Advances in Stochastic Models for Reliability, Quality and Safety
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Contributions motivated by or addressing issues in engineering, industry and natural sciences are particularly welcomed, as several focused sessions are devoted to such topics. The workshop aims at facilitating the exchange of research ideas, stimulating collaboration and promoting young researchers. We invite high quality submissions of extended abstracts on topics related to the workshop including, but not limited to:.
Invited sessions devoted to active research topics have been organized and renowned experts will be give plenary lectures and invited session talks. We also invite proposals for contributed sessions related to those topics. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Competing interests: The authors have declared that no competing interests exist. Underground reinforced concrete tanks are commonly used to store rain water.
Precast structures of this type can be effective due to a short construction period and almost unlimited volume when build form typical segments. However transport requirements set serious limits—the elements cannot be too large and too heavy. Thus precast tanks require a number of joints that can be critical points as presented in the paper. Underground structures are loaded with a soil pressure and in the same time interact with the soil.
In this way the soil reduces deformation of the structure and influences on a mode of failure. Moreover, hard to control and variable soil parameters can play more important role than parameters of the structure. Although failures of various structures are widely studied, few deal with reinforced concrete water tanks [ 1 ],[ 2 ]. This could mean that they seldom collapse, are well designed and constructed or accidents are kept secret.
Much more attention is paid to underground pipes of various cross-section shapes. Cross-sectional behaviour of this type of structures can be similar to the considered tank due to type of loadings, geometry and the interaction with the soil. Although analysis methods of the underground structures are well developed, the use of these methods does not always give unambiguous results [ 3 ]. Analyses of the load capacity of underground structures after collapse have been already carried out. Nonlinear finite element method is frequently used for simulation [ 2 ],[ 4 ],[ 5 ] in this case.
If possible, complexity of the soil—structure interaction and the load capacity are experimentally verified [ 3 ]. Test or field data can be used to calibrate the model [ 2 ],[ 3 ] to make the simulation more accurate. The finite element simulation was found helpful and leading to reconstruction of failure modes, so it was first used to explain the problem presented in this paper. Probabilistic analysis gives new perspective. Thus various kinds of uncertainties are possible to account and reliability can be estimated as presented in [ 6 ],[ 7 ] and [ 8 ].
This can be particularly important for the structures vulnerable to progressive collapse [ 9 ],[ 10 ] and construction errors [ 6 ] as presented example of a structure. Since finite element simulation failed, probabilistic analysis explained the collapse. The mentioned aspects were considered in the presented investigation. Although details of the collapsed structure were known, deterministic reproduction of the failure mode was hard to find. In search for the critical impact, stochastic methods were used to determine an effect of selected properties of soil and structural materials on the tank collapse.
In this way complexity and fuzzy nature of the problem is uncovered. The findings presented in the paper can help to increase robustness of the precast underground tanks by eliminating critical impacts in their structure and construction process. The tank presented in Fig 1 was designed to store rain water m 3 in the area of a shopping centre in the southwest Poland.
The structure was covered by 18 cm thick reinforced concrete circular arch shells and flat slabs at the ends. The designed bottom thickness was mm and vertical walls were 15 cm thick. Horizontal shell to wall joint was the only one used in the structure. The joint was filled with recycled rubber stripe.
All other contacts of elements were leak proofed, so only friction forces provided interaction between adjacent precast elements of the tank. The friction forces might exist due to soil pressure, but their estimation and consideration is doubtful due to possible manufacturing and construction imperfections of surfaces in contact. The tank during construction was covered with the 60—80 cm layer of soil over the shell crown.
Some additions of recycled materials from power plant were used in the backfill and possibly made its properties similar to sandy clay. Two adjoining circular shells collapsed under additional load from small excavator Figs 2 — 4. The collapse was accompanied by failure of the shell to wall joint Fig 4 and excessive cracks in the shells. Fortunately no one was injured in the accident.
Additional information used for model calibration was gained from measurements of the vertical walls deformations presented in Fig 3. For the joint detail after collapse marked with a circle see Fig 4. Initial distance— mm. Although only two of six shells collapsed, the mode of accident is similar to progressive failure. Any secondary structural system protected the structure.
