Division IV
ON CATEGORIZATION OF SEISMIC LOAD AS PRIMARY OR
SECONDARY FOR PIPING SYSTEMS
Pierre Labbé1,2
1
Senior Expert, EDF, Saint-Denis, France
2
Professor, Ecole Spéciale des Travaux Publics, Cachan, France
ABSTRACT
The concept of primary/secondary categorization is first recalled and generalized for its application to a non-linear oscillator subjected to seismic input. Seismic load categorization requires calculating the input level associated to the oscillator ultimate capacity. This non-linear dynamic problem is resolved by assuming that the oscillator behaves like a linear equivalent one, with an effective stiffness (or frequency) and an effective damping. However, as it is not a priori possible to predict the equivalent stiffness and damping, a wide range of possibilities is considered.
It turns out that key parameters for categorization are i) the “effective stiffness index” (varying from 0 for perfect damage behaviour to 1 for elastic-perfectly plastic) and the slope of the response spectrum in the vicinity of the natural frequency of the oscillator. A formula and an associated abacus are presented that enable to calculate the primary part of a seismically induced stress as a function of both the oscillator and input spectrum features.
The actual “effective stiffness index” of piping systems is derived from outputs of experimental research programs carried out in USA, Japan and India. It appears that even when subjected to large beyond design input motions, the observed effective frequencies of piping systems are very close to their natural frequencies. This leads to the conclusion that the seismic load should be regarded as secondary.
INTRODUCTION
Numerous experiments carried out in the past, principally in USA and Japan have revealed the robustness of piping systems under seismic loads [1] [2]. Still ongoing experiments are carried out in India [3], which substantiates this robustness. Concurrently the Kashiwasaki-Kariwa event (2007) and the Fukushima-Daïchi NPP accident (2011) reminded that NPP can be subjected to earthquakes larger than that of the design and renewed interest for evaluation of margins in the seismic design of piping systems.
In this regard categorizing the seismic load either as force-controlled (primary), or displacement-controlled (secondary) is crucial because the margins, and consequently the acceptance criteria, are significantly different. This question was addressed in the past by some authors. For instance, the inelastic response spectrum established by Newmark and Hall in 1978 for the NRC [4] can be regarded as a classification of seismic load: secondary for flexible structures and primary for stiff ones. The purpose of this paper is to bring some clarity on the subject by presenting an objective method for categorization of seismic load or seismically induced inertia stresses.
For this purpose it is worth recalling first some basic background of load/stress categorization by considering a steel whose tensile curve is characterised by its elastic capacity (εe, σe) and ultimate
capacity (εu, σu). Its ductile [hardening] capacity is µ=εu/εe [η=σu/σe]. We consider two identical simple
ties made of this steel. The first one is subjected to a force controlled load so that Fe [Fu] is the force
corresponding to the elastic [ultimate] capacity. The second one is subjected to a displacement controlled load so that De [Du] is the displacement corresponding to the elastic [ultimate] capacity. It is clear that
Fu/Fe=η and Du/De=µ. In simple words, η and µ can be regarded as the margin under force controlled load
We consider now a third identical tie equipped at one end with a mass and subjected at the other end to a seismic input motion G(t). Assuming that we are able to calculate the response of this oscillator, it is possible to identify Ge(t) and Gu(t), for which the elastic and ultimate capacity are reached in the oscillator
under seismic load. We introduce γ=Gu(t)/Ge(t), which can be regarded as the margin under seismic load
of an oscillator designed at the elastic capacity.
In this paper we present an engineering oriented approach for calculating γ as a function of the oscillator and seismic input motion features. We derive a seismic load categorization which is presented in the form of an abacus.
NON-LINEAR OSCILLATORS
This paper considers a family of oscillators, described as follows: The mass is denoted M. For a displacement that does not exceed X0, the oscillator is in the elastic regime with a stiffness k0, a natural
frequency f0 and reduced viscous damping ξ0. It is assumed in this paper that the oscillators do not exhibit
any hardening capacity: once the elastic capacity X0 is exceeded, the associated force does not exceed F0.
This plateau is limited by an ultimate displacement Xu. The ratio µu=Xu/X0 is the ductility - or ductile
capacity - of the oscillator.
