_{e} = \frac{0.9 \times 0.9 \times 0.9}{1.2} f_{u} A_{e} = 0.6075 f_{u} A_{e}

(5)

Where α₁ is the reduction factor considering the non-uniformity of bolt material, taken as 0.9. α₂ is the over tensioning coefficient in order to compensate for the relaxation of bolt pre-tension during construction, taken as 0.9. β₁ is the impact factor of contact surface lubricant on stress when considering bolt tightening, taken as 1.2. f_u is the minimum value of the tensile strengths of HS bolts (MPa). For grade 8.8 heavy hex HS bolts, the value of R_m can be taken from the test results. A_e is the net cross-sectional area at the thread (mm²), calculated by

A_{e} = π d_{e}^{2} / 4

, where d_e is the effective diameter at the thread, d_e = d - 0.9382p, p is the pitch of HS bolts.

The standard tensile strength of grade 8.8 HS bolts is taken as 830 MPa, according to equation (5), the standard values of pre-tension for grade 8.8 HS M36 and M48 bolts are 412 kN and 743 kN, respectively. Based on the experimental results and equation (5), the values of pre-tension P of grade 8.8 heavy hex HS bolts are given in Table 8. For M36 and M48 bolts sampled at different positions, the pre-tension values vary from 420 kN to 460 kN and from 815 kN to 825 kN, respectively, while the pre-tension values of MG36 and MG48 of hot-dip galvanized bolts vary from 440 kN to 500 kN and from 790 kN to 850 kN. It is noted that the pre-tension values obtained from the edge position sampling are significantly higher than those obtained from the standard and central position. Hence, for safety’s sake, the test results obtained from the standard position or central position can be used to determine the pre-tension value. The MG36 specimens sampled at the standard or central positions present slightly higher pre-tension values than those of M36 specimens sampled at the same position, while M48 and MG48 specimens show the opposite case. In summary, it is recommended that the pre-tension value of 420 kN can be used for bolt M36 bolts and MG36 bolts, which is higher than the standard value of 410 kN obtained from the Quality Inspection Report. The recommended pre-tension value is 790 kN for both M48 bolts and MG48 bolts, higher than the standard value of 740 kN obtained from the Quality Inspection Report. It is indicated that using standard values as the pre-tension values for hot-dip galvanized MG36 and hot-dip galvanized MG48 bolts is conservative and safe in practical engineering design.

Table 8.

HS bolt pre-tension calculated from the standard uniaxial tensile test.

Sampling position	Specimen	Bolt diameter d (mm)	d_e (mm)	A_e (mm²)	f_u (MPa)	P (kN)	Mean value P_ave (kN)
Standard position	M36S-1	36	32.247	816.7	869.1	431	432
	M36S-2	36	32.247	816.7	871.3	432	432
	MG36S-1	36	32.247	816.7	904.0	449	446
	MG36S-2	36	32.247	816.7	893.3	443	446
Central position	M36C-1	36	32.247	816.7	842.8	418	422
	M36C-2	36	32.247	816.7	860.6	427	422
	MG36C-1	36	32.247	816.7	902.6	448	440
	MG36C-2	36	32.247	816.7	870.1	432	440
Edge position	M36E-1	36	32.247	816.7	940.8	467	456
	M36E-2	36	32.247	816.7	897.3	445	456
	MG36E-1	36	32.247	816.7	1079.1	535	497
	MG36E-2	36	32.247	816.7	924.1	458	497
Standard position	M48S-1	48	43.309	1473.1	926.6	829	823
	M48S-2	48	43.309	1473.1	912.7	817	823
	MG48S-1	48	43.309	1473.1	892.9	799	805
	MG48S-2	48	43.309	1473.1	907.2	812	805
Central position	M48C-1	48	43.309	1473.1	922.5	826	822
	M48C-2	48	43.309	1473.1	914.4	818	822
	MG48C-1	48	43.309	1473.1	918.0	822	795
	MG48C-2	48	43.309	1473.1	858.4	768	795
Edge position	M48E-1	48	43.309	1473.1	905.7	811	816
	M48E-2	48	43.309	1473.1	917.9	821	816
	MG48E-1	48	43.309	1473.1	874.4	782	841
	MG48E-2	48	43.309	1473.1	1004.6	899	841

