Abstract
An approach to obtain fragility curves taking into account the formulation for shear deformable plate theory with combined geometric and material nonlinearities and the boundary element method is proposed. It is assumed that the material undergoes large deflection with small strains. The von Mises yield criterion is used to evaluate the plastic zone and is supposed to have elastic-perfectly plastic material behaviour. An initial stress formulation is used to formulate the boundary integral equations. The domain integrals are evaluated using a cell discretization technique. A total incremental method is applied to solve the nonlinear boundary integral equations. The approach is illustrated in a plate subjected to incremental load. The uncertainties in both geometric and mechanical properties are considered in order to obtain the structural response. Results show that there are high probabilities of exceeding the damage state, , equal to 0.05 while for the rest of the values of , these probabilities are low.
1. Introduction
Plates are structural elements of great importance because they are used to characterize multiple mechanical conditions. Due to this, their analysis and prediction under different conditions and behaviours are of vital importance in their design. The probability that certain parameters exceed limits that could lead to failure takes an important role in determining such safety indicators in its design, for which probabilistic analyses help to find appropriate values for those parameters.
Kirchhoff [1] developed a theory that is currently widely used and is commonly known as the classical theory of plates. Reissner [2] on the other hand, enriched the classical theory with shear deformation contributions on the plate. In general, the classical theory provides good approximations when analysing some cases; however, for situations where the modelling of the stress concentration is required, this theory is not appropriate because it omits the shear deformations. In the theory proposed by Reissner, the problem is modelled in terms of three degrees of freedom that includes the generalized displacements and tractions.
In the last three decades, the Boundary Element Method (BEM) has emerged as a powerful numerical tool for plate analyses [3]. With this technique, some analyses have been developed in order to solve linear elastic behaviours on plates [4, 5]. The boundary integral equation for the model of the Reissner plate was presented in the work of Weeën [6]. Later, Karam and Telles [7] extended the formulation for infinite regions and reported that this theory is suitable for thin and thick plates. Recent works regarding plates can be found in [8].
A thin plate can suffer large deflections with normal service loads. In such cases, the behaviour cannot be adequately described using the theory of small elastic deformations. When that deflection in the plate is equal or bigger than its thickness, effects of nonlinearity occurs due to the coupling of forces and deflections in equilibrium conditions. Some works that deal with the geometrical linearity in plates using the classical theory can be found in [9–14]. Dirgantara and Aliabadi [15] reported the application of BEM for large deformations in shells. Purbolaksono and Aliabadi [14] used this method in a plate under shear deformation where they showed two methods to calculate the derivatives of the nonlinear terms in the domain integral. To solve the nonlinear problem, they concluded that the most efficient approach is the method of total increase proposed by Wen et al. [13].
Karam and Telles [16] introduced a BEM formulation for the elastoplastic analysis of Reissner plates where the classical theory of plasticity was used. The combination of geometric and material nonlinear analysis in 2D problems using BEM was first presented by Chandra and Mukherjee [17], and in this work he analysed large deformations for an isotropic material. Supriyono and Aliabadi [18] made a new formulation of boundary integral equations for combined geometric and material nonlinearities of shear deformable plate-bending analysis. The cell discretization approach is used for evaluating the domain integrals, and the total increment method is used for the nonlinear boundary integral equations, the same as was presented by [13].
On the other hand, there are different factors that affect the response of a certain structure such as manufacturing errors that could change both mechanical and geometrical properties. So, the loads to which the structure could be exposed are variable in magnitude, occurrence, duration, etc. This implies to consider all possible parameters as stochastic variables that affect the response of the structural element subjected to a certain load. According to this, several authors such as Rahman and Chen [19] and Huang and Aliabadi [20] have used the BEM method for probabilistic analysis in order to solve crack problems, elastic-linear problems [21–23], elastostatic problems [24–27], and optimization problems [28]. Fragility curves have been estimated in some kind of structural systems such as bridges, buildings, and transmission towers under different loads, i.e., seismic [29–33], wind [34, 35], and tsunami [36]. Unfortunately, the mentioned approaches have not used BEM to obtain fragility curves that represent the probability of exceeding a certain damage state considering the uncertainties related to both mechanical and geometrical properties of the structural element.
