# Evolution of EC8 Seismic Design Rules for X Concentric Bracings

^{*}

## Abstract

**:**

## 1. Introduction

_{pl,Rd,i}is the plastic capacity under tension of the brace at the i-th story, and N

_{pl,Ed,i}is the relevant required strength.

## 2. Ductility Classes and Behavioral Factors

#### 2.1. Current EN 1998-1

_{o}is the reference value of the behavior factor for regular structural systems, while a

_{u}/a

_{1}is the plastic redistribution parameter accounting for the system overstrength due to redundancy. EC8 [1] recommends a

_{u}/a

_{1}= 1 for CBFs.

_{gr}, S being the soil factor and a

_{gr}the reference ground acceleration.

#### 2.2. prEN 1998-1-2:2019.3

- q
_{S}is the behavior factor component accounting for overstrength due to all other sources; - q
_{R}is the behavior factor component accounting for overstrength due to the redistribution of - seismic action effects in redundant structures;
- q
_{D}is the behavior factor component accounting for the deformation capacity and energy - dissipation capacity.

_{R}= q

_{D}= 1, while q = q

_{S}= 1.5; structural members and connections are verified according to Eurocode 3 [14].

## 3. Overstrength Factors

#### 3.1. Current EN 1998-1

#### 3.2. prEN 1998-1-2:2019.3

## 4. Design of Dissipative Members

#### 4.1. Current EN 1998-1

_{pl,Rd}is the design plastic strength of brace cross-section, and N

_{Ed}is calculated by linear-elastic analysis on the TO model. Since the braces provide poor energy dissipation in the post-buckling range, the Code states further requirements devoted to limit the global and local slenderness of the diagonal members. The global slenderness, $\overline{\lambda}=\sqrt{\frac{{N}_{pl,\mathrm{br},Rd}}{{N}_{cr,br}}}$ (N

_{cr,br}being the Eulerian critical load), of bracing members must fall in the range (1.3, 2).

#### 4.2. prEN 1998-1-2:2019.3

_{pl,Rd}is the design plastic strength of brace cross-section, and N

_{Ed}is calculated by linear-elastic analysis.

_{Ed,G}being the axial force due to gravity loads in seismic-design situations, q the behavior factor assumed at design stage and N

_{Ed,E}the axial force due to the seismic action.

- Circular hollow sections should verify $D/t\le 19.4\cdot (\epsilon /\sqrt{{\gamma}_{rm}})$, where D is the external diameter and t is the thickness of the cross section, γ
_{mr}is the material randomness coefficient and $\epsilon =\sqrt{235/{f}_{y}}$. - For rectangular hollow sections, the maximum local slenderness c/t should not be greater than $47.4\cdot ({\epsilon}^{2}/{\gamma}_{rm})$, c being the side width and t the thickness of the cross section.

## 5. Design of Non-Dissipative Members

#### 5.1. Current EN 1998-1

- N
_{pl,Rd}(M_{Ed}) is the design resistance to axial force of the beam or column calculated in accordance with EN 1993:1-1 [14], accounting for the interaction with the design value of bending moment; - M
_{Ed}, in the seismic design situation; - N
_{Ed,G}is the axial force in the beam or in the column due to the non-seismic actions in the seismic design situation; - N
_{Ed,E}is the axial force in the beam or in the column due to the design seismic action; - γ
_{ov}is the material overstrength factor; - Ω is the minimum overstrength ratio ${\mathsf{\Omega}}_{i}=\frac{{N}_{pl,Rd,i}}{{N}_{Ed,i}}$

#### 5.2. prEN 1998-1-2:2019.3

_{Ed}, bending moments M

_{Ed}and shear force V

_{Ed}calculated as:

_{Ed,G}, M

_{Ed,G}and V

_{E}

_{d,G}are the axial force, the bending moment and the shear force in the non-dissipative member due to the non-seismic actions in the seismic design situation, and N

_{Ed,E}, M

_{Ed,E}and V

_{Ed,E}are the axial force, the bending moment and the shear force, in the non-dissipative member due to the design seismic action;

