# Study on Hysteresis Model of Welding Material in Unstiffened Welded Joints of Steel Tubular Truss Structure

^{1}

^{2}

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## Abstract

**:**

## Featured Application

**The main purpose of this study is to present a damage constitutive model of welded joints in steel structures, which can provide a basis for the precise design of steel structure joints. It can be popularized and applied in steel structure building, hydraulic steel structure pump house, dike and so on in field of civil engineering and hydraulic engineering. It also can be used for reference in the field of mechanical manufacturing.**

## Abstract

## 1. Introduction

## 2. Experimental Study

#### 2.1. Specimen Design

#### 2.2. Test Procedure

_{max}/ε

_{min}= −1). In order to study the effect of the loading strain amplitude on the hysteretic model, the loading strain amplitude of each group of specimens ranges from 0.2% to 0.5%, that is, the loading strain amplitudes of the specimens numbered (1), (2), (3) are 0.25%, 0.35% and 0.45%, for specimens of weld, steel, and the heat-effected zone. As a result, Figure 4a corresponds to all No. (1) specimens, Figure 4b corresponds to all No. (2) specimens, and Figure 4c corresponds to all No. (3) specimens. Table 2 summarizes the primary mechanical parameters of the specimens, which were obtained by uniaxial tensile test (averaged over three specimens).

## 3. The Test Results and Discussion

#### 3.1. Failure Model and Damage Processes

#### 3.2. Cyclic Behavior and Damage Analysis

#### 3.3. Variation of Cyclic Stress Amplitude

#### 3.4. EnergyDissipation Behavior

_{a}. The expression is shown in Equation (1) [58]. E

_{u}is the ultimate hysteretic energy under monotonic loading, which can be calculated from the area enclosed by the stress-strain curves of monotonic tensile test and the coordinate axis. E

_{i}is the actual dissipated energy of the No. i semi-cycle, which can be calculated from the area enclosed by the cyclic stress-strain hysteretic curves of cyclic loading test and the coordinate axis. Then, ξ

_{a}-η curve of welded material is depicted in Figure 9, it can be seen that the slope of the curves decreases with the increase of cyclic cycles, which means that the rate of cumulative energy dissipation decreases gradually, in other words, the enclosing area of the hysteretic loops decreases with the increase of cyclic cycles, and the energy dissipation capacity decreases gradually. It is due to the continuous damage accumulation of the welding materials under cyclic loading. In a word, the cumulative damage rule of the welding material can be discussed through cumulative energy dissipation behavior is consistent with the law of stress amplitude degradation. That is, with the continuous development of damage accumulation under cyclic loading, the amplitude of cyclic stress, unloading stiffness, and energy dissipation capacity of the welded metal, these degenerate gradually until failure occurs, and compared with the base metal, the welded metal is more prone to damage accumulation.

## 4. An Evolution Equation of Damage Accumulation for Welding Materials

_{m}and δ

_{u}is the maximum deformation with loading and ultimate deformation of material respectively, $\int dE$ is the cumulative dissipation of plastic energy, F

_{y}δ

_{u}is equivalent to the ultimate hysteretic energy under the monotonic load of perfect elastic-plastic condition.

_{i}is the plastic deformation energy dissipation of No. i semi-cycles, which is equal to the area of the hysteresis loop in numerical value, E

_{u}is the ultimate hysteretic energy under monotonic loading, which can be calculated from the area enclosed by the stress-strain curves of the monotonic tensile test and the coordinate axis, λ is a parameter means the weight of cumulative plastic energy dissipation, which can be calculated by the cyclic loading test results. Compared with Equation (2), the energy part of the model in Equation (3) is no longer confined to the perfect elastic-plastic condition, and the fact that the value of damage variable D is always less than or equal to 1 until fractured is ensured.

## 5. A Hysteresis Model with Damage Accumulation of Welding Materials

#### 5.1. Basic Requirements of the Model

#### 5.2. Initial Loading Curve

#### 5.3. Cyclic Stress-Strain Curve Based on Ramberg–Osgood Model

_{0}and K

_{0}are the initial elastic modulus and hardening coefficients, respectively, which can be fitted by the Low-cycle fatigue test of the welding materials. The parameters of Equation (5) are summarized in Table 5.