The tightening process of heavy hex HS bolts with large diameter includes initial tightening and final tightening. The torque of final tightening T_c is given by:

T_{c} = k P d

(6)

Where k is the average torque coefficient of the HS bolt assembly, obtained from the Quality Inspection Report. P is the pre-tension of HS bolt, taken as the average value of HS specimen sampled at the standard position as shown in Table 9. d is the nominal diameter of the HS bolt. The pre-tension and final tightening torque are summarized in Table 9. Obviously, the tightening torques of grade 8.8 HS bolts with large diameters of 36 mm and 48 mm are significantly greater than those with regular diameters specified in JGJ 82-2011 (2011).

Table 9.

Calculated pre-tension and tightening torque of grade 8.8 heavy hex HS bolts with large diameter.

Bolt size	d (mm)	d_e (mm)	A_e (mm²)	k	P (kN)	T_c (N⋅m)
M36	36	32.247	816.7	0.119	432	1851
MG36	36	32.247	816.7	0.119	446	1911
M48	48	43.309	1473.1	0.121	823	4780
MG48	48	43.309	1473.1	0.121	805	4675

Constitutive model

The establishment of the stress-strain constitutive model is critically important for the design and numerical modelling of HS bolts. Based on the measured stress-strain curves, conventional Ramberg-Osgood model as well as Ramussen model and its improved form, a more appropriate stress-strain constitutive model was developed and proposed with adequate accuracy.

Ramberg-Osgood model

The Ramberg-Osgood model (Ramberg and Osgood, 1943) is considered suitable for simulating the non-linear characteristics of metallic materials such as stainless steels and aluminum alloys, expressed as:

ε = \frac{σ}{E_{0}} + 0.002 {(\frac{σ}{R_{p 0.2}})}^{n}

(7)

Where σ, ε are stress and strain. E₀ is the initial Young’s modulus. R_p0.2 is the 0.2% proof stress. n is the strain hardening exponent which determines the sharpness of the knee of the stress-strain curve, and takes the following expressions:

n = \frac{In (20)}{In (R_{p 0.2} / R_{p 0.01})}, for stainless steels

(8)

n = \frac{In (2)}{In (R_{p 0.2} / {R_{p}}_{0.1})}, for aluminum alloys

(9)

Where R_p0.01 and R_p0.1 are 0.01% and 0.1% proof stresses. The Ramberg-Osgood model was developed on the premise that the total strain (ε) is a linear summation of elastic strain (ε_e) and plastic strain (ε_p). Therefore, the total strain ε_0.2 (including 0.002 plastic strain and all elastic strain) can be expressed by (see Figure 17):

ε_{0.2} = \frac{R_{p 0.2}}{E_{0}} + 0.002

(10)

It should be noted that, for aluminium alloys and stainless steels, the Ramberg-Osgood model can accurately simulate the stress-strain relationship within ε_0.2. However, it becomes seriously diverse when the stress reaches beyond R_p0.2, because the extrapolations of Ramberg-Osgood model to over R_p0.2 show significant deviations due to distinct strain hardening for nonlinear metals (Rasmussen, 2003). Nevertheless, for HS bolts, even the stress is lower than R_p0.2, large discrepancy still occurs, as shown in Figure 17. The discrepancy mainly occurs in a large corner region of the entire stress-strain curve, if equation (7) is applied over the full strain range up to the ultimate stress.

