Shear Lag Effect on Box Steel Beams with Wide Curved Flanges

Abstract

For wide flange box sections, conventional Euler–Bernoulli beam theory with maintaining the cross-section planarity may lead to underestimation of axial stresses. Axial stresses in cross-section flanges may have a non-uniform distribution due to shear pliability, decreasing in value from the flange–web junction to the middle area of the flange. This phenomenon leads to the introduction of an effective flange width with a uniform distribution of original maximum stress. Furthermore, the introduction of flange curvature makes it even more complex due to the varying lever arm of each flange part with respect to the neutral bending axis. Because of this, in some cases, it is hard to predict where the flange’s highest normal stress value will appear. In this paper, the shear lag effect on wide curved box sections is analyzed through parametric numerical analysis using the FEA software Dlubal RFEM 5, together with visual programming performed in Rhino Grasshopper. This study investigates the interaction of the shear lag effect and plane section hypothesis, which can be simplistically represented as a reduction in the impact of shear lag and the activation of a larger part of the flange of a wide-flange beam in the structural system of a continuous beam. The results suggest that for higher flange curvature and higher width to length ratio, this effect is more prominent.

1. Introduction

This paper examines the axial stress distribution law in curved flanges, essential for ensuring proper design and structural safety throughout the service life of the structure. Initially, the shear lag phenomenon in general is thoroughly described, discussing findings from the existing research. Subsequently, a parametric analysis is conducted, detailing both the methodology employed and the findings, followed by a discussion of the conclusions drawn from the results.

When a beam with the span–height ratio larger than five is subjected to bending, it is often assumed that the Euler–Bernoulli’s hypothesis of plane sections can be applied—for a linearly elastic material, the plane cross-sections remain plane during deformation, resulting in a linear distribution of normal stresses across the height of the cross-section. This hypothesis is fully satisfied only in the case of “pure” bending, for a constant bending moment value along the beam, without the action of any additional transverse shear force. For cases where the beam is loaded with transverse forces, the shear stresses cause the shear deformation of the cross-sections, thus violating the abovementioned hypothesis.

However, for the aforementioned beam height–length ratio, it is known that the influence of deformation due to shear stresses/deformations can be neglected, and in most cases, the calculation of stresses and deformations of the beam can be reduced to the problem of a one-dimensional element with corresponding flexural stiffness [1]. In such analyses of thin-walled cross-sections, the complex planar behavior of the flanges and webs and their shear deformability are neglected. If we consider a beam loaded with transverse forces applied in the web plane, its in-plane bending yields the longitudinal fiber displacements located at the web–flange junction. This results, then, in longitudinal normal and horizontal shear stresses, and thus transverse normal stresses (because of the prevented displacements due to cross-section symmetry). The distribution of shear loading to the web along the beam is assumed to be affine to the shear force diagram. Now, considering the flange as a separate disk loaded only with edge shear loading, for low half-width to beam span (distance between the zero points of the moment diagram) ratio (b₀/L_e > 0.02 according to [2]), a uniform distribution of longitudinal stresses across the flange width can be assumed (the flange behaves like a shell with load eccentricity being neglected), while for larger ratios, it is necessary to determine the distribution of longitudinal stresses across the width of the flange. Due to the shear deformability, it is expected that the flange fibers closer to the web will be more loaded than those farther away—this phenomenon is called “shear lag” [3]. Therefore, the concept of the effective width b_eff = β × b₀ is introduced, which takes into account the shear pliability by excluding the flange part farther from the junction with the web, satisfying the condition of the equilibrium of longitudinal force by integrating stresses across the flange width, as shown in Figure 1. In the wide-flange beam calculation, not taking into account shear lag leads to a great probability of underestimating the stress values at the junction of the web with the flange. Since large-span steel beams are generally composed of slender plates, a stability issue due to the action of compressive stresses (unless it is a cross-section of class 3 or less) can occur. Such slender plates have longitudinal stiffeners (open or closed) that prevent buckling, and their longitudinal stiffness is taken into account when determining effective width by calculating the orthotropy coefficient of the web α₀. Longitudinal stiffeners only increase the longitudinal stiffness of the web and not the shear stiffness, thereby increasing the effect of shear lag in such beams.

Apart from beam bending theory, the term “shear lag” is also widely inspected in high-rise building systems where, for instance, a tube system consists of multiple very rigid and connected frame panels which, subjected to moment caused by lateral loads, act as the webs and flanges of a tube cross-section. The flexibility of beams in the web and flange frames results in the increase in axial forces in the columns at the end of the frame panels and the decrease in axial forces—shear lag. Along with positive shear lag, the negative one also occurs as a result of the compatibility of displacements [4].

