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The overall stability of a movable high-strength inverted-triangular steel bridge is worth studying because of its new truss structure. In this study, an approach was proposed based on the stiffness equivalence principle in which the inverted-triangle truss structure was modeled as a thin-walled triangular beam. On this basis, the calculation of the critical load of elastic stability of a movable high-strength inverted-triangular steel bridge with variable rigidity at both ends and locally uniformly distributed load was carried out based on the energy theory, which was in good agreement with existing theories. A material performance test at BS700 was carried out to establish the material properties, and then a finite element model of the bridge was established, the results of which were compared with those of the experimental load test, in order to verify the accuracy of the finite element model. Considering material nonlinearity and geometric nonlinearity, nonlinear buckling analysis of the bridge was conducted and the factors influencing the bridge’s ultimate bearing capacity were analyzed.

High-strength steel with a yield strength of 460–690 MPa has been commonly used in construction projects in Japan, the United States, and some other countries [

Apart from BS700 high-strength steel, a new inverted-triangular truss structure was used in this bridge. Two pieces of inverted-triangular truss structures made up two lanes, which were connected and integrated in between on the top using cross beams. Central bridge segments are shown in Figure

The members of a typical bridge segment.

Considerable research on the stability of triangular or inverted-triangular truss structures has been done. The eigenvalue stability of truss structures was studied through critical point theory [

Analysis of the elastic stability load-carrying capacity lays the foundation for analyzing the nonlinear stability load-carrying capacity. To perform the theoretical analysis, the truss structure is usually modeled as a beam, following the principle of equivalent stiffness [

The simplified diagram of the cross section of the movable high-strength inverted-triangular steel bridge is shown in Figure

Schematic diagram of the cross section of the movable high-strength inverted-triangular steel bridge.

There are two different sections on the top chord whose areas are

Schematic diagram of the movable high-strength inverted-triangular steel bridge.

One section of the movable high-strength inverted-triangular steel bridge is taken as a calculation unit and converted into a thin plate whose thickness is

One segment of the movable high-strength inverted-triangular steel bridge.

The internal force generated by a slanted web member with the action of shear force is

Top chord:

Slanted web member:

Bottom chord:

Vertical web member:

The equation for the

The equivalent rigidity can be calculated using MATLAB according to this theoretical derivation [

It is difficult to find the solution to the torsional rigidity if the movable high-strength inverted-triangle steel bridge is simplified into a thin-walled triangular beam, since the bridge’s rigidity varies at the two ends while the central rigidity stays the same. To this end, it is solved using the energy method [

Lateral bending strain energy can be obtained according to the strain energy equation [

Therefore, total strain energy is

Position of the shearing center of the movable high-strength inverted-triangular steel bridge.

Total potential energy is

Thin-walled triangular beam under locally uniformly distributed load.

The expressions of segmented load and rigidity are shown as follows.

According to the minimal potential energy principle, we have

To further verify the accuracy of the equation for the elastic stability ultimate load, a model was built using the finite element method to perform elastic buckling analysis.

The beam element 188 of the ANSYS software, which can import a user’s section data, was adopted in the finite element model. In order to analyze many modes used for studying the influence factors on stability ultimate load capacity, the model uses many parameters which can be easily changed in an APDL document (Analysis Parameter Design Language) [

In order to get the material model data, uniaxial tension tests were carried out as shown in Figure

Material mechanical property test. (a) Test procedure. (b) The members’ failure.

As shown in Figure

MISO data used for ANSYS models.

Strain | 0.001 | 0.00178 | 0.00203 | 0.003 | 0.0037 | 0.00385 | 0.00426 | 0.00451 | 0.008 |
---|---|---|---|---|---|---|---|---|---|

Stress [MPa] | 209 | 370 | 419 | 592 | 689 | 704 | 734 | 754 | 790 |

In order to verify the basic finite element model validity, the load capacity test of the bridge was performed. A 51 m-long bridge, where the bearing length at both ends was 2 m and clearance height under the span was 0.6 m, was constructed. Two sections of planks were placed at the middle of bridge span to simulate two decks. Steel plates weighing 25.42 t→48.97 t→58.42 t→63.15 t→67.87 t→71.17 t→74.56 t→80.84 t (1 t≈10 kN) were loaded to and piled up on the planks. There were three deformation points at

Diagram of test points and loading.

The load test scene.

