Extension of the Effective Thickness Method (ETM)

The new version of the effective thickness approach properly modifies the simplified rules given in the general provisions of EN1991-1-1 [6.5] to account for the occurrence of buckling in the plastic range, as it occurs in the case of sections whose plate elements are characterised by small values of the width-to-thickness ratio.

As reported in Chapter 2 [Eq. (2.11)], the elastic buckling stress of an isolated plate element is given by:

𝜎 _. = 𝑘 𝜋 𝐸

12(1 − 𝜈 )(𝑏 𝑡⁄ ) (6.1)

where 𝐸 is the elastic modulus, 𝜈 is the Poisson’s ratio in the elastic range, 𝑏 is the plate width, 𝑡 is

section and by the longitudinal stress gradient. These effects can be taken into account by modifying Eq. (6.1) using two factors: 𝜁 and 𝛼 . The factor 𝜁 accounts for the interaction between the plate elements composing the section. The factor 𝛼 accounts for the influence of the longitudinal stress gradient occurring in structural members under non-uniform bending. Obviously, in the case of uniform compression, herein investigated, 𝛼 = 1.0.

Therefore, including the effects of interactive buckling and longitudinal stress gradient, the elastic buckling stress can be expressed as:

𝜎 _. = 𝜁 𝛼 𝑘 𝜋 𝐸

12(1 − 𝜈 )(𝑏 𝑡⁄ ) (6.2)

The occurrence of buckling in the plastic range can be accounted for using a correction factor which depends on the non-linear behaviour of the material. By denoting with 𝜉 such correction factor, the buckling stress in the plastic range 𝜎 _. is given by:

𝜎 _. = 𝜉 𝜁 𝛼 𝑘 𝜋 𝐸

12(1 − 𝜈 )(𝑏 𝑡⁄ ) (6.3)

where also the Poisson’s ratio 𝜈 has to be evaluated accounting for the non-linear behaviour of the material. Regarding the 𝜉 factor, many different expressions have been proposed in the technical literature for its evaluation. Some of them will be presented in the following discussion.

Concerning the Poisson’s ratio in the yield region, as shown in Chapter 2, Gerard and Wildhorn [6.8]

have studied the problem in the case of several aluminium alloys and have shown that it is seriously affected by the anisotropy of the material. In the case of materials exhibiting the same properties in the two directions orthogonal to the loading direction, they proposed Eq. (2.39). However, in the case of perfectly plastic material, the condition 𝜈 = 𝜈 is reached only when the strain assumes infinite value (ε → ∞) so that 𝐸 → 0. Therefore, within the framework of a simplified procedure like the effective thickness approach, an alternative relation can be proposed:

𝜈 = 𝜈 − 𝐸

𝐸 𝜈 − 𝜈 (6.4)

which assures 𝜈 = 𝜈 in the elastic range (𝐸 = 𝐸) and the condition 𝜈 = 𝜈 when a plastic plateau is

where 𝜀 is the strain corresponding to buckling. Consequently, the effective ratio 𝑏 𝑡⁄ can be defined as a function of the strain level as:

𝑏

𝑡 = 𝜉𝐸

𝐸 𝜁 𝛼 𝑘 𝜋

12(1 − 𝜈 ) 1

𝜀 (6.6)

The effective thickness can be derived by noting that:

𝑡

𝑏 = 𝐸

𝜉𝐸 1 𝜁 𝛼 𝑘

12(1 − 𝜈 )

𝜋 𝜀 (6.7)

and by introducing a parameter 𝜇, which accounts for the non-linear behaviour of the material:

𝜇 =𝜉𝐸 𝐸

1

1 − 𝜈 (6.8)

so that the following relation is obtained:

𝑡

𝑡 = 𝑏 𝑡

12 𝜋

1 𝜇

1

𝜁 𝛼 𝑘 𝜀 (6.9)

Remembering that:

𝜆̅ =𝑡

𝑡 (6.10)

It means that the normalised slenderness in the non-linear range (elastic-plastic range), depending on the strain level, can be defined as:

𝜆̅ =𝑏 𝑡

12 𝜋

1 𝜇

1

𝜁 𝛼 𝑘 𝜀 (6.11)

To use the buckling curves of EN1999-1-1 with the normalised slenderness corrected to account for the non-linearity depending on the strain level, it has to be considered that:

𝜆̅ = 0.03143 𝛽

𝜖 (6.12)

where 𝜖 = 250 𝑓⁄ _. , therefore:

𝛽

𝜖 = 𝜆̅

0.03143= 1

0.03143 3 𝜋 𝜀

𝜇 𝜁 𝛼 2

√𝑘 𝑏

𝑡 (6.13)

Eq. (6.12) provides:

𝛽

𝜖 = 17.54 𝜂 𝑏 𝑡

𝜀

𝜇 𝜁 𝛼 (6.15)

which is the final expression to compute the slenderness parameter of the plate element to be used, in combination with the buckling curves of EN1999-1-1. This allows computing the effective thickness in the non-linear range as a function of the strain level 𝜀. In fact, according to EN1999-1-1, the reduction factor accounting for local buckling is computed as:

𝜌 = 1 𝑖𝑓 𝛽 𝜖 ≤ 1

2 C + C − C (3 + 𝜓) (6.16)

and:

𝜌 = C

𝛽 𝜖⁄ −C (3 + 𝜓)

4 (𝛽 𝜖⁄ ) 𝑖𝑓 𝛽 𝜖 > 1

2 C + C − C (3 + 𝜓) (6.17) The parameter 𝜓 accounts for the strain distribution along the loaded edge of the plate. It is given by the ratio between the maximum compression strain at one end of the plate and the strain at the second end of the plate element. In the case of uniform compression, it results 𝜓 = 1 while 𝜓 < 0 when the second end of the plate element is subject to tension. While the coefficient C and C are reported in Eurocode 9 and they are defined according to the Buckling Curves:

Table 6.1. Values of the coefficients 𝐶 and 𝐶 reported in the Eurocode 9.