This phenomenon can also occur as a local effect in steel elements, e.g., the connection of angle profiles where welded or bolted connections are used. Here, shear lag effect is evident since for these members only one part of the cross-section (e.g., one flange) is connected to the other element, which leads to unequal longitudinal stresses due to shear pliability [5,6].

In addition to the specific aspects of the beam theory outlined above, this study introduces the novel concept of flange curvature, adding complexity to the topic. The results of the numerical analysis confirm the hypothesis that the impact of shear lag is reduced in curved flanges as a result of the interaction between the plane section theory and the shear lag effect.

2. Wide-Flange Beam with Curved Flanges

A special case of a girder cross-section is a wide-flange section with curved flanges, often used in bridge construction not only due to its favorable aerodynamic shape, but also due to its esthetically pleasing appearance (Figure 2). Let us consider a boxed cross-section continuous girder with an ideally linear stress–strain diagram, subjected to uniform transversal load, applied in the plane of the webs. For such a girder (assuming adequate L/H ratio) where the flanges behave in accordance with the hypothesis of plane sections, there is a stress variation across the width of the flange not only due to the flange shear deformability, but also due to the varying distance of the flange fibers from the neutral axis [7] depending on the degree of curvature. In practice, we mostly encounter convex curvatures of the lower flanges of girders, so it is necessary to determine the distribution of longitudinal stresses across the width of the flange, taking into account the parameters of flange curvature, its width, and the girder spans. For low values of the radius of the curvature of the flange, the impact of the reduction in shear lag (increasing the distance from the web–flange joint increases the distance from the neutral axis) is significant, and for high values (approaching the form of a flat flange), the impact of reduction will be negligible. Therefore, the concept of the “effective angle” (Figure 3) of the flange is introduced. Effective angle defines the part of the arc curvature that will remain “active”, but unlike the case of flat flanges, the axial stresses of the effective flange part (normalized to the maximum stress of the flange) must have an equal resulting bending moment (relative to an arbitrary point) as the moment calculated from the stress state obtained by experimental/analytical/numerical analysis.

3. Review of Previous Research on Shear Lag Effect

Many authors have investigated the phenomenon of shear lag, and with the development of numerical methods, numerous studies have been conducted to compare results with those obtained by analytical and experimental methods. The result was a certain discrepancy in the output data due to certain peculiarities of the individual methods.

Experimental methods provide empirical data that reflect the actual behavior of a specific structure. Although it is the most realistic method, it requires significant resources including structural material, human labor, and time. Physical models also often have to be scaled down (especially in structural engineering) which leads to the uncertainty of full-scale behavior replication. If structural models are examined under a high load rate, several “identical” models may lead to different behavior under the same load rate due to non-linear behavior because of geometric and/or material model uncertainties—local and global imperfections, yield strength, hardening rate, etc. Experimental methods are also limited by the (im)possibility of producing physically and mathematically “ideal” boundary and loading conditions. Analytical (exact or approximate—e.g., Fourier series) methods are precise and simple as they provide exact solutions under specific conditions which are often the result of several simplifications including material and geometry properties, as well as loading and boundary conditions. The less simplified problem leads to more complex equations which are difficult or impossible to solve analytically which leads us to a more versatile method—a numerical one. Numerical methods are adaptable to complex and arbitrary geometry and loading conditions which would be impractical to replicate experimentally, but they are, with certain generalizations compared to analytics, often sensitive to the element mesh and type, as well as to the specifics of the results (stress concentrations—singular points).

The beginnings of research dates back to the first half of the twentieth century when, in [3,9], the phenomenon of shear lag was solved by various analytical methods and described by equations suitable for practical engineering use. In addition to increased stress in the fiber at the web–flange junction, slightly increased deflection of the beam due to shear lag was quantified. In [10], Zhang further analytically divided the (longitudinal) stress state of the cross-section into the first part caused by bending, and the other part caused by the cross-section warping. In [11], an analytical derivation of the expression for the effective width of the cross-section (taking into account the effect of shear lag) was presented, which is today an integral part of European standards for the design of steel continuous beams. Luo et al. [12], considered a box beam as a set of interconnected shells loaded with membrane forces. Analytical solutions for different b/L and b/H ratios, as well as different loading cases, analyzed the occurrence of transverse normal stress due to the prevention of transverse movement of the beam’s midpoint due to symmetry of the beam geometry, loading, and boundary conditions.

