International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Probabilistic capacity assessment of a prestressed concrete pile in a corrosive marine environment Schmuhl, Daniel T.; Shafieezadeh, Abdollah; Hur, Jieun Jul 31, 2015

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata


53032-Paper_429_Hur.pdf [ 4.89MB ]
JSON: 53032-1.0076126.json
JSON-LD: 53032-1.0076126-ld.json
RDF/XML (Pretty): 53032-1.0076126-rdf.xml
RDF/JSON: 53032-1.0076126-rdf.json
Turtle: 53032-1.0076126-turtle.txt
N-Triples: 53032-1.0076126-rdf-ntriples.txt
Original Record: 53032-1.0076126-source.json
Full Text

Full Text

12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1 Probabilistic Capacity Assessment of a Prestressed Concrete Pile in a Corrosive Marine Environment Daniel T Schmuhl  Graduate Student, Dept. of Civil Engineering, Ohio State University, Columbus, Ohio, USA Abdollah Shafieezadeh Assistant Professor, Dept. of Civil Engineering, Ohio State University, Columbus, Ohio, USA Jieun Hur Assistant Professor, Dept. of Civil Engineering, Ohio State University, Columbus, Ohio, USA ABSTRACT: This study aims to provide a comprehensive method for modeling and analyzing prestressed concrete substructure elements, specifically wharf piles, subjected to chloride-induced corrosion.  Fully understanding this process is crucial to capturing uncertainties in structural response and providing accurate predictions of its time-dependent degradation mechanisms.  A probabilistic modeling framework and detailed finite element model and analysis are employed to explore these effects.  Results indicate a high degree of uncertainty captured by the spatial and time-dependent modeling techniques in addition to multiple significant modes of failure represented that were not previously considered in structures of this type.  1. INTRODUCTION Accurately representing and predicting pitting corrosion degradation in steel tendons in prestressed concrete structures is a challenging but vitally important task because it can provide the ability to account for a highly uncertain and time-dependent form of structural deterioration.  The overall goal of this work is to provide a detailed and realistic framework for temporal probabilistic chloride-induced corrosion modeling and concurrent updating finite element structural analysis.  This framework will strive for universal applicability in both research activities and practice due to simple and refined methods accompanied by detailed and accurate modeling procedures. There are multiple considerations in this work that will enable achievement of a realistic and efficient model.  Integration of spatial dependencies, uncertainties, and time-dependent effects with respect to corrosion severity into a single unified numerical and finite element model is a primary focus.  Time-dependent probabilistic modeling is used to represent uncertainty in environmental, structural, and material variables in the corrosion environment.  Recent research on this topic has established a knowledge base that benefits from more focused projects isolating specific issues within the corrosion process, but a lack of comprehensive studies that seek to integrate techniques to create realistic scenarios.  This work provides enhanced procedures that link various probabilistic parameter models to provide accurate field simulations of the chloride corrosion phenomenon. The resulting effects are implemented on a partially prestressed concrete wharf pile in a marine environment.  The structure is modeled in the open source object oriented finite element (FE) platform OpenSees (Open System for Earthquake Engineering Simulation).  The pile is discretized using an automated technique so that spatial variability can be easily incorporated and accurate structural response can be achieved.  The study of the pile considers soil-structure interactions via nonlinear soil springs.  The 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2 numerical modeling and principles used in building the FE model are automated and modular in nature so that the framework is easily portable to other applications or structural elements. 2. METHODOLOGY A probabilistic modeling procedure that incorporates spatial variation and time-dependency of variables is used to simulate the chloride-induced corrosion process.  In addition, a finite element modeling and analysis procedure is used that updates time dependent structural variables and corrosion pit geometry based on realizations from the probabilistic framework.  Those processes are described in the following sections. 2.1. Probability Modeling Procedures There have been a number of valuable insights into the development of pitting corrosion models in prestressed concrete in recent years which this study will draw upon.  