Rasmussen model and its revised form

In order to improve the simulating accuracy of the experimental stress-strain curves for stainless steel alloys, Rasmussen (2003) developed a two-stage constitutive model based on the Ramberg-Osgood model:

ε = {\begin{cases} \frac{σ}{E_{0}} + 0.002 {(\frac{σ}{R_{p 0.2}})}^{n}, for σ \leq R_{p 0.2} \\ ε_{0.2} + \frac{σ - R_{p 0.2}}{E_{0.2}} + ε_{u} {(\frac{σ - R_{p 0.2}}{R_{m} - R_{p 0.2}})}^{m}, for R_{p 0.2} < σ \leq R_{m} \end{cases}

(11)

Where the strain hardening exponent n and 0.2% total strain ε_0.2 are given by equations (8) and (10), respectively. ε_u is the ultimate total strain upon the tensile strength R_m. m is the second-stage strain hardening exponent dependent on the yield ratio and regressed on the basis of test data given as

m = 1 + 3.5 R_{p 0.2} / R_{m}

. E_0.2 is the initial tangent modulus at the 0.2% proof stress, which is obtained from the derivative with respect to ε in equation (7) and takes as:

E_{0.2} = {\frac{d σ}{d ε} |}_{σ = R_{p 0.2}} = \frac{E_{0}}{1 + 0.002 n / e}

(12)

Where e is the nondimensional proof stress defined as

e = R_{p 0.2} / E_{0}

Evidently, the first stage of Rasmussen model is identical to the Ramberg-Osgood expression, retaining its original accuracy and applicable within R_p0.2. The second stage significantly improves the accuracy of simulation from R_p0.2 up to R_m, on the basis of continuity condition at (ε_0.2, R_p0.2). By comparison with a wide range of test data, Rasmussen model is demonstrated as reasonably accurate for all structural stainless steel alloys (Rasmussen, 2003), despite that the entire stress-strain curve does not strictly pass through (ε_u, R_m) (Shi and Zhu, 2017). However, when Rasmussen model is applied to grade 8.8 HS bolts (see Figure 18), large discrepancy still exists around the corner region of the stress-strain curve, with 90.74% and 85.32% goodnesses of fit at each stage, respectively.

On the basis of Rasmussen model, Shi and Zhu (2017) constructed an improved nonlinear stress-strain model applicable to HS steels without apparent yield plateau:

ε = {\begin{cases} \frac{σ}{E_{0}} + 0.002 {(\frac{σ}{R_{p 0.2}})}^{n}, for σ \leq R_{p 0.2} \\ ε_{0.2} + \frac{σ - R_{p 0.2}}{E_{0.2}} + (ε_{u} - \frac{R_{m} - R_{p 0.2}}{E_{0.2}} - ε_{0.2}) {(\frac{σ - R_{p 0.2}}{R_{m} - R_{p 0.2}})}^{m}, for R_{p 0.2} < σ \leq R_{m} \end{cases}

(13)

Where E₀, R_p0.2, R_m and ε_u are obtained from tensile experiment or taken from Table 3 (Shi and Zhu, 2017), if experimental data are unavailable. n and E_0.2 are given by equations (8) and (12). The second-stage strain hardening exponent m is regressed on the basis of experimental data taken as

m = 25.202 - 24.647 R_{p 0.2} / R_{m}

. Shi and Zhu (2017) concluded that, for HS steels, the strain hardening exponents n and m are dependent on R_p0.01/R_p0.2 and R_p0.2/R_m, respectively. The distinction between Shi model and Rasmussen model exists that the former strictly passes through (ε_u, R_m), and to some extent, improves the accuracy of curve fitting for HS steels. Comparison of the test result and proposed stress-strain curves based on Rasmussen model and Shi model for M36S-1 of HS bolt is shown in Figure 18. The curve fitting accuracy of Shi model applied to the second stage (from R_p0.2 to R_m) reaches 96.88%, much higher than that of Rasmussen model (85.32%), but large divergence still exists when the strain goes beyond ε_0.2.

Constitutive model for HS bolts

As discussed above, the existing stress-strain models applied to HS bolts are less accurate around the knee of the stress-strain curve, irrespective of Ramberg-Osgood model, Rasmussen model or Shi model. Based on Shi model, a further modified stress-strain model for HS bolts is proposed as follows:

ε = {\begin{cases} \frac{σ}{E_{0 b}} + α {(\frac{σ}{R_{p 0.2}})}^{n}, for σ \leq R_{p 0.2} Stage 1 \\ ε_{0.2} + \frac{σ - R_{p 0.2}}{χ E_{0.2 b}} + β {(\frac{σ - R_{p 0.2}}{R_{m} - R_{p 0.2}})}^{m}, for R_{p 0.2} < σ \leq R_{m} Stage 2 \end{cases}