In [13], at the end of the last century, a parametric study was conducted using the finite element method with variations in loading conditions, as well as boundary conditions, the ratio of flange width to beam span b/L, and the ratios of the modulus of elasticity to shear modulus E/G in order to consider the orthotropy of the cross-section. Furthermore, the effects of the dimensions of the web (H, t_w) and the thickness of the flange t_f were examined. Since previous research did not achieve a uniform standpoint of all the authors, this numerical research concluded that in slender, thin-walled beams, varying the mentioned parameters (satisfying the conditions H < 0.5 × L and H > 10 × t_f) does not have a pronounced effect. Also, varying the thickness of the flange and web while maintaining a constant ratio (t_f/t_w = const.) does not affect the shear lag effect. The paper confirmed expressions obtained earlier by analytical methods (including those in [3]) and obtained empirical formulas for quantifying the shear lag coefficient, which excludes part of the cross-section to consider the increase in longitudinal stress at the junction of the flange with the web—a principle adopted by Eurocode. The principle was further applied to beams with longitudinal stiffeners by modifying (increasing) the E/G ratio, where the shear lag effect is even more pronounced.

In recent research over the past twenty years, extensive numerical parametric analyses of single-span beams [14] and multi-span beams [15,16] have been conducted, and box beams were analyzed, while the effect of shear lag was described as a “stress concentration factor”, which is a product of stress at the joint with the web and theoretical stress that would have been obtained by the beam theory with Euler–Bernoulli’s hypothesis of plane sections. In these studies, the effects of all the relevant factors on the shear lag of beams were examined: the height of the cross-section h; the thickness of flanges t_f and web t_w (and their ratio); the form of load application (concentrated or distributed loads); the density of the element mesh; and the ratio of flange width to span (b/L).

The problem associated with the finite element method, related to the type and density of the element mesh, was solved by the “multi-mesh extrapolation” method [17], where the size of the finite element was gradually reduced towards a “finer” mesh, and ultimately the stress concentration value was extrapolated to the size of the finite element tending towards zero.

In recent years, several authors extensively elaborated the shear lag effect in composite concrete–steel beams, which are highly cost-effective due to the favorable composite combination of steel and concrete in tension and compression, respectively. In [18], the authors proposed a simplified analysis method considering shear lag effect of steel–concrete composite decks, including the experimental static load test of composite twin I-girder deck and box girder deck, subjecting it to vertical bending and compressive axial loads. Hu et al. [19] investigated stress and deflection behavior in double-box cross-section composite beams through simultaneous analytical and experimental studies. Comparative analysis of the results from both studies showed good agreement within the linear phase, but the ignored influence of relative concrete–steel slips and material nonlinearity needs to be further analyzed. Zhao et al. [20] proposed a beam finite element model considering the slip, shear lag, and time-dependent effects of steel–concrete composite box beams. The results show that concrete rheology including shrinkage and creep significantly influences the structural responses of the composite box beams, affecting (increasing) relative slippage and vertical beam deflection. Zhu et al. [21] assembled a finite beam element with 26 degrees of freedom for curved (in plan layout) considering various effects such as constrained torsion, distortion, shear lag, and biaxial slip. That method represents a significant step forward in comparison to standard FEM models which imply high computational costs for modeling such a complex task.

Since flanged flexural members are likely to have plastic deformation at their ultimate limit state and the study of inelastic shear lag is limited, Lin and Zhao [22] put extensive effort into the analytical modeling of inelastic shear lag, followed by the laboratory tests of two steel box beams. A comparison of the results indicated that the proposed analytical method can predict normal stress and strain distribution and therefore, beam deflection with decent accuracy.

The authors in [23] marked that the beam axial equilibrium condition is not strictly satisfied according to conventional beam methods that imply that the neutral axis of the section coincides with the centroid. It is proposed that three separate functions have to define shear lag in the top slab, the bottom slab, and the cantilever slab in order to calculate the axial stresses and deflections on simply supported box girders more accurately.

Recent research have shown the possibility of combining 1D and 2D FEM analysis in the shear lag of beams of arbitrary cross-sections [24]; the authors of a parametric numerical study in [25] have shown that in the case of a three-span continuous curved steel box girder bridge, the curvature radius and the width–span ratio have a significant influence on the shear lag effect apart from the height–span ratio.

In [26], the authors have pointed out that (compression) plate buckling effects, which result in an effective area of the flange, may occur in addition to the shear lag effects. That was observed in the case of steel box girders of a crane runway, but it may be applied to all thin-walled steel sections, including bridges.

In [27,28,29], special effort has been dedicated to the analysis of the shear lag effect on composite box girder bridges with corrugated steel webs, including the computation of deflections.

Also, significant effort has been put in research regarding the shear lag effect of steel–Ultra-High-Performance Concrete (UHPC) ribbed slab composite structure (SU-RSCS) in the elastic range [30].

