RELIABILITY OF UNREINFORCED MASONRY BRACING WALLS

Transcription

1 RELIABILITY OF UNREINFORCED MASONRY BRACING WALLS Brehm, Eric 1 ; Lissel, Shelley L. 2 1 Dr.-Ing., Forensic Engineer, TÜV SÜD, Munich, Germany, eric.brehm@tuev-sued.de 2 PhD, Associate Professor, Dept. of Civil Engineering, University of Calgary, sllissel@ucalgary.ca Bracing walls are essential members in typical masonry structures. However, design checks are only performed rarely in Germany. The reason for this is a paragraph in the German design code DIN which allows for neglecting of this design check. This mentioned paragraph is based on construction methods different from the current state of the art. Additionally, design codes have mostly been calibrated on the basis of experience. Consequently, the provided level of reliability remains unknown. In this paper, a systematic analysis of the provided level of reliability is conducted. Analytical models for the prediction of the shear capacity of the walls are analyzed and assessed with test data to identify the most realistic model. A complete stochastic model is developed and the reliability of typical bracing walls is determined. The difference between the theoretical level of reliability and the actual level of reliability is evaluated taking into account the realistic utilization of the walls. Subsequently, the derived level of reliability is presented and assessed. Keywords: reliability, masonry, target reliability, probabilistic INTRODUCTION The key objective in structural design is the design of sufficiently safe and reliable structures. While safety commonly refers to the absence of hazards, 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. Recent research (see Glowienka (2007)) showed the necessity for a probabilistic approach to masonry structures. However, most studies are focussed on masonry subjected to axial stress and flexure. Since masonry shear walls exhibit a much more complex load-carrying behaviour and are even more important to structural integrity, this paper focuses on masonry shear walls subjected to wind loading. The structural reliability of these walls will be analysed by assessing different analytical models and probabilistic modelling. RELIABILITY OF STRUCTURES The most important requirement for structures is reliability. The term reliability concerns every aspect of a structure; structures have to be reliable when it comes 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

2 a member. Reliability is linked to the probability that a member will exceed a certain limit state. This can be described by so called limit state functions which can be written as in Eq. (1): Z (R,E) = R - E (1) where R is the resistance and E is the load effect. In the case where R = E, the ultimate limit state is reached. It can be seen from this equation that the safety of a member can be defined as the difference between the resistance and the load effect. It has to be noted that R and E are independent random variables in many cases, so they have to be described by means of stochastics. Therefore a stochastic model, mostly consisting of a probability distribution and the corresponding moments (e.g. mean, standard deviation) is required for every basic variable. Figure 1 presents this for the simplified twodimensional case. safety margin resistance R load effect E failure probability P f Figure 1: Definition of failure probability (Glowienka (2007)) The failure probability can be computed by probabilistic methods such as SORM (Second Order Reliability Method) or Monte Carlo-simulation. For further information see Rackwitz (2004). For the description of the resistance, appropriate models are required that describe the load carrying behaviour realistically. Contrary to design models, a model that underestimates the load carrying behaviour is not appropriate for 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 β R according to Cornell (1969) has proven useful, see Eq. (2). The major advantage of this definition is that only the mean, m z, and standard deviation, z, of the basic variables need to be known. mz R (2) Z