Pitting corrosion is the typical mechanism that occurs in prestressed tendons subjected to chloride ingress and is the focus of this study. It involves the development of deep localized pits that can occur randomly over the length of a typical wire in the presence of chloride ions.  Due to the uncertainty involved in pit development, time-dependent probabilistic modeling of the actual pit depth caused by steel and chloride ion interaction has been studied based on accelerated laboratory corrosion tests on prestressed concrete structural elements.  The results of one such study by Darwaman and Stewart present a predicted distribution for maximum pit depth on a given length of wire based on experimental findings.  Their suggested distribution will be integrated into this project framework.  Pit depths are modeled using the Gumbel (EV-Type I) distribution, which in other previous works by Stewart, Darwaman and Stewart, and Stewart has also been found to adequately model pit depth or parameters related to directly describing pit depth.  The distribution function for the predicted Gumbel distribution is defined in Eq. (1) below for the maximum pit depth a [mm], in a wire of length L [mm], at time T [years], and for corrosion rate icorr(1) [µA/cm2] at the beginning of corrosion propagation. Fa(T,icorr,L)=1-e-e-α(aλ0.54 - μ)  (1) where 𝜆 =[𝐷02−(𝐷0−0.232𝑖𝑐𝑜𝑟𝑟(1){1+𝜅𝜃+1[(𝑇−𝑇𝑖)𝜃+1−1]})2][𝐷02−(𝐷0−0.232𝑖𝑐𝑜𝑟𝑟(1){1+𝜅𝜃+1[𝑇0𝜃+1−1]})2]  (2) 𝑇0 =exp [1(𝜃+1)ln ((𝜃+1)(𝑖𝑐𝑜𝑟𝑟−𝑒𝑥𝑝𝑇0−𝑒𝑥𝑝)+(𝜅−𝜃−1)(𝑖𝑐𝑜𝑟𝑟(1))𝜅𝑖𝑐𝑜𝑟𝑟(1))]  (3) 𝜇 = 𝜇0−𝑒𝑥𝑝 +1(𝛼0−𝑒𝑥𝑝)ln (𝐿𝐿0−𝑒𝑥𝑝)   (4) 𝛼 = 𝛼0−𝑒𝑥𝑝   (5) 𝑖𝑐𝑜𝑟𝑟(𝑇 − 𝑇𝑖) = 𝑖𝑐𝑜𝑟𝑟(1)𝜅(𝑇 − 𝑇𝑖)𝜃,  𝑇 − 𝑇𝑖 ≥ 1 𝑦𝑒𝑎𝑟  (6) and D0 is initial wire diameter in [mm], Ti is the corrosion initiation time [years], κ = 0.85, θ = -0.29, and α0-exp and µ0-exp are the Gumbel distribution parameters drawn from the accelerated corrosion tests which are listed in the original work along with the other experimental variables.  The complete geometry of the pit calculated using pit depth as the known variable is discussed by Val and Melchers.  This approach is extended to each of the outer six wires of the 7-wire prestressed tendon where corrosion will almost exclusively occur. A number of other distributions have been developed for additional important corrosion parameters and are built in to this framework to account for uncertainty in varying field conditions.  However, these distributions have generally been developed and utilized with a slight degree of integration in tandem with more simplified FE models with a number of restricting assumptions.   Initiation time (Ti), which is a key component in determining the corrosion process, describes the amount of time required in given conditions for chloride ions to penetrate the concrete, reach 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3 the surface of the steel tendon, and begin the corrosion degradation process.  It has typically been described by a rearrangement and solution to the differential equation representing Fick’s Second Law of diffusion assuming the concrete matrix behaves as a semi-infinite solid.  The final result of this prescribed solution is discussed by (Padgett 2010) and is shown below in Eq. (7). 𝑇𝑖 =𝑥24𝐷𝑐[erf−1 (𝐶0−𝐶𝑐𝑟𝐶0)]−2 (7) where x is concrete cover [cm], Dc is the diffusion coefficient [cm2/year], C0 and Ccr are the surface and critical chloride concentrations, respectively [% wt. of concrete]. Use of this approach is based on the assumption that C0 is time invariant which is acceptable and supported for structures exposed to de-icing salts.  However, for structures in marine environments, this assumption may not be acceptable as chlorides accumulate on the concrete surface over time.  Assuming that C0 increases with the square root of time, Duprat derived the following relationship for the chloride content 𝐶(𝑥, 𝑡) = 2𝐹0 × √𝑡𝜋𝐷𝑎[exp (−𝑥24𝐷𝑎𝑡) −𝑥2√𝜋𝐷𝑎𝑡erfc (𝑥2√𝐷𝑎𝑡)]  (8) where C(x,t) is the chloride concentration at depth x and time t [years] and F0 is the chloride diffusion flux of the concrete surface.  In order to utilize the model for generating initiation times for any given situation, the equation is solved for t using numerical algorithms in MATLAB.  The given values for the other variables correspond to conditions at initiation time, so that the output is t = Ti.  This implies that the value of C(x,t) must represent the critical chloride concentration at the cover depth required to begin corrosion, C(x,Ti).  In order to enforce this condition, a value for critical concentration is probabilistically generated from a uniform distribution proposed by Duprat assuming good quality concrete whose mean is 2 [kg/m3].  The use of a uniform distribution is generally considered a good choice for critical concentration because of the parameter’s heavy dependence on difficult to obtain chemical and mechanical properties within the concrete that are not readily available for sampling in the field. In order to provide the necessary information for solution, each of the other parameters in the model are also probabilistically generated for each simulation.  