The test results compared with finite element analysis (FEA) results are shown in Table

Comparison of the displacement results of the test points and FEM calculation.

| | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|

254.2 | 38 | 40 | 5.3 | 97 | 105 | 8.2 | 40 | 42 | 5.0 |

489.7 | 73 | 78 | 6.8 | 202 | 211 | 4.5 | 80 | 83 | 3.6 |

584.2 | 89 | 93 | 4.5 | 247 | 262 | 5.7 | 95 | 100 | 5.0 |

631.5 | 95 | 100 | 5.3 | 272 | 282 | 3.5 | 105 | 108 | 2.8 |

678.7 | 102 | 107 | 4.9 | 292 | 304 | 3.9 | 115 | 116 | 0.9 |

711.7 | 107 | 112 | 4.7 | 312 | 319 | 2.2 | 120 | 121 | 0.8 |

745.6 | 112 | 117 | 4.5 | 327 | 334 | 2.1 | 130 | 127 | -2.4 |

808.4 | 122 | 127 | 4.1 | 342 | 362 | 5.5 | 135 | 138 | 2.2 |

According to Table

The typical equation for eigenvalue buckling analysis was [

Without considering self-weight, the eigenvalue

First-order eigenvalue buckling mode of the movable high-strength inverted-triangular steel bridge.

Values of bridge’s elastic ultimate bearing capacity obtained by theoretical derivation and eigenvalue buckling analysis were usually so large that they could only be used for qualitative analysis of the structure’s load-carrying capacity instead of genuinely reflecting it. Therefore, nonlinear buckling analysis was required. Considering factors such as the structure’s geometric defect and the nonlinearity of materials, a nonlinear buckling analysis was able to more truthfully present the structure’s load-carrying capacity. In consideration of the material-geometry dual nonlinearity, the basic buckling equation could be converted into

The geometric deflection was introduced to the analysis model by the consistent-deflection-mode method, which simulates deflection distribution by the lowest buckling mode shape. The maximal displacement of buckling analysis was multiplied by a coefficient to amplify to the 1/1000 span of bridge.

Based on the above finite element model and considering material and geometric dual nonlinearity factors, the stability ultimate load of the bridge can be obtained by using the arc-length method [

Load-displacement at middle span. (a) Load-vertical. (b) Load-lateral.

From Figure

There are many factors which can affect the stability ultimate load capacity of a movable high-strength inverted-triangular steel truss bridge, such as bridge span, bridge height, track width, and member’s section. In order to study the influence of these factors on the stability ultimate capacity of the bridge, many FEA (finite element analysis) models with different parameters were analyzed.

The height and track width are two important parameters for design. When the analysis was performed, one parameter was changed each time. Through FEA analysis the stability ultimate load capacity

Change percentage of the bridge’s ultimate load capacity when track width varies.

From Figure

The top and bottom chord are the important members in the bridge. How a chord influences the stability ultimate load was also studied in this paper. The top chord has two different box sections, which were labeled chord I and chord II. The top chord area changed at the same time of the width and height changes of the section. The multiplied area of the top chord can be found through adjusting the width and height of section gradually. The bottom chord is a kind of complex section. The multiplied area of the bottom chord can be found through adjusting the main dimensions of the section gradually. The results are shown in Figure

Change percentage of the bridge’s ultimate load capacity when chord area varies.

From Figure

The web member mainly bears shear force in the bridge structure but also provides some bending stiffness. There are two different web members shown in Figure

Change percentage of the bridge’s ultimate load capacity when web member area varies.

According to Figure

This paper investigated the stability of a movable high-strength inverted-triangular steel bridge by the theoretical, experimental, and finite element methods. From this study the following conclusions can be drawn:

Axial rigidity

In-plane flexural rigidity

Out-plane flexural rigidity

Torsional rigidity

The top chord I area

The top chord II area

The bottom chord area

The distances between the centroid and the top chord

The distances between the centroid and the bottom chord

The thickness of equivalent thin-walled beam

The intersegment length

The intersegment height

The strain energy of each piece of thin plate

The total shear force that results from shear flow

The top chord strain energy of each piece of truss unit

The bottom chord strain energy of each piece of truss unit

The vertical web member strain energy of each piece of truss unit

The total strain energy of the

Lateral bending strain energy

In-plane bending strain energy

Pure torsion strain energy

The total strain energy

Total potential energy

The lateral displacement

The vertical displacement

The torsion angle

The potential energy of external force

The section asymmetry coefficient

The distance between the shearing center and the top side of equivalent thin-walled beam

The width of the local uniformly distributed load

The length of the segment with variable rigidity

The intensity of uniformly distributed load

The span of the simply supported bridge

The parameters related to

The shape function parameters of the lateral displacement and the torsion angle

The parameters related to

The parameters related to

The critical load

The small-displacement elastic stiffness matrix

The initial stress stiffness matrix

The initial strain matrix

Nodal displacement vector

Nodal load vector

The stability ultimate load capacity of the bridge.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors wish to acknowledge the financial support from the project of the Major State Basic Research Development of China (973 Program, No. 2014CB046801) and China Postdoctoral Science Foundation (Grant No. 2017M623403).