In case the displacement, x, comprises between X0 and Xu, unloading results in a range of possible
responses which lie between two bounder cases of constitutive relationship:
-The “perfect damage” model: when unloaded, the oscillator’s stiffness equals k’, with k’=F0/x. The
oscillator behaves elastically with the stiffness k’ until a larger displacement is attained. In case the oscillator undergoes a series of cycles of variable amplitudes, its stiffness at the end of the series is the one corresponding to the largest cycle, denoted k*.
-The “perfect non-damage” model or “elastic perfectly plastic”, with k0 stiffness when unloaded.
In practice concrete structures trend to behave in a manner close to the first type and steel structures to the second type.
When excited by a seismic input motion, the dynamic response (relative displacement) is x(t) and its maximum absolute value is denoted X.
-If X ≤ X0 the oscillator is in elastic regime.
-If X0 ≤ X < Xu, the oscillator is in post-elastic regime. The ductile demand is µ=X/X0 (1≤µ≤µu). It is
assumed that an effective frequency f≤f0, and an effective viscous damping ξ≥ξ0 , can be determined.
Possible values of f and ξ are discussed hereunder. -If Xu,<X the oscillator is deemed collapsed.
Figure. 1: Considered range of possible effective stiffness
In post-elastic regime, the effective stiffness k is assumed to take a value between k* and k0 , where
k*=F0/X, as illustrated in Fig. 1. The associated effective frequency, f, is so that (2πf)²=k/M. In order to
cover the full range of possibilities, an “effective stiffness index” τ is introduced so that:
F0
X X
k* k k0
k = (1-τ) k* + τ k0 ; with 0 ≤ τ ≤ 1. (1)
The value of τ is expected to depend on the ductile demand µ. However in the framework of this paper we are mainly interested in the specific index, designated by τu, associated to the ultimate ductile
demand, µu. For reasons that will appear later, a structural index λ is also introduced, which combines the
ductile capacity of the oscillator and its associated effective stiffness factor, τu, as follows:
1. > µ , )
( Ln
)) 1 ( 1 ( Ln
u u
u u
µ − µ τ + =
λ (2)
Note that regardless of the µu value, τu=0 corresponds to λ=0 while τu=1 corresponds to λ=1.
Regarding damping, it is strongly related to the area of the cycle exhibited by the constitutive relationship under cyclic loading conditions. For the family of oscillators considered in this paper, the cycle area is nil for τ=0 while it is maximum for τ=1. We therefore adopt of a pragmatic approach taking into account that
-ξ should be an increasing function of τ and µ, and -ξ should equals ξ0 when either τ=0 or µ=1.
Meeting these criteria, the following mathematical formula is adopted for the sake of clarity and simplicity of the final outputs presented later (β values are discussed further):
ξ=ξ0 (1+τ (µ -1))β. (3)
In particular, the damping value of an oscillator that reaches the ultimate displacement (µ=µu) reads:
ξu=ξ0 (1+τu(µu-1))β. (3’)
In case the reader has at her or his disposal other documented damping values or is willing to test other options, replacing the above formula with their own will not impede the following development.
In the concept of oscillator, the couple {displacement-force} can be replaced by other variables such as {curvature-bending moment} or {strain-stress} that make sense for the case under consideration. X0, F0
and Xu should then be replaced by the relevant values of yield and ultimate capacity.
SEISMIC INPUT MOTION
0 1 10
1 10 100
frequency
Example of non dimensional response spectrum
Figure 2 Example of evaluation of p value
S(f,ξ) = Γ s(f,ξ), (4)
For frequencies in the vicinity of f0, typically between f0 and f0/2, it is assumed that s(f,ξ) reads:
s(f,ξ) = s(f0,ξ0) (f/f0)p / (ξ/ξ0)q (5)
In a conventional log-log scheme, and for a given damping value, it means that the response spectrum is regarded as a straight line with a slope p. For example, a non-dimensional response spectrum is presented in Fig. 2, for which a slope p=0.54 is associated to f0=6 Hz. Design spectra are generally
smooth curves, allowing an easy evaluation of the p value.
Dependence on damping is controlled by the factor q. The random vibration theory leads to adopt q = 0.5, which is retained in the present paper for numerical applications. Some authors consider effects of damping on spectra that correspond to slightly different values [5].