(14)

Where E_0b is the initial Young’s modulus for grade 8.8 HS bolts. R_p0.2 is the 0.2% proof stress (equivalent to R_p0.2). n and m are the strain hardening exponents. α and β are the regression parameters from experimental data. R_m is the ultimate tensile strength (equivalent to R_m). E_0.2b (equivalent to E_0.2) is the bolt initial tangent modulus of the stress-strain curve at R_p0.2, given by equation (12). χ is the shape optimization factor for improving the accuracy of the curve simulation at Stage 2, in view of the sharp change in slope around the knee region of the stress-strain curve (all experimental values of E_0.2b/E_0b are less than 0.1, leading to n > 10, see Tables 10 and 11). Rasmussen model for stainless steel alloys is basically applicable to the stress-strain curve with blunt knee (for most cases n ≤ 10), because the initial tangent modulus E_0.2 is introduced in developing the second-stage stress-strain curve between 0.2% proof stress and the ultimate tensile strength, considering that the stress-strain curve in this range is similar in shape to that of the first stage (Rasmussen, 2003). In the case of the n-value greater than 10 (with remarkable sharp knee), the maximum difference between the measured stress-strain curve and Rasmussen model would be larger. Therefore, it is possible that minimization of this difference requires the addition of at least one more parameter (χ) (Ramberg and Osgood, 1943). Although Shi model for HS steels (with sharp knee in general) was constructed on the basis of Rasmussen model and strictly passes through (ε_0.2, R_p0.2) and (ε_u, R_m), the goodness of fit for the second-stage curve is not adequate if directly applied to the HS bolts due to the inadequate consideration of the sharp knee effect.

Table 10.

Measured and regressed values of the proposed model for grade 8.8 HS specimens.

Specimen	E_0b (MPa)	R_p0.01 (MPa)	R_p0.2 (MPa)	R _p0.01 /R _p0.2	R_m (MPa)	R _p0.2 /R _m	E_0.2b (MPa)	E _0.2 /E _0b	ε_0.2 (‰)	ε_u (‰)	ε_0.2 /ε_u (%)	ε_m (‰)
M36S-1	206,196	688.3	708.2	0.972	869.1	0.815	3133	0.02	5.638	65.87	8.6	85.37
M36S-2	199,590	682.8	709.4	0.962	871.3	0.814	4578	0.02	5.716	61.50	9.3	73.34
M36C-1	199,336	629.5	684.9	0.919	842.8	0.813	8944	0.04	5.493	85.37	6.4	127.73
M36C-2	208,840	679.1	719.3	0.944	859.8	0.837	6686	0.03	5.429	48.54	11.2	70.05
M36E-1	231,782	768.4	790.6	0.972	940.8	0.840	3571	0.02	5.441	47.59	11.4	72.28
M36E-2	208,355	709.4	747.8	0.949	897.3	0.833	5855	0.03	6.412	53.98	11.9	95.10
MG36S-1	209,001	708.3	749.4	0.945	904.0	0.829	6740	0.03	5.529	52.45	10.5	67.43
MG36S-2	219,645	691.4	740.3	0.934	893.3	0.829	7961	0.04	5.665	53.12	10.7	72.39
MG36C-1	208,698	723.1	761.3	0.950	902.6	0.843	6395	0.03	5.740	47.94	12.0	57.25
MG36C-2	213,193	687.3	727.0	0.945	870.1	0.836	6608	0.03	5.428	51.96	10.4	70.05
MG36E-1	200,066	893.7	925.8	0.965	1079.1	0.858	5471	0.03	6.876	73.76	9.3	93.79
MG36E-2	205,886	767.7	786.5	0.976	924.1	0.851	3578	0.02	5.463	48.76	11.2	53.88
M48S-1	208,390	715.8	776.4	0.922	926.6	0.838	9491	0.05	5.588	45.27	12.3	57.81
M48S-2	211,666	683.9	750.0	0.912	912.7	0.822	11,003	0.05	5.641	56.88	9.9	86.39
M48C-1	219,668	692.0	757.4	0.914	922.5	0.821	10,639	0.05	5.551	54.58	10.2	77.97
M48C-2	199,333	652.4	730.5	0.893	914.4	0.799	12,837	0.06	5.579	58.80	9.5	79.69
M48E-1	214,835	664.9	737.7	0.901	905.7	0.815	12,000	0.06	5.415	59.09	9.2	84.75
M48E-2	216,153	686.3	755.0	0.909	917.9	0.823	11,403	0.05	5.779	53.97	10.7	82.23
MG48S-1	209,428	652.4	738.2	0.884	892.9	0.827	14,189	0.07	5.622	49.60	11.3	69.70
MG48S-2	206,691	665.5	748.5	0.889	907.2	0.825	13,784	0.07	5.631	56.41	10.0	82.72
MG48C-1	209,548	668.7	759.4	0.881	918.0	0.827	14,895	0.07	5.607	54.22	10.3	77.01
MG48C-2	206,971	613.7	694.0	0.884	858.4	0.808	13,176	0.06	5.412	59.22	9.1	72.32
MG48E-1	212,303	660.5	701.6	0.941	874.4	0.802	6762	0.03	5.314	53.44	9.9	76.14
MG48E-2	212,761	728.5	852.9	0.854	1004.6	0.849	19,275	0.09	6.433	51.12	12.6	71.32
Mean value	209,931	696.4	752.2	0.926	908.7	0.827	9124	0.04	5.683	55.98	10.3	77.36
Standard deviation	7354	55.9	51.4	0.034	49.3	0.015	4211	0.02	0.371	8.88	1.4	14.91

