Seismic Design of Industrial Facilities || Reliability Analysis on Capacity Design Rules for Steel Frames

337 S. Klinkel et al. (Eds.), Seismic Design of Industrial Facilities, DOI 10.1007/978-3-658-02810-7_28, © Springer Fachmedien Wiesbaden 2014

Reliability Analysis on Capacity Design Rules for Steel Frames

Max Gündel1, Benno Hoffmeister1, Markus Feldmann1 1 Institute of Steel Construction, RWTH Aachen University

Mies-van-der-Rohe-Str. 1, 52074 Aachen, Germany [email protected]

ABSTRACT:

Capacity design rules are applied to ensure the intended plastic behaviour of structures subjected to earthquakes. They need to cover scattering due to the seismic action and material strength. Recent evaluations of data of structural steel from European producers show for low steel grade an overstrength, which is higher than covered by the recommended value in DIN EN 1998-1. In this paper results of reliability analysis performed on this topic are presented. For this purpose a stochastic model for the seismic action was derived, which enables to perform the investigations via push-over analysis instead of time-consuming non-linear time history analysis. The main findings are discussed in the view of adjustments of design rules in DIN EN 1998-1 and European product standards, respectively.

Keywords: steel frame, capacity design, reliability analysis, probabilistic seismic load model

1 Introduction

Steel frames resist earthquake very efficiently by means of energy dissipation due to plastic deformation. However, the design according to current seismic standards is usually carried out by elastic analysis with equivalent static loads based on the elastic response spectra and the effective mass of the building. Here, energy dissipation due to plastic deformation is considered by force reduction factors (named q in DIN EN 1998-1), where the elastic response spectra are reduced in relation to the plastic deformation behaviour and capacity of the structure. To guarantee the expected and required plastic behaviour capacity design rules are applied in the design procedure. These contain for steel frames following verifications (Figure 1):

(i) Weak storey failure has to be prevented and the desired global plastic mechanism should form. To guarantee this, the moment capacity of columns needs

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338 M. Gündel et al.

a sufficient overstrength compared to the moment capacity of the adjacent beams at each node (DIN EN 1998-1):

COFMM RbRc ≥ ( 1 )

(ii) Dissipative elements (structural members where plastic hinges are intended) need a sufficient rotation capacity to ensure the considered plastic moment capacity even under strong plastic deformations and repeated loading. For this purpose sections need to be semi-compact for medium dissipative design (1.5 < q ≤ 4) and compact for high dissipative design (4 < q ≤ 6.5·α, where α considers the global structural overstrenght).

(iii) Buckling of columns (non-dissipative elements) has to be prevented, as this failure mode provides only little deformation capacity and the failure consequences are considerable. Hence, design forces resulting from seismic actions are increased by an amplification factor considering structural and material overstrength:

EEdovGEdEd NNN ,, 1,1 ⋅Ω⋅⋅+= γ ( 2 )

(iv) Finally, failure of non-dissipative connection has to be prevented to achieve the desired global plastic deformation capacity. Therefore, joints are designed with a sufficient overstrength considering the plastic section capacity of the attached structural member and expected material overstrength:

fyovEd RR ⋅⋅≥ γ1,1 ( 3 )

Material scattering and overstrength is considered in the capacity design rules for steel structures in DIN EN 1998-1 by the material overstrength factor 1.1·γov,

Figure 1: Capacity design of steel frames acc. to DIN EN 1998-1

(iii)

M Rb

M Rc

M Rb

M Rc

b

t

(ii)

(i)

N Ed

M Ed

(iv)

Fließgelenkplastic hinge

Reliability Analysis on Capacity Design Rules for Steel Frames 339

where the recommend value of γov is 1.25. However, recent statistic evaluations of material data from European steel producers show that for steel grades with low nominal strengths even the mean value of yield strength is higher than the recommended material overstrenght factor (Figure 2 left). On the other hand, the overstrength of high strength steel grades is very small (Figure 2 right).

The influence of material strength distribution representative for current structural steel production in European on the capacity design rules for steel frames according to DIN EN 1998-1 has been investigated by seismic reliability analysis. Methods and results of this study are presented in this paper and are discussed in view of possibly necessary adjustments of production and/or seismic design standards.

