Yield Displacement Charts for performance-based seismic design

doi:10.21203/rs.3.rs-1439436/v1

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Yield Displacement Charts for performance-based seismic design

https://doi.org/10.21203/rs.3.rs-1439436/v1

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A new methodological concept is presented for seismic design, called Yield Displacement Charts (YDC). As with its predecessors, the Yield Point Spectra (YPS) and the Yield Frequency Spectra (YFS), the YDC takes advantage of the simple features of yield displacement (uy), to use uy in a performance-based design versus a force-based period-dependent approach. A self-contained and comprehensive approach to YPS and YFS is presented, enabling the novel aspect of the YDC to be introduced: a tool for a multi-performance objective design that only depends on the location of the structure. Once the YDC has been calculated for a particular place, it can be used for the preliminary design of any structure. For a given value of the yield displacement, the YFS is obtained from the YDC. The suitability of the methodology proposed is illustrated by using a simple case study for a concrete bridge column.

Performance-based design

yield displacement

yield frequency spectra

preliminary design

Preliminary design is the first step in a seismic design process (Aschheim et al. 2019) (Palermo et al. 2016). The accuracy of a preliminary design is of the greatest importance. Due to the fact that, in most structures, displacement caused by earthquakes is mainly in the first mode of vibration, preliminary seismic designs are usually based on the response of single degree of freedom systems.

The first section of the article is dedicated to describing, in a mathematically compact and simple way, the response of a single degree of freedom system in the linear elastic domain and in the plastic domain. The second section presents the seismic hazard curve (peak ground acceleration -PGA- versus probability of occurrence), particularized for Granada –Spain-. PGA has been chosen to measure the intensity of the ground motion, despite the existence of better period-dependent describers (e.g. the spectral acceleration, S_a(T), (Luco and Bazzurro 2007)) and other non-period-dependent describers (e.g. peak ground velocity or peak ground displacement (Palermo et al. 2014)).

The main reason for not using a period-dependent variable is that yield displacement is going to be used in the presentation of the Yield Displacement Charts (YDC), the main novel aspect of this work. The use of the period has been avoided throughout the article since the period loses its physical meaning in the plastic range.

Several spectral representations of the seismic demand of single degree of freedom (sdof) systems are available in the literature. The most common representation is pseudo-acceleration versus the period of the sdof system. The yield point spectra (YPS) uses yield displacement (u_y) as an alternative to the period (T) (Mark Aschheim and Black 2000). The great advantage of yield displacement is that, for a given structural configuration, it is a more stable parameter than the period, which is an important property in preliminary designs. The properties of yield displacement are explained in the third section of this paper.

Performance-based seismic designs that consider several performance objectives corresponding to different seismic hazard levels are now more commonly used, and they are recommended in (SEAOC 1999), (Fib 2012). In this context, Yield Frequency Spectra (YFS) (Vamvatsikos and Aschheim 2016) are regarded as an efficient tool for dealing with this type of analysis. YFS are a graphical representation that link the mean annual frequency/rate (MAF) of exceeding a ductility threshold (i.e. a performance objective) with the system yield strength coefficient (C_y). YFS are plotted for a given yield displacement (u_y), and they allow the demand of the system for different performance objectives to be visualized. The fifth section deduces the YFS from the theoretical framework of the conditional probability, and it presents the Yield Displacement Charts (YDC). Yield Displacement Charts (YDC) pursue the same objective as the YFS, but they are obtained by considering yield displacement as a variable, so the result is a performance-based seismic design tool that only depends on the geographical location of the structure. The YDC is calculated for Granada city in the sixth section. Finally, a simple example of structural design is developed.

