Numerous experimental and numerical studies on prismatic plate girders subjected to shear can be found in the literature. However, the real structures are frequently designed as non-uniform structural elements. The main objective of the research is the development of a new proposal for the calculation of the ultimate shear resistance of tapered steel plate girders taking into account the specific behaviour of such members. A new mechanical model is presented in the paper and it is used to show the differences between the behaviour of uniform and tapered web panels subjected to shear. EN 1993-1-5 design specifications for the determination of the shear strength for rectangular plates are improved in order to assess the shear strength of tapered elements. Numerical studies carried out on tapered steel plate girders subjected to shear lead to confirm the suitability of the mechanical model and the proposed design expression.
En la literatura pueden encontrarse numerosos estudios experimentales y numéricos sobre vigas armadas de acero de canto constante sometidas a cortante. Sin embargo, muchas estructuras de acero se proyectan, frecuentemente, con elementos de canto variable. El objetivo del artículo es ofrecer una formulación para el cálculo de la resistencia última a cortante de paneles de alma de canto variable, basada en un nuevo modelo mecánico, que es utilizado para mostrar las diferencias de comportamiento entre paneles de alma de canto constante y variable, sometidos a cortante. Las reglas de cálculo de EN 1993-1-5 para determinar la resistencia a cortante de paneles de alma de canto constante se mejoran para poder evaluar la resistencia a cortante de paneles de canto variable. Los numerosos estudios numéricos llevados a cabo sobre vigas armadas de acero de canto variable confirman la idoneidad del modelo mecánico y de la nueva expresión propuesta para el cálculo.
The behaviour of rectangular steel plates subjected to shear load was deeply studied in the last century. In consequence, several theories to predict the ultimate shear capacity of such members were developed. Some of them were taken as a reference and evolved in time and other ones were implemented in design codes. The most important methods to be mentioned are: Basler’s model
Almost all ultimate-shear-strength models for tapered plate girders proposed in literature are derived from the previous presented models for rectangular plates. Several models for tapered girders were developed by: Falby and Lee
On the other hand, the current European design norm EN 1993-1-5
In this research an attempt of an extension of the existing design rules included in EN 1993-1-5 for non-prismatic panels were done. Various geometrical parameters such as: the aspect ratio of the panel, the slope of the inclined flange and the web and flange slenderness were taken into account. Moreover, a new mechanical model for assessing the influence of the Resal effect was developed and included in the final version of the proposal for calculating the ultimate shear resistance of tapered steel plate girders subjected to shear and shear-bending interaction.
Different theories have been developed to analyse the behaviour of steel plated girders subjected to shear, and some of them are used nowadays in design codes. The Rotated Stress Field Method
The results obtained by Bedynek
For the girders belonging to typologies I and III the diagonal tension field develops on the shortest diagonal of the web panel and the inclined flange is under compression or tension, respectively. For typologies II and IV the diagonal tension field appears on the longest diagonal of the web and the inclined flange is under tension or compression, respectively. Different behaviour observed within each typology of tapered panels also influences their ultimate shear resistance.
Moreover, research conducted by Zárate and Mirambell
For those cases where the moment of inertia of the cross-section increases with the increase of internal forces (typologies I and II), the vertical component of the Resal effect acts against shear force and reduces the shear force design value; in other words, consideration of the vertical component of the Resal effect increases the ultimate shear resistance of tapered girders (positive influence). For typologies III and IV the opposite situation is observed. Due to their smaller bearing capacity, these two typologies are not as common as typologies I and II, but sometimes might be required in a case of non-standard structural solutions where the geometry plays an important role.
Since there is still observed a shortage in bibliography focused on the Resal effect in tapered steel plate girders, additional studies are needed to better understand its actual role.
The study on the influence of the Resal effect starts from defining a simplified two-dimensional mechanical model with simplified boundary conditions in some characteristic nodes (
The main objective of this section is to find a direct correlation between Resal force and the shear load acting on the girder. The model represents a rigid frame consisting of flanges and transversal stiffeners only. It is assumed that all the internal forces are transmitted only by surrounding rods of the frame (excluding contribution from the web). In fact, this simplification is used in order to find an approximated value of the Resal force, not its exact value. The boundary conditions reproduce the real ones: simply supported on the one end without bending moment along the shortest depth and with maximum constant bending moment along the rod representing the central cross-section (the largest depth for typologies I and II). Similar idea of simplification of tapered plate girder by rod structure (in that case by truss) was proposed by Davies and Mandal
Solving the hyperstatic frame with two unknowns X_{1} and X_{2}, it is possible to find a contribution of the internal forces from the flanges and from the transversal stiffeners. Consequently, the axial force
Full procedure of determining the Resal force will be explained for the typology I, however, only the most important steps will be pointed out. Other typologies were analysed in the same way and their final expressions for calculating the Resal force will be presented.
