The study reports investigations into shear buckling resistance of the corrugated web of cantilever SIN girders. Experimental tests were conducted on ten SIN girders with the web height of 500, 1000, 1250 and 1500 mm made to 1:1 scale. The tests confirmed the advantageous effect of support stiffeners of cantilever girder parts on shear buckling resistance. Load-displacement paths of global displacements in cantilevers of SIN girders were analysed. The Finite Element Method was used to construct models that simulated the behaviour of the experimental models. The numerical analysis of FEM girders was conducted for twelve models with the web height ranging from 500 to 1500 mm, and web thickness of 2, 2.5 and 3 mm. In the FEM analysis, different modes of the web failure, namely local and interactive ones, were taken into account. Based on experimental investigations and the FEM analysis, a method for estimating design shear buckling resistance of the corrugated web in cantilever girders with support stiffener was proposed. The method was based on the determination of interactive buckling resistance. It was demonstrated that support stiffeners in cantilever girders produced an advantageous effect on increase in shear buckling resistance. The solution developed was compared with the methods currently employed to determine buckling resistance. Conclusions and recommendations were drawn on dimensioning of cantilever girders with support stiffener.

Welded plate girders with thin-wall corrugated webs, lower in weight than conventional plate girders, have gained in popularity in 90’s. Currently, SIN girders available on the market have three basic web thicknesses of 2.0, 2.5 and 3.0 mm and heights ranging from 333 to 1500 mm. Guaranteed by the manufacturer, yield resistance of the corrugated web steel is _{y}

Due to the web thickness, SIN girders are more stressed in shear compared with flat webs. The buckling mechanism in sinusoidal corrugated web under shear load is still classified separately as a local and global instability [

In the author’s studies [

Based on preliminary investigations, it is known that for cantilever girders under two-sided symmetrical load on cantilevers (Fig.

Cantilever SIN girders: a) static scheme, b) flat plates on the support in the SIN girder

This study reports investigations into shear buckling resistance of the corrugated web of SIN girders with one-sided cantilever (Fig. _{1} in Fig.

Experimental investigations on end-loaded one-sided cantilever were conducted using ten girders with the web height of 500, 1000, 1250 and 1500 mm, composed of three pre-assembled units. Girders, the loading diagram of which corresponded to a simply supported beam with one-sided cantilever, were constructed from pre-assembled units, butt-connected using HV bolts.

The Finite Element Method was used to simulate experimental investigations [_{w}_{w}

For girders with trapezoidal profile of the web folds, the estimation of design shear buckling resistance was based on the computation of the interactive buckling resistance. The general form of the equation describing interactive buckling resistance _{crI}

Equation (_{crL}_{crG}_{y}. That can be found in studies [

In 2009, Moon [_{i} [_{s}_{I}_{y}.

In 2011, Sause and Braxtan presented their solution [

In the solution of concern, local and global slenderness was determined from formula (_{G}

All available solutions concerning the computation of design shear buckling resistance relate to webs with trapezoidal shape of folds. In 2009, Eldib [

As regards SIN girders utilizing sine-shaped folds, the solution currently used can be found in EC3 [_{crL}_{crG}_{w}_{w}_{w}

Next, the dependence describing shear buckling stress at global instability based on the stiffness relation of the orthotropic plate [_{y}_{z}

The solution offered by EC3 [

In order to determine shear buckling resistance of cantilever girders with support stiffeners, experimental investigations were conducted. They covered ten SIN girders with a loading diagram that corresponded to a simply supported beam with one-sided cantilever. (Fig.

Cantilever girders with corrugated web a) models M 1.12, M 2.12 b) models M 1.22, M 1.32, M 1.42, M 1.52, M 2.22, M 2.32, M 2.42, M 2.52; c) d) end stiffeners

Girders were made from pre-fabricated pre-assembled units. Girders with the web height of _{w}_{w}

In WTA 500 girders (the first two letters WT mean the girder with corrugated web, the next letter means the thickness of the web, that is: A – 2 mm, B – 2.5 mm, C – 3 mm), 20 mm thick end plates were used (Fig.