Table 11.

Regressed parametric values of the proposed model.

Specimen	Stage 1 (S1) of equation (14)			Stage 2 (S2) of equation (14)					Overall R² (%)
Specimen	α	n	R₁² (%)	χ	χE_0.2b (MPa)	Β	m	R₂² (%)	Proposed equation (14)	Shi equation (13)	Rasmussen equation (11)
M36S-1	0.0022	111.30	99.99	1.632	5113	0.0275	6.059	99.75	99.92	99.04	95.77
M36S-2	0.0019	75.70	99.92	1.143	5233	0.0219	8.199	99.64	99.89	99.68	89.97
M36C-1	0.0022	36.57	99.96	0.538	4812	0.0409	8.673	99.21	99.70	98.44	92.50
M36C-2	0.0019	52.08	99.97	0.808	5403	0.0157	5.262	99.85	99.96	99.76	90.76
M36E-1	0.0021	109.00	99.98	1.610	5749	0.0142	6.570	99.80	99.96	95.05	92.25
M36E-2	0.0028	62.07	99.94	0.900	5269	0.0170	7.459	99.71	99.92	99.73	89.97
MG36S-1	0.0020	53.80	99.97	0.872	5877	0.0180	6.447	99.70	99.91	99.65	89.45
MG36S-2	0.0022	44.81	99.89	0.717	5708	0.0185	6.656	99.73	99.92	99.45	90.27
MG36C-1	0.0019	57.70	99.97	0.928	5934	0.0160	5.816	99.66	99.90	99.70	88.15
MG36C-2	0.0020	53.30	99.98	0.796	4612	0.0167	6.759	99.68	99.91	99.50	88.78
MG36E-1	0.0019	82.30	99.90	0.843	5188	0.0273	8.766	98.52	99.46	98.65	88.25
MG36E-2	0.0024	108.00	99.95	1.450	5260	0.0149	6.149	99.84	99.93	98.07	91.45
M48S-1	0.0022	39.04	99.96	0.873	8285	0.0190	4.346	99.68	99.90	99.67	90.66
M48S-2	0.0020	32.31	99.95	0.596	6558	0.0234	5.877	99.54	99.85	99.21	91.80
M48C-1	0.0021	33.87	99.94	0.730	7767	0.0247	5.162	99.51	99.84	99.41	92.04
M48C-2	0.0020	26.62	99.91	0.570	7317	0.0244	6.225	99.53	99.84	98.98	91.62
M48E-1	0.0020	29.02	99.98	0.573	6876	0.0252	6.303	99.40	99.78	98.78	91.08
M48E-2	0.0019	31.36	99.96	0.660	7526	0.0232	5.311	99.46	99.84	99.28	92.05
MG48S-1	0.0020	24.25	99.96	0.515	7308	0.0208	4.598	99.68	99.90	99.39	92.68
MG48S-2	0.0019	25.34	99.96	0.470	6479	0.0230	5.830	99.47	99.83	98.95	92.39
MG48C-1	0.0020	23.68	99.93	0.544	8103	0.0255	5.097	99.40	99.83	99.20	93.37
MG48C-2	0.0021	24.66	99.94	0.510	6720	0.0255	5.746	99.38	99.77	98.78	92.15
MG48E-1	0.0021	50.23	99.83	0.994	6721	0.0195	5.940	99.66	99.89	99.73	90.05
MG48E-2	0.0024	20.12	99.97	0.370	7132	0.0208	4.856	99.61	99.89	99.17	93.38
Mean value	0.0021	50.30	99.95	0.818	6290	0.0218	6.171	99.56	99.86	99.05	91.29
Standard deviation	0.0002	28.19	0.04	0.344	1086	0.0057	1.169	0.27	0.10	0.96	1.77