It is important to take into account since the load is increasing, and the probability of exceeding a certain established damage threshold increases. When the probability of exceeding a certain threshold is known before all possible loads, decisions can be made on the structural element through redesign or, if appropriate, verifying that the proposed element will develop a proper behaviour.
This research presents an approach to obtain fragility curves using BEM that takes into account the combination of large deformations and plasticity using the formulations shown in the work of Supriyono and Aliabadi [18]. The uncertainties related with mechanical (modulus of elasticity, yield stress, and ultimate stress) and geometric (thickness, base, height) properties were considered. So, different damage states were selected to estimate the probability of being exceeded. A simple supported plate at its ends with uniformly distributed load increased in every step was used for the analysis.
2. Formulation
2.1. Governing Equations
The relationships between stress resultants and strains by using the Reissner’s variational theorem of elasticity are shown in [29]. The development to get this formulation can be reviewed in [18]. Then, the relationship can be written aswhere , , and
The equilibrium equations arewhere , , and are the moment stress resultants, the shear stress resultants, and membrane stress resultants, respectively. is uniform load per unit area in the direction.
2.2. Displacement Integral Equations
The following are the displacement boundary integral equations for the membrane and the plate, see [18]. These can be written for the plate as
Also, for the membrane, these can be written aswhere denotes a Cauchy principal value integral, and are the jump terms. Equations (3) and (4) constitute the boundary displacement integral equations for plate bending problem.
2.3. Stress Integral Equations
As is shown in [18], the stress integral equations for moment stress resultants can be stated as
Also, the shear stress resultants can be written as
Finally, membrane stress resultants can be expressed aswhere the kernels and are linear combination of the first derivatives of and , respectively. The kernels and are the linear combination of the first derivatives of and , respectively. The kernels and are the linear combination of the first derivatives of and , respectively. The free terms appear in equations (5) and (7) arising from using Leibnitz formula. The expressions of the kernels are listed in [37].
2.4. Discretization and System of Equations
The displacement boundary integral equations for membrane and plates mentioned above are discretized, which makes possible to analyze the problem applying the boundary element method. Due to plasticity, it is necessary to discretize the domain Ω into cells and the boundary Γ into boundary elements. For the discretization of the domain nine nodes, quadrilateral quadratic cells were used and the boundary was divided by quadratic isoparametric elements. The corner problems are solved using semidiscontinuous boundary elements, and for the coincident side problems between the boundary and domain, semidiscontinuous cells were applied. In order to avoid the calculation of the deflection derivative on the boundary nodes, the internal values in the nodes of the cells were utilized. To compute the nonlinear terms due to large deflection, it was necessary to use the derivative of the deflection.
Dividing the boundary in quadratic elements and the domain in cells and using the collocation point method, the equations (3) and (4) can be written in a matrix form as follows:where the variables and are the matrices of influence of boundary elements and and are the influences for the case of large deflection and plasticity. The plate and the in-plane mode are considered by using and . The displacement and traction rate vectors are represented by , , , and . The variable that represents the load rate vector is . The stress resultant terms for bending and the membrane are denoted by and , respectively. Applying boundary conditions, we obtain the following equation:where is the system matrix, is the unknown vector, and is the vector of prescribed boundary values. Analogously, the stress integral equations can be presented in a matrix form aswhere , , and are the vectors of bending stress resultants, shear stress resultants, and membrane stress resultants, respectively. Superscripts and denote the bending and shear modes, respectively.
2.5. Elastoplastic Constitutive Equations
A yield function for the elastoplastic analysis is considered here. In terms of the hardening parameter k and the stresses , such yield function is stated, while in the loading process there is yielding and the stresses must remain at the yield surface. We satisfy the following equation:where is the uniaxial yield stress and is the equivalent stress using von Mises yield criteria. In our case, for an elastic-perfectly plastic material, . Recalling that we are applying the Reissner theory for the case when the membrane and moment stresses exist at the same time, the aforementioned values can be written as [38]
In the abovementioned equations (12) and (13), the equivalent membrane and moment stress are represented by and , respectively. The uniaxial membrane and moment stress are designated by the letter and , respectively. Equations (12) and (13) are shown in [18]. When the equivalent stress reaches the yield stress, there is yielding in the whole cross section at the same time. After yielding, the behavior of the stress-strain relationship is characterized incrementally aswherewhere, represents the components of fourth order isotropic tensor of elastic constants, and , which are given byfor moment and for membrane, and can be stated aswhere for moment and for membrane. is called the slope of the stress-plastic strain curve.
