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Steel structures are usually damaged by disasters. According to the influence law of the damage on the elastic modulus of steel obtained by the mechanical test of damaged steel, the average elastic moduli of H-section steel members were analyzed. The equations for calculating the average elastic moduli of damaged H-section steel members at different damage degrees were obtained. By using the analytical cross-sectional method, the cross-sectional

Natural disasters or other factors often lead to local damage or total damage of steel members. Whether damaged members can meet the bearing capacity and stiffness requirements will directly affect the safety and reliability of damaged steel structures. A large number of scholars around the world have conducted numerous researches on this subject.

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Currently, only few studies on the mechanical properties of damaged steel beams and columns have been done. In this study, the calculation method of the average elastic modulus of the damaged cross section was studied according to the influence law of damage on the elastic modulus of steel. The

The mechanical properties of damaged steel were acquired by the tensile coupon tests. The initial elastic modulus of the steel is _{0}. The specimens were unloaded when plastic deformation occurred in different degrees, and the slope of unloading line was recorded as unloading elastic modulus _{.} Then a secondary loading test is performed on the damaged specimens, and the slope of loading line was recorded as loading elastic modulus

In this paper, Q345 steel samples are divided into 8 groups, and each group has 3 specimens. The size of the specimen is shown in Figure

Tensile specimens.

The first group of specimens was subjected to uniaxial tensile test in the testing machine, and when the specimens reached the ultimate strength, the loading ended. It is found that the deformation at the end of the flow molding phase (the end point of the yield stage) is about 0.8 mm, and that at the end of the strengthening phase is about 8.4 mm.

According to the results of the first set of test data, the deformation control values of the second group to the eighth group were 0.8 mm, 2 mm, 3 mm, 4 mm, 5 mm, 7 mm, and 8.4 mm, respectively.

The second to eighth groups of specimens were loaded to the deformation control value, then unloaded, and then loaded into each group of specimens to destroy. The test process and the contrast of the deformation before and after tensile test are shown in Figures

Universal material testing machine.

Contrast of elongation (initial elongation is 3 mm and 8.4 mm).

The tensile test results of 8 groups of specimens are shown in Table

Data of tensile test.

Q345 steel plate test | ||||||
---|---|---|---|---|---|---|

Group number | Specimen number | Initial deformation (mm) | Initial strain (%) | Initial elastic modulus _{0} (GPa) | Unloading elastic modulus | Loading elastic modulus |

The first group | X-01 | 0 | 0 | 189.406 | -- | -- |

X-02 | 0 | 0 | 194.075 | -- | -- | |

X-03 | 0 | 0 | 186..831 | -- | -- | |

The second group | X-04 | 0.8 | 1.6 | 189.914 | 152.243 | 159.828 |

X-05 | 0.8 | 1.6 | 176.957 | 157.844 | 161.121 | |

X-06 | 0.8 | 1.6 | 190.728 | 155.084 | 157.661 | |

The third group | X-07 | 2 | 4 | 180.096 | 148.282 | 148.913 |

X-08 | 2 | 4 | 184.590 | 151.591 | 168.217 | |

X-09 | 2 | 4 | 194.798 | 150.475 | 152.643 | |

The fourth group | X-10 | 3 | 6 | 205.387 | 142.254 | 147.342 |

X-11 | 3 | 6 | 175.724 | 134.118 | 137.034 | |

X-12 | 3 | 6 | 193.402 | 136.921 | 151.378 | |

The fifth group | X-13 | 4 | 8 | 182.236 | 133.788 | 134.963 |

X-14 | 4 | 8 | 191.110 | 135.149 | 138.170 | |

X-15 | 4 | 8 | 174.076 | 141.097 | 148.569 | |

The sixth group | X-16 | 5 | 10 | 176.064 | 135.568 | 146.122 |

X-17 | 5 | 10 | 182.819 | 134.673 | 133.307 | |

X-18 | 5 | 10 | 179.192 | 135.745 | 140.715 | |

The seventh group | X-19 | 7 | 14 | 207.494 | 120.060 | 121.416 |

X-20 | 7 | 14 | 164.472 | 120.332 | 121.737 | |

X-21 | 7 | 14 | 189.840 | 120.994 | 122.178 | |

The eighth group | X-22 | 8.4 | 16.8 | 190.598 | 127.443 | 125.962 |

X-23 | 8.4 | 16.8 | 178.959 | 118.255 | 124.386 | |

X-24 | 8.4 | 16.8 | 178.563 | 115.233 | 119.571 |

As can be seen from Table

The relationship between the initial elastic modulus, the unloading elastic modulus, and the loading elastic modulus and the deformation control value of the eight groups of specimens is plotted into curves, as shown in Figure

Comparison of elastic modulus during the three stages.