The pre-assembled units of the girders of concern were butt-connected using HV M20 (_{w}_{w}_{j}

The girders were constructed from pre-assembled units. Openings in the connection end plates were adjusted at the experimental stand, which limited the occurrence of imperfections in end-plate connections. A frame (FR) (Fig.

The following quantities were measured in the investigations: reaction

a) Girder 2.32 on the test stand; b) steel frame FR

Location of strain gauges on the girder web M 2.52

In order to establish the point of the corrugated web instability, the profiles of strains were determined for all strain gauges glued onto the web. Based on the analysis of graphs for diagonal strain gauges, the onset of instability of the corrugated web was specified. The first buckling load _{eB}

Strain in the direction 60° relative to the axis of the web a) girder M 1.32; b) girder M 2.12

Based on the global displacement y measured at the end of the girder support (Fig.

Diagram of global displacements y of tested girders

Figures

Load – displacements paths

Load – displacements paths

The upper boundary of the rectilinear part of global displacement (point _{1}(_{eB}

Characteristic co-ordinates _{1}(_{eB}_{2}(_{uRd}

_{1}(_{eB}_{eB}

_{2}(_{uRd}_{uRd}

_{3}(_{3}) – girder unloading.

The boundary of the range of displacements resulting from the effect of bending moments and shear forces was marked in the global LDPs _{1}(_{eB}_{1}(_{eB}_{2}(_{uRd}_{1}(_{eB}_{2}(_{uRd}

In cantilever girder with corrugated web, in which support stiffener made from two connected sheets is applied, a large range of elastic strains 0 – _{1}(_{eB}

Table _{uRd}_{eB}

Experimental results of girders

Girder | Web _{w} x t_{w} |
Flange [mm] | Suport stiffener | Failure modes | Limit load _{eB} |
First buckling load _{u,Rd} |
_{eB}_{u,Rd} |
---|---|---|---|---|---|---|---|

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

M 1.12 | 500×2 | 300×15 | 2×300×20 | L | 184 | 147 | 0.80 |

M 2.12 | 500×2 | 300×15 | 2×300×25 | L | 181 | 151 | 0.82 |

M 1.22 | 1000×2 | 300×20 | 2×300×25 | I | 342 | 298 | 0.87 |

M 2.22 | 1000×2 | 300×15 | 2×300×25 | L | 343 | 296 | 0.86 |

M 1.32 | 1000×2.6 | 300×20 | 2×300×25 | I | 478 | 380 | 0.79 |

M 2.32 | 1000×2.6 | 300×15 | 2×300×25 | I | 492 | 390 | 0.79 |

M 2.42 | 1000×3 | 300×15 | 2×300×25 | I | 694 | 510 | 0.73 |

M 1.42 | 1250×2 | 300×15 | 2×300×25 | I | 348 | 304 | 0.87 |

M 1.52 | 1500×2 | 300×15 | 2×300×25 | I | 468 | 400 | 0.85 |

M 2.52 | 1500×2 | 300×15 | 2×300×25 | I | 459 | 399 | 0.87 |

Failure of the examined girders occurred in the cantilever part, in the area affected by a load induced by a constant shear force. The failure took place suddenly.

The process of the corrugated web instability started from local instability of the sinusoidal panel. In the first stage of girder failure, plastic strains occurred in the corrugated web, adjacent to the tension flange, which led to formation of the web yield zone (1). The latter took the form of diagonal tension line (local instability – L) (Fig. _{w}

Failure modes of cantilever SIN girders: a) M 1.12 (2×300×20 mm); b) M 2. 52 (2×300×25 mm)

In all tested girders, support stiffener stayed straight and did not bend after the yield zone was formed. The end plate that makes the edging of the stiffeners and end – plate connections of the middle segment also remained intact in all girders.

The above indicates that support stiffener in cantilever girders restricted the action of the tension field and the resulting change in the interaction of compression and shear components along the generatrix of the web. That led to a narrowing of the range of plastic strains _{1}(_{eB}_{2}(_{uRd}

As regards cantilever girders shown in Fig.