4. Numerical Parametric Analysis of Shear Lag in Steel Beams

4.1. Introduction to Numerical Parametric Analysis

A review of the available literature and published scientific papers reveals considerable effort invested in studying the impact of shear lag in girders of various structural systems by varying a large number of the parameters previously mentioned.

The research in this paper comprises a numerical analysis of a “family” of wide-flange beams with curved flanges.

Numerical simulations were carried out using linear plate elements with ideally elastic structural steel, specified by an elastic Young’s modulus of 2.1 × 10⁵ MPa and a Poisson’s ratio of 0.3, implying also shear modulus with a value of 8.1 × 10⁴ MPa.

Although higher-order finite elements provide more accurate results, in this numerical analysis the model is densely discretized, allowing linear elements to also produce results of sufficient accuracy. The potential sources of numerical error are not apparent in this case, as the study handles a geometrically and materially linear model, making it numerically simple and allowing for smooth solver convergence.

Below is a summary of the conducted analysis (

Table 1. Display of sets of conducted numerical analyses.

Analysis Type/Static System	Load Type	(b/2)/Le	(b/2)/R	Total Number of Models 1
3D FEM Analysis-Multi-span continuous beam	uniform	0.10, 0.20, 0.30, 0.40, 0.50, 0.60	0.05, 0.20, 0.40, 0.60	6 × 4 = 24

¹ the total number of models is the number of combinations resulting from varying the parameters given in the table.

) where the parameter sets are visible. For all the sets, uniform load along the entire girder is considered. The first varying parameter is the ratio of the effective span to flange width (b/2)/L_e, which is later shown as the horizontal axis in the graphical presentation of results. Next, the impact of flange curvature on the effect of shear lag is also introduced as a second varying parameter (b/2)/R. It is verified that the ratio of web height to beam span has a negligible impact on the results, as does the ratio of flange thickness to web thickness, so varying these parameters is not in the scope of this parametric analysis.

Extensive analysis demonstrated that the results for a flange curvature ratio of (b/2)/R = 0.05 align with those obtained for a flat flange (where the radius approaches infinite value). Consequently, the results for the flat flange are not presented in the subsequent sections.

Despite the fact that the model is entirely geometrically and materially linear, in the observed cases involving significant curvature of the flanges, there is an exclusion of the “non-bearing” middle part of the cross-section due to buckling caused by the action of skewed (due to shear stresses) principal compressive stresses. In engineering practice, this problem is avoided by adding longitudinal and transversal stiffeners to slender flanges (and/or webs). In this article, the plate elements are modeled with increased longitudinal bending stiffness (stiffness multiplier equals 100) to avoid the buckling of the middle part of the flange caused by the action of compressive longitudinal stresses on the pre-deformed system (rounded flange).

It is also evident that the gradual rounding of the flange to a certain extent neutralizes the reduction in the cross-section caused by the shear deformability of the flange.

4.2. Parametric Numerical Analysis Using Rhino 3D, Grasshopper, B + G Toolbox, and Colibri

The parameter analysis is conducted using several computer programs (Figure 4)—using visual programming in the Grasshopper interface [31], which is part of Rhino 8 software [32]. The Parametric FEM Toolbox [33] is an add-on in Grasshopper that generates a numerical model in Dlubal RFEM and allows communication with it—this includes the subsequent modification and reading results after the analysis has been conducted. Additionally, with the Colibri add-on, numerical models are generated by varying certain parameters by predefined values.

As visible in Figure 5, the parameters flange width B and B/2R ratio are defined, the combinations of whose values will determine the geometry of the numerical models and will be associated with the results obtained from the FEM analysis and subsequent analytical processing of the data.

Pre- and post-processing was conducted using B + G Toolbox components and visual programming, as shown in Figure 6 and Figure 7.

4.3. Numerical Analysis of Continuous Beam with Box Cross-Section

A numerical model of a steel beam (using the Dlubal RFEM 5 software package [34]) with a box cross-section, continuous over spans of L₁ = 8.0 m and L₂ = 10.0 m, and with an overhang of L₃ = 4.0 m (model analog to one in [2]) was created. As visible in Figure 8, the beam is supported on hinged supports in the middle of the web and loaded with continuous loading in the plane of the web along the entire beam. The cross-section of the beam is a box with a web height of 1000 mm and a thickness of 10 mm, and a variable width of the flange with a constant thickness of 10 mm. The model was automatically discretized (“meshed”) with linear plate finite elements of a minimum size of 100 × 100 mm. As a part of the parametric analysis, the mesh density was evaluated in order to optimize both the model performance and accuracy. A selected mesh size emerged as the optimal value from a mesh sensitivity and a computational efficiency standpoint.