2 Probabilistic seismic load model

2.1 Basics

The structural behaviour of steel frames subjected to earthquakes can be simulated by non-linear time history analysis with very high accuracy. The scattering of seismic actions is considered by repeating the calculations with a number of (scaled) recorded or artificial accelerograms which fulfil the design response spectra representative for the location of the building. However, in reliability analysis - even with advanced methods - hundreds of calculations are required to determine the failure probability. In this context non-linear dynamic calculations are disadvantageous, as they are rather time consuming. Furthermore, there are difficulties to derive direct relationships between seismic action and structural performance, as the scattering of the action is considered inherently by a number of accelerograms. Simplified and more efficient reliability methods specifically developed for seismic design (e. g. the SAC2000/FEMA procedure [2]) are

(a) S235 (b) S460

Figure 2: Distribution of yield strength of structural steel [1]

0.0

0.1

0.2

0.3

0.4

0.5

245 265 285 305 325 345 365 385 405

rela

tive

fre

que

ncy

[-]

yield strength [MPa]

1,1∙γov

0.0

0.1

0.2

0.3

0.4

0.5

470 490 510 530 550 570 590 610 630

rela

tive

fre

que

ncy

[-]

yield strength [MPa]

1,1∙γov


inadequate for this study, as they do not consider the influence of material scattering on the structural performance (demand) as well as system failure (system of limit state functions).

Figure 3: Schematic description of deterministic and probabilistic push-over analyses procedure

g (2. Fließgelenkkette)2


2. Mode

1. Mode



2. Mode

1. Mode

nichtlineare statische Berechnung

2.3.

...

(a) deterministisch (b) stochastisch

Monte-Carlo-Simulation

2.

1. 1.

max(1., 2.)

uDF

deterministic stochastic

simulation

plastic mechanism)

plastic mechanism)

push-over analysis

plastic mechanism)

plastic mechanism)


Therefore, non-linear static analyses (push-over analysis) were used in the reliability analysis instead of non-linear dynamic analyses. Push-over analyses enable to evaluate the plastic behaviour of structures with a sufficient accuracy and are much less time consuming than time history analysis. In this method horizontal forces at each storey are increased until the target displacement of the reference point (usually the top of the building) is reached, while gravity loads are kept constant. The target displacement is determined on the basis of an equivalent single degree of freedom system (SDOF) and the displacement response spectra. Higher mode effects are considered in DIN EN 1998-1 by repeating the calculation with two different distributions of the horizontal forces over the height of the building: one distribution is based on the fundamental mode shape and one is based on a constant displacement over the height of the building. Obviously, this procedure does not consider the scattering of the seismic actions is in a probabilistic way; a probabilistic model for the horizontal seismic forces is not available so far.

2.2 Stochastic model for distribution of horizontal seismic forces

The description of seismic actions consists of the probability of occurrence of earthquakes with a specific intensity as well as the variability of the acceleration time history itself. The first part is described by a hazard functions, which connects the mean value of the target displacement with the probability of occurrence of a seismic action with a specific intensity. The corresponding distribution of the horizontal forces over the height of the building is related to the fundamental modal shape and period. The variation of acceleration time history yields to scattering of the target displacement and variations of the horizontal forces due to higher modes.

The probabilistic model for horizontal seismic forces is derived based on non-linear and linear time history analyses for a number of case studies and 20 artificial accelerograms each. For this purpose, analyses with elastic material behaviour are beneficial, as there is a direct relationship between deformation and horizontal forces as well as models can be derived by the linear random processes theory. The evaluation of non-linear (with nominal material properties) and linear time history analysis in the critical time step shows that the distribution of plastic hinges is very similar (Figure 4). Non-linear material behaviour does not lead to a considerable redistribution of internal forces and different plastic mechanism. Therefore, it can be assumed that the distribution of the horizontal forces in structures with elastic and plastic material behaviour is very similar and the stochastic model can be derived based on linear time history analyses.

The evaluation of horizontal deformations of the steel frame in the critical time step for different accelerograms shows that the scattering is very small and close to the fundamental eigenmode (Figure 5 left). By subtracting the deformation of the fundamental eigenmode from the total deformation, the influence of higher modes becomes visible (Figure 5 right).