As can be seen from Figure

The uniaxial tensile test of damaged Q235 steel has been carried out in [

Comparison of elastic modulus.

As can be seen from Figure

The loading elastic modulus curve of Q345 steel in Figure

The percentage decline of elastic modulus in unloading stage.

As can be seen from Figure _{y} = 350 MPa and the initial elastic modulus _{0} = 1.85 × 105 MPa.

The stress-strain relationship of steel in different part of the cross section changes greatly after the cross section of a steel member is damaged. Steel exhibits varying degrees of damage and under elastic stress state. To facilitate the calculation of ultimate bearing capacity and deformation, the average elastic modulus of damaged cross section is used to replace the elastic modulus of undamaged steel.

For the H-section, when the steel damage occurs only in the flange, partial flange damage does not cause a major change in the average elastic modulus of the entire cross section due to the thinness of the flange. Hence, this situation can be assumed as slight damage of the cross section. At this time, the average elastic modulus of the cross section is taken as the elastic modulus of the steel.

When the flange and part of the web of the cross section are damaged, the mechanical properties of the cross section change obviously. Hence, this situation can be assumed as general damage of the cross section.

When the damage degree is further aggravated, some strains in the damaged area of the section exceed 1.6%; that is, serious damage is present in the section. Hence, this situation can be assumed as serious damage of the cross section.

Given that slight damage has minimal effect on the structure, this study only considers general damage and serious damage of the cross section. The distribution characteristics of loading elastic modulus of the generally damaged and seriously damaged cross section are shown in Figures

Distribution characteristics of elastic modulus of generally damaged cross section.

Distribution characteristics of elastic modulus of seriously damaged cross section.

_{1} = section area of undamaged zone, _{2} = section area of generally damaged zone, _{3} = section area of seriously damaged zone, _{1} = height of undamaged cross section, and

To simplify the calculation, the following assumptions are made:

Damage mainly refers to the phenomenon of partial or total yield of the cross section caused by the bending moment generated by the external load on the H-section.

Assumption plane section is satisfied after the cross section is damaged; that is, the cross section remains in the same plane after the cross section is damaged.

The maximum strain value in the cross section is less than 18%. The physical meaning of this assumption is that all materials in the damaged cross section have not overstepped the strengthening stage, and no serious nonuniform deformation occurs in the necking stage.

The strain distribution characteristics of generally damaged cross section and seriously damaged cross section can be obtained as shown in Figures

Strain distribution characteristics of generally damaged cross section.

Strain distribution characteristics of seriously damaged cross section.

According to the static equilibrium condition [

After Equations (

Equation of generally damaged cross section:

Equation of seriously damaged cross section:

To study the mechanical properties of steel members subjected to interaction of axial force and bending moment, the relationships between bending moment

Stress-strain relationship of steel.

Figure

The characteristics of undamaged H-shaped cross section when the value of axial force is constant: (a) H-shaped cross section; (b) internal force of the cross section; (c) full-sectional elastic stress distribution; (d) single-sided elastic stress distribution; (e) double-sided elastic stress distribution.

According to the static equilibrium condition, the axial force and bending moment of the cross section can be calculated as follows:

According to the method of material mechanics [

To facilitate numerical computation, the

When the axial force and moment are relatively small, under the interaction of the bending moment and the axial force, the maximum stress in the cross section of the member does not exceed the yield strength of the steel, and the whole cross section of the member is in the elastic state. The stress distribution of the section is shown in Figure _{y}_{p}

The following can be obtained from Equations (

The following can be obtained from Equations (

With the increase of bending moment, the cross section of the member comes into the single-sided plastic distribution, and the stress distribution of the section is shown in Figure _{1} and −_{1} are the ordinates of the outer edges of the upper and lower flanges of H-shaped cross section, respectively. _{1} = _{1/2}. _{2} and −_{2} are the ordinates of the inner edges of the upper and lower flanges of H-shaped cross section, respectively. _{2} = _{1/2}. _{3} is the ordinate of the junction of the plastic zone and the elastic zone at the upper end of the cross section. In this case, the ultimate stress state of the section is that the tensile side edge of the cross section reaches the yield stress _{y}.