#### 5.4. A Model of Hysteretic Curve with Damage Accumulation

_{n}-C

_{n}-B

_{n}is the loading curve of No. n semi-cycle, which is constructed by considering nonlinear loading. Point A

_{n}is the starting point of the n semi-cycle loading curve, and B

_{n}is the final point of the loading curve, but is the starting point of the n semi-cycle unloading curve. ${\sigma}_{m}^{D(n)}$ is the stress amplitude of the No. n semi-cyclic stress-strain curve and the ordinate of point B

_{n}. The stress amplitude of each semi-cycle will be degraded with damage accumulating, so ${\sigma}_{m}^{D(n)}$ is thought of as the damage stress amplitude, which can be calculated by Equation (8) based on damage mechanics theory, ${\sigma}_{m}^{}$ is the initial stress amplitude, ${D}_{n}$ is the cumulative damage variable after nth semi-cycle, which can be calculated by Equation (3),η and ξ are the damage parameters that can be fitted by the low-cycle fatigue test results of the welding material. Curve B

_{n}-A

_{n+}

_{1}is the unloading curve of No. n semi-cycle, and the unloading process is approximately elastic. ${E}_{n}^{D}$ is the elastic modulus with damage accumulation, which in respect of ${D}_{n}$ can be calculated by Equation (9) based on damage mechanics theory and Shen [30,31]. ${E}_{0}$ is the original elastic modulus, g and h are the damage parameters like η and ξ.

- (1)
- unloading curve (Linear)

_{n}and ε

_{n}are stress and strain values of point at nth semi-cycle curves, t

_{n}is the parameter for determining the shape of the curve, ε

_{B}is the strain amplitude.

- (2)
- loading curve (nonlinear)

#### 5.5. Parameter Fitting of the Model

#### 5.5.1. The Parameter λ of EvolutionEquation of Damage Accumulation

#### 5.5.2. The Parameters η, ξ, g, h, K’ and n’

- (1)
- Damage Parameters η, ξ, g, h

- (2)
- K’ and n’

#### 5.6. Comparison between the Test Results and Results Calculated by the Proposal Model

## 6. Conclusions

- (1)
- The cumulative damage process of welding materials can be divided into three stages, which are presented as the stage of crack initiation, the stage of crack propagation, and the stage the specimenis fractured, and when the specimensare destroyed along the maximum main crack.
- (2)
- As the imposed displacement cycles increased, the cyclic softening behavior of the welding materials of the welded joints in a tubular truss structureis apparent, and the steel hardening behavior is shown. Furthermore, with the increase of cyclic cycles, the effect of damage accumulation on the welding materials is obvious, that is, the cyclic stress amplitude, unloading stiffness, and energy dissipation capacity of the weldingmaterials degenerate gradually. Furthermore, the larger the controlled loading strain amplitude, the faster the rate of damage accumulation. The amplitude of the loading strain has a greater effect on the welding material of base metal Q235 than that of base metal Q345.
- (3)
- Based on the test results and Park–Ang model, an evolutionequation of damage accumulation for welding materials is established considering both energy and deformation comprehensively, in which the energy part is no longer confined to the perfect elastic-plastic condition. Then, the value of parameter λ is fitted by the test data.
- (4)
- It is revealed by the experimental research that the cyclic characteristic and constitutive behavior of the welding material is quite different from the base metal, and the constitutive model used for monotonic loading cannot simulate the damage degradation behavior of the welding materials under cyclic loading. Therefore, a hysteresismodel with damage accumulation of welding materials is constructed based on the Ramberg–Osgood model and the experimental results, which includedthe initial loading curve, cyclic stress-strain curve, and a model of the hysteretic curve. The model can reveal the effects of damage accumulation and the nonlinear constitutive relation of the plasticstage.
- (5)
- The damage parameters and model parameters η, ξ, g, h, K’ and n’ are fitted by the test results. The hysteretic curves calculated by the hysteretic model are compared with the test results, which show that the model has good applicability and the cumulative damage evolution law of the welding materials reflected by the model is basically consistent with the test results.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Acquisition of the standard specimen. (