Apparent diffusion coefficient is generally dependent on concrete properties, which are considered constant throughout a given structural element.  As such, each value generated for a simulation is applied throughout the pile with no spatial dependency.  The distribution is assumed lognormal as proposed by Duprat with mean 10-12 [m2/s] and a coefficient of variation 0.7.  Cover depth is similarly modeled with a lognormal distribution with mean 0.0381 [m] and a coefficient of variation of 0.2 as noted in the work by Ghosh and Padgett as a single value for each simulation. Diffusion flux on the surface is dependent on ambient conditions and other various exposure parameters and should therefore be modeled as a spatially correlated variable based on locations along the pile.  Three different exposure zones are considered based on the availability of chloride ions on various areas of the structure as discussed in Val and Stewart:  Submerged zone: concrete is continuously underwater and corrosion risk is low due to lack of oxygen availability  Splash/tidal zones: concrete is subject to wetting and drying cycles.  Highest risk for chloride-induced corrosion  Atmospheric zone: concrete is not exposed directly to seawater and is only affected by wind carried chloride in sea spray Diffusion flux can be modeled based on these zones in a pile structure.  The original parameters for the lognormal distribution proposed by Duprat are altered to reflect the severity of exposure conditions in a given zone.  These relationships are derived from work the previous work by Val and Stewart for surface chloride concentration and are here also applied to diffusion flux given 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4 the parallel characteristics of both parameters in a corrosive marine environment.    Splash zone: No alteration to mean  Atmospheric zone: mean*0.4  Submerged zone: mean*0.1  The original splash zone diffusion flux mean is taken as 3.5×10-10 [kg/m2] and the coefficient of variation (which does not change between zones) is 0.6.  The given factors can be used to calculate the means for the other zones.  In addition, each probabilistically generated diffusion flux value at each discrete element is correlated to the other values in that zone through the use of a correlation coefficient block matrix.  The coefficient calculation is presented in Eq. (9) 𝜌 =1𝛾|𝑖−𝑗|+1  (9) where i and j represent positions of discrete elements in a particular zone, and γ denotes the strength of correlation: γ=0 provides perfect correlation, while γ=∞ represents the case of no correlation. In this study, γ is assumed one. Chloride concentration of different zones is assumed uncorrelated because the physical properties and ambient conditions are considered to change between the zones and thus significant correlated relationships between external condition-dependent variables are expected not exist between the zones. Pit depths are directly modeled to be fully correlated between all wires within a discrete element but completely uncorrelated between elements.  This approach is taken because all parameters that would affect corrosion severity (e.g. diffusion flux, cover depth, etc.) are the same within each element, but can change between elements based on the different zone conditions.       2.2. Finite Element Model   The OpenSees platform provides a powerful framework within which complex modeling can be done with custom materials and user-defined algorithms.  A number of available material models are combined to create a complex custom steel wire material that exhibits realistic behavior when embedded in the concrete structure.  This material was created through the use of nested and in-series modeling of multiple materials with properties that allowed for conditions such as no resistance to compression, fixed failure strain, and multiple alternate failure criteria to be imposed simultaneously. Stress corrosion cracking (SCC) and brittle wire fracture are two failure mechanisms of 7-wire prestressed tendons that are taken into account in this study.  They are serious failure criteria that can develop as a result of corrosion degradation and are modeled based on probabilistically generated variables.  Based on previous work (Darwaman and Stewart, 2007) in which these modes of failure were characterized through the use of a Linear-Elastic Fracture Mechanics approach, lognormal probability distributions were utilized to obtain spatially dependent but uncorrelated samples of the threshold stress intensity factors KSCC and Kbrittle.  The means and coefficients of variation for KSCC and Kbrittle are 43 [MPa m0.5] and 0.1, and 86 [MPa m0.5] and 0.05, respectively.  The overall stress intensity factor, KI, for a given wire with pit depth a, pit width b, and initial diameter D0 can be calculated using Eq. (10) 𝐾𝐼 = ∑ ∑ 𝐶𝑖𝑗 (𝑎𝐷0)𝑖3𝑗=0 (𝑎𝑏)𝑗4𝑖=0,𝑖≠1 𝜎√𝜋𝑎 (10) where σ is the applied stress and Cij are obtained from Astiz.  