Only the values of the longitudinal unit forces obtained by integrating the normal stresses of the cross-sections with maximum/minimum bending moments were considered—cross-sections A-A, B-B, and C-C (Figure 9). Effective widths were not calculated at the edge cross-sections (beginning and end of the beam) since the values of longitudinal stresses are negligible (the bending moment is practically zero) and integrating them yields nonsensical effective width factor values. Effective widths were also not sought in the cross-sections in the area between the points of bending moment extreme values (near the areas where the null points are located), which can be partially in tension and compression, because due to the redistribution of longitudinal stresses due to shear lag (and thus the redistribution of bending moments) there is no ideal null point, i.e., a cross-section perpendicular to the axis of the beam where all the longitudinal stresses are zero. Integrating the stresses of opposite signs (tension and compression) would also yield nonsensical values of the effective width factor β.

The input parameters (visually shown in Figure 10) are as follows:

• Constant flange and web plate thickness t_f = t_w = t = 10 mm;
• Web height h = 1000 mm;
• Flange width b—variable (dependent on κ = b₀/L_e ratio);
• Flange curvature radius R—variable (dependent on b/2R ratio);
• Height difference Δ between flange end and midpoint (1).

$$∆ =\mathrm{R}\times \left(1-\cos {\displaystyle \frac{\mathsf{\alpha }}{2}}\right)$$

(1)

The data are processed in such a way that the results from certain number of finite elements N were analyzed—the total number of mesh elements depends on the width of the flange. For these elements, the unit axial forces n_x,i are extracted from the results and the vertical eccentricity e_i,A is measured in relation to the web–flange junction (2) (point A in Figure 11). Then, the total resulting moment is determined (by summing the partial moments) with the actual state of stress obtained by numerical analysis (3). After that, constant axial stress (with maximum value—point A) with varying lever arm (again related to point A) is assumed along the flange and partial moments are integrated (4) along the flange up to the total value obtained by the original stress distribution, where integration stops and the number of effective finite elements N_eff is obtained and therefore, effective width (angle) factor β can be calculated (5).

Vertical eccentricity—lever arm of bending moment can be obtained as follows:

$${\mathrm{e}}_{\mathrm{i},\mathrm{A}}=\mathrm{R}\times \left[\cos \left({\displaystyle \frac{\mathsf{\alpha }}{2}}-\mathrm{i}{\displaystyle \frac{\mathsf{\alpha }}{2\mathrm{N}}}\right)-\cos {\displaystyle \frac{\mathsf{\alpha }}{2}}\right]$$

(2)

Equal resulting moment condition arises from the equality of Equations (3) and (4), leaving N_eff as the only unknown value:

$$\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{m}}_{\mathrm{x},\mathrm{i},\mathrm{A}}=\mathrm{t}\times \sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathsf{\sigma }}_{\mathrm{x},\mathrm{i}}\times {\displaystyle \frac{\mathrm{R}\mathsf{\pi }\frac{\mathsf{\alpha }}{2}}{180°\times \mathrm{N}}}\times \mathrm{R}\times \left[\cos \left({\displaystyle \frac{\mathsf{\alpha }}{2}}-\mathrm{i}{\displaystyle \frac{\mathsf{\alpha }}{2\mathrm{N}}}\right)-\cos {\displaystyle \frac{\mathsf{\alpha }}{2}}\right]$$

(3)

$$\sum _{\mathrm{i}=1}^{{\mathrm{N}}_{\mathrm{eff}}}{\mathrm{m}}_{\mathrm{x},1,\mathrm{A}}=\mathrm{t}\times {\mathsf{\sigma }}_{\mathrm{x},1,\max }\times {\displaystyle \frac{\mathrm{R}\mathsf{\pi }\frac{\mathsf{\alpha }}{2}}{180°\times \mathrm{N}}}\times \mathrm{R}\times \sum _{\mathrm{i}=1}^{{\mathrm{N}}_{\mathrm{eff}}}\left[\cos \left({\displaystyle \frac{\mathsf{\alpha }}{2}}-\mathrm{i}{\displaystyle \frac{\mathsf{\alpha }}{2\mathrm{N}}}\right)-\cos {\displaystyle \frac{\mathsf{\alpha }}{2}}\right]$$

(4)

$$\mathsf{\beta }={\displaystyle \frac{{\mathrm{N}}_{\mathrm{eff}}}{\mathrm{N}}}={\displaystyle \frac{\frac{{\mathsf{\alpha }}_{\mathrm{e}\mathrm{f}\mathrm{f}}}{2}}{\frac{\mathsf{\alpha }}{2}}}={\displaystyle \frac{{\mathsf{\alpha }}_{\mathrm{eff}}}{\mathsf{\alpha }}}$$

(5)

In the case of the equidistant meshing of the flange (as conducted in this parametric analysis), the ratios N/N_eff and α_eff/α yield the same result—effective angle factor β.