𝑩𝒖𝒄𝒌𝒍𝒊𝒏𝒈 𝑪𝒖𝒓𝒗𝒆 𝑰𝒏𝒕𝒆𝒓𝒏𝒂𝒍 𝒑𝒂𝒓𝒕 𝑶𝒖𝒕𝒔𝒕𝒂𝒏𝒅 𝒑𝒂𝒓𝒕

𝐂_𝟏 𝐂_𝟐 𝐂_𝟏 𝐂_𝟐

A 32 220 10 24

B 30.5 209 9.5 22

C 29 198 9 20

Regarding the 𝜉 factor needed in Eq. (6.8) as already stated, a variety of different formulations have been proposed in the technical literature [6.9]-[6.12].The most commonly used formulations are:

 tangent modulus theory: 𝜉 =𝐸

𝐸

 secant modulus theory: 𝜉 =𝐸

The use of the tangent modulus theory provides the smallest value of the buckling load. Conversely, the secant modulus theory provides the greatest value of the buckling load. While the relationships provided by Stowell-Bijlaard and Li-Reid provide intermediate values. However, aiming to improve the accuracy of the results obtained when the effective thickness approach is applied to predict to ultimate resistance of aluminium members subjected to local buckling under compression and non-uniform bending, other two formulations are proposed. The first one derives by the theoretical results obtained in Chapter 2. In particular, as seen in Section 2.36, this is obtained by combining the Eqns.

(2.108) and (2.109) derived by a single plate simply supported along four edges under uniform compression:

𝜉 = 0.42𝐶 + 0.58 𝐶 𝐶 1 − 𝜈 1 − 𝜈

𝐸

𝐸 (6.18)

where the expressions of the plastic coefficients 𝐶 are reported in Eq.(2.76). It is possible to observe that this formulation is similar to the relations proposed by Stowell-Bijlaard and Li-Reid, however in this case the variability of Poisson’s ratio is accounted. While, in the case of aluminium beams in bending, the following relationship is suggested:

𝜉 =𝐸 𝐸

𝑛 − 8 8 +8

𝑛 𝐸

𝐸 (6.19)

It can be recognised that Eq. (6.19) is a combination of the secant modulus theory with the Gerard formula. In particular, for small values of the Ramberg-Osgood exponent 𝑛, Eq. (6.19) tends to provide values close to those given by Gerard. Conversely, for high values of 𝑛, Eq. (6.19) tends to provide values close to the secant modulus theory. In the following, referring to the application of the effective thickness approach, Eqns. (6.18) and (6.19) are used.

Regarding the factor accounting for the influence of the longitudinal stress gradient, the following relations can be adopted [6.13],[6.14]:

 for a flat internal compression cross-section part:

𝛼 = 1 +1 4

1.70 𝐿

𝑏

.

− 0.20 (6.20)

 in case of flat outstand compression elements:

1 1.70

where 𝐿 is the shear length and 𝑏 is the plate width in compression. The shear length is defined as the distance between the point of zero bending moment and the section where the maximum bending moment occurs. In the case of uniform compression, 𝛼 = 1.0.

The correction factor 𝜁 for interactive buckling can be evaluated taking into account that it represents the ratio between the buckling factor 𝑘 accounting for interactive buckling and the buckling factor 𝑘 evaluated for the isolated plate element, i.e. 𝜁 = 𝑘 𝑘⁄ .

 in the case of plate elements, acting as flange, connected to webs on both edges (curve 1 of Figure 6.2)

𝜁 = 1.75 − 0.45 𝑏 𝑏⁄

0.15 + 𝑏 𝑏⁄ − 0.02275(𝑏 𝑏⁄ ) ≥ 1 (6.22) which is derived from the expression of 𝑘 given by BS5950-5 [6.15] considering that, in this case, 𝑘 = 4.

 in the case of plate elements, acting as flange, connected to the web on one edge and the lip on another edge (curve 2 of Figure 6.2)

𝜁 = 1.35 − 0.35 𝑏 𝑏⁄

0.60 + 𝑏 𝑏⁄ − 0.005(𝑏 𝑏⁄ ) ≥ 1 (6.23) which is derived from the expression of 𝑘 given by BS5950-5 [6.15] considering that, in this case, 𝑘 = 4.

 in the case of unstiffened elements, acting as a flange (Figure 6.2) 𝜁 = 3.00 −1.882 𝑏 𝑏⁄

2.0 + 𝑏 𝑏⁄ − 0.0059(𝑏 𝑏⁄ ) ≥ 1 (6.24) which is derived from the expression of 𝑘 given by BS5950-5 [6.15] considering that, in this case, 𝑘 = 0.425.