The Use of Probabilistic Methods in Assessing the ...

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ScienceDirect Procedia Engineering 193 (2017) 160 – 167

International Conference on Analytical Models and New Concepts in Concrete and Masonry Structures AMCM’2017

The use of probabilistic methods in assessing the reliability of masonry structures Izabela Skrzypczaka, Joanna Kujdaa ,Lidia Buda-OĪóga*, a

Rzeszow University of Technology, 2 PoznaĔska, Rzeszów 35-084, Poland

Abstract Most of the construction works carried out on masonry objects are associated with the strengthening of existing elements or assessment of their load capacity to additional load resulting from the change of use. In this situation, the proposed design solutions should always be part of a comprehensive design assessment taking into account the relevant inputs and including information on the actual material properties and the effects of interactions. The use of probabilistic design to evaluate masonry structures enables more economical approach to the issue of possible repairs and reinforcement than the commonly used method of partial factors. In the paper the normalized procedure of semi-probabilistic calculation of load capacity of masonry units and full probabilistic design, recommended in Eurocode 0 [1] are presented and compared. Because of the natural variability of the masonry structures, the information on its real mechanical properties should be obtained from in situ studies. Due to the lack of sufficient number of tests carried out on the masonry structures as well as unknown variation coefficients and distribution parameters, the analyses conducted in the article are limited to basic variables adopted for the probabilistic model based on experimental results described in the literature. The influence of each variable on the reliability index by providing sensitivity coefficients was determined. The basic element of masonry structures such as the lintel masonry was the subject of the analysis. Keywords: masonry structures, full probabilistic and semi-probabilistic design, reliability index

Published by Elsevier Ltd.Published This is anby open access article © 2017 The Authors. Elsevier Ltd. under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the International Conference on Analytical Models and New Peer-review under responsibility of the scientific committee of the International Conference on Analytical Models and New Concepts in inMasonry ConcreteStructures and Masonry Structures. Concepts Concrete and

* Corresponding author. Tel.: +48-17-7432402; fax: +48-17-854 29 74. E-mail address: [email protected]

1877-7058 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the International Conference on Analytical Models and New Concepts in Concrete and Masonry Structures

doi:10.1016/j.proeng.2017.06.199

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1. Introduction The basis in structural design lies in sufficiently safe and reliability structures’ design. While the safety commonly refers to the absence of hazards, the reliability is a quantifiable value and can be determined by probabilistic methods. In current structural design codes, the demands of safety are incorporated through the use of partial safety factors which can be derived from probabilistic analysis. Unlike other materials in construction, the reliability of masonry members has not been subjected to extensive research in the past. The recent research [2,3] showed the necessity for a probabilistic approach to masonry structures. However, most studies are focused on masonry subjected to axial stress and flexure. Since masonry shear walls exhibit a much more complex load-carrying behavior and they are even more important to structural integrity [4,5], this paper focuses on a masonry beam at shear. The structural reliability of the lintel masonry were analyzed by assessing different analytical models and probabilistic modeling. A lintel is a structural member placed over an opening in a wall. In the case of a brick masonry wall, lintels may consist of reinforced brick masonry, brick masonry arches, precast concrete or structural steel members. Regardless of the material chosen for the lintel, its prime function is to support the loads above the opening and so it has to be designed properly. Several cracks which appear in masonry walls over openings are caused by excessive deflection of the lintels resulting from improper or inadequate design. In the paper, the analysis of the provided level of reliability was conducted. Analytical models for the prediction of the bearing capacity of the lintel masonry structures were analyzed. A complete stochastic model was developed and the reliability of the lintel masonry was determined. The difference between the theoretical level of reliability and the real level of reliability was evaluated taking into account the realistic utilization of the lintel masonry. Subsequently, the derived level of reliability was presented and assessed. 2. Reliability Reliability is the most important requirement for structures. It concerns every aspect of a structure so structures have to be reliable according to load bearing capacity as well as serviceability. In design, every parameter is uncertain to some extent. The uncertainty may be in the strength of materials as well as in dimensions and quality of workmanship. All parameters, further referred to as basic variables, influence the properties of a member. Reliability is linked to the probability that a member will exceed a certain limit state. It can be defined by limit state functions which can be described by the formula (1). Z = NR - NE

(1)

