EFFECTIVE MODULUS METHOD (EMM) CONCEPT APPLIED TO THIN‐WALLED STEEL BEAMS AFFECTED BY LOCAL‐DISTORTIONAL INTERACTION

The Effective Modulus Method (EMM), which simulates the effects of local and distortional post‐buckling to establish simplified design methodologies concept, is applied to the determination of the beam strength. This methodology was presented before [1], where the authors tested the results in columns, in comparison with experimental tests. In this approach, local and distortional post‐buckling effects are simulated by changing the mechanical characteristics (namely the elasticity modulus) of the cross‐section of the members. Specifically, the paper presents (i) the proposal of a new design methodology for beams affected by local and distortional buckling modes interaction and (ii) the comparison with experimental tests available in the literature.


Introduction
This paper proposes the use of the Effective Modulus Method (EMM) to evaluate the strength of thin-walled steel beams affected by local and distortional buckling mode interaction.Previous work of the authors [1] presented the concept of effective modulus for a plate and for a cross-section was presented and a new methodology was proposed for computing the strength of columns according to the EMM .The results obtained with the new proposal were compared with experimental data available in the literature.
The EMM postulates that the effects of local (LM) and distortional (DM) instability modes are indirectly considered in the global strength of uniformly compressed columns by reducing their nominal modulus of elasticity (E).This methodology has already been suggested by Rasmussen [2][3][4] and later studied by Camotim and Prola [5,6].The authors [1] proposed simplified expressions for columns leading to consistent results.The present paper adapts this methodology for cold-formed beams affected by local and distortional buckling modes interaction.

XIV Conference on Steel and Composite Construction Portugal
The reduction of the nominal modulus of elasticity, affects the value of the local, distortional and global critical strength of the beam, which in turn has an influence on its resistant bending moment.This way, there is a change in the mechanical properties of the material, which now has a different modulus of elasticity, defined by the effects of local and distortional instabilities.It is also verified that, regardless of the instability modes (critical bending moment characteristic of thin-walled steel members), the critical load is linearly dependent on the modulus of elasticity, thus enabling the calculation of the critical stresses of the beam as given by Eq. ( 1): where   !  " is the local, distortional or global critical stress of the beam with an effective modulus of elasticity   ,   () is the critical stress of the beam with nominal modulus of elasticity E and c  is the reduction factor that considers local and distortional mode effects, given by Eq. ( 2): The post-buckling stress distribution of plates, simply supported on all edges, subjected to uniform compression, obtained by the Koiter's theory [7,8], is shown in Fig. 1.In this figure, s  the critical stress, s  the maximum (edge) stress, s  the average stress,  the width and   the effective width of the plate.By equilibrium of the stress resultants, shown in Fig. 1, Von Kármán [9] derived Eq. ( 3) to define the effective width of a plate without initial imperfections, characterized by the reduction coefficient c  .
In Eq. ( 3), l  is the slenderness, which is given by Eq. ( 4): In Fig. 1, it can also be found Eq. ( 5) (see equation (6.101) in reference [10]) that relates the edge, mean and critical stresses of the plate, which can be presented in the form of Eq. (6).
Accordingly, let one define an effective modulus of elasticity   relating the limit strain e  correspondent to the mean stress s  , as shown in Fig. 2. That is, the effective modulus of elasticity is the one that takes the plate to its maximum strain.Thus, by considering s  =   e  and s  = e  and by replacing in Eq. ( 6), the reduction factor of the modulus of elasticity c  is derived, see Eq. ( 7): The effective width formulas for the constituent elements of the cross-section (flanges, webs, stiffeners) must account the effects of applied stress distribution, the initial imperfections of the member and the level of restriction to the rotation of adjacent plates (web-flange, flange-stiffener).This way, the behaviour of all constituent plates acting simultaneously is considered.
The formula adopted by the AISI S100-16 [11] for the load reduction factor in the Direct Strength Method (DSM), based on the Winter-type formulae [12,13] for whole section, was initially proposed by Shaffer and Peköz [14,15], and is defined in Eq. ( 8): where ML is the resistant bending moment for local mode, My is elastic/yield bending moment and l  = ,    ⁄ , where   is the local critical bending moment of the cross-section.For the distortional mode, Eq. ( 9) has been used in the DSM [15][16][17][18]:

+
with l  > 0.561 (9) where MD is the resistant bending moment for distortional mode, l  = ,    ⁄ , and   is the distortional critical bending moment of the cross-section.
It would be appropriate to adapt the coefficients of Eq. ( 7) to apply the EMM to whole crosssections.For this purpose, a campaign of comparisons between experimental and numerical data will be necessary, which the authors intend to perform in the future.For the present paper, the EMM will use the same type of reducing coefficients as the DSM (see Eq. ( 8) and ( 9)), applying it to the modulus of elasticity.

Effective modulus method: methodology
Two methodologies used within the scope of the DSM [16] are proposed in the EMM to evaluate the effects of the interaction between local and distortional modes using (i) a local slenderness affected by the distortional mode (LD) and (ii) distortional slenderness affected by the local mode (DL).The EMM calculation scheme is represented in Fig. 3 for both methodologies.
Note that no formulation is proposed to evaluate the interaction between local and distortional modes with the global modes, which will be the subject of another publication.In other words, in the present work, only the influence of the interaction between the local and distortional modes on the resistance of the members is considered.

Comparison with experimental results
Two sets of results available in the literature were chosen for comparison with the EMM results.These experimental results refer to pure (uniform) [17,18] and non-uniform [19] bending.