**a**) Sampling of fillet weld(mm); (

**b**) Sampling of butt weld (mm).

**Figure 3.**Experimental specimen and setup. (

**a**) Dimensions of specimen (mm); (

**b**) Experimental setup and instrumentation.

**Figure 4.**Cyclic loading patterns. (

**a**) For all No. (1) specimens; (

**b**) For all No. (2) specimens; (

**c**) For all No. (3) specimens.

**Figure 5.**Damage process of welded specimen in low cycle fatigue test. (

**a**) Crack appeared in specimen; (

**b**) Crack develop; (

**c**) Specimen fractured at the maximum main crack.

**Figure 6.**Stress-strain (σ-ε) hysteresis curve of the welded specimen. (

**a**) WA1 (1) ε

_{max}= 0.25%; (

**b**) WA1 (2) ε

_{max}= 0.35%; (

**c**) WA1 (3) ε

_{max}= 0.45%; (

**d**) WA2 (1) ε

_{max}= 0.25%; (

**e**) WA2 (2) ε

_{max}= 0.35%; (

**f**) WA2 (3) ε

_{max}= 0.45%; (

**g**) WA3 (1) ε

_{max}= 0.25%; (

**h**) WA3 (2) ε

_{max}= 0.35%; (

**i**) WA3 (3) ε

_{max}= 0.45%; (

**j**) WB1 (1) ε

_{max}= 0.25%; (

**k**) WB1 (2) ε

_{max}= 0.35%; (

**l**) WB1 (3) ε

_{max}= 0.45%; (

**m**) WB2 (1) ε

_{max}= 0.25%; (

**n**) WB2 (2) ε

_{max}= 0.35%; (

**o**) WB2 (3) ε

_{max}= 0.45%; (

**p**) WB3 (1) ε

_{max}= 0.25%; (

**q**) WB3 (2) ε

_{max}= 0.35%; (

**r**) WB3 (3) ε

_{max}= 0.45%.

**Figure 7.**Stress-strain (σ-ε) hysteresis curve of the steel (base metal) specimen.(

**a**) SA1 (1) ε

_{max}= 0.25%; (

**b**) SA1 (2) ε

_{max}= 0.35%; (

**c**) SA1 (3) ε

_{max}= 0.45%; (

**d**) SA2 (1) ε

_{max}= 0.25%; (

**e**) SA2 (2) ε

_{max}= 0.35%; (

**f**) SA2 (3) ε

_{max}= 0.45%; (

**g**) SB1 (1) ε

_{max}= 0.25%; (

**h**) SB1 (2) ε

_{max}= 0.35%; (

**i**) SB2 (1) εmax = 0.25%; (

**j**) SB2 (2) ε

_{max}= 0.35%; (

**k**) SB2 (3) ε

_{max}= 0.45%.

**Figure 8.**Stress amplitude-dimensionless (σ

_{m}-η) half cycle of welding material and steel. (

**a**) WA1; (

**b**) WA2; (

**c**) WA3; (

**d**) WB1; (

**e**) WB2; (

**f**) WB3; (

**g**) WA; (

**h**) WB; (

**i**) SA1; (

**j**) SB2.

**Figure 9.**ξ

_{a}-η curve of welding material. (

**a**) WA1ξ

_{a}-η; (

**b**) WA2ξ

_{a}-η; (

**c**) WA3ξ

_{a}-η; (

**d**) WB1ξ

_{a}-η; (

**e**) WA2ξ

_{a}-η; (

**f**) WA3ξ

_{a}-η.