Numerical models

Their thickness corresponded to that of connection sheets, i.e. 50 or 40 mm (in models with _{w}

The FEM analysis was conducted using 12 numerical models. All supports were modelled to have the length of _{w}_{w}

Current numerical program

_{w} |
_{w} |
Flange [mm] | Suport stiffener [mm] | Number of models | ||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 7 | 8 |

500 | 2; 2.6; 3 | 300×15 | 2×300×20 | 1500 | 6350 | 3 |

1000 | 2; 2.6; 3 | 300×15 | 2×300×25 | 1500 | 6350 | 3 |

1250 | 2; 2.6; 3 | 300×15 | 2×300×25 | 1500 | 6350 | 3 |

1500 | 2; 2.6; 3 | 300×15 | 2×300×25 | 1500 | 6350 | 3 |

Materials tests on steel in the experimental cantilever girders were conducted using samples cut out of flanges and the web acc. EN [

Material properties

Girder | Percentage total elongation at maximum force (_{m} |
Percentage total elongation at fracture [%] | Percentage total elongation at maximum force (_{m} |
Percentage total elongation at fracture [%] | E [GPa] | ||||
---|---|---|---|---|---|---|---|---|---|

web | flange | ||||||||

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

M 1.12 | 334.7 | 430.6 | 16.1 | 22.1 | 297.5 | 443.1 | 23.9 | 30.3 | 196 |

M 2.12 | 337.9 | 429.1 | 15.7 | 20.8 | 311.2 | 476.7 | 24.5 | 30.8 | 188 |

M 1.22 | 339.4 | 435.4 | 15.9 | 22.2 | 287.2 | 448 | 24.1 | 31.4 | 200 |

M 2.22 | 336.3 | 427.3 | 14.9 | 22.5 | 323.5 | 461.4 | 23.1 | 29.6 | 234 |

M 1.32 | 312.5 | 453.9 | 13.8 | 20.2 | 264.2 | 460.9 | 22.6 | 28.0 | 199 |

M 2.32 | 324.6 | 451.5 | 13.6 | 19.2 | 293.9 | 442.1 | 25.1 | 32.3 | 241 |

M 2.42 | 445.9 | 544.1 | 10.7 | 14.6 | 301.5 | 439.8 | 21.3 | 27.7 | 205 |

M 1.42 | 267.2 | 360.9 | 13.4 | 19.9 | 302.8 | 440.4 | 22.9 | 30.8 | 199 |

M 1.52 | 299.1 | 380.5 | 16.8 | 23.3 | 312.5 | 445.6 | 24.1 | 30.0 | 200 |

M 2.52 | 281.0 | 375.5 | 17.2 | 24.6 | 306.7 | 449.3 | 21.9 | 28.6 | 203 |

In the materials tests conducted on girders, a very

In all numerical models (Fig. _{z}_{x}_{y}_{z}_{y}_{y}_{x} = 0) was averted.

Boundary conditions and load application to numerical model

Support conditions adopted in numerical investigations represent those used in actual structures. That caused slight extension of the cantilever part in numerical models. However, it did not affect the values of the limit load, buckling load, or shear buckling resistance. Conversely, in experimental girders support conditions included a roller bearing support, and also hinge support. The latter also supported the end of the girder.

The load (Fig.

In the numerical analysis reported in this study, the Riks method was used. In this method, the load is proportionally applied in individual steps. The control parameter is the so-called path parameter. The Riks method allows finding a solution to a task regardless of the web failure mode. That is related to identifying load-displacement equilibrium at the end of each iteration step. While seeking load-displacement equilibrium, the load can be increased or decreased until the limit load is reached acc. [

The numerical model was validated in two stages. The first stage of the model validation involved the use of a “perfect model”, in which no imperfections occurred, and the geometry of webs, flanges and stiffeners from measurements was accurately represented. For M 2.52 (1500×2) and M 1.42 (1250×2) girders, the ratio of first buckling load _{eBINV}/P_{eBFE}

In the second stage of the model validation, an “imperfect model” was considered. The imperfection consisted in the web thinning by 1/20 of its thickness acc. [_{uRd}_{eB}

Numerical results of girders

Girder _{w} x t_{w} |
Support Stiffener | Failure modes | Limit load _{u,Rd} |
First buckling load _{eB} |
_{eB}_{u,Rd} |
---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 |