Through nonlinear regression analysis of the experimental data of HS bolts by using the curve fitting toolbox in MATLAB, the values of relevant parameters are summarized in Tables 10 and 11. The values of the regression parameter α at Stage 1 for all test specimens are roughly 0.002, which are consistent with the 0.2% plastic strain given in the original Ramberg-Osgood model. The effect of parameter α is to obtain an accurate n value with the maximum goodness of fit R₁² at Stage 1. Table 11 indicates that the values of R₁² and R₂² are basically above 99.9% and 99.0%, both showing high accuracies of simulation. In addition, the values of α suggest that the two-stage curves in equation (14) pass through (ε_0.2, R_p0.2) and meet the continuity condition at this point. However, the proposed curve does not strictly pass through (ε_u, R_m) due to the introduction of χ and β, but presents acceptable tolerances which will be discussed in Section “Discussion”. Comparisons of the measured stress-strain curves and simulated curves based on the proposed constitutive model for all tested HS bolts are shown in Figure 19. Good agreement is reached between the experimental data and proposed constitutive model for all tested HS bolts, with overall goodness of fit generally above 99.5%. The comparison of the overall goodness of fit on the basis of different constitutive models (equations (14), (13) and (11)) is summarized in Table 11. The relatively lower accuracies of the simulation of the second-stage curve lead to poorer overall goodness of fit for HS bolts according to Shi model and Rasmussen model (see Figure 20). The improvement of the simulating accuracy based on the proposed model is mainly reflected by the introduction of the shape optimization factor (χ). As shown in Figure 21, the above-mentioned three constitutive models (i.e., the proposed model, Shi model and Rasmussen model) unexceptionally demonstrate that the stress-strain equations are simply linear combinations of the linear and nonlinear items, which justifies the introduction of the shape optimization factor (χ) to the proposed model.

Discussion

By performing the nonlinear regression analysis of the strain hardening exponent n (from Table 11, which is based on the proposed model and test results) versus the stress ratio R_p0.01/R_p0.2 (from Table 10), the following expression is established:

n = \frac{2.972}{In (R_{p 0.2} / R_{p 0.01})} \approx \frac{In (20)}{In (R_{p 0.2} / R_{p 0.01})}

(15)

Where R_p0.01 is the 0.01% proof stress. As shown in Figure 22, good agreement (with R² = 98.27%) between the proposed constitutive model and the fitting curve is reached for grade 8.8 HS bolts. The values of the strain hardening exponent n for M36 HS bolts are generally larger than those for M48 HS bolts, because the latter present lower stress ratio (R_p0.01/R_p0.2).