**Figure 10.**Hysteresis curves under different stress-strain constitutive relations. (

**a**) Bilinear model; (

**b**) Basic requirements of the hysteresis model.

**Figure 11.**Fitting results of the initial loading curve. (

**a**) The schematic diagram of the initial loading curve; (

**b**) Fitting result of the WA1 specimen.

**Figure 15.**Parameter fitting of WB1 (2). (

**a**) Curve $\frac{{\sigma}_{m}^{D(n)}}{{\sigma}_{m}}-{D}_{n}$; (

**b**) Curve $\frac{{E}_{n}^{D}}{{E}_{0}}-{D}_{n}$.

**Figure 16.**Comparison of hysteretic curves calculated from welding hysteresis models and test results.(

**a**) WA1(1)σ-ε; (

**b**) WA1(2)σ-ε; (

**c**) WA1(3) σ-ε; (

**d**) WA2(1) σ-ε; (

**e**) WA2(2) σ-ε; (

**f**) WA2(3) σ-ε; (

**g**) WA3(1) σ-ε; (

**h**) WA3(2) σ-ε; (

**i**) WA3(3) σ-ε; (

**j**) WB1(1) σ-ε; (

**k**) WB1(2) σ-ε; (

**l**) WB1(3) σ-ε; (

**m**) WB2(1) σ-ε; (

**n**) WB2(2) σ-ε; (

**o**) WB2(3) σ-ε; (

**p**) WB3(1) σ-ε; (

**q**) WB3(2) σ-ε; (

**r**) WB3(3) σ-ε.

Type for Base Metal | Steel Specimen before Welding | Welding Material | Steel of Heat-Effected Zone | |||||
---|---|---|---|---|---|---|---|---|

Parallel Rolling Direction | Vertical Rolling Direction | Butt Weld | Fillet Weld of T Type | Butt Weld | Fillet Weld of T Type | |||

Left | Right | Left | Right | |||||

Q235 | SA1 | SA2 | WA1 | WA2 | WA3 | HA1 | HA2 | HA3 |

(1)–(3) | (1)–(3) | (1)–(3) | (1)–(3) | (1)–(3) | (1)–(3) | (1)–(3) | (1)–(3) | |

Q345 | SB1 | SB2 | WB1 | WB2 | WB3 | HB1 | HB2 | HB3 |

(1)–(3) | (1)–(3) | (1)–(3) | (1)–(3) | (1)–(3) | (1)–(3) | (1)–(3) | (1)–(3) |

Specimens | f_{y}/MPa | f_{u}/MPa | ε_{y}/% | ε_{u}/% | E/GPa |
---|---|---|---|---|---|

SA1 | 268.8 | 430.1 | 0.15 | 16.3 | 207.1 |

SA2 | 268.1 | 429.7 | 0.15 | 16.4 | 195.7 |

WA1 | 391.7 | 497.6 | 0.16 | 13.0 | 239.7 |

WA2 | 401.6 | 486.8 | 0.17 | 12.7 | 233.6 |

WA3 | 402.3 | 497.8 | 0.17 | 11.8 | 213.0 |

HA1 | 254.7 | 433.5 | 0.13 | 15.8 | 223.3 |

HA2 | 259.2 | 423.6 | 0.13 | 15.5 | 203.6 |

HA3 | 255.4 | 422.9 | 0.13 | 15.6 | 204.5 |

SB1 | 365.6 | 532.6 | 0.16 | 16.0 | 216.1 |

SB2 | 385.1 | 540.1 | 0.15 | 15.2 | 217.9 |

WB1 | 420.1 | 498.1 | 0.15 | 12.7 | 235.5 |

WB2 | 426.3 | 508.5 | 0.17 | 11.7 | 251.0 |

WB3 | 431.5 | 525.4 | 0.17 | 11.3 | 217.6 |

HB1 | 365.2 | 531.0 | 0.16 | 16.0 | 235.0 |

HB2 | 357.8 | 530.5 | 0.15 | 15.3 | 225.1 |

HB3 | 358.5 | 527.6 | 0.16 | 15.2 | 218.3 |

Specimens | Welded Specimens | Steel Specimens | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

WA1 | WA2 | WA3 | WB1 | WB2 | WB3 | SA1 | SA2 | SB1 | SB2 | |

(1) (0.25%) | 3091 | 3810 | 3430 | 4351 | 4205 | 4293 | 4321 | 3789 | 4766 | 4703 |

(2) (0.35%) | 1865 | 1801 | 1831 | 2223 | 1992 | 1973 | 2109 | 1983 | 2525 | 2607 |

(3) (0.45%) | 1089 | 994 | 977 | 1559 | 1165 | 1106 | 1301 | 976 | 1639 |

Base Metal | Welding Material | E/Gpa | σ_{y}/MPa | K/Gpa | m | ε_{y}_{1}/% | ε_{y}_{2}(ε_{0})/% | ε_{c}/% | b | D_{C}/% |
---|---|---|---|---|---|---|---|---|---|---|