500×2 | 2×300×20 | L | 154.1 | 149.1 | 0.97 |

500×2.5 | 2×300×20 | L | 192.2 | 186.8 | 0.97 |

500×3 | 2×300×20 | L | 231.3 | 224.6 | 0.97 |

1000×2 | 2×300×25 | L | 308.1 | 297.2 | 0.96 |

1000×2.5 | 2×300×25 | I | 384.3 | 371.3 | 0.97 |

1000×3 | 2×300×25 | I | 462.2 | 445.3 | 0.96 |

1250×2 | 2×300×25 | I | 384.6 | 360.1 | 0.94 |

1250×2.5 | 2×300×25 | I | 480.4 | 449.9 | 0.94 |

1250×3 | 2×300×25 | I | 576.5 | 539.5 | 0.94 |

1500×2 | 2×300×25 | I | 460.8 | 419.2 | 0.91 |

1500×2.5 | 2×300×25 | I | 575.6 | 531.9 | 0.92 |

1500×3 | 2×300×25 | I | 687.5 | 635.7 | 0.92 |

In all remaining numerical models, because of lower yield strength than that in experimental girders, the results of first buckling load turned out to be slightly smaller. The effect of variable yield strength on the results of the design buckling resistance was accounted for in the theoretical solution adopted (Chapters 5 and 6).

Load-displacement paths LDPs

Figures

Comparison of LDPs

Comparison of LDPs

In load-displacement paths, characteristic coordinates _{1}(_{eB}_{2}(_{uRd}

For each numerical model of cantilever girders, the web instability took place at point _{1}(_{eB}_{2}(_{uRd}_{1}(_{eB}_{2}(_{uRd}

The FEM analysis confirmed that in cantilever SIN girders, a large range of elastic strain 0 – _{1}(_{eB}

Table _{uRd}_{eB}_{eB}_{uRd}

Figures

Comparison of failure modes: a) experimental girder M 1.12 (500×2: stiffener 2×300×20); b) numerical model 500×2 (stiffener: 2×300×20)

Failure modes of numerical models: a) 1000×2 (stiffener 2×300×25); b) 1000×2.5 (stiffener 2×300×25)

Comparison of failure modes: a) experimental girder M 1.52 (1500×2: stiffener 2×300×25); b) numerical model 1500×2 (stiffener: 2×300×25)

For girders with the web height of _{w}_{w}

In numerical models of cantilever girders with _{w}_{w}_{w}

In girders with the web height of _{w}_{w}

For cantilever girders, shear buckling resistance at the point of instability _{1}(_{eB}_{eB}_{w}_{w}

In cantilever girders of concern, similar to girders with the loading diagram of a simply supported beam, two modes of instability were found: local one and interactive one. Failure modes in cantilever girders indicate that the use of support stiffeners in cantilever girders produces an effect on the web failure mode similar to that brought about by rigid stiffeners in girders with the loading diagram of a simply supported beam. Stiffeners contribute to increase in shear buckling stress which leads to higher shear buckling resistance.

Thus, to estimate design shear buckling resistance _{n}_{crI,6}

Interactive shear buckling resistance _{crI,6}

It was based on estimating local _{cr,L}_{cr,G}

In the solution adopted, the value of coefficient _{L}_{w}

In addition, the value of coefficient _{G}_{I,6}_{y} and the interactive shear buckling resistance _{crI,6}

In the case of girders where only a local instability takes place, the slenderness should be determined as dependent on the critical stresses at the local loss of stability _{cr,L}

As for cantilever girders with corrugated web, they were girder cantilever parts that suffered failure. The value of design buckling resistance of cantilever girders was affected by the web failure mode. For the height of _{w}_{w}

Table _{y}_{INV}_{n,SB}_{n,EC}_{n,BA}

Comparison investigations, FEM analysis with design [

Girder _{w} x t_{w} |
_{INV} |
_{FEM} |
|||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

