09_Nowak_A_S_i_inni__Stany_graniczne-nosnosc_czy_uzytkowalnosc.pdf

XXIV

awariebudowlane

Prof. dr inŜ. A NDRZEJ S. N OWAK , anowak2@unl.edu

Mgr inŜ. P IOTR P ACZKOWSKI , ppaczkowski2@unl.edu

University of Nebraska – Lincoln

STANY GRANICZNE: NO Ś NO ŚĆ CZY U ś YTKOWALNO ŚĆ ?

ULTIMATE OR SERVICEABILITY LIMIT STATES?

Streszczenie Nowa generacja norm projektowych jest oparta na stanach granicznych. Współczynniki obciąŜeń

i nośności są ustalane poprzez kalibracje i analizę niezawodności konstrukcji. Metody obliczania niezawodności

zostały opracowane dla stanów granicznych nośności dlatego jest potrzeba wypracowania obliczania

niezawodności dla stanów granicznych uŜytkowalności. W referacie przedstawiono propozycje kalibracji na

przykładzie stanu granicznego ugięć oraz rozwarcia rys (dekompresji).

Abstract New generation of design codes is based on limit states. Load and resistance factors are determined in

the reliability-based calibration. However, the reliability analysis procedures are available for the ultimate limit

states and, therefore, there is a need to develop practical tool for calibration of the serviceability limit states.

The paper present such a procedure and it is illustrated for SLS-deflection and SLS-decompression.

1. Introduction

The notion of limit state is fundamental in the LRFD design code. A limit state is defined

as the boundary between acceptable and unacceptable performance of the structure or its

component. However, for any structure or structural component, there can be many different

limit states. For example, the limit states for a prestressed concrete beam include the moment

carrying capacity, shear capacity, torsion capacity, and also deflection, tensile stress at the

bottom, and cracking among others.

The AASHTO code [1] is calibrated but for the strength (ultimate) limit states of moment

carrying capacity and shear capacity [2] with the limit state function in the simple form,

g = R – Q

(1)

where R = resistance (load carrying capacity) and Q = load effect (sum of dead load, live

load and dynamic load).

Both R and Q can be treated as random variables with the statistical parameters assessed

from load survey, material tests, etc. It was assumed that R and Q are uncorrelated random

variables. Furthermore, R was treated as constant in time and Q was calculated as the extreme

expected value in the economic lifetime of bridge, i.e. 75 years. The major time varying load

component is live load. The extreme 75 year live load was obtained by extrapolation of the

distribution function obtained in the truck survey representing a two week heavy traffic.

XXIVKonferencjaNaukowoTechniczna

SzczecinMiędzyzdroje,2629maja2009

Referaty problemowe

b T = 3.5 [3].

The service limit states (SLS) represent the boundary of desirable performance. When the

SLS is exceeded, the result can be a need for repair or replacement of components, repeated

exceeding of SLS can lead to deterioration and eventually collapse or failure (ULS). ULS

absolutely cannot be exceeded as it can result in a total failure and collapse. In general, SLS

can be exceeded but the frequency and magnitude have to be within limits.

Therefore, consideration of service limit states requires a different input data than ULS.

In ULS, the limit state function is defined with two variables, R and Q (see Eq. 1), where R = re-

sistance (constant in time) and Q = extreme load effect specified in the code. In case of SLS:

The definition of resistance can be very difficult.

Acceptable performance can be subjective (full life-cycle analysis is required).

Resistance and load effects can be and often are correlated.

Load is to be considered as a function of time, described by magnitude and frequency

of occurrence.

Resistance and loads can be strongly affected by quality of workmanship, operation

procedures and maintenance

Resistance can be a subject to changes in time, mostly but not only deterioration, with

difficult to predict initiation time and time-varying rate of deterioration (e.g. corrosion,

accumulation of debris, cracking)

· Resistance can depend on geographical location (climate, exposure to industrial

pollution, exposure to salt as deicing or proximity to the ocean)

An example of the difficulties in the approach to SLS can be treatment of cracking in the

design of prestressed precast concrete girders. The design codes limit occurrence of the tensile

stress at the bottom of the girder. However, even for a properly designed girder, the proba-

bility of exceeding the tensile strength of concrete is very high. Under heavy traffic, there is

50% probability that the crack will open once every few weeks. Frequent opening of the crack

can facilitate penetration of salt water (pumping action) and corrosion of prestressing strands.