The failure probability can be computed by probabilistic methods such as FORM (First Order Reliability Method), SORM (Second Order Reliability Method) or Monte Carlo-simulation [2,6-9]. For the description of the resistance, appropriate models are required that describe the load carrying behavior realistically. A model that underestimates the load carrying behavior is not appropriate for a probabilistic analysis. To find a measure for reliability that can be defined independently from the type of distribution of the basic variables, the reliability index ȕ according to Cornell [6,10] has proven useful. The major advantage rely on that only the mean, mz, and standard deviation, ı z, of the basic variables need to be known. The target reliabilities depending on the failure consequences are given by the Joint Committee on Structural Safety (JCSS) 0 and in ISO 2394 [12], and they are considered to be sufficient for most cases and so they were taken as reference for further calculations. Another recommendation is given by Eurocode 1990. where the value of the reliability index for a 50 year observation period is ȕlim = 3.8. In this concept, the partial safety factors for different basic variables allow to find different scatter of the variables. A typical application of partial safety factors is presented by the semi- probabilistic method in Eurocodes. The better knowledge of the existing infrastructure is, the more precise evaluation of its real performances can be realized. In addition, the management of the infrastructure may be refined all the more, with the aim of getting closer to the best balance between the control of the structure’s performance (safety, reliability, durability, quality) and the constraints (costs, risk) necessary for receiving this control level. A structure is designed and built in order

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to reach a level of performance. However, the environment as well as operating conditions and materials properties can evolve over time. Then, it is necessary to assess the reliability of a structure by considering the current state (ageing and/or more important loads) on one hand and the required performances of the existing structure on the other hand. Checking the infrastructure safety, during design or assessment phases, consists of making sure that the resistance (NR) of a structural element is higher than the action (NE) to which the element is subjected with a prescribed safety margin. By considering the problem from the opposite way, a question arises if the probability of NR < NE (or NR NE < 0 ) is low enough to be considered as acceptable. This probability represents the probability of failure of the structure. The input variables of resistance and stresses can be considered as random variables and described using statistical distributions. Then, the probability of failure of the structure can be calculated by probabilistic calculations based on the statistical distributions of input data, and therefore allows estimating the safety level of the structure. In practice, it is still quite rare to verify the safety of a structure with probabilistic calculations. In most design codes, for example in Eurocodes, the verification of the safety of a structure is carried out by deterministic calculations, which take into account input data uncertainties and modelling approximations by using partial safety factors [13]. The level of uncertainties consideration can vary from deterministic approaches to probabilistic approaches. The input data are: • Standard or design values; The values of input data are taken identical to the values which have been used for design, or they are determined as standard values from guidelines or from design codes. • Estimated values; The values of input data can be updated from feedback or estimations which take into account available information about the structure, for example a feedback about compressive strength of concrete after a given lifetime and given environmental conditions. The simulations which allow to estimate the residual section of steel reinforcements can also be used. • Measured in situ values; The values of input data can be measured on the existing structure, for example geometrical measures on the structure. Experimental measurements, both destructive or non-destructive, for materials can be applied. • Live loads can also be measured in situ [14]. There is a multitude of different levels, which provide a large number of possible methods of an assessment of construction [1,6,7,9,14]: • Level 0: The calculation is purely regulatory, by using standard or design input data. It can be necessary to adjust design input data, in case of a change of code between design and current assessment. The calculation is purely regulatory and allows to take into account operating modifications (change of nature or value of loads) or regulation modifications (new standards). • Level 1: The calculation is regulatory and uses updated input data which take into account the real state of existing structures (structural abnormality, ageing). The calculation is regulatory and allows to take into account structural problems such as designed such as built. The input data (loads, section of steel reinforcement, compressive strength of concrete) can be updated from: inspections, in-situ measures and testing on the existing structure, estimations and numerical simulations with data on the existing structure. • Level 2: The calculation is probabilistic (out of classical regulations) and is based on the knowledge of statistical characteristics of a few input data. If the classical regulatory calculations conclude about a nonconformity of the structure and it is possible to use more precise methods which are out of classical regulations allowing to assess the importance of the difference between the current structure. • Level 3: Calculations are carried out through a modelling for which input data could be considered as probabilistic. The choice of the assessment method can be made according to the context and the expected objective. According to the results of the first assessment levels, it is possible to refine the calculations by moving to higher levels. The consideration of a maximum of data about the properties of the actual existing structure can allow in certain cases the structure to demonstrate sufficient reliability, in spite of ageing or operating change. The interest of

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the various levels of assessment is to be able to take into account the various levels of information on the existing structure, and so to be able to get closer to the real reliability of the structure. 3. Case study Reliability index of the masonry member of structure such as the lintel was calculated according to [6,15,16]. The specified the limit state function, parameters and the type of distribution of the variables should be established for the analyzed member. The reliability index was obtained on the basis of FORM, SORM and Monte Carlo simulation method for the analyzed limit state function.