In this case, the applied stress is initially unknown, but the SCC and brittle fracture stress limit states are desired so they can be compared to the applied stress once it is known in the FE analysis.  Therefore, Eq. (10) is rearranged to Eq. (11) to solve for the stress limit at the corresponding value of the critical K value.  𝜎𝑐𝑟𝑖𝑡 =𝐾𝑐𝑟𝑖𝑡∑ ∑ 𝐶𝑖𝑗(𝑎𝐷0)𝑖3𝑗=0 (𝑎𝑏)𝑗4𝑖=0,𝑖≠1 √𝜋𝑎 (11) where 𝐾𝑐𝑟𝑖𝑡 = min(𝐾𝑆𝐶𝐶 , 𝐾𝑏𝑟𝑖𝑡𝑡𝑙𝑒) (12) In this manner, the maximum stress that can be applied to the wire is calculated based on the stress intensity factor derived from the minimum SCC or brittle fracture failure criterion.  It is also 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5 worth noting that this is a very important variable to consider over time, because although only one stress threshold factor is calculated per element per simulation, the increase in pit depth (a) over time dictates a decrease in the stress limit for these additional failure modes which may cause significant danger for sudden failure late in the structure’s service life.     The complex combination of materials ends by modeling each of the 7 wires in a strand in parallel so that behavior of the strand as a whole can be analyzed while each of the independent wires may change independently if specified to do so based on the statistical parameters that are input.  In the current setup, it was previously mentioned that the wires within a strand and the strands within a single discrete element are considered totally correlated. Another important aspect of prestressed concrete FE modeling is the consideration of bonding between the strand and the surrounding concrete and how this relationship and the materials involved are affected by the prestressing load in addition to behavior under large external loads.  The typical representation of reinforcement in the OpenSees environment entails allowing the bars or strands to be a part of a predefined cross-section that models all concrete, mild steel reinforcement and prestressed tendons at one time.  This method is straightforward and presents a lower risk for numerical convergence issues in the analysis, but it assumes complete bonding between steel and concrete throughout all simulations which is not a true representation of in-situ conditions for prestressed structures.  In this study, each tendon in a concrete segment is modeled by an individual element to allow for concrete to tendon bonding interactions to be explicitly accounted for.  By modeling the tendons individually, the interface between concrete and steel can be defined as separate zero-length elements which link two coincidental nodes; one of which represents a point on a tendon and the other a point in the concrete matrix.  These elements are defined by specific shear stress and displacement parameters that determine the behavior of the bonding interaction between the two materials as the pile is loaded.  In addition, the parameters have the capacity to change as the tendons corrode, creating a more realistic representation of bonding.  The interface parameters and approach for creating the interface elements were taken from Markovic, et al. Soil-structure interactions are crucial to consider when modeling pile or column structures under any sort of hazard event that provides extreme lateral loading.  Also, even typical lateral loads for piles that are severely degraded may become a danger to serviceability.  A soil profile was built for this study out of uniform clay materials derived from a wharf pile soil profile originally used by Shafieezadeh, et al. that begins at a depth of about 6 [m] from the top of the pile.  Soil springs are utilized at each node located within the soil profile to simulate the interaction between the pile and the surrounding soil materials in all three degrees of freedom.  OpenSees supports the use of various types of soil springs to represent vertical shear forces along the surface of the pile, horizontal normal forces around the pile perimeter, and end bearing reactions on the bottom of the pile. 3. APPLICATION 3.1. Wharf Pile Description The structure considered in this study is a wharf structure typical on the West Coast of the United States designed in the late 1960s or early 1970s.  This is an ideal structure to analyze because it has theoretically resided in a severely corrosive salt water environment for many years, and many are still in use and within their service lives. One partially prestressed circular concrete pile is isolated from the overall wharf structure and analyzed.  The selected pile retains its geometric properties, materials, soil profile, ambient conditions, and boundary conditions, but physical connections to other structural components are not modeled.  The pile has a total length of 21 [m] and was discretized into 1 [m] elements for analysis procedures.  