# On the Use of the Effective Width for Simply Supported Generally Loaded Box-Girders
## 1 Introduction  
It is well-known that shear effects play an important role on the mechanical response of frames behaving as beam-like structures [1; 2; 3]. However, they also affect single thin-walled members, especially when the flanges are wide compared to the length of the member. Among these latter structures, the box-girders are particularly prone to the shear-lag phenomenon, which is caused by shear strains in the flanges. This topic is of some technical importance, since box-girders are widespread in civil engineering applications, such as highway bridges, railway bridges and footbridges. Indeed, in box-girders in bending, it happens that longitudinal membrane stresses, developed at the intersection between the flange and web, cause warping of the cross-section and alteration of the normal stresses distribution, with respect to those predicted by the classic [PERSON] and [PERSON] theories. In particular, the latter assumes that the cross-sections remain planar and predict normal stress constants on the flanges and linear on the webs, while in practice, they are non-uniformly distributed along the flange, with a maximum value at the web intersection, and this value is typically larger than those predicted by the classic Euler-Bernoulli and [PERSON] theories. The shear-lag effect in box-girders has been largely studied both in the past [4; 5] and in recent years [6; 7], with the aim of evaluating the maximum value of the normal stresses in the cross-sections. The first person to study the shear-lag effect of rectangular box-girders was [PERSON] [8] in 1946, by using a variational method based on the principle of minimum potential energy. There is a vast amount of literature on this subject. The shear-lag effect has been studied by analytical methods [9; 10; 11], numerical methods [12; 13; 14] and experimental tests [11; 15; 16]. In particular, [PERSON] and [PERSON] [17] opted for empirical equations to obtain normal stresses with the finite-stringer method, [PERSON] and [PERSON] [18] adopted the finite strip method, [PERSON] et al. resumed the biharmonic analysis of isolated flanged plates previously applied by [19], and [PERSON] and [PERSON] [20] studied box-girder bridges under uniform load by the Symplectic Elasticity method. Moreover, several researchers adopted the finite-segment method [21; 22], the finite element method [23; 24; 25] and the energy-based variational method [4; 26; 27]. A simple but efficient method to consider the shear-lag effect of box-girders in bending is the _effective flange width method_. An effective flange width is defined as the length of the flange in which the normal stresses are assumed constant and equal to the maximum stress arising at the intersection with the web, under the assumption that the cross-section remains planar. This is a simplified approach, but it usually involves reasonable errors in design. The use of the effective width method was originally proposed in 1924 by [PERSON] [28] for aeronautical applications. Later on, it was adopted by [PERSON] and [PERSON] [29], who gave a separate determination of the effective widths for bending and torsion, [PERSON] and [PERSON] [18], who also investigated the effective width of postcritically loaded stiffened plates, and [PERSON] et al. [30], who proposed an analysis of the ultimate load obtaining formulae for flange effective width ratio. Moreover, the idea of effective flange width has been featured in many design codes and standards, such as Eurocode 4 [31], the AASHTO LRFD Bridge Design Specifications [32], European Committee for Standardization (CEN) [33] and British Standards Institution (BSI) BS 5400 [34], where specific formulas for load type are provided. A drawback in using effective width lies in the fact that it depends on the load distribution. Therefore, the effective width of a beam loaded by forces uniformly distributed on its whole length is different from that relevant to a segment of distributed loads, or even from that related to a concentrated load, applied, e.g., at different abscissas.