In the development of the Ontario Highway Bridge Design Code (Lind and Nowak 1978),

predecessor to the AASHTO LRFD Code, it was decided that opening of the crack once every

three weeks is acceptable, with the probability of 50%, but more often than that is not accep-

table. Therefore, the limit state function was formulated as in Eq. 1, but with R = decom-

pression moment for concrete and Q = maximum three week moment due to trucks, and in the

design formula, R = mean decompression moment and Q = mean maximum three week

moment. The mean values were used because of probability of occurrence = 50%, which

corresponds to b T = 0.

2. Ultimate Limit States

Examples of the ultimate limit states include:

Moment carrying capacity of a beam

Shear capacity of a beam

Tensile strength of a cable

Torsional capacity of a beam

Overall bucking capacity of a column

134

The ultimate limit states (ULS) represent the boundary of the load carrying capacity [2] and

acceptable performance. Exceeding the strength limit state results in a collapse or failure,

an event that should not occur any time during the lifetime of the structure. Therefore, there is

a need for an adequate safety margin expressed in form of a target reliability index, b T .

For bridge girders, the target reliability is taken as,

Nowak A.S. i inni: Stany graniczne: no ś no ść czy u Ŝ ytkowalno ść ?

Local bucking of the flanges or web

Loss of stability of the structure

The reliability analysis starts with the formulation of a limit state function also known as a

performance function. In a special case, when the effect of all loads can be represented by one

variable, Q, and the resistance of the structure can also be represented by one variable, R, then

the limit state function, g(R, Q), can be expressed as in Eq. 1. In a case when random

variables R and Q are independent, the probability of failure, P , and reliability index, b, can

be defined as:

+¥

( ) ( )

(

)

∫

(2)

= F

( )

(3)

where Φ is the standard normal distribution function,

f Q

( x

)

is the probability density function

F R is the cumulative function of resistance. In general, however, the limit state

function is a function of many variables (load components, influence factors, resistance

parameters, material properties, dimensions, analysis factors, etc.). Consequently, the

computations using Eq. 2 become very complex. Therefore, the probability of failure can be

determined indirectly by calculation of the reliability index,

( x

)

, and then from Eq. 3:

( )

(4)

The available procedures vary with regard to accuracy, required input data and computing

cost. Parameters of R and Q , or even the limit state function g , can also be obtained by

Monte Carlo simulations [2].

3. Serviceability Limit States

The major service limit state problems are related to function and maintenance of the

structure. The LRFD service limit states include limits on:

live-load deflection,

cracking of reinforced-concrete components,

tensile stresses of prestressed-concrete components,

compressive stresses of prestressed-concrete components,

permanent deformations of compact steel components,

slip of slip-critical friction bolted connections,

settlement of shallow and deep foundations,

· gradual degradation of steel (corrosion)

· gradual degradation of concrete (alka-silica reaction)

among others. Some of these service limit states may relate to a specified design life; others

do not. Many are presently very deterministic, such as some owners’ wish to limit the tensile

stresses in prestressed-concrete components to ensure a crack-free component. This service

limit state could be calibrated to achieve a certain probability of a crack-free component, but

this calibration includes a service life only in the determination of the live-load the component

must resist, for example, a 75-year live load.

135

of load and

Referaty problemowe

For bridges, the serviceability can be affected by bridge bearings, joints, water drainage

and steel coating. The current practice indicates that their performance can be strongly

dependent on:

Design parameters (dimensions, material properties, connections)

Type and model (joint, bearing, drainage system, steel coating)

Location (winter/freeze-and-thaw cycles, urban/rural, industrial pollution, exposure to

salt water)

Traffic volume and magnitude

Quality of workmanship (construction, operation, maintenance)

Correlation between bearing, joint, drainage system, coating (no-joint, leaking joint) and

other parameters.

The designer has control over the first two items (design parameters and selection of the type

and model). Based on the past practice, the designer can make assumptions with regard to

location characteristics and traffic parameters. However, the prediction of the quality

of workmanship involves a considerable degree of uncertainty, and yet it has a significant

impact on the long term performance. The last item, the development of correlations, requires

a considerable data base. In the proposed study, the available information will be collected

and utilized to develop the interaction models for the considered items and parameters.