The relationship between the second-stage strain hardening exponent m and the yield ratio R_p0.2/R_m is plotted in Figure 23(a). Apparently, the correlation between the two parameters is too weak, and the relationship between β and R_p0.2/R_m is also ambiguous (see Figure 23(b)). However, if defining $ζ = ε_{u} - \frac{R_{m} - R_{p 0.2}}{χ E_{0.2 b}} - ε_{0.2}$ , the correlation between β and ζ becomes considerably strong with the coefficient of correlation at 97.55% and goodness of fit at 95.16% (see Figure 23(c)). Therefore, the second stage in equation (14) actually becomes a two-parameter-controlled expression (χ and m) and the proposed stress-strain model can be approximately expressed as:

ε = {\begin{cases} \frac{σ}{E_{0 b}} + α {(\frac{σ}{R_{p 0.2}})}^{n}, σ \leq R_{p 0.2} \\ ε_{0.2} + \frac{σ - R_{p 0.2}}{χ E_{0.2 b}} + (0.7844 ζ + 0.0024) {(\frac{σ - R_{p 0.2}}{R_{m} - R_{p 0.2}})}^{m}, R_{p 0.2} < σ \leq R_{m} \end{cases}

(16)

The influence of the regression parameters χ, β and m on the goodness of fit for the second-stage curve (denoted as “S2”) and for the entire proposed model (denoted as “overall”) of M36S-2 is shown in Figure 24. Obviously, the exact values for the parameters corresponding to the optimal model are consistent with the values in Table 11. In general, when χ, β and m take average values of 0.818, 0.022 and 6.171 (see Table 11), the most accurate model for the second-stage expression of all tested HS bolts is obtained. With respect to the first-stage expression, the optimal value for parameter α takes as 0.002 (see Table 11), leading to the same expression as the Ramberg-Osgood model in equation (7).

It should be stressed that all constitutive models mentioned above (i.e., equations (11), (13), (14), and (16)) are used for characterizing the ascending part (up to the tensile strength) of the engineering stress-strain relationship. With respect to the post-necking stage, the engineering stress-strain curve descends until the fracture occurs with a relatively small plastic extension from the peak strain (see Figure 11), it is not discussed herein due to its minor significance in practical design philosophy. Besides, it should be clarified that all constitutive models and corresponding parameters were proposed on the basis of the engineering stress-strain curves, as done by most mathematical models and numerical simulations. Therefore, a certain deviation does exist between the constitutive models and the true stress-strain curves, as emphased in Section “Stress-strain curves”. Finally, from the perspective of material mechanical properties, the new constitutive model was proposed on the basis of the standard tensile test results of grade 8.8 heavy hex large diameter bolt material, thus it can be used with the effective diameter at the threaded part (since the sampled part and the threaded part are the same material). Because the standard tension coupons were sampled from the interior parts of the bolts, all along the length direction of the bolt rods, the influence of the thread can not be considered in the constitutive model. However, its anti-slip resistance and tensile strength will be reflected in the future full-scale tests on the bolt-group flange/gusset plate connections.

Conclusions

A series of standard tensile tests on two type of grade 8.8 heavy hex large diameter HS bolts were carried out to obtain the stress-strain curve and fundamental mechanical properties. The effects of important factors such as sampling positions and surface treatment methods on mechanical properties were analyzed. Based on the experimental results, the pre-tensions and tightening torques for two types of large diameter HS bolts were obtained. According to the experimental results, regression analysis was carried out to obtain a reasonable stress-strain constitutive model and relevant parameters. The proposed constitutive model was compared with those in existing literatures, and the following key conclusions are summarized:

(1) All test specimens exhibited similar deformative process under monotonic tension, and showed apparent necking phenomenon. The hot-dip galvanizing and sampling position produced a certain impact on the material plasticity and fracture ductility of large diameter HS bolts. The hot-dip galvanizing significantly shortened the peak strain and the fracture ductility at the necking stage compared with Zhang et al. (2024).

(2) According to the engineering stress-strain curves, the yield plateau hardly occurs, and all stress-strain curves experience three stages: linear ascending stage, nonlinear strain hardening stage, and nonlinear necking stage. The percentage elongation generally varies between 5% and 15%. When the average strain relative to the gauge length reaches 5% (i.e., peak strain), the axial tensile force almost attains the tensile strength.