4.4. Results of Numerical Analysis

After close inspection of the result stresses obtained by the parametric numerical analysis, it can be seen that one can divide systems into 3 separate cases (one result set from each case group is shown in

Table 2. Display of certain sets of conducted numerical analyses.

Variable	Case 1	Case 2	Case 3
h [m]	1.00	1.00	1.00
B [m]	1.36	2.72	2.72
B/2 [m]	0.68	1.36	1.36
B/(2R) [/]	0.40	0.20	0.05
R [m]	1.70	6.80	27.2
κ [/]	0.10	0.20	0.20
α/2 [°]	23.6	11.5	2.9
Δ [m]	0.134	0.136	0.034

):

• Narrow flange box section beam with a dominant Euler–Bernoulli hypothesis effect and not negligible flange curvature—maximum axial stress value in the flange middle;
• Beams where the Euler–Bernoulli hypothesis and shear lag effect are “neutralized by each other”, which results in constant axial stress along flange width;
• Wide flange box section beam with a low curvature effect where dominant shear lag leads to maximum axial stress value in the edge of the flange.

Figure 12, Figure 13 and Figure 14 show the corresponding distribution of longitudinal stresses on one quarter (due to double symmetry) of the cross-sectional box section, where the red hatched diagram represents the “actual” state of stress obtained by the numerical analysis, and the black dashed line represents the distribution obtained analytically by the beam theory (Euler–Bernoulli’s hypothesis of plane sections).

Figure 12 shows the stress distribution with a maximum at the apex of the arc of the curved flange because it represents a narrow-flange beam with a large ratio of span length to the cross-sectional width of the box section.

Figure 13 shows a transitional area where the stresses are approximately equal across the width of the flange, which results from the ‘nullification’ of the effect of stress increase (from web–flange junction towards the middle point of the flange) according to the plane section hypothesis and the decrease in stress due to the impact of shear lag.

Figure 14 shows the distribution of longitudinal stresses for a wide-flange beam where the longitudinal stresses decrease as they move away from the junction of the flange with the web due to the shear flexibility of the curved flange of the box beam.

In

Table 3. Tabular display of one analyzed result set.

I [/]	xi,A [m]	nx,I [kN/m]	ei [m]	|mx,i,A| [kNm/m] 1	|mx,1,A| [kNm/m] 2	Σ|mx,1| [kNm/m]
1	0.000	−30.770	0.005	0.144	0.144	0.144
2	0.105	−29.450	0.009	0.266	0.277	0.422
3	0.209	−28.230	0.013	0.367	0.400	0.822
4	0.314	−27.110	0.017	0.451	0.512	1.334
5	0.419	−26.080	0.020	0.520	0.614	1.948
6	0.523	−25.150	0.023	0.576	0.704	2.652
7	0.628	−24.330	0.025	0.620	0.784	3.436
8	0.733	−23.610	0.028	0.655	0.854	4.290
9	0.837	−23.000	0.030	0.682	0.913	5.203
10	0.942	−22.490	0.031	0.702	0.961	6.163
11	1.047	−22.100	0.032	0.717	0.998	7.161
12	1.151	−21.820	0.033	0.727	1.025	8.186
13	1.256	−21.650	0.034	0.732	1.041	9.226
14	1.361	−21.590	0.034	0.734	1.046	10.272
N = 14				Σmx,i,A = 7.89		Neff = 11
β = Neff/N = 11/14 = 0.79

¹ |m_x,i,A| = n_x,i × e_i—partial moment of i-th element with real stress and lever arm related to point A. ² |m_x,1,A| = n_x,1 × e_i—partial moment of i-th element with maximum stress (from element i = 1—point A) and lever arm related to point A.