It is also possible to disregard any additional damping effect by selecting q=0. However, such an assumption leads to considerably underestimate margins of those oscillators with a significant effective stiffness factor.
MARGIN AND LOAD/STRESS CATEGORIZATION FORMULA
We establish now formulas for calculating γ. A first generic formula (9) is derived from (6) and (7). The selected form of the non-dimensional spectrum turns it into (10). Then fu/f0, it is calculated as
(ku/k0)1/2, where ku/k0 can be expressed as ku/k0=(1-τu)/µu+τu, which results in (11). Eventually, deriving
ξu/ξ0 from (3) by setting µ =µu and τ=τu, leads to (12):
γ=µu s(f0,ξ0)/s(fu,ξu) (fu/f0)². (9)
γ=µu (fu/f0)2-p (ξu/ξ0)q. (10)
γ=µup/2 (1 + τu (µu −1))1-p/2 (ξu/ξ0)q. (11)
γ = µup/2 (1 + τu(µu−1))1-p/2+βq. (12)
More convenient than the γ margin for the purpose of stress categorization, a margin index m is now introduced. This index is so that m=0 corresponds to γ=1, (for m=0 the seismic input should be regarded as a primary load), while m=1 corresponds to γ=µu, (for m=1 the seismic input should be regarded as a
secondary load). This m index is defined by setting: γ = µu
m
(16)
Introducing this definition into the above formulas (11) and using the logarithm leads to:
m= (1-λ) p/2 + λ + q Ln(ξu/ξ0)/Ln(µu). (17)
In case the reader has a formula providing ξu as a function of ξ0 and/or µu and/or τu, she or he may
introduce it into (17) to derive the corresponding m value. Another option is to identify β so that the formula (3’) holds and the formula (17) reads more simply as:
m= (1-λ) p/2 + λ (1+βq). (18)
Formula (18), which calculates m as a function of the structural features (λ and β) and seismic input features (p and q), establishes the aimed load/stress categorization:
-If the calculated m value equals 0, the seismic load should be regarded as a typical force-controlled load (primary load in mechanical engineering terminology). If m is smaller than 0, the seismic load is even more damaging than anticipated by elastic analysis for a force-controlled load.
-If the calculated m value lies between 0 and 1, it should be considered that a seismically-induced stress σ has a primary part σP so that σP = σ/µu
m .
In application of formula (18), we present below an abacus that provides m as a function of p and λ for the entire range of possible λ values (0≤λ≤1), a large range of realistic p values (−3≤p≤3) and a
realistic value of βq. This abacus shows a visual categorization of seismic loads or seismically-induced stresses.
PIPING SYSTEMS EXPERIMENTAL OUTPUTS
Extensive experimental research programs were carried out in the USA in the nineties under the leadership of the Electric Power Research Institute [1] and in Japan during the first decade of this century under the leadership of the Nuclear Power Engineering Corporation (NUPEC) [6]. Both programs incorporated tests of components and of piping systems and concluded that the nuclear industry’s design practice for piping systems incorporates large margins.
Tests carried out in USA
Regarding the EPRI pressurized piping components, every specimen presented a clear SDOF response and, interestingly, experimental outputs of some specimens were processed so as to identify their effective frequency and effective damping [1]. These documented outputs are summarized in Fig.3.
-Fig. 3-a: Component for which damping and peak cyclic strain are concurrently available. -Fig. 3-b: Component for which damping and frequency shift are concurrently available. In a first step, these results are processed as follows:
-Fig. 3-a: Considering that the plastic yield is classically associated to a 0.2% strain, recorded strain values are divided by 0.2 to be converted into ductile demand (µ values). Then an empirical linear relationship is established between µ and ξ, which reads ξ(%)=2.5µ.
-Fig 3-b: The frequency shifts are converted into k/k0 by k/k0 = (f/f0)² and the damping values are
converted into µ values through the above empirical formula.