(3) All measured Young’s moduli vary between 195 GPa and 235 GPa, with an average value of 210 GPa. The deviation of all Young’s moduli is closely related to the sampling location and surface treatment method. Overall, the Young’s modulus increases with the increase of the distance between the sampling location and the center of the bolt surface. In addition, hot-dip galvanizing significantly weakens the Young’s modulus at three sampling positions of M48 bolt and the value at the edge position of M36 bolt, but produces a strengthening effect at the standard position and central position of M36 bolt. Obviously, the heat input of hot-dip galvanizing at 600°C impairs the rigidity of the bolt surface. The temperature away from the surface would be lower than 600°C, which might produce a decaying effect or strengthening effect on the rigidity of interior steel depending on the specific temperature.

(4) The measured proof strengths and tensile strengths are greater than the corresponding standard values. In general, the hot-dip galvanizing treatment of the bolt surface has a limited impact on the proof strength and tensile strength. The proof strength and tensile strength sampled at the central position are the smallest, then at the standard position, and the largest values occur at the edge of the bolt rod, thus the most rational sampling position is the standard position.

(5) The percentage elongation and reduction area after fracture for most specimens fall within the limiting values specified in the relevant standard. Difference in bolt size, sampling positions and surface treatment methods has a slight impact on post-fracture properties. However, this does not mean that these factors can be neglected when the mechanical properties of the whole bolt are taken into account, which is another research topic under progress. Besides, there are many other research issues related to the size effect of the bolt, such as fatigue performance, brittle fracture, pre-tension control, stress concentration, joint rigidity, contact surface slip, torque coefficient, etc.

(6) The actual bolt pre-tensions based on the measured tensile strengths for all M(G)36 and M(G)48 specimens are greater than 420 kN and 790 kN, respectively. Therefore, the use of nominal values of 410 kN and 743 kN as the pre-tension design values for hot-dip galvanized MG36 and MG48 bolts is conservative and safe. The tightening torques of grade 8.8 bolts with large diameters are significantly greater than those with regular diameters.

(7) A stress-strain constitutive model applicable to the grade 8.8 heavy hex HS bolts with large diameter, is proposed based on the existing models. By thorough comparison between the proposed model and existing models in terms of goodness of fit for different segments of the stress-strain curve, the proposed constitutive model presents much higher accuracy in simulating the knee of the stress-strain curve (with overall R² greater than 99.8%). Therefore, the proposed model is sufficiently accurate and reliable in predicting the HS bolt’s stress-strain relationship, which can be used in the design and numerical modelling of HS bolts.

As concerned, all existing mathematical models for numerical analyses are used to simulate the full-range engineering stress-strain curves, but the simulation of the true stress-strain curves (especially at the necking stage) is yet to be validated. Besides, the proposed Ramberg-Osgood model in this paper is accurate enough to simulate the engineering stress-strain curves before necking. At the post-necking stages of the engineering and true stress-strain curves, the simulation becomes less significant due to the ambiguous accuracy of existing mathematical models as well as the absence of numerical validation.

Footnotes

Acknowledgements

Sincere gratitude is given to the China Construction Third Engineering Bureau Group Co., Ltd. for supplying the test specimens. Furthermore, many thanks are delivered to the Engineering Structure Laboratory of Tsinghua University for providing test assistances. Any standpoints, insights, findings and conclusions elaborated in this paper are those of the authors and do not necessarily reflect the viewpoints of sponsors.

ORCID iD

Xianglin Yu

Author contribution

Jiangtao Wen: Conceptualization, Methodology, Resources and Funding, Project Coordination. Xianglin Yu: Conceptualization, Data analysis, Investigation, Visualization, Writing – original draft. Dong Liu: Experiment preparing and conducting, Data curation, Data analysis, Visualization, Writing – review & editing. Shihua Zhao: Checking and proofing, Writing – review & editing. Yongjiu Shi: Conceptualization, Methodology, Supervision, Writing – review & editing. Kui Yin: Test Specimens Preparation.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors gratefully acknowledge the support from the National Natural Science Foundation of China (Grant No. 52378163).

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

All experimental or analysed data relevant to this article are available from the corresponding author upon reasonable request.*

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