3 With this measure, target reliabilities can be defined. Ideally, target reliabilities are based on a complex optimization process accounting for aspects of safety as well as economic requirements. In the past, target reliability has mostly been determined on an empirical basis. Since the target reliability has a major influence on safety factors, setting too large of a target reliability will lead to uneconomic design. More information can be found in Brehm (2011), GruSiBau (1981) and Rackwitz (2004). The Joint Committee on Structural Safety (JCSS) gives the target reliabilities depending on the failure consequences as shown in Table 1. Table 1: Target reliabilities according to JCSS (2001) for an observation period of 50 years Relative cost for enhancing the structural reliability Failure consequences Minor a) Average b) Major c) large β=1.7 (P f ) β=2.0 (P f ) β=2.6 (P f ) medium β=2.6 (P f ) β=3.2 (P f ) d) β=3.5 (P f ) small β=3.2 (P f ) β=3.5 (P f ) β=3.8 (P f 10-5 ) a) e.g. agricultural building b) e.g. office buildings, residential buildings or industrial buildings c) e.g. bridges, stadiums or high-rise buildings d) recommendation for regular cases according to JCSS 2001 These target reliabilities are considered to be sufficient for most cases and will be taken as reference for further calculations. Another recommendation is given by the German code DIN There, a value of β target = 3.8 is given for a 50 year observation period. A full probabilistic approach for design is difficult since stochastic models have to be known for all basic variables and good prediction models are required. To simplify design, the semiprobabilistic partial safety concept is applied in most design codes. In this concept, the partial safety factors for different basic variables make it possible to account for different scatter of the variables. A typical application of partial safety factors is presented by the following equation: R E E (3) R where E is the load effect and R is the resistance; both are usually defined as characteristic values. The safety factors, which are greater than unity, are represented by γ i. LOAD-CARRYING BEHAVIOUR OF URM BRACING WALLS To be able to determine the resistance and capacity of a member, the load-carrying behaviour has to be known. Masonry members subjected to shear exhibit a complex load-carrying behaviour. There is, however, a general consensus on the 3 main in-plane failure modes in masonry which include: flexural failure (tension at the heel or crushing at the toe), sliding failure in one or multiple bed joints, and diagonal tensile failure of the panel, which may be combined with sliding failure of the joints. Cracks are typically diagonal and stepped in nature but may also transverse through units as shown in Figure 2.

4 Tension Crushing Sliding Diagonal Tension (a) Figure 2: Typical (b) (a) failure modes for (b) (c) in-plane shear failure of (c) masonry Many models have been developed in the past for the prediction of the shear capacity of Unreinforced Masonry (URM) walls. In this study, a variety of models was compared to test data aiming at the identification of the most realistic model. For further information on the load-carrying behaviour of masonry shear walls and shear models, see Kranzler (2008). STRUCTURAL MODEL To apply the different shear models and to assess the test data, an appropriate structural model is required. The model is considered appropriate, if it allows for the modelling of a large number of walls with variation of only a few parameters and takes into account the coupling moments of the concrete slabs. Such a model has been proposed by Kranzler (2008) and is presented in Figure 3. The shear slenderness v can then be computed from Eq. (4). Figure 3: Structural Model (4) SHEAR CAPACITY PREDICTION / MODEL UNCERTAINTIES The previously mentioned structural model was combined with a number of prediction models to assess a test database. The goal was the determination of the uncertainties linked to

5 the different prediction models to finally be able to identify the most accurate model. This model should then be used for the analysis of the reliability of the walls. To achieve more realistic results and to eliminate stochastic uncertainties, a Bayesian update has been performed taking into account prior information. This prior information mainly consisted of expert opinions found in the literature. More information on this can be found in Rackwitz (2004). In Table 2, the results in form of the mean and scatter of the test-to-prediction ratio are presented for Calcium Silicate (CS), Autoclaved Aerated Concrete (AAC), and Clay Brick (CB) walls. For detailed information about the different models, please see Brehm (2011). Table 2: Stochastic moments of test-to-prediction ratios for different models after Bayesian Update Failure mode Model Unit m CoV Mann & Müller a AAC % CS % CB % CS % DIN EN /NA AAC % CB % Diagonal tension Sliding shear Flexure Kranzler DIN CS % AAC % CB % CS % AAC % CB % Mann & Müller a AAC % CS % CB % DIN EN /NA DIN ideal-plastic a ideal-plastic stress-strain relationship b updated with a mean of 1.0 (Likelihood) CS 1.15 b % AAC 1.21 b % CB % CS % AAC % CB % CS % AAC % CB % It was found that every model performed differently depending on unit material and possible failure mode. Thus, just a single most appropriate model could not be determined. Therefore, a combination of models was chosen for the most realistic representation of the behaviour of masonry subjected to in-plane shear. The corresponding model uncertainties are summarized in Table 3.