0 5 10 15 20 25
0 0,5 1 1,5
D
a
m
p
in
g
%
1/2 peak-peak strain %
Elb 3
Red 15
Elb 35
Tee 38
0 5 10 15 20 25
0 5 10 15 20 25 30 35
D
a
m
p
in
g
%
Frequency shift (abs. values) %
Tee 14 Red 15 Elb 35
Tee 38
Reformulating formula (1) as 1+τ(µ -1)=µ k/k0, and using formula (2) it is now possible to calculate
the λ value of every mark in Fig. 3-b and to plot {λ, µ} couples (Fig. 4). It appears that for these EPRI piping components, an empirical expression of λ could be:
λ= 1- 0.044(µ -1) (19)
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
0 1 2 3 4 5 6 7 8 9 10
λλλλ
µ
Observations
Formula 19
0 1 2 3 4 5
0 1 2 3 4 5 6 7
ξ/ξ ξ/ξξ/ξ ξ/ξ0
1+ττττ(µ-1)
Observations
Formula 3 (β=0.8)
Figure 4: λ values versus ductile demand for EPRI piping components
Figure 5: Calibration of formula (3) against EPRI experimental data
Similarly it is possible to derive ξ and to plot ξ/ξ0 versus 1+τ(µ-1), as presented in the Fig. 5, where
the conventional 5 % value is adopted for ξ0. A conclusion is that the proposed formula (3) with β = 0.8
fits reasonably the EPRI experimental data.
Tests carried out in Japan
Outputs of the NUPEC/JNES research program were processed and presented by DeGrassi & al. [2]. The large scale piping system under consideration consisted of a 200 mm (8 inch) nominal diameter schedule 40 carbon steel pipe. It included straight pipe, nine elbows, a tee, a 1000 kg added mass representing a valve, and was pressurized to a design stress intensity of Sm. According to the Japanese
Code, primary stress is limited to 3Sm. Two series of tests were performed, on the large Tadotsu shaking
table: Design Method confirmation (DM) test and Ultimate Strength (US) test.
In the DM test, the piping system had a first natural frequency of 6.3 Hz with a damping of 2.1%. A series of seismic input motions were applied so that the corresponding primary stresses were respectively calculated at 4.5 Sm, 6 Sm, 10.5 Sm, and 13.5 Sm. Processing outputs of this last run resulted in a 5.9 Hz
effective frequency and a 3.4% effective damping.
For the US test, the piping system was modified by addition of another 1000 kg mass and removal of a lateral support. The natural frequency dropped to 3.8 Hz and the damping to 0.9%. The seismic input motion was calibrated so that the corresponding primary stress was calculated at 24 Sm. Under these
conditions the observed effective frequency was estimated at 3.6 Hz and the damping at 4.5%. (The test was repeated until failure, which occurred during the fifth run in a fatigue-ratcheting mode).
during these tests, the frequency shift is very limited. This is likely due to the fact that the plastic yielding is limited to elbows, the vast majority of the system remaining in elastic regime. A consequence of the very small frequency shift is that the τ value of this system is close to 1 as well as its λ value.
A tentative interpretation of the US test could be: k/k0 can be estimated from the frequency shift by
k/k0=(3.6/3.8)²=0.90. The test being designed so that the primary stress is 24 Sm, we may assume that the
ductile capacity of the system is at least 16. These concurrent values of k/k0 and µu result in τu = 0.89 and
λ = 0.96. Regarding damping, the experimental output is such that ξu/ξ0 = 4.5/0.9=5. Eventually, applying
the above formula (17) drives to the conclusion that m is not sensitive to p, varying from m=1.2 for p=-3 to m=1.3 for p=3. This result is consistent with the formulas (14) and (15), which state that, for τu=1, γ is
at least equal to µu and does not depend on p. As a conclusion, it seems that for such systems, the
response of which is practically elastic-perfectly plastic, the seismic input motion should be regarded as a secondary load regardless of the p value, as long as a realistic ξ0 value is selected.
Tests carried out in India
An important research program, still in progress, has been carried out in India, funded by Bhabha Atomic Research Centre, Mumbai, implemented by the Central Power Research Institute (CPRI), in Bangalore [3]. It encompasses simple component tests (pressurized straight pipes and elbows in cyclic pure bending) as well as different pressurized piping configurations tested on shaking table. It includes two sets of stainless steel 304L and two sets of carbon steel systems with the same layout, tested between Feb. 2012 and Oct. 2014. A difference with the previous tests is that the failure mode (fatigue-ratcheting) was known in advance. Consequently the test designers decided to focus on investigating this failure mode instead of trying to identify the available seismic margin for a single seismic event.