When the bending moment _{1}≥_{3}≥_{2}. When the bending moment _{3}<_{2}.

According to the coordinates _{y}_{3}) and _{y}_{1}) of the two endpoints of the stress curve in the elastic area, the stress equation _{z} =

When _{3} = _{2}, the following can be obtained from Equations (

When _{and}

We have

The following can be obtained from Equations (

The following can be obtained from Equations (

The following can be obtained from Equations (

By substituting Equation (

_{f}) ≤ _{3}≤ _{2}, the cross-sectional damage is general damage, and the following can be obtained from Equations (_{1} ≤ _{3} ≤ –(0.29_{f}_{4} = 1.855_{3}; _{4} is the ordinate at the junction of the generally damaged area and the seriously damaged area of the cross section; that is, the ordinate of the sectional strain is −1.6%.

With the increase of bending moment limit, the cross section of the member comes into the double-sided plastic distribution, and the stress distribution of the cross section is shown in Figure _{pc}.

When _{3} = −_{2}, the following can be obtained from Equations (_{2} ≤ _{3} ≤ 0, the following can be obtained from Equations (

The following can be obtained from Equations (

Substituting Equation (

The following can be obtained from Equations (

After summarizing the above equations, the calculation expressions of

Calculation expressions of M-Φ-P of H-section steel members.

Dimensionless variable | Range of | Equation |
---|---|---|

_{1} | 0≤ | Equation ( |

_{1} | 0≤ | Equation ( |

_{2} | Equation ( | |

Equation ( | ||

_{2} | Equation ( | |

Equation ( | ||

Equation ( | ||

_{pc} | Equation ( | |

Equation ( |

However, the elastic modulus _{1}, _{2}, _{pc}, _{1}, and _{2} are dimensionless variables. Whether undamaged or damaged cross section, _{y} can be eliminated in the derivation. Hence, the above derivation and results are applicable to the undamaged and damaged cross sections of the H-shaped steel members.

The cross-sectional sizes of test steel columns are as follows [_{1} = 113.09 mm, _{1} = 59.45 mm, _{f} = 5.98 mm, _{w} = 6.17 mm, ^{2}, and _{x} = 6.21 × 10^{6} mm^{4} were analyzed. From the equations in Table

Dimensionless parameters of steel column cross sections.

0 | 1 | 1 | 1.000 | 1.000 | 1.500 |

0.1 | 0.9 | 0.9 | 1.035 | 1.046 | 1.480 |

0.2 | 0.8 | 0.8 | 1.002 | 1.025 | 1.418 |

0.3 | 0.7 | 0.7 | 0.920 | 0.966 | 1.316 |

0.4 | 0.6 | 0.6 | 0.804 | 0.876 | 1.165 |

0.5 | 0.5 | 0.5 | 0.675 | 0.759 | 0.985 |

0.6 | 0.4 | 0.4 | 0.543 | 0.632 | 0.763 |

0.7 | 0.3 | 0.3 | 0.411 | 0.496 | 0.608 |

0.8 | 0.2 | 0.2 | 0.279 | 0.349 | 0.411 |

0.9 | 0.1 | 0.1 | 0.147 | 0.191 | 0.208 |

1 | 0 | 0 | 0 | 0 | 0 |

Considering the damage of member cross section, the relationships between _{2}, _{2}, _{pc}, and

Correlation curves between (m)_{2}, _{2}, (m)_{pc}, and (p) of column sections.

Approximate expressions of M-Φ-P relationship of steel column sections.

0 ≤ | 0.4 ≤ | |
---|---|---|

_{1} | ||

_{1} | ||

_{2} | ||

_{2} | ||

_{pc} |

Once the

The data of steel column test [

Schematic diagram of the loading setup.

Photograph of the loading setup.

Comparison of failure mode. (a) Failure mode of finite element model. (b) Failure mode of the specimen.