, the number i represents each finite element (i = 1 … N) with x_i,A (auxiliary parameter) as the horizontal distance from point A, while e_i represents the y ordinate, (measured from point A), as well as the lever arm of the axial forces n_x,i, which are introduced by assuming constant stresses through shell thickness. Therefore, comprehensively described expressions (3) and (4) may be, by the introduction of expressions for lever arm (2), as well the expressions for unit axial forces (per unit length) (6) and (7), respectively, concisely described by expressions (8) and (9), respectively.

$${\mathrm{n}}_{\mathrm{x},\mathrm{i}}=\mathrm{t}\times {\mathsf{\sigma }}_{\mathrm{x},\mathrm{i}} [\mathrm{kN}/{\mathrm{m}}^1]$$

(6)

$${\mathrm{n}}_{\mathrm{x},\mathrm{A}}=\mathrm{t}\times {\mathsf{\sigma }}_{\mathrm{x},1,\mathrm{m}\mathrm{a}\mathrm{x}} [\mathrm{kN}/{\mathrm{m}}^1]$$

(7)

$$\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{m}}_{\mathrm{x},\mathrm{i},\mathrm{A}}=\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{n}}_{\mathrm{x},\mathrm{i}}\times {\mathrm{e}}_{\mathrm{i},\mathrm{A}}$$

(8)

$$\sum _{\mathrm{i}=1}^{{\mathrm{N}}_{\mathrm{e}\mathrm{f}\mathrm{f}}}{\mathrm{m}}_{\mathrm{x},1,\mathrm{A}}=\sum _{\mathrm{i}=1}^{{\mathrm{N}}_{\mathrm{e}\mathrm{f}\mathrm{f}}}{\mathrm{n}}_{\mathrm{x},1}\times {\mathrm{e}}_{\mathrm{i},\mathrm{A}}$$

(9)

The product of finite element unit axial forces n_x,i and n_x,1 and corresponding lever arm (with respect to point A) represent the partial moments m_x,i,A and m_x,1,A, which were then summed according to (8) and (9), respectively.

5. Influence of Flange Curvature on Shear Lag Effect

On the following diagrams (Figure 15, Figure 16 and Figure 17), the graphical representation of the result sets can be found. The results are shown as curves (polygon curves connecting discrete result points) on separate diagrams for each section A-A, B-B, and C-C. With variable κ [/] on the horizontal abscissa axis, each b/2R ratio is represented by one curve which shows the correlation of κ with vertical ordinate—effective width (angle) factor β [/]. Width b₀ (or b/2) to span L_e ratio κ [/] is calculated separately for the field (A-A and B-B) and support area (C-C) cross-sections using Formulas (10) and (11) provided by Eurocode [2]:

$${\mathrm{L}}_{\mathrm{e}}=0.85\times {\mathrm{L}}_1 \mathrm{and} {\mathrm{L}}_{\mathrm{e}}=0.70\times ({\mathrm{L}}_1+{\mathrm{L}}_2) - \text{for sagging bending}$$

(10)

$${\mathrm{L}}_{\mathrm{e}}=0.25\times ({\mathrm{L}}_1+{\mathrm{L}}_2) - \text{for hogging bending}$$

(11)

On each diagram blue smooth line represents the analytical correlation between κ and β using expressions (12)–(16) provided by [2]

$$\mathrm{For} \mathsf{\kappa } \le 0.02; \mathsf{\beta }=1,0$$

(12)

$$\mathrm{For} 0.02 < \mathsf{\kappa } \le 0.70; \mathsf{\beta }={\mathsf{\beta }}_1={\displaystyle \frac{1}{1+6.4{\mathsf{\kappa }}^2}} - \text{for sagging bending}$$

(13)

$$\mathsf{\beta }={\mathsf{\beta }}_2={\displaystyle \frac{1}{1+6.0\left(\mathsf{\kappa }-\frac{1}{2500\mathsf{\kappa }}\right)+1.6{\mathsf{\kappa }}^2}} - \text{for hogging bending}$$

(14)

$$\mathrm{For} \mathsf{\kappa } > 0.70; \mathsf{\beta }={\mathsf{\beta }}_1={\displaystyle \frac{1}{5.9\mathsf{\kappa }}} - \text{for sagging bending}$$

(15)

$$\mathsf{\beta }={\mathsf{\beta }}_1={\displaystyle \frac{1}{8.6\mathsf{\kappa }}} - \text{for hogging bending}$$

(16)

In Figure 15 and Figure 16, there is minimal deviation in the values of the effective width factor β from the curve given by the expressions in the Eurocode for flanges with a small b/2R ratio—practically flat flanges. It is also evident that the gradual rounding of the flange and the distancing of the fibers from the neutral axis (which results in increased stress according to the Euler–Bernoulli hypothesis) to a certain extent reduces the effect of the cross-sectional reduction caused by the shear deformability of the flange. It was observed that this effect is less pronounced in the case of the middle support where the bending moment is of negative sign with the presence of a concentrated support reaction (large shear force).

6. Result Summary

The result summary for sections A, B, and C is shown in

Table 4. Deviation percentage [%] for Δβ at section A-A (middle of the first span).