Uniform bending experimental results
To test simple "C" sections (named LC) subjected to uniform and non-uniform bending moments, Chen et al. [17,19] conducted experimental studies using the scheme represented in the Fig. 4. When length Ls1 is equal to Ls2, the bending moment diagram between two applied loads is uniform [17]; the results are presented in this section (3.1).Otherwise, when Ls1 is different from Ls2, the bending moment is non-uniform [19]; for this case, results will be presented in section 3.2.Table 1 shows the dimensions of the LC-section, the column specimens' uniform bending moment resistance obtained from the experimental results (Mexp), the critical bending moment associated with LM (McrL) and DM (McrD) (see [17,20]) and the bending moment obtained by the EMM: (i) LD (MR-LD) and (ii) DL (MR-DL).Fig. 5 presents the relation between EMM and experimental results.The results in Table 1 and in Fig. 5 (MEXP/MRD) show that, for uniform bending moment, the values of the DL methodology are very close to the experimental ones (mean 0.94 and standard deviation 0.047), however showing non-conservative values (MEXP/MRD<1).The LD methodology presents reliable results although with significantly lower precision (average 1.4 and standard deviation 0.1).Pham and Hancock [18] conducted experiments on simple "C" sections (C) and web stiffened "C" sections (SC) subject to uniform bending (see Fig. 6, obtained from [20]).The dimensions and the of the C-section and SC-section are shown in Table 2. Table 2: C and CS-beams specimens' dimensions and comparison of results for non-uniform bending [18] With the purpose to force the members to buckle locally rather than distortionally, in the experiments, Phan and Hancock [18] used the steps showed in the Fig. 7. Thus, Ms is used to named members with straps, and Mw, without straps.The bending moment distribution between two support is uniform.Fig. 7: Tests with and without straps made by [18].Table 2 compares the column specimens' uniform bending moment resistance obtained from the experimental results (Mexp) [18] with the bending moment obtained by the two of methodologies of the EMM:  From Table 2 and Fig. 8 it can be concluded that the EMM results show a good correlation with the experimental ones: (i) for DL methodology, with average equal 1.44 and standard deviation equal 0.198, for Ms SC20015 specimen (all results are on the safe side) and (ii) for LD methodology with average equal 1.2 and standard deviation equal 0.105, for Ms C20019 specimen (only one specimen, Mw SC15012, presents unsafe result -about 3%).

Non-uniform bending experimental results
To test simple "C" sections (named LC) subjected to non-uniform bending moments, Chen et al. [19] conducted experimental studies using the same scheme represented in the Fig. 4. When length Ls1 is different from Ls2, the bending moment diagram between the two applied loads is non-uniform.
Table 3 shows the dimensions of the LC-section, the column specimens' bending moment resistance obtained from the experimental results (Mexp), the critical bending moment associated with LM (McrL) and DM (McrD) (see [18,19]) and the bending moment obtained by the EMM: (i) LD (MR-LD) and (ii) DL (MR-DL).Fig. 9 presents the relation between EMM and experimental results.
The results in Table 3 and in Fig. 9 show that, for non-uniform bending moment, the values of the DL methodology practically coincide with the experimental results, with some unsafe values (average equal 0.98 and standard deviation equal 0.032).The LD methodology produces secure estimated values (with average equal 1.34 and standard deviation equal 0.057).

Comments on the DL and LD methodologies results
It can be concluded that the DL is the methodology presents results closest to the experimental ones, although with unsafe estimates for some specimens.On the other hand, the LD methodology presents always safe results, but with less accuracy.The effective modulus reduction coefficients are based on the concept that the average stress is limited by the yield deformation.Therefore, it is a criterion for limiting deformations, that has implications for the resistance of structural elements.The fact that the same DSM reduction coefficients (which are directly based on resistive efforts) are being used, may explain the unsafe results of the DL methodology.This suggests that parametric studies must be carried out to adjust the coefficients of Eq. ( 7).

Conclusion Remarks
The paper presents the concept of the EMM of a plate and suggests a new methodology for evaluating the strength of beams affected by local-distortional interaction.This methodology applies the concept of effective modulus using correction factors like those used in the DSM to reduce the strength due to post-buckling behaviour.
The EMM assumes that the post-buckling effect of the local and distortional modes interact with each other and with the global mode by the loss of stiffness.This effect is modelled with a reduction in the modulus of elasticity of the material, which manifests itself in a direct relationship with the reduction of the critical stresses and, consequently, the reduction of the compressive strength.
In this paper, results obtained by the new proposal for uniform and non-uniform bending moment are compared with data available in the literature.From the tests carried out, it was concluded that DL is the methodology that presents the most accurate results, although it makes some estimates (very close to experimental results) that are unsafe.On the other hand, the LD methodology presents always safe results, but with less accuracy.
The unsafe results provided by the DL methodology can be explained by using the reduction coefficients of the DSM elasticity moduli.The effective modulus reduction coefficients are based on the concept that the average stress is limited by the yield deformation, while the DSM uses the stress limitation criterion.This suggests that parametric studies must be carried out to adjust the coefficients of Eq. (7).
Cases of interaction of local, distortional and global modes for beams should also be performed by means of the EMM.In this context, it is planned to carry out parametric studies to better adjust the equations of the effective modulus associated with LM and DM methodologies.IN addition, there are ongoing studies to find new formulations for reducing the coefficient of elasticity modulus for columns and beams.
Another potential use of the EMM, to be tested, is in the determination of member deflection, as it is enough to change the modulus of elasticity of the material.

Fig. 3 :
Fig. 3: The EMM flowchart for DL and LD methodologies

Fig. 6 :
Fig.6: a) C-section and b) SC-section[20] (i) LD (MR-LD) and (ii) DL (MR-DL).The minimum value of critical bending moments for local (McrL) and distortional (McrD) modes are considered in calculations, as usual in DSM, and the same values considered by Pham and Hancock [18].Fig. 8 presents the relation between the experimental and EMM results.