Using the principle of virtual works, two unknowns X_{1} and X_{2} can be found and expressed by
where a is the frame length and ϕ is the angle of the inclined flange (
From the equilibrium of upper right hinge, the axial force
Next, the Resal force was calculated as the vertical component of axial force
As it was mentioned before, the Resal effect for tapered plate girders from typologies I and II is favourable. Thus, reduced shear force V_{red} should be calculated according to
For typologies III and IV the Resal effect has a negative influence and V_{red} is given by
Parametric studies were conducted on 85 tapered panels belonging to four typologies. The girders varied by geometric parameters such as the aspect ratio α = a/h_{1}, the slope of the inclined flange ϕ and the web thickness t_{w}. The range of dimensions for the studied models is presented in
Variable | Symbol | Unit | Range |
---|---|---|---|
web thickness | t_{w} | [mm] | 2 to 8 |
smaller depth | h_{0} | [mm] | 300 to 1235 |
larger depth | h_{1} | [mm] | 700 to 3000 |
panel length | a | [mm] | 800 to 8220 |
aspect ratio | α | - | 1 to 4 |
tangent | tan(ϕ) | - | 0.0 to 0.5 |
angle | ϕ | [°] | 0 to 26.6 |
In the initial part of the numerical study several nonlinear analyses were performed on the tapered plate girders with use of quad-dominant 4-node shell element S4R5. This finite element, with 5 degrees of freedom (in each node) and linear shape functions, is especially suitable for modelling shell surfaces where large rotations and displacements are expected. A convergence analysis allowed setting the mesh density with a mesh size of 20 mm for all modelled cases. The nonlinear analyses were conducted with the Modified Riks algorithm implemented in ABAQUS
All numerical simulations were done using the same bilinear material model with kinematic hardening: steel S275 with yield stress f_{y} = 275 MPa, Young’s modulus E = 210 GPa and Poisson ratio ν = 0.3.
The beams are modelled with a full 3D model for tapered plate girders to reflect the actual boundary conditions between web and flange panels in the continuous span of the girder. The numerical model is thought to be rigid end-post. The modelled girder is simply-supported on two edges and its boundary conditions are presented in
u_{x} | u_{y} | u_{z} | θ_{x} | θ_{y} | θ_{z} | |
---|---|---|---|---|---|---|
L | 0 | 1 | 1 | 1 | 1 | 0 |
R | 1 | 1 | 1 | 1 | 1 | 0 |
* 0 – free movement; 1 – restraint.
Two different analyses have been considered in this study. First an eigen-value analysis to obtain the critical shear load as the first eigen-value and to obtain the deformed shape for the critical shear load as the first eigen-vector. The second analysis is a geometric and material nonlinear analysis to reproduce the postbuckling behaviour of tapered plate girders up to failure. Geometric imperfections should be introduced in the model in order to develop the second order effects in the web panel. In this case, the deformation corresponding to the first eigen-mode was used.
The numerical model validation based on four experimental tests was presented in Bedynek
EN 1993-1-5 suggests calculating the ultimate shear resistance of tapered panels treating them as rectangular ones with their largest depth h_{1}. Unfortunately, for some particular geometries of non-rectangular plates this approach leads to significant overestimation of the results what means that the obtained values of the ultimate shear resistances do not fulfil the safety requirements. It happens especially for the typologies III and IV with larger slopes of the inclined flange. This situation can be partially caused by Resal effect, an additional vertical force derived from the axial force that appears in the inclined flange, which for typologies III and IV is unfavorable and should be taken into account. On the other hand, results obtained for typologies I and II are a bit conservative for some cases. Nevertheless, in general this approach gives a good estimation of the ultimate shear resistance of tapered panels with less than 18% difference between the analytical (EN 1993-1-5) and numerical solution (see
In order to illustrate the differences when ultimate shear resistance of tapered panels is calculated using EN 1993-1-5 simplification for tapered panels, results for some of the studied girders are presented in
New approach which gave better assessment of the ultimate shear resistance for typologies III and IV was presented by Bedynek
A new proposal for determining the design shear resistance for tapered steel plate girders based on the existing design rules in EN 1993-1-5
The new proposal is valid for slender, rigid end-posted girders, whose slenderness parameter
According to EN 1993-1-5
where contribution from the web is
being χ_{w} the shear buckling reduction factor, f_{yw} the yield stress of the web, h_{w} the depth of the web panel, t_{w} the web thickness and γ_{M1} the partial safety factor.
The contribution from the flanges is given by:
where b_{f} is the flange width, t_{f} is the flange thickness and f_{yf} is the yield stress of the flanges.
In the last expression
M_{Ed} is the design bending moment calculated at mid-span cross-section M_{Ed}= a V_{Ed}, and M_{f,Rd} is the moment of resistance of the cross-section consisting of the effective area of the flanges only.