- (i)
- The resistance and stability of both beams and columns should be verified in compression against the following actions:$$\begin{array}{l}{N}_{Ed}={N}_{Ed,G}+{\gamma}_{rm}\cdot {\gamma}_{sh}\cdot {\mathsf{\Omega}}_{d}\cdot {N}_{Ed,E}\\ {M}_{Ed}={M}_{Ed,G}+{\gamma}_{rm}\cdot {\gamma}_{sh}\cdot {\mathsf{\Omega}}_{d}\cdot {M}_{Ed,E}\\ {V}_{Ed}={V}_{Ed,G}+{\gamma}_{rm}\cdot {\gamma}_{sh}\cdot {\mathsf{\Omega}}_{d}\cdot {V}_{Ed,E}\end{array}$$
_{rm}is the material randomness coefficient (depending on the steel grade); γ_{sh}is the factor accounting for hardening of the dissipative zone; and Ω_{d}is the minimum design overstrength ${\mathsf{\Omega}}_{i}=\frac{{N}_{pl,Rd,i}}{{N}_{Ed,i}}$. - (ii)
- The internal forces are calculated by mean of plastic mechanism analysis, namely considering a free-body distribution of axial forces in both tension and compression diagonals with values equal to their expected buckling resistance equal to ${\gamma}_{rm}\cdot \chi \cdot {N}_{pl,Rd}$;
- (iii)
- The internal forces are calculated by mean of plastic mechanism analysis, considering a free-body distribution of axial forces in which the braces under tension transmit a force equal to ${\gamma}_{rm}\cdot {\gamma}_{sh}\cdot {N}_{pl,Rd}$ and the braces under compression attain their post-buckling resistance equal to ${\gamma}_{rm}\cdot 0.3\cdot \chi \cdot {N}_{pl,Rd}$.

## 6. Design of Brace-to-Frame Connections

#### 6.1. Current EN 1998-1

_{j.Rd}is the joint axial strength, N

_{pl,Rd}is the brace plastic axial strength and γ

_{ov}is the material randomness

#### 6.2. prEN 1998-1-2:2019.3

- N
_{pl,Rd}is the design plastic resistance of the brace in tension; - N
_{b,Rd}is the design buckling resistance of the brace in compression; - M
_{pl,Rd}is the design plastic moment of the cross section of the brace; - γ
_{rm}and γ_{sh}are the material randomness and the hardening factors, respectively.

## 7. Conclusive Remarks

- the use of a tension-only diagonal scheme;
- the requirement of diagonal slenderness;
- the homogeneity condition of the overstrength factor;
- the design of brace-to-frame connections.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Gusset plate connections accommodating brace buckling with the brace ending before a linear clearance corresponding to the theoretical yielding line of the gusset plate [92].

**Figure 2.**Gusset plate connections accommodating brace buckling with the brace ending before an elliptical clearance corresponding to the theoretical yielding line of the gusset plate [92].

Current EN 1998-1 | Next EN 1998-1 | ||
---|---|---|---|

Ductility Class | Reference q | Ductility Class | Reference q |

DCL | 1.5–2 | DC1 | 1.5 |

DCM | 4 | DC2 | 2.5 |

DCH | 4 | DC3 | 4 |

Current EN 1998-1 | Next Rules | ||
---|---|---|---|

DCM | Resistance: ${N}_{Ed,i}\le {N}_{pl,Rd,i}$ | DC2 (TO) | Resistance: ${N}_{Ed,i}\le {N}_{pl,Rd,i}$ |

Global slenderness: $1.3\le \overline{\lambda}\le 2$ | Global slenderness: $1.3\le \overline{\lambda}\le 2.5$ | ||

Local slenderness: class 1, 2 | Local slenderness: class 1, 2 | ||

Overstrength variation: $\mathsf{\Omega}=\mathrm{min}\left({\scriptscriptstyle \raisebox{1ex}{${N}_{pl,Rd,i}$}\!\left/ \!\raisebox{-1ex}{${N}_{Ed,i}$}\right.}\right)\in \left(\mathsf{\Omega},1.25\mathsf{\Omega}\right)$$i\in \left(1,n\right)$ | Overstrength variation: none | ||

DCH | Resistance: ${N}_{Ed,i}\le {N}_{pl,Rd,i}$ | DC3 | Resistance: ith story: ${N}_{Ed,i}\le \chi \cdot {N}_{pl,Rd,i}$ * roof story: ${N}_{Ed,G}+q\cdot {N}_{Ed,E}\le \chi \cdot {N}_{pl,Rd,i}$ |

Global slenderness: $1.3\le \overline{\lambda}\le 2$ | Global slenderness: $\overline{\lambda}\le 2$ | ||