The specific calculation process of the steel column deformation using the Newmark method [

The steel column was divided into four equal parts, and the number of each piecewise point is shown in Figure

During the test, the steel column was loaded twice, and the steel column reached the ultimate bearing capacity for each loading. The residual deformation of the steel column at each piecewise point after the first loading was taken as the initial defect value

Calculate the bending moment _{k} of each piecewise point,

According to the known bending moment–curvature–axial force relationship of the cross section, calculate the curvature Φ_{k} of each piecewise point,

A new group of piecewise point displacement values of the steel column was calculated by using the conjugate beam method, and the curvature distribution between the piecewise points was assumed to be distributed according to the characteristics of Figure

Compare the deflections calculated in step 5 with the deflections assumed in step 2. If the difference can be neglected, then the solution is considered to be obtained. Otherwise, take the calculated deflections as a new group of hypothetical deflections, and repeat steps 3 to 5 until convergence.

Piecewise points of steel column.

Equivalent nodal loads.

By using this method, the deformations of damaged steel columns under the effect of each test load after secondary loading are calculated, respectively. Table

Deformation calculations of steel columns.

Load | ||||||||||
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Piecewise point | 0 | 1 | 2 | 3 | 4 | 0 | 1 | 2 | 3 | 4 |

Initial moment _{1} | 0.43 | 0.43 | 0.43 | 0.43 | 0.43 | 0.75 | 0.75 | 0.75 | 0.75 | 0.75 |

Initial imperfection | 0 | 0.0072 | 0.0144 | 0.0235 | 0.0325 | 0 | 0.0072 | 0.0144 | 0.0235 | 0.0325 |

Assuming additional deflection | 0 | 0.000687 | 0.00248 | 0.00528 | 0.00897 | 0 | 0.002 | 0.0068 | 0.0137 | 0.0224 |

Moment of | 0.00802 | 0.00649 | 0.00475 | 0.00245 | 0 | 0.0187 | 0.0155 | 0.0115 | 0.006 | 0 |

Changing common factor | 0.179 | 0.145 | 0.106 | 0.0547 | 0 | 0.416 | 0.346 | 0.256 | 0.134 | 0 |

Total moment _{k} | 0.610 | 0.576 | 0.537 | 0.486 | 0.431 | 1.166 | 1.096 | 1.006 | 0.884 | 0.75 |

_{k} | 0.665 | 0.611 | 0.553 | 0.496 | 0.431 | 2.055 | 1.512 | 1.178 | 0.960 | 0.755 |

_{k} | 0.379 | 0.610 | 0.553 | 0.495 | 0.262 | 1.108 | 1.530 | 1.188 | 0.961 | 0.475 |

_{k} | 0.379 | 0.989 | 1.543 | 2.038 | 1.108 | 2.638 | 3.826 | 4.787 | ||

0 | 0.379 | 1.368 | 2.911 | 4.949 | 0 | 1.108 | 3.746 | 7.572 | 12.359 | |

Calculating additional deflection | 0 | 0.00069 | 0.00248 | 0.00528 | 0.00897 | 0 | 0.002 | 0.00678 | 0.0137 | 0.0224 |

Table

Comparison of horizontal displacements at the top of columns.

Load (kN) | Displacement (mm) | ||||
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Test result (mm) | Theoretical result (mm) | Relative error (%) | FEM result (mm) | Relative error (%) | |

260 | 20.80 | 22.40 | 7.69 | 17.59 | 18.25 |

240 | 17.98 | 17.10 | 5.15 | 15.39 | 16.83 |

210 | 14.30 | 13.40 | 6.29 | 13.06 | 9.49 |

180 | 12.07 | 10.90 | 9.69 | 11.16 | 8.15 |

150 | 9.91 | 8.97 | 9.48 | 9.31 | 6.44 |

Table

Comparison of load-displacement curves of steel columns under secondary loading.

The analysis above shows that the theoretical calculation method for damaged steel columns used in this study is in good agreement with the test results.

According to the above analysis, the main conclusions are drawn as follows:

The average elastic modulus is used to consider the effect of different degrees of damage on the mechanical properties of H-shaped cross section. The damage of H-shaped cross sections is divided into three levels: slight damage, general damage, and serious damage. The calculation equations of average elastic moduli under different degrees of damage are derived.

By using the cross-sectional analysis method, the equations for calculating

The Newmark method is used to analyze the deformations of damaged steel columns, and the results are in good agreement with the test results. The equations and method derived in this study have high accuracy and can be applied to the calculation of

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was supported by the Natural Science Foundation of Hunan Province of China (2020JJ4943) and Science and Technology Program of Hunan Housing and Urban Rural Development Department (KY202002).