6 Load Resistance Table 3: Model uncertainties Failure mode Unit Model Dist. m CoV CS Mann & Müller % Diagonal tension Θ dt AAC % DIN EN /NA CB % Sliding shear Θ s AAC Mann & Müller % LN CB DIN EN /NA % CS DIN EN /NA % CS % Flexure Θ f AAC fully-plastic a % CB % Shear crushing Θ c all DIN EN /NA % a stress-strain relationship STOCHASTIC MODEL For the reliability analysis, every significant basic variable has to be modelled as a random variable. In this study, every parameter related to shear design of masonry walls subjected to wind load has been assessed thoroughly to obtain the relevant stochastic moments. Note, that these variables are related to both sides of the limit state function, resistance and loads. A summary of the stochastic model is presented in Table 4. Note that the model uncertainties for the resistance are displayed in Table 3. The model uncertainties are modelled differently for every unit material and failure mode according to the previously presented assessment of the test database. Table 4: Stochastic model Basic variable Material Distr. m X,i /X k,i CoV CS % Compressive strength of masonry f m AAC % CB % CS % Tensile strength of unit f bt AAC % CB LN % Cohesion f v0 TLM % GPM % Friction Coefficient all % Model uncertainty on the shear load E,v % Model uncertainty on the axial load E,a % Wind load v a,b all Weibull % a Live load n Q Gumbel % Dead load n G N % a observation period of 50 yrs, b = 0.073

7 RELIABILITY ANALYSIS The reliability analysis has been performed for selection of over 400 walls, which were previously designed according to German design codes. This made it possible to determine the realistically provided reliability of shear walls in Germany. The reliability has been determined by SORM. Basically, the failure probability for every possible failure mode for every wall has been determined and then the obtained failure probabilities for every failure mode have been summed up. Since one failure mode clearly always governs the analysis, the effects of possible overlap are small (< 1%). Figure 4 illustrates the determination of the failure probability. By applying this method, timeconsuming Monte-Carlo-Simulation could be avoided. The reliability of the walls was determined for full utilization of the cross-section to model walls in ultimate limit state and to obtain comparable results for the different unit materials. Figure 4: Approach for Probabilistic Analysis RESULTS: THEORETICAL RELIABILITY It quickly became obvious that the walls do not fulfil the code requirements in terms of target reliability when the examined walls were slender. Squat walls, however, reached very large values of the reliability. Note that every wall assumed to be 100% utilized. The reason for the differences in reliability between slender and squat walls is the eccentricity. Slender walls reach a much larger eccentricity than squat walls. In the following the axial and shear loads according to Eq. (5) and Eq. (6) will be used. N Gk n Gk t l f (5) v w V m Ek Ek t lw f (6) m

8 where f m is the masonry compressive strength, N Gk is the axial dead load and V Ek is the shear load. Figure 5 shows the reliability and eccentricity for a slender CS wall related to the axial dead load n Gk. Figure 5: Reliability vs. Eccentricity for a slender CS wall ( v = 3.0) and full utilization of the cross-section (100%) In general, a large eccentricity leads to small reliability. In Figure 6, the reliability for a squat CB wall is shown. It can be seen that the reliability is consistently very high and above the target region. Figure 6: Reliability vs. Eccentricity for a squat CB wall ( v = 0.5) and full utilization of the cross section (100%) Figure 7 shows a comparison of the reliabilities obtained for different unit materials and slender walls. It can be seen that the reliabilities are similarly distributed and stay below the target values. The larger reliability for AAC walls is due to a peculiarity in the German code DIN where the tensile strength of AAC units was massively underestimated.