The water level 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  6 and soil surface are assumed to coincide at a depth of 6 [m].  It is worth noting that the location of this pile relative to the original wharf structure it is modeled from dictates that 15 [m] of the pile’s 21 [m] length is fully embedded in soil.  The pile has a diameter of 46.4 [cm] and is reinforced by a circular layer of 12 Grade 270 7-wire prestressed steel tendons along its entire length and 4 #10 mild steel Grade 60 vertical bars that extend from the top of the pile to about 1 [m]. 3.2. Analysis Procedure and Model Linking Static pushover analyses are used to study structural behaviors of the pile as it degrades over time.  Simulation of corrosion degradation in the wharf pile and the analysis itself is carried out in three main steps.  First, calculation of reduced wire and rebar areas for each strand in each discrete pile element for all time intervals takes place in MATLAB.  The results are output as text files.  Second, OpenSees reads the resulting cross-sectional strand area files into a matrix and builds the FE model of the structure based on the methods outlined above. Within the model-building algorithm at each time step, the correct reduced reinforcing element areas from the MATLAB simulation are automatically applied to their designated pile element.  Third, both gravity and pushover analyses are performed on the pile at each time interval.  These procedures repeat for any designated number of simulations. 4. RESULTS A total of 100 simulations at time steps of 0 years (pristine conditions) 25 years, 50 years, and 75 years were performed on the pile with variables generated from the probability distributions and related equations  outlined in the previous sections.  The mean initiation times for the various exposure zones are 31.7 [years] for the splash zone, 60.6 [years] for the atmospheric zone, and 372.0 [years] for the submerged/embedded zone.  These results are important indicators of the performance of the integrated probabilistic modeling framework because many of the probabilistically generated variables were included in the model to determine initiation time.  The atmospheric zone contained only one discrete element of 1 [m] length, the splash zone consisted of five elements and the submerged zone was comprised of the remaining 15 elements.  The mean Ti for the splash zone was a factor of 0.52 times that of the atmospheric zone mean and 0.085 times that of the embedded zone mean.  This follows logically and shows expected behavior given that the mean values for diffusion flux (which was the main spatially dependent variable) used in its distribution were multiplied by factors of 0.40 and 0.1 with respect to the splash zone mean for the atmospheric and embedded conditions, respectively.  It should be noted that changing tide line and water level with respect to the pile was not explicitly modeled. The pushover analyses showed interesting results with respect to pile capacity and strength under increasing deterioration over time.  The average force-deformation relationship of the pile for the 100 simulations and the individual simulation results for each time step can be found in Figures 1-5.  It should first be noted that since there is no corrosion in the pristine pile and that there is no other source of uncertainty considered in this study, the force-deformation relationship of the pile is represented by only one pushover curve.  These results highlight the importance of capturing adequate sample sizes and of the impact of corrosion deterioration on potential failures in the later stages of aging.  It also demonstrates the uncertainty in conditions over time.  As time-dependent probabilistic values for pit depth  Figure 1: Force-deformation relationships for wharf pile in pristine condition with respect to corrosion  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  7  Figure 2: Force-deformation relationships for wharf pile after 25 years of exposure to corrosive conditions   Figure 3: Force-deformation relationships for wharf pile after 50 years of exposure to corrosive conditions   Figure 4: Force-deformation relationships for wharf pile after 75 years of exposure to corrosive conditions.  increase, yield strength of the tendons degrade, and stress corrosion cracking and brittle failure become more likely over time, there are an increasing number of variables that dictate pile behavior and failure conditions, leading to a wide  Figure 5: Average force-deformation relationships and standard deviations for wharf pile capacity  range of failure possibilities being captured.  Another observation is the increase in sudden failures that occur over time.  The pristine pile reaches the maximum specified displacement in the analysis without showing signs of failure, while analyses in subsequent time steps appear to show an increasing number of abrupt loss of ability to carry force, resulting in an immediate analysis failure.  