The time-history displacement response, u(t), of a linear-elastic SDOF system, for a fixed value of damping, subjected to a given ground motion, $\ddot{u}$_g(t), is characterized by only one variable, which is usually period T. From this period, ω = 2π/T and k = ω²m can be computed, with k as the initial stiffness, m the mass and ω the natural circular frequency of the system (see Fig. 1). The maximum displacement of a linear-elastic SDOF for a given ground motion is called elastic spectral displacement S_d = max[u(t)], for which the SDOF develops an elastic force F_e = k S_d. Typically, this elastic force is presented in the literature as being divided by mass m, and then it is called spectral pseudo-acceleration: S_a=F_e/m. From the simple relationships above, it can be demonstrated that S_a = ω²S_d. Since T characterizes the SDOF response for a given ground motion, all the variables defined above can be expressed as functions of T, i.e.: S_a(T), S_d(T) and F_e(T). For example, the (pseudo-acceleration) response spectrum of a ground motion can be represented by S_a(T) versus T, as shown in Fig. 2.

In structural engineering design, the seismic action is not unique (note that Fig. 2 corresponds only to one seismic accelerogram, $\ddot{u}$_g(t)). Furthermore, a probabilistic treatment of the seismic action is required. Simple envelope approximations of uniform-hazard response spectra derived from probabilistic seismic hazard analysis (PSHA, Cornell 1968) are used in professional standards for structural design purposes: the design spectra. For a SDOF system of period T and mass m, the design spectrum gives the maximum elastic force that can be developed by the SDOF system, S_a(T)m, for a certain probability of occurrence, e.g. 10% in 50 years.

Likewise, a non-linear SDOF system modeled as elasto-plastic can be characterized by only two variables, usually C_y and u_y, with u_y as the yield displacement and C_y as the yield strength coefficient defined as C_y = F_y/mg where F_y is the yield strength (see Fig. 3). Note the absence of a period in this characterization, which is defined via C_y and u_y as

$$T=2\pi \sqrt{\frac{{u}_{y}}{{C}_{y}g}}$$

If the maximum displacement of this non-linear system for a given ground motion is u_max so that if u_max > u_y, then the maximum displacement is in the plastic range. It is important to note that once the system is in the yield plateau, the main recovery factor of the system in terms of displacement is the cyclical nature of the seismic action itself, whereby the system alternates between unloading/reloading at elastic stiffness (or near-elastic for non-kinematic-hardening hysteresis rules), and it sustains further damage along the yield plateau. In other words, the importance of a period as a performance-prediction variable is limited when the system enters the post-yield range. The reader should be aware that the authors are looking for a simple preliminary design tool, not a complete and comprehensive description of the response.

In the time-history analysis, ductility (µ) is defined as u_max/u_y (i.e., ductility demand). Given that demand should not overcome capacity (D < C), it is necessary to ensure that the ductility demanded by the earthquake is at least equal to the ductility capacity in the structure. In other words, when designing for a certain level of ductility, the designer needs to ensure that the structure is going to allow reliable plastic behavior in the areas of (pre-designated) structural detailing where plasticity is going to take place (i.e., ductility capacity). Although it is a little unconventional, for the sake of mathematical completeness, values of µ smaller than 1 are going to be considered (even if they are not proper ductility) in order to also include the cases where u_max is smaller than u_y in the study. Even though the SDOF is characterized by one or two variables depending on whether it is linear or elasto-plastic, the (maximum or worst-case) response of the system can be described by only one variable: the maximum displacement, S_d or u_max. The inelastic displacement ratio C₁ is defined to relate S_d(T) and u_max, so that u_max = C₁S_d(T), FEMA 440(FEMA 2005).

A simple way to describe the intensity of a ground motion is S_d, because it shows, as no other parameter does, the effect of a specific ground motion on an elastic SDOF of period T. Notice that, as mentioned, S_a = ω²S_d, lending the same properties of S_d to S_a. As discussed, the strength of this relationship is weakened when entering the non-linear range. This is quantified by the use of inelastic displacement ratios C₁(C_y, Τ), also known as R-µ-T strength ratio/ductility/period relationships (where R is essentially C_y, see Fig. 3). Since the connection is no longer deterministic, these relationships convey the statistics of inelastic displacement u_max for systems of a given C_y and T, typically offering the mean estimate, $\stackrel{-}{{u}_{\text{m}\text{a}\text{x}}}$, for use in tandem with the static pushover (e.g., (FEMA 2005), (EN1998-3 2005)):

$$\stackrel{-}{{u}_{\text{m}\text{a}\text{x}}}={C}_{1m}\left({C}_{y},T\right){S}_{d}\left(T\right)$$

where C_1m is the mean inelastic displacement ratio function for a given C_y and T.