Fig. 1. Geometry of lintel

Lintel with full brick with ݂௕ = 10 N/mm2 for the cement - lime mortar class M5 with a width 24 cm. The load from the roof of a span 6.0 m was assumed. Table 1. Material properties (according to [17]) Material

Full brick wall of cement- lime mortar

Reinforcing steel

Basic variables

Value

Units

Ȟm

19%

-

ˆ୩

3.66

N/mm2

ˆ୫

5.32

N/mm2

ˆୢ

0.08

N/mm2

Ȟvm

19%

-

ˆ୴୩

0.20

N/mm2

ˆ୴୫

0.29

N/mm2

ˆ୴ୢ

1.46

N/mm2

Ȟy

8%

-

ˆ୷୩

500

N/mm2

ˆ୷୫

575

N/mm2

ˆ୷ୢ

420

N/mm2

(characteristic values - fk, fvk, fyk , mean values – fm, fvm, fym, coefficient of variation – Ȟm, Ȟvm, Ȟy,)

Load: Gk = 13.8 kN/m, ȞG = 10%, G = 11.85kN / m , Sk = 3 kN/m (snow), Ȟs = 40%, S = 1.81kN / m ഥ ത, coefficient of variation - ȞG, ȞS) (characteristic values - Gk, Sk, mean values - ǡ

Geometrical dimensions: height of lintel h= 300 mm (24 cm brick + 6 cm), d = h – a1 = 300 – 30 = 270 mm, width of lintel t = 240 mm, coefficient of variation - Ȟ = 5%, span of lintel l = 1040 mm - constant values, area of reinforcement As = 235.5 mm2 - constant value

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3.1. Semi-probabilistic method calculations The value from accepted combination of loads: q = 18.63 kN/m Calculation span: leff = 1.04 m Internal forces: MEd = 2.52 kNm; VEd = 9.69 kN •

Checking the flexural capacity of a lintel: MEd

≤ MRd

Capacity according to concrete in compression: ‫ܯ‬ோௗ ൌ ͲǤͶ ή ݂ௗ ή ܾ ή ݀ ଶ = ͲǤͶ ή ͳǤͶ͸ ή ͲǤʹͶ ή ͲǤʹ͹ଶ ൌ ͳͲǤʹʹ݇ܰ݉ ൐ ‫ܯ‬ாௗ  ൌ ʹǤͷʹ݇ܰ݉- Condition fulfilled •

Checking the shear capacity of lintel: ܸாௗ Pfd = 7.23E-05.

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The analysis performed using the probabilistic method has confirmed that it is not the condition of reliability.

Fig. 4. Density functions for the effects of capacity (NR), the effects of interactions (NE) and the condition of the boundary condition (Z = NR NE) due to the shear excluding the favourable impact brick forces due to the use of longitudinal reinforcement.

4. Conclusions The main task in the event of design or strengthen the structures of masonry is to determine the reasonable value of the likelihood of achievements of the limit state function by the construction. The calculations provided by probabilistic methods have shown that the proposed lintel masonry under bending satisfies the requirements of reliability class RC2 of the ultimate limit state due to the strength of the masonry for the compression zone, the strength of the steel reinforcement in the tensile zone and the strength of the shear assuming that strength of the wall shear is constant. In the case of calculations, when the strength of the shear wall is a LN random variable, the received reliability index value for the condition of the shear strength of the wall was lower than the target level of reliability ȕlim = 3.8. For a quantitative assessment of the reliability of the constructional elements are decisive parameters of a random variable, and adopted distribution type, which is confirmed by the results of calculations carried out for the brick beam under bending (Table. 3). Table 3. Summary values obtained indicator of reliability for the calculation of the ultimate limit state brick beam bending (reinforced masonry lintels) with probabilistic methods Calculation method