Q235 | Butt weld | 239.70 | 391.70 | 2.73 | 0.77 | 0.16 | 1.50 | 21.25 | 0.065 | 59.24 |

Fillet weld | 223.30 | 401.94 | 1.82 | 0.54 | 0.18 | 1.50 | 21.34 | 0.046 | 50.03 | |

Q345 | Butt weld | 247.71 | 420.07 | 2.52 | 0.65 | 0.17 | 1.70 | 17.71 | 0.052 | 57.64 |

Fillet weld | 241.07 | 433.92 | 2.17 | 0.60 | 0.18 | 1.70 | 16.97 | 0.044 | 53.11 |

With Q235 Base Metal | With Q345 Base Metal | ||||||
---|---|---|---|---|---|---|---|

Specimens | E_{0}/GPa | K_{0}/MPa | n_{0} | Specimens | E_{0}/GPa | K_{0}/MPa | n_{0} |

WA1 | 239.700 | 1266 | 4.784 | WB1 | 247.710 | 953 | 8.099 |

WA2 | 223.000 | 1331 | 5.531 | WB2 | 241.070 | 939 | 8.610 |

WA3 | 223.000 | 1466 | 6.203 | WB3 | 241.070 | 885 | 9.272 |

Specimens | Welding Material of Q235 Steel | Welding Material of Q345 Steel | ||||
---|---|---|---|---|---|---|

Butt Weld | Fillet Weld | Butt Weld | Fillet Weld | |||

WA1 | WA2 | WA3 | WB1 | WB2 | WB3 | |

(1) | 0.02529 | 0.02609 | 0.02871 | 0.02311 | 0.02156 | 0.02176 |

(2) | 0.02233 | 0.02538 | 0.02613 | 0.01905 | 0.01909 | 0.02433 |

(3) | 0.02905 | 0.03043 | 0.03280 | 0.02071 | 0.02326 | 0.02230 |

Model Parameters | 0.0256 | 0.0273 | 0.0294 | 0.0209 | 0.02130 | 0.0227 |

Specimens | I | II | III | ||||||
---|---|---|---|---|---|---|---|---|---|

ξ_{1} | η_{1} | D_{1} | ξ_{2} | η_{2} | D_{2} | ξ_{3} | η_{3} | ||

Butt weld of Q235 steel | WA1(1) | 0.509 | 0.961 | 0.228 | 0.099 | 0.873 | 0.897 | 1.799 | 2.350 |

WA1(2) | 0.475 | 0.998 | 0.239 | 0.105 | 0.875 | 0.866 | 1.879 | 2.355 | |

WA1(3) | 0.485 | 1.001 | 0.231 | 0.087 | 0.897 | 0.860 | 1.770 | 2.362 | |

Model Parameters | 0.489 | 0.986 | 0.233 | 0.097 | 0.882 | 0.874 | 1.816 | 2.356 | |

Left fillet weld of Q235 steel | WA2(1) | 0.530 | 1.001 | 0.247 | 0.057 | 0.858 | 0.883 | 1.191 | 1.871 |

WA2(2) | 0.455 | 0.997 | 0.241 | 0.058 | 0.901 | 0.886 | 2.158 | 2.790 | |

WA2(3) | 0.456 | 0.998 | 0.226 | 0.059 | 0.899 | 0.859 | 2.157 | 2.788 | |

Model Parameters | 0.480 | 0.999 | 0.238 | 0.058 | 0.886 | 0.876 | 1.825 | 2.483 | |

Right fillet weld of Q235 steel | WA3(1) | 0.445 | 1.002 | 0.251 | 0.086 | 0.887 | 0.863 | 1.167 | 1.843 |

WA3(2) | 0.500 | 1.000 | 0.243 | 0.045 | 0.889 | 0.858 | 2.078 | 2.678 | |

WA3(3) | 0.525 | 0.999 | 0.229 | 0.077 | 0.896 | 0.886 | 1.996 | 2.675 | |

Model Parameters | 0.490 | 1.000 | 0.241 | 0.069 | 0.891 | 0.869 | 1.747 | 2.399 | |

Butt weld of Q345 steel | WB1(1) | 0.468 | 0.999 | 0.235 | 0.058 | 0.902 | 0.899 | 2.551 | 3.151 |