(b/2)/R	0.10	0.20	0.30	0.40	0.50	0.60
0.05	−6.9%	−1.3%	+0.3%	+1.2%	+4.0%	+2.3%
0.20	+6.4%	+16.6%	+21.8%	+22.9%	+26.3%	+33.7%
0.40	+6.4%	+25.6%	+50.1%	+51.8%	+56.0%	+65.2%
0.60	+6.4%	+25.6%	+50.1%	+59.0%	+70.9%	+80.9%

,

Table 5. Deviation percentage [%] for Δβ at section B-B (support).

(b/2)/R	0.10	0.20	0.30	0.40	0.50	0.60
0.05	−6.9%	−1.3%	+7.5%	+8.4%	+11.4%	+18.0%
0.20	+6.4%	+16.6%	+28.9%	+44.6%	+48.6%	+57.3%
0.40	+6.4%	+25.6%	+57.6%	+73.5%	+85.7%	+96.7%
0.60	+6.4%	+25.6%	+57.6%	+73.5%	+93.1%	+112.4%

, and

Table 6. Deviation percentage [%] for Δβ at section C-C (middle of the second span).

(b/2)/R	0.15	0.30	0.45	0.60	0.76	0.91
0.05	+20.2%	+47.1%	+45.5%	+31.6%	+33.3%	+46.5%
0.20	+44.2%	+47.1%	+45.5%	+50.4%	+52.4%	+46.5%
0.40	+68.3%	+76.5%	+71.4%	+69.2%	+52.4%	+46.5%
0.60	+68.3%	+89.1%	+71.4%	+50.4%	+52.4%	+46.5%

, respectively. The results are given as a percentage of value differences using the following expression):

$$\mathsf{\Delta }\mathsf{\beta } ={\displaystyle \frac{\mathsf{\beta }(\mathsf{\kappa };\frac{\mathrm{b}}{2\mathrm{R}})-{\mathsf{\beta }}_{\mathrm{E}\mathrm{C}}\left(\mathsf{\kappa }\right)}{\mathsf{\beta }(\mathsf{\kappa };\frac{\mathrm{b}}{2\mathrm{R}})}}[\mathrm{\%}]$$

(17)

where β(κ, b/2R) and β(κ) (both shown on graphs in the previous chapter) represent a set of the results obtained by the parametric numerical analysis and the effective width factor calculated with the expressions provided by Eurocode, respectively.

A couple of (low) negative values represent that no Euler–Bernoulli effect is present (narrow flange beams with practically flat flanges), while increasing κ and flange curvature leads to higher β values up to more than 100% in some cases.

From Table 4, Table 5 and Table 6, it is visible that for middle span sections A-A and B-B, low curvature values show minor value differences even for wide flanges. For wide flanges with moderate and high curvature values, the effective width relative difference Δβ reaches into the value range of 70–110%, which makes them extremely sensitive to changes in the key parameters. For section C-C above the support, the presence of a high shear force and a concentrated support force in the center of the web leads to a general effective width factor in the range of 30–70% with some deviating values.

7. Conclusions

The phenomenon of shear lag leads to the uneven distribution of longitudinal stresses within the cross-section of the beam, occurring due to the shear flexibility of the flanges. The problem of quantifying the impact of shear lag has been addressed for decades through analytical, experimental, and numerical methods, and a divergence in results can be observed depending on numerous assumptions that need to be adopted when simplifying this complex problem. Initial analyses conducted by analytical methods had to introduce a series of simplifications to the problem to derive expressions suitable for practical engineering use. With the later introduction of numerical methods, extensive parametric analyses were conducted, quantifying the impacts of individual parameters.

In the case of a wide-flange box beam with curved flanges, observing flange fibers more distant from the junction with the web and thus also from the neutral axis, there is a problem in quantifying the superposition of the effects of Euler–Bernoulli’s hypothesis of plane sections and the effect of shear lag. Various parameters and their ratios were considered in setting up the parametric analysis and curves were obtained showing this interaction, represented as a reduction in the impact of shear lag and the activation of a larger part of the flange of a wide-flange beam in the static system of a continuous beam. The analysis concluded that this reduction is smaller at the support of the continuous beam than in the middle of the span, and for mild curvatures of the flange, the curve largely coincides with the one given in the current Eurocode standard.

The results show that steel beams with low flange curvatures behave like those with flat flanges, while the gradual rounding of flanges leads to an increase in effective width factor up to about double its value. The presented analysis, thus, confirms the hypothesis of the activation of a larger ratio of flange due to the interaction of shear lag and plane section hypothesis for box beams with wide curved flanges.