FORM

Monte Carlo

Index of reliability (ȕ)

the strength of the compression

4.055

9.416

the strength of steel reinforcement in the tension zone

7.03

16.611

2.67*

3.158*

Shear strength

Limit State Function

without considering the brick strength, shear strength of masonry is a random variable with considering the brick strength, shear strength of masonry is a constant variable

6.72

9.364

with considering the brick strength, shear strength of masonry is a random variable

2.75*

3.173*

The values - z*, values that not fulfill the requirements target reliability index for Class RC2 (ȕdop = 3.8)

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Shown in Table 3 the differences result from non-linear and heterogeneous forms of reliabilities. Another reason is the accepted basic variables that have different probability distributions. The calculated results for level 3 overlap the results for level 2 in two cases. Firstly, when random variables are those of normal distribution and limit state function is linear. Secondly, when random variables are logarithmic-normal and limit state function is expressed as a monomial. References [1] EN 1990:2002 Eurocode – Basis of structural design. 2002 CEN 15 EN 1996-1-1:2004 Eurocode 6: Design of masonry structures - Part 1-1: General rules for reinforced and unreinforced masonry structures. [2] S. Glowienka, Zuverlässigkeit von Mauerwerkswänden aus großformatigen Steinen”, Doctoral Thesis, Technische Universität Darmstadt, Darmstadt, Germany, 2007. [3] L. Schueremans, and D. Van Gemert, Evaluating the reliability of structural masonry elements using the response surface technique, Durability of Building Materials and Components 8. (1999) Edited by M.A. Lacasse and D.J. Vanier. Institute for Research in Construction, Ottawa ON, K1A 0R6, Canada, pp. 1330-1342. [4] U. Andreaus, A Failure Criteria for Masonry Panels under in plane loading, Journal of Structural Engineering, 122(1) (1996) 37-46. [5] H. Derakhshan, M. Griffith, J. Ingham , „Out-of-plane behavior of one-way unreinforced masonry walls”, ASCE Journal of Engineering Mechanics. 139(4) (2013) 409-417. [6] A. Biegus, Probabilistyczna analiza konstrukcji stalowych, Wydawnictwo Naukowe PWN, Warszawa – Wrocáaw, 1999. [7] à. Drobiec, R. JasiĔski, A Piekarczyk „ Konstrukcje murowe wedáug Eurokodu 6 i norm powiązanych” tom 1 , PWN, Warszawa, 2013. [8] R. Rackwitz, Zuverlässigkeit und Lasten im konstruktiven Ingenieurbau, Technical University of Munich, Munich, Germany, 2004. [9] Sz. WoliĔski, K. Wróbel, NiezawodnoĞü konstrukcji budowlanych, Oficyna Wydawnicza Politechniki Rzeszowskiej, Rzeszów, 2000. [10] C.A. Cornell, A reliability-based structural code, ACI Journal. 66(12) (1969). [11] Joint Committee on Structural Safety (JCSS), Probabilistic Assessment of Existing Structures, RILEM publications, S.A.R.L., 2001 [12] ISO 2394:2015 General principles on reliability for structures. 2015 ISO. [13] V. Tur , V. Nadolski , Target values of reliability indices within the concept of reliability adopted by the European norms (codes), Republican Unitary Scientific-Research, Enterprise for Construction, “Institute BelNIIS”, http://belniis.by/en. [14] R. Goy, G. Thillard, H. Yanez-Godoy, Contribution of knowledge of real properties of existing civil engineering structures in refining the structures reliability assessment, 9-th IPW, eds H. Budelmann, A. Holst, D. Proske, Branscheig , Germany, 2011, pp. 127-138. [15] EN 1996-1-1:2004 Eurocode 6: Design of masonry structures - Part 1-1: General rules for reinforced and unreinforced masonry structures. [16] J. Hoáa, P. Pietraszek, K. Schabowicz, Obliczanie konstrukcji budynków wznoszonych tradycyjnie, DolnoĞląskie Wydawnictwo Edukacyjne, Wrocáaw, 2010. [17] E. Brehm, S.L. Lissel., “Reliability of unreinforced masonry bracing walls”, 15th International Brick and Block Masonry Conference Brazil 2012.

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