WB1(2) | 0.534 | 1.002 | 0.228 | 0.065 | 0.893 | 0.869 | 1.832 | 2.427 | |

WB1(3) | 0.508 | 0.989 | 0.236 | 0.075 | 0.887 | 0.889 | 2.235 | 2.912 | |

Model Parameters | 0.503 | 0.997 | 0.233 | 0.066 | 0.894 | 0.885 | 2.206 | 2.830 | |

Left fillet weld of Q345 steel | WB2(1) | 0.535 | 0.998 | 0.233 | 0.068 | 0.889 | 0.865 | 1.622 | 2.233 |

WB2(2) | 0.625 | 1.002 | 0.241 | 0.077 | 0.868 | 0.887 | 1.620 | 2.237 | |

WB2(3) | 0.612 | 1.000 | 0.231 | 0.095 | 0.881 | 0.859 | 1.634 | 2.210 | |

Model Parameters | 0.592 | 1.000 | 0.235 | 0.080 | 0.879 | 0.865 | 1.625 | 2.227 | |

Right fillet weld of Q345 steel | WB3(1) | 0.505 | 0.998 | 0.219 | 0.064 | 0.902 | 0.875 | 2.279 | 2.841 |

WB3(2) | 0.446 | 0.997 | 0.241 | 0.048 | 0.901 | 0.887 | 2.587 | 3.153 | |

WB3(3) | 0.455 | 0.998 | 0.246 | 0.063 | 0.902 | 0.878 | 2.284 | 2.854 | |

Model Parameters | 0.469 | 0.998 | 0.235 | 0.058 | 0.902 | 0.880 | 2.383 | 2.949 |

Specimens | I + II | III | ||||
---|---|---|---|---|---|---|

h_{2} | g_{2} | D_{2} | h_{3} | g_{3} | ||

Butt weld of Q235 steel | WA1(1) | 0.045 | 0.991 | 0. 889 | 2.083 | 2.466 |

WA1(2) | 0.024 | 0.982 | 0. 881 | 1.694 | 2.433 | |

WA1(3) | 0.024 | 0.989 | 0. 879 | 1.668 | 2.453 | |

Model Parameters | 0.031 | 0.987 | 0. 883 | 1.815 | 2.446 | |

Left fillet weld of Q235 steel | WA2(1) | 0.020 | 0.997 | 0. 875 | 1.751 | 2.553 |

WA2(2) | 0.021 | 0.998 | 0. 882 | 1.751 | 2.503 | |

WA2(3) | 0.020 | 0.997 | 0. 879 | 1.709 | 2.513 | |

Model Parameters | 0.020 | 0.997 | 0. 879 | 1.737 | 2.523 | |

Right fillet weld of Q235 steel | WA3(1) | 0.056 | 1.000 | 0.887 | 1.345 | 2.134 |

WA3(2) | 0.020 | 0.997 | 0.886 | 1.384 | 2.196 | |

WA3(3) | 0.113 | 1.000 | 0.886 | 0.981 | 1.796 | |

Model Parameters | 0.063 | 0.999 | 0.886 | 1.225 | 2.044 | |

Butt weld of Q345 steel | WB1(1) | 0.109 | 0.999 | 0.890 | 1.519 | 2.269 |

WB1(2) | 0.111 | 1.000 | 0.882 | 1.807 | 2.499 | |

WB1(3) | 0.103 | 1.002 | 0.886 | 1.445 | 2.196 | |

Model Parameters | 0.107 | 1.000 | 0.886 | 1.509 | 2.321 | |

Left fillet weld of Q345 steel | WB2(1) | 0.113 | 0.997 | 0.879 | 1.005 | 1.825 |

WB2(2) | 0.094 | 0.999 | 0.881 | 1.083 | 1.878 | |

WB2(3) | 0.104 | 1.001 | 0.877 | 1.054 | 1.883 | |

Model Parameters | 0.104 | 0.999 | 0.879 | 1.047 | 1.862 | |

Right fillet weld of Q345 steel | WB3(1) | 0.083 | 0.999 | 0.857 | 1.198 | 1.956 |