Figure 4: Distribution of plastic hinges in non-linear and linear time step analyses

Figure 5: Horizontal deformation in the critical time step (left) deformation of higher modes (right)

To separate the deformations coming from the fundamental and from higher modes, linear time-step analyses with equivalent SDOF for each relevant eigenmode are carried out. The time step of the maximum displacement of the frame is consistent with that of the SDOF for the first eigenmode (Figure 6 left, displacement of SDOFs without participation factors). The first eigenmode represents the mean value and the corresponding target displacement is the ordinate of the displacement spectra (Equ. 4). The standard deviation of the first eigenmode between different accelerograms is zero at the time step of the maximum displacement.

The amplitude of higher eigenmodes at the time step of the maximum displacement is random, if the eigenmodes can be considered as stochastically independent

≤ji TT 0.9). The standard deviation of the amplitude of higher modes can be

determined by linear random process theory, as the displacement history of SDOFs can be described by a stationary Gauss process. The mean value of higher modes is

0

2

4

6

8

10

12

14

16

1 2 3 4 5

Häu

figke

it

aktivierte Stockwerke l [-]

nichtlinearelastisch

activated storeys l

... ... ..1 2 3

0

1

2

3

4

5

0.0 0.1 0.2 0.3 0.4 0.5

stor

ey [-

]

u [m]

fundamental mode shape scaled to mean value of roof drift

0

1

2

3

4

5

-0.02 -0.01 0.00 0.01 0.02

stor

ey

[-]

u [m]

second mode shape

plastic elastic

activated storeys l [-]

frequenc


Figure 6: Displacement histories of equivalent SDOFs (left); resulting standard deviation over the height of the building (right)

zero and the standard deviation results from the ordinates of displacement spectrum divided by the peak factor (Equ. 5). The peak factor represents the ratio between maximum value and standard deviation of a random process and is about 3 for earthquakes [3].

( ) ( ) 1u φ⋅⋅Γ= 11 TSμ d ( 4 )

( ) ( ) ( )nnnu φφ ⋅⋅Γ≈⋅⋅Γ=

3nd

nn

ndn

TS

r

TSσ ( 5 )

With nφ the eigenvector, nΓ participation factor, ( )nd TS ordinate of the

displacement response spectra at eigenmode n . The horizontal forces can be determined by the condensed stiffness matrix or alternatively directly by the acceleration response spectra:

( ) ( )1 1aS Tμ = Γ ⋅ ⋅ ⋅1F M φ ( 6 )

( ) ( ) ( )3

nan

n

nan

TS

r

TSσ ⋅⋅⋅Γ≈⋅⋅⋅Γ= nnn MMF φφ ( 7 )

With ( )na TS ordinate of the acceleration response spectra at eigenmode n , M mass matrix. The total standard deviation resulting from all higher modes is determined with the SRSS-rule. A very good correlation of the standard deviation of the horizontal forces based on time-step analysis and the probabilistic model (Equ. ( 6 ) to ( 7 )) can be observed (Figure 6 right).

2.3 Stochastic model for target displacement

Besides the scattering of distribution of horizontal forces also the target displacement itself is subjected to stochastic variations. The deviations result from

-0.4

-0.2

0.0

0.2

0.4

14 15 16 17 18

disp

lace

men

t u [

m]

t [s]

1. mode3. mode 2. mode

t(umax)t(umax)

frame

0

100

200

300

400

1 2 3 4 5 6 7 8 9 10

σ(F

i) [k

N]

storey

time step analysisstochastic model


higher modes (elastic part), which can be determined by Equ. 8, and from non-linear material behaviour (plastic part). The latter is investigated by non-linear time history analysis with different accelerograms. In Figure 7 left the mean value of maximum roof displacement over intensity level (scaled to the ductility μ respectively deformation ductility dμ ) is shown. The behaviour of different structures is very similar. Plastic material behaviour reduces the maximum deformation due to energy dissipation. At the same time the variation of the maximum roof displacement increases (Figure 7 right). This can be described by following empiric linear equations:

( ) 11max, Γ⋅= TSu dD for ovΩ⋅≤ 0,1μ ( 8 )

( )μ

μ ovdD TSu

Ω⋅+⋅⋅Γ⋅= 3,07,011max, for ovov Ω⋅≤<Ω⋅ 0,80,1 μ ( 9 )

( ) 0max, =Duσ for ovΩ⋅≤ 0,1μ ( 10 )

( )

−

Ω⋅⋅= 06,006,0max,max,

ovDD uu

μσ for ovov Ω⋅≤<Ω⋅ 0,60,1 μ ( 11 )

( ) 30,0max,max, ⋅= DD uuσ for μ>Ω⋅ ov0,6 ( 12 )

Global structural overstrength ovΩ delays yielding and leads to higher mean target displacement and less scattering.