500×2 | 147.0 | 149.1 | 0.80 | 0.92 | 0.79 | 0.86 | 0.86 | 1.09 | 0.93 |

500×2.5 | 149.4 | 0.92 | 0.79 | 0.91 | 0.86 | 0.93 | |||

500×3 | 149.7 | 0.92 | 0.79 | 0.95 | 0.86 | 0.93 | |||

1000×2 | 149.0 | 148.6 | 0.78 | 0.92 | 0.79 | 0.83 | 0.86 | 1.10 | 0.94 |

1000×2.5 | 152.0 | 148.5 | 0.90 | 0.92 | 0.79 | 0.88 | 0.86 | 0.96 | 0.94 |

1000×3 | 170.0 | 148.4 | 0.75 | 0.91 | 0.79 | 0.93 | 0.86 | 1.26 | 0.94 |

1250×2 | 121.6 | 144.0 | 0.86 | 0.89 | 0.79 | 0.82 | 0.85 | 1.01 | 0.96 |

1250×2.5 | 144.0 | 0.89 | 0.79 | 0.88 | 0.85 | 0.96 | |||

1250×3 | 143.9 | 0.89 | 0.79 | 0.92 | 0.86 | 0.96 | |||

1500×2 | 133.0 | 139.7 | 0.88 | 0.86 | 0.79 | 0.82 | 0.84 | 0.96 | 0.98 |

1500×2.5 | 141.8 | 0.87 | 0.79 | 0.87 | 0.85 | 0.97 | |||

1500×3 | 141.3 | 0.87 | 0.79 | 0.92 | 0.85 | 0.97 | |||

AVG. |

The comparison of normalized buckling resistances as a function of slenderness obtained on the basis of experimental investigations (_{INV}_{y}_{FEM}_{y}_{n,SB}_{y}_{n,EC}_{y}_{n,BA}_{y}

Normalized shear resistance in function of the slenderness

Cantilever girders with corrugated web analysed in the study lose stability in the elastic-plastic range. The results of normalized shear resistance obtained from the FEM analysis (_{FE}_{y}_{INV}_{y}

The results of normalized resistance (_{n,SB}_{y}

Among all solutions analysed in the study that concern design shear buckling resistance, the solution put forward by the author gives the results that are closest to the experimental ones. The solution acc. equation (_{w}

Similar to girders with the loading diagram of a simply supported beam, the experiment and FEM analysis show that cantilever SIN girders start losing stability below shear yield strength.

It should be noted that the formulas used for estimating interactive shear buckling resistance acc. equation (

Cantilever girders with corrugated web are internally statically indeterminate systems. The failure of cantilever girders with corrugated web is related to the occurrence of tension line that affects the creation of the yield zone or the yield zone associated with the snap-through of the neighbouring waves of the web.

Shear buckling resistance depends on the web thickness and height. The resistance varies non-linearly with a change in the web height, due to local or interactive instability. Webs of cantilever girders with the height of _{w}_{w}

Shear buckling resistance of the web in cantilever girders can be affected by the use of support stiffeners. They increase shear buckling resistance and the range of linear elastic displacements. The web shear buckling resistance with appropriate reserve constitutes a limit on the resistance of supports in SIN girders.

Additionally, increasing shear buckling resistance of supports in SIN girders with support stiffener reduces the need to utilise flat transition sheets that are applied for that purpose.

Behaviour of cantilever girders with corrugated web and support stiffener is similar to that of girders with the loading diagram of a simply supported beam, ending in a rigid stiffener.

Based on laboratory tests and the FEM analysis, a solution was proposed for estimating design shear buckling resistance of cantilever girders with corrugated web and support stiffener. The solution put forward (

The solution accounts for the effect of the mutual correlation between local and global instability of the corrugated web in cantilever girders, and also for the beneficial influence of the support stiffener on shear buckling resistance. In addition, the solution provides a better representation of the design shear buckling resistance than it is the case with EC3-based approach [

It should be mentioned that in the tests on the supports of SIN girders, shear displacements of the cantilever ends were found to occur. They substantially exceed the displacements induced by bending. Significant scatter of shear displacements and global displacements of the support ends indicates that a need may arise to apply tension diagonal stiffeners acc. [

The research is financed by National Science Centre based on grant no. No. N N506 072538.