Design against localization of strains

Clauses 6.5.5(I), In Eurocode 8, design against localization of strains is stated as a general requirement for 6.5.4(1), 7.5.4(1) joints in clause 6.5.5(1). No imposed design of connections is provided. Some explicit rules are related to the mitigation of strain localization. Conformity to standards on steel material is one of them:

The development of a dissipative zone involves a 'spreading' of yield, which requires strain hardening. As the material becomes harder with plastic strains, strain-hardened sections become more resistant than adjacent non-hardened zones; as a result, the latter yield and harden, which generates a progressive extension of the yielding zone until a plastic zone is formed.

Fig. 6.3. Evaluation of the required elongation capacity d/ of a diagonal d I

Fig. 6.3. Evaluation of the required elongation capacity d/ of a diagonal

Al

i . \

L

y

d/2

UJ

/

v H

— /

M

-► X

Strain-hardening of the material corresponds to fjf > 1, and is a necessary property to propagate yield and to avoid yielding all taking place in the narrow section where yielding first occurred. Steels conforming to EN 10025 have/u//y > 1.4.

Clause 6.5.4(1), which refers to a Eurocode 3 rule for bars in tension, also aims at the mitigation of a localized 'brittle' failure.

It is worth recalling this typical 'capacity design' condition, which requires that the ultimate strength of the 'brittle' section with holes Anet should be greater than the plastic strength of the full ductile section A (no holes, no stress concentration), so that yielding of the section without holes takes place before failure of the section with holes:

AfyhM0<AnJuhM2 (D6.3)

Applied to a bar, equation (D6.3) guarantees that yielding can affect the whole length of the bar.

To achieve the absolute deformation capacity required in dissipative zones, yielding must take place in zones that are large enough. The meaning of 'large enough' depends on the absolute deformation required and on the deformation scheme envisaged in the dissipative zone.

A dissipative zone can have a large enough dimension, if yielding does not take place in a short-length zone surrounded by a bigger section. If that were the case, yielding would not propagate, and the ductility of the element would appear much lower than that of the material. That situation, in which all yielding takes place in a narrow zone, is called

'localization of strains', and should be avoided, as required in clause 6.5.5(1)P. Localization of strains is generally due to design 'details'. Clause 6.6.4(3) Design against localization of strains is best explained by considering an example, referring to the design of a dissipative zone at a beam end in a moment-resisting frame, in the case of a rigid connection between the beam and the column. Due to the shape of the seismic bending moment diagram, the beam ends in moment-resisting frames are inevitably the sites for dissipative zones. Many design details are possible for the connection of the beam to the column:

• in design a of Fig. 6.4, yielding can only develop in a narrow zone of length L , because the value of MEd/MRd in the beam section with the cover plate or further in the beam section alone is smaller than at the connection close to the column face

9 in design b of Fig. 6.4, the length of flange that may potentially yield is not limited, and it can extend, for instance, up to a length Ly equal to the depth d of the member.

The plastic rotation capacity of these two details can be estimated in two realistic examples: consider a profile with a depth d = 400 mm made of steel S500 (/ = 500 MPa) with an elongation corresponding to the yield plateau equal to e max= 10e = 10/ /E = 10 X 500/210000 = 2.38%. This corresponds to an elongation at failure above 20%. The plastic rotation is found by the relationship 0 = Al/(d/2), with A/ = le max (see Fig. 6.4).

Ly = 10 mm, ey>max - 2.38% => AZ = 0.0238 x 10 = 0.238 mm, 6 = 0.238/(400/2) = 1.2 mrad 25 mrad

Ly = 400 mm, ey>max = 2.38% =» A/ = 9.52 mm, 0 = 9.52/(400/2) = 47.6 mrad » 35 mrad

Design b offers a significant plastic rotation, greater than the 35 mrad requirement for DCH in clause 6.6.4(3) of EN 1998-1. Design a is unable to provide the required plastic rotation, even for DCM.

The practical conclusions of this simple example are straightforward:

• a length of yielded zone of the order of the depth of the section is needed for an effective plastic hinge to form. This requires avoiding localization of strains

9 material with high enough values of ey max and/u//y are required

9 for a given material, higher beam depth d means smaller plastic rotation capacity since 6 = Al/(d/2).

Further explanation on moment connections is given in Section 6.9.

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