Figure 7: Mean value (left) and coefficient of variation (right) for target displacement over ductility

0

4

8

12

0 4 8 12

μ d[-

]

μ [-]

elasticplastic

5-storey

10-storey15-storey

0.0

0.1

0.2

0.3

0.4

0.5

0 4 8 12

ν(u D

,pl)

[-]

μ [-]

time step analysisstochastic model

5-storey

10-storey

15-storey


3 Reliability analysis

Based on the probabilistic model for seismic actions presented in section 2 reliability analyses on the capacity design rules for steel frames are carried via push-over analysis. For the yield strength following probabilistic model is used [1]:

( ) ,0.83 125 / ²y y nomf f N mmμ = ⋅ + ( ) ²/25 mmNf y =σ ( 13 )

Reliability analyses on the buckling of columns are not performed, as non-linear time history analyses have shown only negligible influence of variations caused by the seismic action as well as the material strength scattering. The reference case is a 5-storey-3-bay steel frame in steel grade S355 designed for a PGA of 0.25 g.

3.1 Weak storey failure

The plastic performance of steel frames is investigated by evaluating the safety index of each possible plastic mechanism by first order reliability methods (FORM) analyses. Obviously, the overdesign of columns (COF) shifts the probability of occurrence (equal to the inverse of reliability index) from storey mechanism to global mechanism (Figure 10). Therefore, the required COF is defined in such a way that the probability of storey failure has to be smaller than the probability of global mechanism. The results for different steel grades and standard deviation of seismic action (mainly influenced by the first eigenperiod) on the COFreq is shown in Figure 9. The influence of the steel grade is small, while the influence of the scattering of seismic action is dominant. The determined required COF with values up to 4 would govern the design and would lead to uneconomic structures. Hence, in the following investigations COF is kept to 1.3 (recommended value acc. to DIN EN 1998-1), even if the probability for storey mechanism increases the rotation demand in plastic hinges of beams and columns.

Figure 8: Reliability index βj of plastic mechanisms of a steel frame: COF = 1.0 (left) and COF = 1.3 (right)

12

3

45

6

7

8

9

10

11

12

13

1415

0.0

0.5

1.0

1.5

2.0

2.5

1 2 3 4 5 6 7 8 9 101112131415

β j/

β glo

bal[-

]

activated storeys l [-]1 53 42

plastic mechanism j

1

3

4

5

7

8

9

10

11

12

13

1415

0.0

0.5

1.0

1.5

2.0

2.5

1 2 3 4 5 6 7 8 9 101112131415

β j/

β glo

bal[-

]

activated storeys l [-]1 53 42

plastic mechanism j


Figure 9: Required overdesign factor COF for steel frames under seismic action: variation of yield strength (left) and of action (right)

3.2 Rotation capacity

The available rotation capacity is evaluated by mechanical models presented by Feldmann (moment-rotation curve) and Ibarra-Medina-Krawinkler (cyclic degradation). The coefficient of variation is rather constant for different sections and steel grades (COV = 0.3). Failure is defined, if the rotation capacity is smaller than the maximum rotation in on of the outer columns or both internal columns or of all beams in one of the storeys. The rotation capacity check is analysed by fragility curves. Repeating fragility analysis for different rotation capacities yields the required rotation capacity required to reach the target reliability index of 2.0 (following [2]).