The predictive ability of S_d and S_a, the intensity measures (IMs) within the framework of Performance-Based Earthquake Engineering (Cornell and Krawinkler 2000) can be characterized by implementing the associated dispersion of the distribution of C₁(C_y,Τ), as offered by e.g. (Vamvatsikos and Cornell 2006) or (Ruiz-García and Miranda 2007). This is the so-called IM efficiency (Luco and Cornell 2007) and the lower the dispersion, the higher the efficiency and associated predictive capability. The issue of IM sufficiency also arises (Nicolas Luco and Cornell 2007), which characterizes the capability of S_d and S_a to render the distribution of C₁(C_y,Τ), or equivalently of u_max conditioned on S_d or S_a, independent of other seismological characteristics.

Sufficiency is a particularly useful property, as it negates (or generally reduces) bias when assessing performance (see Section 4). S_a(T) is moderately efficient and sufficient, but clearly imperfect whenever large excursions into non-linearity or significantly higher-mode effects are involved (e.g. (Luco and Bazzurro 2007). At the same time, S_a(T) is clearly better than peak ground acceleration PGA (e.g., (Kazantzi and Vamvatsikos 2015) ) for everything but the shortest periods, in which the two parameters are practically identical. Nevertheless, in the Yield Displacement Charts (YDC) presented, the authors are going to use PGA as the intensity measure, emphasizing the fact that it is not period dependent, which makes it more versatile than S_a(T) at the cost of reducing efficiency and sufficiency. The authors believe this tradeoff is acceptable for practical design applications, where simplicity is important, and conservativeness can be added as needed to make up for the probabilistic deficiencies of PGA.

The intensity of the ground motion at a certain site can be measured by IMs that, among other parameters, may be represented, as mentioned, by peak ground acceleration PGA or pseudo-acceleration S_a(T). The level of ground motion intensity expected at the site is associated to a certain frequency/rate of occurrence, and this is calculated by using the total probability theorem, combining several probabilistic distributions, and conveying information about the seismic sources (faults), the site (e.g., soil type), and the source-to-site propagation (Baker 2013). This is known as probabilistic seismic hazard analysis (PSHA), and a representation of the intensity of ground motion, measured by the PGA versus its MAF is the seismic hazard curve λ(PGA).

The site-specific seismic hazard curve, λ(PGA), is obtained by using the seismic source model (Giardini et al. 2014) as estimated by (EFEHR, 2021) for Granada (latitude 37.18̊, longitude − 3.60̊), considering a soil with an average shear wave speed in the upper 30m of V_s30=250m/s. The seismic hazard curve is approximated by using a continuous smooth interpolation curve (splines), as shown in Fig. 4.

Preliminary design usually begins with the period of the first mode of the structure, or a combination of some of the modes, which is known as the equivalent single degree of freedom (ESDOF) system (Aschheim et al. 2019).

As an alternative, some displacement-based methods for seismic design (e.g. (Mark Aschheim and Black 2000) and (Paulay 2002) ) use an estimate of the yield displacement for establishing values of the base shear strength. The main advantage of yield displacement, instead of the period, is that it is very stable for stiffness changes. This stability has been confirmed at both a sectional level and at a global system level (e.g. Hernández-Montes and Aschleim 2003, Priestley et al. 2007, Hernández-Montes et al 2019). As an illustration of this feature, Fig. 5 shows the base shear resultant of two buildings subjected to a monotonic lateral force profile, as a function of the displacement at the roof. The only difference between the two models is the amount of reinforcement. Figure 5 shows that the yield displacement (D_y) remains stable (gray bar in Fig. 5).

So, the stability of the yield displacement converts it into a key performance variable for the seismic design of structures (Aschheim and Montes, 2003). Estimations of the yield displacement for several structural systems and different cross-sections can be found in Chap. 9 of (Aschheim et al. 2019).