For typologies III and IV its influence is unfavourable increasing the external shear load (
For tapered plate girders the depth h_{w} used to calculate the distance
For typologies I and II, the best assessment of their ultimate shear capacity is obtained when in
From the comparison of the results given in
The web contribution V_{bw,Rd} for tapered plate girders has been evaluated by analysing the shear buckling reduction factor χ_{w} from new 85 numerical simulations. The numerical values of χ_{w} have been obtained from
Here, it is important to point out that for all typologies of tapered plate girders, the slenderness parameter
Numerical results for the shear buckling reduction factor are presented in
As it can be observed in
On the other hand, from the results shown in
Based on the same examples of the girders presented in
The results for the 85 girders obtained according to three methods: EN 1993-1-5, the proposal by Bedynek
In
The verification of the new proposal was also conducted using the experimental data from three of the four tests under pure shear load presented in Bedynek
For the cases where the design bending moment M_{Ed} is higher than the moment of resistance of the cross-section consisting of the effective area of the flanges M_{f,Rd} and less than the plastic moment of the resistance of the cross-section M_{pl,Rd} (M_{f,Rd} < M_{Ed} < M_{pl,Rd}), the shear-bending interaction should be considered. According to EN 1993-1-5, if V_{Ed} > 0.5 V_{bw,Rd} the combined effects of bending and shear in the web of a I girder should satisfy
Some numerical analyses considering shear-bending interaction in tapered steel plate girders for the four different typologies studied in this research have been conducted and the results are presented in the next section.
In order to check the validity of the new proposal for tapered plate girders under shear-bending interaction, twelve new cases, three for each typology were examined. It is necessary to mention that the situation where influence of bending moment on the ultimate shear is significant and may lead to its reduction is not very common. It happens due the fact that design bending moment M_{Ed} derives from the vertical reaction in the support (ultimate shear force) and in order to achieve its significant level, the span length of the tapered girders should be long enough and the flexural resistance of the flanges should be relatively low, which is not usually recommended from the design point of view.
In this section, the geometries of the studied girders were designed in order to fulfil condition about the design bending moment at the mid-span cross-section (calculated as the distance
The values of V_{u red} (
In the first three columns of
Relevant improvement for typologies III and IV is observed when the contribution from the web is calculated according to
In general, it can be observed that results obtained according to the new proposal give a very good agreement with the numerical ones, and only for typology III, the values of the ultimate shear resistance are slightly overestimated (not greater than 2%). When calculating V_{u,red}, the partial safety factor, included in design codes, was not taken into account, so its application may bring unsafe values of V_{u,red} on the safe side.
In
On the other hand, for typologies III and IV the use of the new proposal improves the results and gives a good assessment for the ultimate shear resistance.
In this paper the structural response of tapered steel plate girders subjected to shear and shear-bending interaction has been studied. Numerical research has considered four different typologies of such girders which have been studied separately.
A new proposal for the assessment of the ultimate shear resistance of tapered steel plate girders on the existing design rules in EN 1993-1-5 and considering the Resal shear value obtained by a simplified model has been presented.
A simplified model to obtain the value of the shear load produced by the Resal effect has been presented, then the shear force acting in the web can be reduced in typologies I and II or increased in typologies III and IV when verifying the shear resistance of the web.
Based on the results obtained from 85 numerical simulations (approx. 20 for each typology) some modifications in expressions proposed by EN 1993-1-5 for rectangular plate girders were made for their application to tapered plate girders. First, the value of the depth used when determining the contribution of the web and used to calculate the distance
Moreover, a new adjustment of the reduction factor χ_{w} has been presented for typologies III and IV.
A very good agreement between the results obtained according to the new proposal V_{u (proposal)} and numerically V_{u (FEM)} is observed. In the case where the shear-bending interaction is not considered, the differences between the values of the ultimate shear strength V_{u (FEM)} and V_{u (proposal)} do not exceed 33%, whereas in most cases these differences are less than 15%.
The same design proposal was applied for tapered steel plate girders subjected to shear-bending interaction. Due to presence of the significant bending moments at the mid-span cross-section, an additional reduction of the ultimate shear resistance had to be done. Validation of the new proposal was carried out for 12 girders, 3 from each typology. Also here, a good agreement between the results obtained numerically and these calculated according to the proposal was observed. A significant improvement of the results was especially visible for typologies III and IV.
Presented design proposal not only provides a satisfactory tool for the assessment of the ultimate shear resistance of tapered steel plate girders but also reveals its physical interpretation and respects individual contributions of all parts. Therefore, with use of the proposed method all components of the ultimate shear strength of the whole girder such as: the contribution from the web and flanges or Resal force can be found. This improvement was possible especially thanks to division of tapered plate girders into four different typologies which required an individual analysis according to their geometry.
It is important to point out that the main objective of this proposal is not thought to substitute or change the existing rules EN 1993-1-5 for rectangular plate girders, but only to give a simplified algorithm to assess an approximated value of the ultimate shear resistance of tapered members with considerable slope of the inclined flange. It is strongly recommended to treat the proposal as a reference and for the cases where the high precision is required an additional numerical study of specific girder is needed.
The authors wish to express their gratitude to Spanish Ministry of Science and Innovation for the financial support provided as a part of the Research Project BIA2008-01897, and for a doctoral scholarship for PhD student Agnieszka Bedynek. Additionally, the authors would like to thank Universitat Politècnica de Catalunya for the grant awarded in 2009 to the first author.