The formulation of the limit state function for SLS is much more complex than for the

ULS. In many cases, the occurrence of load exceeding the resistance is not unaccepted as long

as it does not occur frequently. Therefore, the frequency of occurrence, or return period, has to

be considered. If the limit state function can be formulated in terms of load and resistance, the

actual limit state can be the critical frequency of occurrence or critical return period.

4. Target Reliability for ULS vs. SLS

The reliability analysis can be performed for both ULS and SLS using similar procedure

and formulas. This analysis will lead to determination of the probability of exceeding the

formulated limit state. For ULS, the acceptable probability level is very small, as failures are

not tolerated during the economic life time of the structure. For SLS, the probability can be as

high as 50% for some limit states, because it is more a question of return period (or frequency

of occurrence) rather than the load exceeding the resistance.

In general, the consequences of exceeding SLS are an order or even orders of magnitude

smaller that those associated with ULS. Therefore, an acceptable probability of exceeding a

SLS is much higher than for ULS. If the target reliability index for ULS is b T = 3.5 to 4.0,

then for SLS,

136

b T = 0 to 1.0.

The load and resistance factors are determined in the calibration process, with the objective

of maintaining closeness to the target reliability index. The calibration procedure for

Serviceability Limit States (SLS) is different than for Ultimate Limit States (ULS). For ULS

the procedure was presented in the available literature, e.g. [4] or [2]. For SLS, the procedure

will be demonstrated for SLS-deflection and SLS-decompression for prestressed concrete

girders.

Nowak A.S. i inni: Stany graniczne: no ś no ść czy u Ŝ ytkowalno ść ?

5. Calibration for SLS-Deflection

For SLS, the limit state function can be formulated as shown in Eq. 1. For the SLS of

deflection, R can represent the maximum acceptable deflection, and Q can represent the

deflection caused by loads. The corresponding design formula can have a format similar to

L ( L n + I n ) ≤

R n

(5)

in terms of frequency – how many times per certain time period?

· in terms of return period – what is the mean period of time between occurrences of

such events?

In addition, the selection of R n involves a considerable degree of subjective judgment.

The actual role of the SLS of deflection is to provide adequate stiffness to the bridge as

excessive vibration can lead to gradual deterioration and limitation of ability to carry the

traffic. Therefore, the following procedure is being developed for SLS of deflection.

Step 1 Representative Components and Structures

Representative components and structures will be identified and selected to be considered

in the development of code provisions for the SLS-Deflection.

Step 2 Load Model

For each considered component and structure, values of load components will be deter-

mined, including nominal values as well as the statistical parameters for loads as random

variables. The parameters of time-varying loads will be determined for various time periods.

The analysis will be performed for various traffic parameters (ADTT, legal loads, multiple

presence, traffic patterns). The truck database will include the available sources including

recent weigh-in-motion (WIM) data. The load frequencies will serve as a basis for determi-

nation of acceptability criteria in Step 4.

Step 3 Actual Deflections

For each considered component and structure, the actual deflections will be calculated

using advanced (linear) finite element method (FEM). The calculations will be performed for

single lane loaded, two adjacent lanes loaded, and more lanes loaded if applicable (see Fig.1

to Fig.3). The results will serve as a basis for the development of probability density functions

(PDF) of deflection, representing values of deflection vs. frequency of occurrence. It is

expected that these PDF’s will be structure-specific and strongly site-specific.

The PDF type can be assumed as normal, and for each considered case, there will be two

parameters defined: the mean (or bias factor) and coefficient of variation. Bias factor is the

ratio of the mean-to-nominal value, and coefficient of variation is the ratio of standard

deviation and the mean value.

137

where L n and I n are deflections due to nominal live load and nominal dynamic load,

respectively, and R n represents the maximum allowable nominal deflection. Load and resis-

tance factors are selected so that the corresponding reliability index is close to the target value,

or that the probability of exceeding the maximum allowable deflection does not exceed

the target value. However, it is acceptable that the maximum allowable deflection be exceeded

during the life time of the bridge (75 years). The question is how often can it be exceeded?

This can be formulated two ways:

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