The performance of a SDOF system under a seismic action, formulated in terms of conditional probability, can be described as the MAF (λ) by which the maximum displacement (u_max) exceeds a certain limit displacement (δ):

$$\lambda \left(\delta |{C}_{y}, {u}_{y}\right)=P\left[{u}_{\text{m}\text{a}\text{x}}>\delta |{C}_{y}, {u}_{y}\right]={\int }_{0}^{\infty }P\left[{u}_{\text{m}\text{a}\text{x}}>\delta |PGA, {C}_{y}, {u}_{y}\right]\left|\text{d}\lambda \left(PGA\right)\right|$$

where λ(∙) denotes the MAF of its argument, P[u_max > δ|PGA,C_y,u_y] is the probability of u_max exceeding a threshold δ conditioned on the yield strength coefficient (C_y), the yield displacement (u_y), as well as on the peak ground acceleration (PGA). The second term inside the integral is the differential of the PGA seismic hazard curve, the latter being the function of the MAF of exceeding a given value of PGA, λ(PGA). Note that, by assuming the sufficiency of the IM (here PGA), any conditioning on seismological characteristics, such as magnitude or distance has been dropped. This is a key assumption of the conditional approach (Bazzurro et al. 1998) that allows this standard treatment of Performance-Based Earthquake Engineering.

The integral above can be solved numerically as:

$$\lambda \left(\delta |{C}_{y}, {u}_{y}\right)=\sum _{j=1}^{N}P\left[{u}_{max}>\delta |{PGA}_{j}, {C}_{y}, {u}_{y}\right]\varDelta \lambda \left({PGA}_{j}\right)$$

where Δλ (PGA_j) = λ (PGA_j) – λ (PGA_j+1), with PGA_j+1 > PGA_j.

Since the integral (or summation) is calculated over the IM, here, with the IM as PGA, the variable PGA disappears when the integral is completed.

For a particular system, given by C_y and u_y, the above equation can be represented as a curve of λ-δ, see Fig. 6a, where it can be observed that when displacement threshold values (δ) increase, the probability that u_max exceeds δ decreases. Additionally, C_y can be considered to be a variable. Curves of constant value of C_y can be plotted together, see Fig. 6b. It can be observed that as the value of C_y increases, the probability of exceedance decreases.

The limit displacement (δ) divided by u_y is the ductility threshold, µ (i.e. µ = δ/u_y). If the horizontal axis of Fig. 6b is changed from δ to µ ( see Fig. 6c) a new representation, known as the Yield Frequency Spectra (YFS), is obtained (Vamvatsikos and Aschheim 2016).

The YFS relate the probability of exceedance of a given displacement threshold at a certain place (where the seismic hazard and the soil characteristics are known) for an elasto-plastic sdof system, half-defined, because the yield displacement u_y has to be known.

As already mentioned, the yield displacement u_y is very stable for stiffness changes, which makes YFS the perfect tool for starting a multi-objective design.

By forcing u_y to be a variable, a new design tool can be created: The Yield Displacement Charts (YDC), see Fig. 7.

For a series of values of C_y and u_y within a feasible range, the resulting curves λ-δ provide charts that are only dependent on the location of the structure, i.e., the charts are independent of the characteristics of the system. In the next step, once the value of u_y for a particular structural system has been discovered (Chap. 9 of (Aschheim et al. 2019)), a YFS is obtained from the YDC. The designer is able to obtain the base shear for different performance objectives from the YFS.

Unlike the work of (Ruiz-García and Miranda 2007), in the method presented here, the YDC is computed directly from a particular earthquake database instead of using the statistical information from the previously deduced maximum inelastic displacement (Ruiz-García and Miranda 2003). Thus, from Eq. (4), the values of λ as function of C_y and u_y are obtained for a given place (meaning that soil conditions and seismic hazard are constant).

For a comprehensive suite of elastic-hardening bilinear SDOF systems defined by C_y and u_y (see Fig. 3), the maximum displacement values u_max are obtained by using a database of 980 earthquakes. For all the SDOFs, 5% damping, 3% of post-yield stiffness ratio and elastic stiffness degradation defined by α = 0.5 have been considered. The degraded elastic stiffness (k_u) is defined as:

$${k}_{u}=k{\left(\frac{{u}_{y}}{{u}_{max}}\right)}^{\alpha }$$

All the ground motions used in this study have been obtained from (RESORCE 2020). No additional corrections were applied to the downloaded ground motion records. The filtering parameters applied in the process of record selection are as follows:

• Records with a magnitude greater than 5 are selected.