WB3(2) | 0.057 | 1.000 | 0.887 | 2.365 | 3.043 | |

WB3(3) | 0.089 | 0.999 | 0.890 | 1.789 | 2.513 | |

Model Parameters | 0.076 | 0.999 | 0.878 | 1.784 | 2.504 |

Specimens | ε_{B}/% | K’ | n’ | |
---|---|---|---|---|

WA1 | WA1(1) | 0.25 | 2.26 × 10^{7} | 0.546 |

WA1(2) | 0.35 | 2.65 × 10^{7} | 0.499 | |

WA1(3) | 0.45 | 3.05 × 10^{7} | 0.459 | |

Model Parameters | ${K}^{\prime}=3.95\times {10}^{9}{\epsilon}_{B}+1.27\times {10}^{7}$ | ${n}^{\prime}=-43.5{\epsilon}_{B}+0.654$ | ||

WA2 | WA2(1) | 0.25 | 2.23 × 10^{7} | 0.547 |

WA2(2) | 0.35 | 2.64 × 10^{7} | 0.504 | |

WA2(3) | 0.45 | 3.05 × 10^{7} | 0.461 | |

Model Parameters | ${K}^{\prime}=4.10\times {10}^{9}{\epsilon}_{B}+1.21\times {10}^{7}$ | ${n}^{\prime}=-43.0{\epsilon}_{B}+0.655$ | ||

WA3 | WA3(1) | 0.25 | 2.24 × 10^{7} | 0.546 |

WA3(2) | 0.35 | 2.64 × 10^{7} | 0.505 | |

WA3(3) | 0.45 | 3.04 × 10^{7} | 0.464 | |

Model Parameters | ${K}^{\prime}=4.00\times {10}^{9}{\epsilon}_{B}+1.24\times {10}^{7}$ | ${n}^{\prime}=-41.0{\epsilon}_{B}+0.649$ | ||

WB1 | WB1(1) | 0.25 | 1.40 × 10^{7} | 0.567 |

WB1(2) | 0.35 | 1.63 × 10^{7} | 0.521 | |

WB1(3) | 0.45 | 1.86 × 10^{7} | 0.475 | |

Model Parameters | ${K}^{\prime}=2.30\times {10}^{9}{\epsilon}_{B}+8.25\times {10}^{6}$ | ${n}^{\prime}=-46.0{\epsilon}_{B}+0.628$ | ||

WB2 | WB2(1) | 0.25 | 1.41 × 10^{7} | 0.566 |

WB2(2) | 0.35 | 1.63 × 10^{7} | 0.521 | |

WB2(3) | 0.45 | 1.85 × 10^{7} | 0.476 | |

Model Parameters | ${K}^{\prime}=2.25\times {10}^{9}{\epsilon}_{B}+8.46\times {10}^{6}$ | ${n}^{\prime}=-45.0{\epsilon}_{B}+0.679$ | ||

WB3 | WB3(1) | 0.25 | 1.40 × 10^{7} | 0.566 |

WB3(2) | 0.35 | 1.64 × 10^{7} | 0.520 | |

WB3(3) | 0.45 | 1.86 × 10^{7} | 0.475 | |

Model Parameters | ${K}^{\prime}=2.30\times {10}^{9}{\epsilon}_{B}+8.28\times {10}^{6}$ | ${n}^{\prime}=-45.5{\epsilon}_{B}+0.680$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Suo, Y.; Yang, W.; Chen, P.
Study on Hysteresis Model of Welding Material in Unstiffened Welded Joints of Steel Tubular Truss Structure. *Appl. Sci.* **2018**, *8*, 1701.
https://doi.org/10.3390/app8091701

**AMA Style**

Suo Y, Yang W, Chen P.
Study on Hysteresis Model of Welding Material in Unstiffened Welded Joints of Steel Tubular Truss Structure. *Applied Sciences*. 2018; 8(9):1701.
https://doi.org/10.3390/app8091701

**Chicago/Turabian Style**

Suo, Yaqi, Wenwei Yang, and Peng Chen.
2018. "Study on Hysteresis Model of Welding Material in Unstiffened Welded Joints of Steel Tubular Truss Structure" *Applied Sciences* 8, no. 9: 1701.
https://doi.org/10.3390/app8091701