Each shell was a separate system where one joint failure would lead to a collapse. So even though the structure was statically indeterminate, the system was a chain in a reliability sense. The joints between adjacent shells would help in this case. After the collapse the vertical walls were shifted back by approximately 5 cm according to Fig 3. Thus indicating that circular arch shells played key role in the overall stiffness of the structure and joint failure would also lead to the vertical wall damage.
The mode of collapse shows that at first the shell to wall joint was damaged due to horizontal forces or shift in the joint. Then capacity of the shell was exceeded. The reversed scenario is unlikely because one need horizontal forces to damage the joint and after the damage of shell horizontal forces in the joint would be reduced.
Advances in Stochastic Models for Reliablity, Quality and Safety
Possibility of this scenario was initially checked with linear finite element FE model. Two types of soil were analysed to obtain extreme modes of failure: soft and stiff. With the soft soil a failure appeared in the key of circular arch shell. However with the stiff soil the possible failure zone occurred in the horizontal shell to wall joint which could confirm the observed mode of collapse leading to development of a non-linear model presented in the next section.
The model was developed to analyse the structure in a complex realistic conditions. It is based on FE method. Plane stress and bar elements with nonlinear properties of concrete, steel, rubber and soil are used. Geometrical nonlinearities are taken into account via co-rotational formulation for large displacements, large rotations and small strains except rubber where large strains were considered. RC—reinforced concrete. Jefferson concrete model [ 11 ] implemented in Lusas software [ 12 ] was used for the computations. The model employs damage planes, where stresses are calculated with a local constitutive relationship.
The local stresses are the transformed components of the global stress tensor. It uses contact mechanics to simulate crack opening—closing and shear contact effects. The process of cracking is reversible in the model.
Planes of degradation can undergo damage and separation cracking but can also regain contact according to a contact state function. The crack model simulates normal and shear degradation as well as crack closure effects. In the crack plane it is assumed that a material is represented with two components, i. Crack closure is possible due to both shear and normal displacements, and thereby includes aggregate interlock. Three states are defined for a crack plane: open, interlock and closed.
In the open state the stress in the debonded component is assumed zero. In the interlock state the debonded stress is derived from a contact law in which the stress is assumed to depend upon the distance to the contact surface.
12th Workshop on Stochastic Models, Statistics and Their Applications - Statistics Views
In the closed state, the crack plane strain vector is equal to the local strain vector since the contact point coincides with the origin of the origin of local coordinate system. As presented in [ 11 ] the initial position of the yield surface in compressed concrete depends on the degree of triaxial confinement. After the calibration a value of between 0. Reinforcement was modelled as bars of symmetric in tension and compression elastic—plastic material with isotropic hardening. Continuity of displacements between concrete and steel was assumed and the interaction between them after cracking was applied in a simplified manner with softening of cracked concrete.
The uniaxial steel behaviour was represented by bilinear relationship where material behaves elastically up to yield stress. Two soil types cohesive and cohesionless with varying elasticity modulus E b were used to verify hypothesis that soil type played a key role. Details of the finite element mesh are presented in Fig 5 and material properties in Table 1. Concrete properties were set typical values for normal weight concrete and are based on compressive strength tests performed on core samples.
The tensile uniaxial stress—strain response of concrete is linear elastic up to tensile strength, f ct After cracking, the descending branch, which represents formation of microcracks, is modelled by an exponential softening. The end of the damage softening curve is computed from fracture energy and the characteristic element length which is related to the element area and its smallest diagonal [ 12 ]. Linear 2D four node quadratic and three node triangular isoparametric finite elements were used with respectively 4 and 3 point integration rules for concrete, rubber pad and soil while the reinforcement was modelled using 2-node bar elements.
In addition, perfect bond between concrete and the steel bars was assumed. In this method the load level is modified during the iteration procedure so that convergence near limit points may be achieved. Two load cases were considered: self weight the tank and soil and load from excavator approximate mass kg. Wall displacements presented in Fig 3 were intended to be used for model calibration.
While computed displacements 1. However according to the manufacturer information the elements were cracked after formwork removal as presented in Fig 6 which supports hypothesis on the stiffness degradation. The cracks could appear due to incorrect manufacturing or transport Fig 7 , so we decided to abandon FE analysis and prepare another model which is robust and includes wall stiffness degradation and random variation of basic properties.