The trend of increasing occurrences of sudden failures demonstrates the importance of including effects such as stress corrosion cracking, brittle fracture, and individual wire modeling.  Because stress cracking and brittle fracture are by definition non-ductile failure criterion, they were specified to cause immediate failure when one was the governing failure mode.  The distribution used to model each in addition to the dependence of each on pit geometry for failure modeling captured both a high degree of uncertainty in addition to time-dependency due to modeled increases in pit depth over time.  This directly relates to the large amount of spread in sudden failures in addition to an increase in occurrences as time increases. A final observation on overall capacities observed in this analysis yields interesting results.  The average responses drawn do not indicate a qualitatively large amount of capacity loss when moving from pristine to 75 year conditions when abrupt failure does not occur.  Excluding abrupt failures, the maximum average force resisted at the highest displacement is roughly 14% percent higher at pristine conditions than at 75 year conditions.  However, as discussed earlier, sudden 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  8 failures were clearly a major issue in this scenario given that many of the simulations at later time steps failed before reaching maximum specified displacement.  Therefore, the small differences noted here should not be viewed with as high of an importance as the number of sudden failures that occur with increases in time.    5. CONCLUSIONS The aim of this study was to create a realistic modeling and implementation framework for structural response to spatial and time-dependent probabilistic modeling of chloride corrosion in a marine environment.  Results from the study showed significant impacts with respect to multiple alternate failure criterion in addition to time-dependent uncertainties captured in the modeling process.  However, overall capacity data did not show exceedingly significant reductions in this study. The probabilistic modeling component of this framework yields enables capturing field uncertainties and yields a move towards a better understanding of long-term corrosion degradation effects in real structures.  The detailed FE model provides novel approaches to make the integration of modeled structures with corrosion effects more realistic, and allows for updating at discrete time-points to capture the time-component of corrosion deterioration. 6. REFERENCES Darwaman, M., and Stewart, M. (2007). “Spatial time-dependent reliability analysis of corroding pretensioned prestressed concrete bridge girders.” Structural Safety, 29, 16-31. Stewart, M. (2004). “Spatial variability of pitting corrosion and its influence on structural fragility and reliability of RC beams in flexure.” Structural Safety, 26, 453-470. Darwaman M., and Stewart, M. (2007). “Effect of pitting corrosion on capacity of prestressing wires.” Magazine of Concrete Research, 59(2), 131-139. Stewart, M. (2009). “Mechanical behavior of pitting corrosion of flexural and shear reinforcement and its effect on structural reliability of corroding RC beams.” Structural Safety, 31, 19-30. Val, D., and Melchers, R. (1997). “Reliability of deteriorating slab bridges.” Journal of Structural Engineering, 123(12), 1638-1644. Duprat, F. (2007). “Reliability of RC beams under chloride-ingress.” Construction and Building Materials, 21, 1605-1616. Ghosh, J., and Padgett, J. (2010). “Aging considerations in the development of time-dependent seismic fragility curves.” Journal of Structural Engineering, 136(12), 1497-1511. Val, D., and Stewart, M. (2003). “Life-cycle cost analysis of reinforced concrete structures in marine environments.” Structural Safety, 25, 343-362. Astiz, M. (1986). “An incompatible singular elastic element for two- and three-dimensional crack problems.” International Journal of Fracture, 31, 105-124. Markovic, M., Krauberger, N., Saje, M., Planinc, I., and Bratina, S. (2013). “Non-linear analysis of pre-tensioned concrete planar beams.” Engineering Structures, 46, 279-293. Shafieezadeh, A., DesRoches, R., Rix, G., and Werner, S. (2012). “Seismic performance of pile-supported wharf structures considering soil-structure interaction in liquefied soil.” Earthquake Spectra, 28(2), 729-757. McKenna, F., Scott, M. H., and Fenves, G. L., (2010). “Nonlinear finite-element analysis software architecture using object composition.” Journal of Computing in Civil Engineering, 24, 95–107. 


Citation Scheme:


Citations by CSL (citeproc-js)

Usage Statistics



Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            async >
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:


Related Items