9 Figure 7: Reliability of slender masonry walls for different materials ( v = 3.0) and full utilization of the cross section (100%) RESULTS: ACTUAL RELIABILITY The reliabilities obtained for slender walls are small, much smaller than one would expect considering the fact that there are actually no structural failures of masonry shear walls due to wind load reported. The most likely reason for this is the low actual level of utilization in reality. In this study, every wall was designed for full utilization. However, CS and CB walls provide much larger strength than AAC walls, so that higher loads are required to reach full utilization. The wind load acting on a structure in reality will be more or less the same no matter which material is used. Thus, levels of utilization are different for every unit material. The typical effect is shown in Figure 8. Figure 8: Reliability vs. Eccentricity for a slender CS wall ( v = 3.0) and different utilization levels of the cross section

10 To assess this effect, a typical town house was modelled using FE software. Different unit materials were investigated and the actual level of utilization due to wind and axial load was determined for every wall. Subsequently, the reliability taking into account the actual level of utilization, has been determined. It was found that the reliability reaches similar values for every wall material and that these values are significantly higher than the theoretical values of reliability. The results are summarized in Table 5. The actual value of reliability, determined on the basis of DIN , does not reach the target values of DIN Nonetheless, it matches the recommendations of JCSS (2001), see Table 1. However, the target values have to be analysed and assessed since there is no difference in target reliability depending on the failure consequences linked to a structure. These values are valid for a skyscraper, as well as for agricultural buildings. In a future study, a full-probabilistic optimization of typical masonry structures will be conducted to determine the socio-economic optimum target reliability for masonry structures. Table 5: Results for average theoretical and actual reliability Average reliability index provided in common masonry construction a Material theoretical b DIN DIN DIN EN /NA actual c CS (20/TLM) d CB (12/GPM IIa) e AAC (4/TLM) f a corresponding to v = 3.0 and n Gk = b full (100%) utilization of the wall c determined on the basis of DIN d corresponding to a utilization of 70% e corresponding to a utilization of 80% f corresponding to a utilization of 100% SUMMARY The reliability of masonry shear walls subjected to wind loading has been determined. The walls were designed according to German design codes so that the provided level of reliability in Germany could be derived. It was shown that the theoretical reliability of slender masonry walls falls short of the target values of DIN if full utilization of the cross-section is assumed. For realistic values of the utilization, reliability is much larger and the three investigated unit materials reach similar values. REFERENCES Brehm, E. Reliability of Unreinforced Masonry Bracing Walls Probabilistic Approach and Optimized Target Values, Doctoral Thesis, Technische Universität Darmstadt, Darmstadt, Germany, 2011 Cornell, C.A. A reliability-based structural code, ACI Journal, Vol. 66, No. 12, 1969

11 Glowienka, S. Zuverlässigkeit von Mauerwerkswänden aus großformatigen Steinen, Doctoral Thesis, Technische Universität Darmstadt, Darmstadt, Germany, 2007 GruSiBau Grundlagen zur Festlegung von Sicherheitsanforderungen für bauliche Anlagen, NA Bau, Beuth Verlag, Berlin, Germany, 1981 Joint Committee on Structural Safety (JCSS) Probabilistic Assessment of Existing Structures, RILEM publucations, S.A.R.L., 2001 Kranzler, T. Tragfähigkeit überwiegend horizontal beanspruchter Aussteifungsscheiben aus unbewehrtem Mauerwerk, Doctoral Thesis, Technische Universität Darmstadt, Darmstadt, Germany, 2008 Mann, W., Müller, H. Schubtragfähigkeit von Mauerwerk, in: Mauerwerk-Kalender 3, Ernst & Sohn, Berlin, 1978 Rackwitz, R. Zuverlässigkeit und Lasten im konstruktiven Ingenieurbau, Technical University of Munich, Munich, Germany, 2004