Table 1. BARC experimentation: Frequency shift under beyond design seismic input
Specimen SS 1 SS 2 CS 1 CS 2
f(2.5 g)/f0 0.96 0.94 0.99 0.97
The systems under consideration [3] consisted of 6’’ schedule 40 pipes (168 mm outer diameter). They included straight pipe, six long radius elbows, a tee, and two 250 kg masses. They were pressurized at 12 MPa so as to induce a 1 Sm (140 MPa for SS304L) hoop stress. The seismic input consisted of 5
runs at 1 g ZPA, corresponding to a maximum stress 1.72 Sm in the piping system. Then the ZPA was
increased by steps of 0.25 g, with 5 runs at each step, up to 2.5 g (ultimate capacity of the shaking table). Eventually the internal pressure was increased at 14 MPa for 30 runs at 2.5 g and increased again at 16 MPa for seven runs. At the end of the tests the seismic input motions had lasted for more than 40 minutes.
In the most affected elbow, the hoop gauge was lost during the runs at 2.0 g. At that moment the recorded cyclic hoop strain was ± 0.35% while the accumulated hoop strain was close to 2%. [7]
The natural frequency of the system in elastic regime, f0, and its effective frequency in plastic
regime, f, were identified by the wavelet method, in particular for the highest input level (2.5 g). The four ratios f(2.5 g)/f0 obtained for the four specimen are presented in the Table 1.
These experimental results confirm those obtained by NUPEC: Even when experiencing plastic strains induced by beyond design input motions, the frequency of piping systems is only slightly affected. In the present case, assuming an average value 0.96 for f(2.5 g)/f0 and assuming that under this input the
cyclic strain would be around 0.4% (µ=2), we obtain τ=0.84 and λ=0.88. According to our abacus, it means that in this case the seismic input motion should again be regarded as a secondary load regardless the slope of the input motion.
Effective frequency and effective damping versus ductile demand
In this section, we discuss the observed effective stiffness and effective damping of piping systems in plastic regime. It should be first mentioned that, according to experimental evidences presented by Raganath [1] the EPRI components actually exhibit typical elasto-plastic behaviour with elastic discharge and hysteretic loops. Component steel has a very hardening capacity (less than 1%) but the components (e.g. an elbow) have a hardening stiffness that varies between 15% and 30% of the elastic stiffness. A 20% hardening modulus can therefore be regarded as representative of the EPRI components.
Regarding these components, it has been established above that an empirical relationship could be established, which reads ξ(%)=2.5µ, between the ductile demand and the effective damping. It means that a ductile demand of 8 is corresponding to an effective damping of 20%. This is apparently consistent with the Chopra and Goel estimate [8] for a SDOF system with 20% hardening modulus. However it means also for instance that a 7.5% damping is associated to µ=3, which is approximately one third of the value calculated by Chopra and Goel. The approach developed by Labbé [9] on the basis of the random vibration theory leads to a damping value of 9.5% for µ=3 and 19,5% for µ=6, which fits better with the experimental outputs.
Considering now effective frequency versus ductile demand, combination of experimental outputs presented in Figures 3-a and 3-b result in Figure 6. It is clear on the figure that for small ductile demands the effective frequency remains very close to the natural one. This experimental output is reasonably predicted by [9], while on the opposite it is overestimated by [8], which predicts that small ductile demands should correspond to a significant drop of the effective eigenfrequency. Although it presents a better prediction it seems that the approach presented in [9] still trends to overestimate the frequency shift.
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
0 2 4 6 8
f/f
0
Ductile demand µ Tee 14
Red 15
Elb 35
Tee 38
Figure 6 : Effective frequency / natural frequency ratio versus ductile demand
Figure 7: Margin index m and load/stress categorization of a seismic load, plotted for βq=0.5
CATEGORIZATION ABACUS
Derived from formula (18) an abacus (Figure 6) was established to calculate m as a function of λ and p for βq=0.5 (β=1 and q=0.5) and make visible the corresponding load/stress categorization:
-The thick dotted curve (m=0) corresponds to γ=1, meaning that for every {λ, p} couple on this curve the seismic load should be regarded as a force-controlled load (primary load). For cases below this dotted curve, the seismic load is more damaging than expected in case of a force-controlled load.