Instead of vertical support in the crown bottom ends should be restricted horizontally. The analysis is based on the concept presented in [ 22 ] and generalized in [ 23 ],[ 24 ]. It was used for estimation of actions and deformations of horizontally loaded piles [ 24 ] and retaining walls also in unsaturated conditions [ 25 ]. The method is referred as the Characteristic Load Method CLM and was improved in [ 26 ],[ 27 ],[ 28 ] to include new features, i. The hyperbolic limit state model presented in Fig 8 was used to determine the forces in the analyzed system.
It is based on the limit state theory and the maximum values are derived from Mohr-Coulomb's limit condition and Eq 2 where the value of the passive soil pressure is given in Eq 3. With the increasing deformation, the initial maximum stiffness of the system decreases asymptotically to zero Fig 8.
Uncertainty Quantification (UQ)
Horizontal asymptote reduces a value of compressive stresses between the soil and the tank structure according to Eq 3. Model parameters are based on previous research presented in [ 23 ],[ 29 ] and [ 30 ]. A retaining wall displacement mode presented in [ 31 ] was extended to the one presented in Fig 9A. The extended version of the mode includes an extra hinge in the bottom part of the wall to account for horizontal cracks in the walls Fig 6A. In the assumed geometry two other joints appear: the sliding one at the geometrical symmetry point of the tank Fig 9A and the shell to wall joint allowing to rotation and limited displacement.
Analytical model a scheme of the system b resultant displacements c ultimate values of passive horizontal pressure. The depths z 1 to z 4 Figs 9 and 10 represent the following features of the model:. The system of forces acting on the shell to wall joint: a cross section through the tank, b the shell to wall joint. Directions of forces are presented in Fig 10 and their origins are the following:.
Due to the low value of R t its influence was omitted in the limit analysis of the force balance in the joint. The assumed joint failure mode is presented in Fig 11 with the following notation:. Geometry of: a the tank, b the shell to wall joint with assumed failure modes. The limit value of the passive horizontal soil pressure given directly in the CLM hypothesis is obtained from Eqs 2 and 3 : 2 3 where the pressure p ult [FL -2 ] for cohesive soil is a function of the backfill depth z.
R f and K max can be derived from filed tests. K max according to [ 32 ] and modified in [ 38 ] is obtained from Eq 5 : 5 where: D is a width of the wall [L] assumed in the paper 1. The resultant force of soil passive pressure P max acting on the wall and the shell is obtained from the equation similar to 6 : 7 Finally, taking into account slide in the joint of two rigid bodies the shell and the wall the possible distribution of the displacement s and the passive pressure p are presented in Fig Development of the limit state in concrete cross-section, i.
A proportion of the components in 8 gives utilisation ratio: To perform sensitivity and reliability analysis of the model the following significant geotechnical parameters of the backfill and concrete as random variables were selected:. Limits and mean values of the random variables are given in Table 2 along with the beta distribution parameters assumed for them.
The analysis was carried out iteratively in order to achieve importance of variables. A dimensionless index I i [ 39 ],[ 40 ] was introduced to expresses sensitivity and was defined as a difference quotient after the variable i Table 2. Reliability simulations were performed using only the direct methods, as for example presented in [ 41 ].
The Monte Carlo algorithm was used and a number of samples was determined based on the formulas: 11 12 where:. The probabilistic approach was based on an analytical solution, which forced the use different failure estimation than in the FE analysis. A limit state approach based on the statically admissible stress field method with defined stress discontinuity surfaces was applied. With these assumptions, the model parameters in the numerical and analytical models do not fully coincide.
This is related to the simplification of the analysis. At the same time, departing from the deterministic solution and extending the variability of parameters gives the analytical model a large potential in the search for sources of failure. In any of the analysed combination of properties presented in Table 1 the real failure mode was not reproduced, so only representative results are presented.
This failure mode is different from the observed one.
Horizontal force is too small do damage the shell to wall joint. Since FE results almost tripled the weight of the excavator and failure mechanism Fig 4 was not reproduced, the analysis was found ineffective in search for failure causes. Distributions of the sensitivity indexes I i obtained in the simulations are presented as a pie chart in Fig Although failure probability for cohesionless soil is not very small it appear to be satisfactory for construction stage. All random variables had a significant impact on the value of the reliability index, with the greatest impact of the depth of the backfill, followed by the soil cohesion, tensile strength of concrete, friction angle of the backfill material, horizontal force from the excavator and the smallest impact of the rubber stripe thickness.
Taken together backfill material properties had the biggest impact.