Figure 11 shows that a number of design parameters related to the seismic action and the type of the structure has a strong influence on the required rotation capacity. In contrast, the influence of the steel grade is rather small. It has to be

Figure 10: Fragility function (left), det. of required rotation capacity (right)

1

2

3

4

5

6

0.00 0.10 0.20 0.30

CO

Fre

q[-

]

σ(fy) [-]

σ(F) = 0,15σ(F) = 0,29

S460 S235

1

2

3

4

5

6

0.0 0.2 0.4 0.6

CO

Fre

q[-

]

σ(F) [-]

upper boundmean values

T1= 0.5s 2.0s

T1= 0.5s 2.0s

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5

prob

abili

ty [-

]

ag [g]

hazard function

fragility function

discrete values

0.0

1.0

2.0

3.0

0 50 100 150

relia

bili

ty in

dex

β[-

]

θreq [mrad]

β0


Figure 11: Required rotation capacity for various design parameter

mentioned that in the case studies the storey drift limitation according to DIN EN 1998-1 are not meet. The correlation between this design requirement and rotation demand is poor, which is an indication for the necessity to improve design rules for rotation capacity checks.

3.3 Non-dissipative connections

Statistical evaluation of tests on various components of connections leads to the conclusion that joints with failure of bolts in tension are the most crucial, brittle connections. Hence, each connection is described by a random variable representative for this failure mode ( =μ 1.38, =σ 0.14). Failure is defined, if the strength of the connection of any beam, of one of the outer column base or both internal column bases is smaller than its connection force. The fragility curve of non-dissipative joints is considerably different than fragility curves of the rotation capacity check (Figure 13). While for rotation capacity the failure probability increases continuously with the intensity level, for non-dissipative connections a

Figure 12: Fragility function (left) and required joint overstrength (right)

0.000

0.040

0.080

0.120

0.160

S23

5S

355

S46

01

.01

.32

.03

.0 5 10 15 1 2 3

θ req

[rad

]totalcolumnsbeams

COFsteel storey bay

0.000

0.040

0.080

0.120

0.160

0.1

00

.25

0.4

01-

A1-

B1-

D 11

.5 2 2 4 6 8

θ req

[rad

]

totalcolumnsbeams

agR T1 μspectr.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

prob

abili

ty [-

]

ag [g]

hazard functiondiscrete values

1,1·γov = 1,1

0.0

0.5

1.0

1.5

2.0

2.5

3.0

1.0 1.1 1.2 1.3 1.4 1.5

relia

bilit

y in

dex

β[-

]

1,1 · γov [-]

β0

1,1 · γov,erf


Figure 13: Required joint overstrength for various design parameter

plateau of failure probability is reached, which corresponds to the plastification of the adjacent member. The consequence is that all design parameters related to the seismic action has only a small influence on the reliability level. Hence, for non-dissipative connections the steel grade is the governing parameter.

4 Conclusions

In this paper results of reliability analysis on the capacity design rules in DIN EN 1998-1 for steel frames are presented. For this purpose a stochastic model for the seismic action in push-over analysis was derived. The probabilistic model for yield strength is based on current data from European steel. The main findings are:

• The type of plastic mechanism as well as the rotation demand is mainly influenced by the variation of the seismic action and not by the material strength scattering.

• The design of non-dissipative joints is mainly governed by the material strength distribution; the influence of the seismic action is negligible.

The recommend values to consider material overstrength in the design of non-dissipative connections are 1.05 for S460, 1.20 for S355 and 1.40 for S235.

REFERENCES

[1] Gündel, M.: Zuverlässigkeitsanalysen zur Kapazitätsbemessung von Stahlrahmen, PhD-thesis, Institute for Steel structures, RWTH Aachen, 2013

[2] FEMA 350: Recommended Seismic Design Criteria for New Steel Moment-Frame Buildings. Federal Emergency Management Agency, 2000

[3] Pinto, P. et al: Seismic reliability analysis of structures. Pavia: IUSS press, 2004

1.0

1.1

1.2

1.3

1.4

1.5

1.6

S2

35

S3

55

S4

60 1.0

1.3

2.0

3.0 5

10

15 1 2 3

1,1

.γov

,erf

[-] 1,1.γov = 1,375

COFsteel storey bay

1.0

1.1

1.2

1.3

1.4

1.5

1.6

0.10

0.25

0.40 1-A

1-B

1-D 1.0

1.5

2.0 2 4 6 8

1,1

.γov

,erf

[-]

b) b)

a)a)

1,1.γov = 1,375

agR T1 μspectr.

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Seismic Design of Industrial Facilities || Reliability Analysis on Capacity Design Rules for Steel Frames