Displacement controlled
Force controlled [8] α=0.2
-The thick solid curve (m=1) corresponds to γ=µu , implying that for every {λ, p} couple on this curve,
the seismic load should be regarded as a displacement-controlled (secondary) load. For cases above this solid curve, the seismic load is less damaging than expected for a secondary load.
-For a {λ, p} couple lying between these two curves, it should be considered that a seismically-induced stress σ has a primary part σP so that σP = σ/µum . Interpolation between the two curves would be
easy, taking into account that m is linear in λ and p. The thin curve in the picture corresponds to m=0.5. For example, if the ductile capacity of an oscillator is µu=9, then for a {λ, p} couple on this
line, the primary part of a seismically induced stress is σP=σ/3.
A detailed form of the abacus is presented in Annex A, covering a wide range of realistic βq values and enabling to interpolate between m values. It appears that a significant change of βq has a relative effect on the categorization; it does not change the global pattern.
CONCLUSIONS
It appears that a seismic load or a seismically-induced stress can be of any type from force-controlled, or primary, to displacement-force-controlled, or secondary. Features of the seismic input motion and features of the system or component under consideration should be concurrently taken into account when making a decision on categorization of seismic loads or seismically-induced stresses.
A margin index m has been introduced that enables this load/stress categorization. Formulas have been presented for the calculation of this index, which is mainly controlled by:
-The slope, p, of the input spectrum in the lower frequency vicinity of the dominant natural frequency of the system or component under consideration, and
-The structural index, λ, which encompasses the ductile capacity, µu, of the system or component and its
associated effective stiffness factor, τu.
Once m has been calculated, the load/stress categorization is derived as presented above. The most interesting case, is when the m value lies between 0 and 1. It should then be considered that an inertial seismically-induced stress σ has a primary part σP so that σP =σ/µum .
Mechanical engineers are very familiar with the concept of primary and secondary stresses. However they are reluctant to consider that a seismically-induced stress could be not fully primary, likely because they have doubt about how to calculate its primary part. As it provides a rational way for such a calculation, we hope that the procedure presented in this paper will facilitate establishing a practice consistent with the physical phenomena that control the seismic response of piping systems.
In practice, there are experimental evidences that piping systems i) exhibit large ductile capacity and ii) exhibit effective frequencies that are not sensitive to the ductile demand. This two features result in very large λ values, close to 1. A consequence is that the seismic load should be regarded as secondary for practically any type of in-structure response spectrum.
It should be mentioned here that, because of their λ values that are rather low, such a generic conclusion cannot be derived for concrete structures [10].
Some improvements of the proposed method are still possible. An approach including some hardening capacity of the considered family of oscillator would be useful and would enable better interpretation of some experiments carried out on stainless steel components.
REFERENCES
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[4] Newmark N.M., Hall W.J., 1978. Development of Criteria for Seismic Review of Selected Nuclear Power Plants, NUREG/CR-0098, US-Nuclear Regulatory Commission, Washington DC.
[5] Shibata A. Sozen M., 1976. Substitute Structure Method for Seismic Design in R/C. Journal of the Structural Division, ASCE, 102(ST1): 1-18.
[6] Suzuki Kenichi., Namita Y., Abe H., Ichihashi I., Suzuki Kohei, Ishiwata M., Fujiwaka T., Yokota H., 2002. Seismic Proving Test of Ultimate Piping Strength, 10th International Conference on Nuclear Engineering (ICONE10), pp. 573-580, Arlington, Virginia, USA.
[7] A. Ravikiran, P. N. Dubey, M. K. Agrawal, G. R. Reddy, 2015, Experimental and numerical studies of ratcheting in pressurized stainless steel piping systems under seismic loads. BARC Report 2015/E/018;
[8] A.K. Chopra, R.K. Goel, 2001, Direct Displacement-Based Design: Use of Inelastic vs. Elastic Design Spectra. Earthquake Spectrra 17(1) 47-64.
[9] P. Labbé, 1994, Ductility demand and design of piping systems, 10th European Conf. Earthquake Engineering, Vienna.
[10] P. Labbé, 2013. Categorization of seismic loads for civil engineering and mechanical engineering. 22nd SMiRT Conference, Division IV, San Francisco, California, USA.