Local slenderness: class 1 | Local slenderness: class 1 CHS: $D/t\le 19.4\cdot (\epsilon /\sqrt{{\gamma}_{rm}})$ SHS: $c/t=47.4\cdot ({\epsilon}^{2}/{\gamma}_{rm})$ | ||

Overstrength variation: $\mathsf{\Omega}=\mathrm{min}\left({\scriptscriptstyle \raisebox{1ex}{${N}_{pl,Rd,i}$}\!\left/ \!\raisebox{-1ex}{${N}_{Ed,i}$}\right.}\right)\in \left(\mathsf{\Omega},1.25\mathsf{\Omega}\right)$$i\in \left(1,n\right)$ | Overstrength variation: ${\mathsf{\Omega}}_{b}=\mathrm{min}\left({\scriptscriptstyle \raisebox{1ex}{$\chi {N}_{pl,Rd,i}$}\!\left/ \!\raisebox{-1ex}{${N}_{Ed,i}$}\right.}\right)\in \left({\mathsf{\Omega}}_{b},1.25{\mathsf{\Omega}}_{b}\right)$$i\in \left(1,n-1\right)$ |

Current EN 1998-1 DCH/DCM | |||
---|---|---|---|

DCM/DCH | Beams and columns should be designed to verify: ${N}_{pl,Rd}({M}_{Ed})\ge {N}_{Ed,G}+1.1\cdot {\gamma}_{ov}\cdot \mathsf{\Omega}\cdot {N}_{Ed,E}$, where $\mathsf{\Omega}=\mathrm{min}({\mathsf{\Omega}}_{i})=\mathrm{min}\left(\frac{{N}_{pl,Rd,i}}{{N}_{Ed,i}}\right)$ | ||

Next rules | |||

DC2 | Beams and columns should be designed to withstand: $\begin{array}{l}{N}_{Ed}={N}_{Ed,G}+\mathsf{\Omega}\cdot {N}_{Ed,E}\\ {M}_{Ed}={M}_{Ed,G}+{M}_{Ed,E}\\ {V}_{Ed}={V}_{Ed,G}+{V}_{Ed,E}\end{array}$ where $\mathsf{\Omega}=1.5$ | DC3 | Beams and columns should be designed to withstand: (i) $\begin{array}{l}{N}_{Ed}={N}_{Ed,G}+{\gamma}_{rm}\cdot {\gamma}_{sh}\cdot {\mathsf{\Omega}}_{d}\cdot {N}_{Ed,E}\\ {M}_{Ed}={M}_{Ed,G}+{\gamma}_{rm}\cdot {\gamma}_{sh}\cdot {\mathsf{\Omega}}_{d}\cdot {M}_{Ed,E}\\ {V}_{Ed}={V}_{Ed,G}+{\gamma}_{rm}\cdot {\gamma}_{sh}\cdot {\mathsf{\Omega}}_{d}\cdot {V}_{Ed,E}\end{array}$ where ${\mathsf{\Omega}}_{d}=\mathrm{min}({\mathsf{\Omega}}_{i})=\mathrm{min}\left(\frac{{N}_{pl,Rd,i}}{{N}_{Ed,i}}\right)$ (ii) Plastic mechanism analysis with: ${N}_{T}={N}_{C}={\gamma}_{rm}\cdot \chi \cdot {N}_{pl,Rd}$ (iii) Plastic mechanism analysis with ${N}_{T}={\gamma}_{rm}\cdot {\gamma}_{sh}\cdot {N}_{pl,Rd}$ and ${N}_{c}={\gamma}_{rm}\cdot 0.3\cdot \chi \cdot {N}_{pl,Rd}$ |

Columns should also verify combined action of $\left[{N}_{Ed},({M}_{Ed}=0.2{M}_{pl})\right]$ |

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**MDPI and ACS Style**

Campiche, A.; Costanzo, S.
Evolution of EC8 Seismic Design Rules for X Concentric Bracings. *Symmetry* **2020**, *12*, 1807.
https://doi.org/10.3390/sym12111807

**AMA Style**

Campiche A, Costanzo S.
Evolution of EC8 Seismic Design Rules for X Concentric Bracings. *Symmetry*. 2020; 12(11):1807.
https://doi.org/10.3390/sym12111807

**Chicago/Turabian Style**

Campiche, Alessia, and Silvia Costanzo.
2020. "Evolution of EC8 Seismic Design Rules for X Concentric Bracings" *Symmetry* 12, no. 11: 1807.
https://doi.org/10.3390/sym12111807