• Any source-to-site distance

• Both pulse-like and non-pulse-like motions

• Records with V_S30 between 180 and 360 m/s.

• Both of the horizontal component recordings available are considered.

As a sample, Fig. 8 shows the u_max obtained for the 980 ground motions (490 recordings times with two components each) considered for the case of C_y = 0.3 and u_y = 0.025m. A high dispersion is observed for PGA > 0.2g, where high structural nonlinearity appears, which is captured by few records. Specifically, there are 952 motions in 0 < PGA ≤ 0.2g whereas there are only 28 motions in 0.2g < PGA ≤ 0.9g. Employing scaled ground motion can better populate the higher range of response, but this is at the expense of introducing issues of scaling bias, given the use of PGA. Accounting for the above, the authors preferred to continue using with unscaled motions.

The results are treated using cloud analysis (Cornell et al. 2002). ^{The cloud of data} u_max versus PGA, see Fig. 8 left, are related by using linear regression in the logarithmic space: u_max=a·PGA^b, or by applying logarithms: Ln[u_max] = Ln[a] + b·Ln[PGA]. The constants are calculated by using regression analysis for each particular system (i.e. C_y and u_y are given), see Fig. 8 on the right for the particular case that is being shown, for which Ln[a] = − 2.12 and b = 0.92, with 0.6 as the standard deviation, σ.

The regression analysis provides, the mean and the standard deviation for each value of PGA. In this sample, the mean value is µ=−2.12 + 0.92·Ln[PGA] and σ = 0.6. Assuming that Ln[u_max] follows a normal distribution, see Fig. 8 on the right, then u_max follows a lognormal distribution. Note that the lognormal distribution characterizes variables with a tendency to produce frequent extremely high values, and it is often used in earthquake engineering, especially when modelling seismic demand.

The lognormal cumulative distribution function gives the probability of occurrence of a given displacement threshold, say δ, for given values of PGA, C_y, and u_y: i.e. P[u_max ≤ δ | PGA, C_y, u_y]. Figure 9 shows the lognormal cumulative density function for the sample case (i.e., C_y = 0.3 and u_y = 0.025m) and for six different values of PGA.

Finally, considering that P[u_max > δ | PGA_j,C_y,u_y] = 1 − P[u_max ≤ δ | PGA_j,C_y,u_y], the integrand of the Eq. (3) is obtained.

The λ-δ curve (see Fig. 6a) is obtained by numerically applying Eq. (4). The seismic hazard curve λ(PGA) used in Eq. (4) is the one shown in Fig. 4 (i.e., for the case of Granada).

The displacements (u_max and δ) are divided by the yield displacement (u_y) obtaining the ductility demand and the ductility threshold (i.e., a threshold limit in terms of ductility, also called ductility capacity), respectively. Figure 10 shows the curve λ-µ for the sample system (u_y=0.025m and C_y=0.3) and the location in Granada.

The above methodology is applied for different SDOF systems after the selection of values of C_y (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1) and u_y (0.025 m, 0.05 m, 0.1 m). The obtained u_max values versus the PGA are represented in Fig. 11, and the obtained Yield Displacement Chart for the city of Granada is originally shown in Fig. 12.

In order to illustrate a design case with multiple performance objectives based on yield displacement, a simple example has been developed, see Fig. 13. The example consists of a single-column pier of a highway bridge located in Granada (Spain). In accordance with (EN1998-1 2004), the design has to fulfill the two following performance objectives:

1. The life safety (LS) requirement, associated to a probability of exceedance of 10% in 50 years (return period of 475 years), i.e. λ_LS = − ln(1 − 0.1)/50 = 2.11×10^− 3 years^− 1.

2. The damage limitation (DL) requirement, associated to a probability of exceedance of 10% in 10 years (return period of 95 years), i.e., λ_DL = 0.0105 years^− 1.

The effective yield curvature (ϕ_y) for circular column cross-sections is estimated (Hernández-Montes and Aschheim 2003) as:

$${\varphi }_{y}=2.3 \frac{{ϵ}_{y}}{d}$$

where ε_y is the yield strain of the reinforcing steel and d is the depth of the extreme tension reinforcing bar. For steel B-400 the characteristic strength is f_yk=400 MPa, leading to ε_y = 400/200000 = 0.0020. The column has a circular cross section with a diameter of 1.2 m. The cover chosen is2.5 cm, thus d = 1.2 − 0.05 = 1.15 m. The above results in an effective yield curvature ϕ_y = 0.00397 rad/m.

The yield displacement (u_y) for a cantilever column subjected to a lateral load when shear deformation is neglected can be deduced by integrating the curvature along the length of the column twice with ϕ = ϕ_y for x = 0, see Fig. 14 and Eq. 7.

$${\varphi \left(x\right)=\varphi }_{y}\frac{L-x}{L}\to {u}_{y}=\int \left(\int \varphi \left(x\right)dx\right)dx={\varphi }_{y}{\left.\left(\frac{{x}^{2}}{2}-\frac{{x}^{3}}{6L}\right)\right|}_{0}^{L}={\varphi }_{y}\frac{{L}^{2} }{3}=0.048 m$$

The DL drift ratio, assumed to be of 2%, results in a displacement of 0.02∙6.00 m = 0.12m, with the ductility associated to this displacement as µ = 0.12/0.048 = 2.5.

The target ductility for LS has been chosen following (Mander 1983). According to Figs. 7.10 and 7.11 of (Mander 1983), the ductility capacity goes from 3.0 to 10.0, depending on the confinement and the longitudinal steel ratio. A value of 5.0 has been adopted here for the preliminary design.

The YFS for u_y = 0.048 m (see Eq. 7) is calculated by the interpolation between u_y = 0.025 m and u_y = 0.05 m in the YDC (Fig. 12). This YFS and the two performance objectives are represented in Fig. 15. Both performance objectives were expressed in terms of target MAF and ductility: a ductility of 2.5 associated to a MAF of 0.0105 years^− 1 (which, from Fig. 15, leads to C_y = 0.51) and a ductility of 5.0 associated to a MAF of 2.11×10^− 3 years^− 1 (for which C_y = 0.49). The highest value of C_y is the design value, that in this case is governed by the first performance objective.

As the pier is subjected to a dead load of 6500 kN (see Fig. 13), the column has to be designed for a lateral seismic load of 0.51·6500 kN = 3315 kN.

The Yield Displacement Chart (YDC) approach is introduced. YDCs are based on the fact that the yield displacement remains constant with changes in stiffness, at the sectional level and at the structural level. By further replacing the spectral acceleration basis with a peak ground acceleration one, the YDC requires more compact hazard and response information, compared to the Yield Frequency Spectra (YFS). YDC is essentially a property of a given site, in this sense the engineers can use the same YDC for all the design purposes within the site. Despite the loss of some efficiency and sufficiency, this S_a(T₁)-by-PGA substitution offers increased practicality and conceptual simplicity, turning YDC into a unique tool for performance-based design.

Acknowledgments

This work is part of the HYPERION project (https://www.hyperion-project.eu/). HYPERION has received funding from the European Union’s Framework Programme for Research and Innovation (Horizon 2020) under grant agreement no. 821054. The content of this publication is the sole responsibility of the authors and does not necessarily reflect the opinion of the European Union.

The authors would like to thank Prof. Dimitrios Vamvatsikos from the National Technical University of Athens for his valuable discussion during the development of his article.

The authors declare that there is no conflict of interest.

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Reviews received at journal
12 Apr, 2022
Reviewers invited by journal
12 Apr, 2022
Editor invited by journal
13 Mar, 2022
Editor assigned by journal
11 Mar, 2022
First submitted to journal
10 Mar, 2022

You are reading this latest preprint version

Yield Displacement Charts for performance-based seismic design

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Version 1

Abstract

Figures

Introduction

The Response Of The Single Degree Of Freedom System

The Seismic Hazard

The Use Of Yield Displacement For Design