The additional equations required to calculate the nonlinear term due to large deflection are given by
2.6. Solution Algorithm
By applying the total increment method, as Wen et al. [13] proposed, it is possible to linearize the nonlinear integral equations. An incremental procedure for the method is divided in several steps as follows: firstly, the terms considering plasticity and large deflections are zero, and then in the th step, the approximations of the nonlinear terms are calculated for the th step using the following equations:
To evaluate the plastic zone of the model, the von Misses criterion is used.
By considering that,
Equation (10) can be written as where and are elastic moment stress resultant and elastic membrane stress resultant, respectively and is an identity matrix.
After calculating all the matrices and known vectors for every load step, the following system matrices are solved.where and denote the increment plastic resultants.
The nonlinear terms due to plasticity can be calculated as shown by Karam and Telles [16]. Assuming an incremental fictitious “elastic moment and membrane” we have
Taking into account equations (14), (15), (24), and (25), the folloeing equationscan be written as
A procedure to compute the nonlinear terms due to plasticity is showed in the following flowchart (see Figure 1).
3. Verification
To verify the results obtained with BEM, we compared the results against FEM, specifically with the ABAQUS software. So, we analyzed a square plate with dimensions of 1.0 m per side, thickness of 0.05 m an elastic modulus of 200,000 MPa, and a Poisson’s ratio of 0.3. The plate was simulated simply supported on all its sides with a uniform load of 100 ton-force applied on the complete surface of the squared plate of 1.0 m (see Figure 2). In this plate, we did the analysis for the combination of plasticity and large deformation. Both analyzes according to the graphs are in good agreement as shown in Figure 2. The results correspond to the node in the middle of the plate which is the most critical.
After comparing these two results, we conclude our code is reliable and good enough to do any kind of analysis for such combination.
4. Fragility Curves
The structural elements are made to resist a certain design load which is commonly an extraordinary load established by the technical regulation of the site. The response of the structural element due to this load can be expressed in terms of displacements, shear, fatigue, stresses, and so on, which commonly are known as damage indicators. In the case of structural elements subjected to loads that could be variable in duration, magnitude, occurrence, direction, and so on, it is important to know the probability of exceeding certain damage state. This allows the redesign of the structural element in order to maintain the structural element in acceptable performance levels.
A fragility curve can be defined as a graphical representation that relates the probability of exceeding a certain damage state for a given load. Both parameters, load, , and damage state, , are defined in accordance to the specific problem. Then, the structural response for a given load is defined in this study as the structural demand, . Considering that the structural demand, , follows a lognormal distribution [39], the representation of the fragility curve is as follows:where is the standard normal cumulative distribution function; is the mean value of the natural logarithm of the structural demand; and is the standard deviation of the natural logarithm of the structural demand.
5. Illustrative Example
Fragility curves of a plate are estimated considering the uncertainties in its geometrical properties (thickness, length, and width) and mechanical properties (yielding, ultimate stress, and modulus of elasticity). The nominal geometric properties are the dimensions of the plate are 1.0 m wide, , by 1.0 m long, , with a thickness, , equal to 0.05 m (see Figure 3). In case of the nominal mechanic properties such as the yield stress, , is 250 MPa, the ultimate stress, , is 400 MPa, the Poisson coefficient, , is 0.3, and the modulus of elasticity, , equal to 200,000 MPa. Based on the mentioned nominal properties and the statistical parameters reported by [40, 41] (see Table 1), it is possible to simulate different cases by taking into account the mean and the coefficient of variation for each parameter that is associated to a different type of distribution. The Monte Carlo method was used to simulate the study cases [42]. Moreover, it is noticed in Table 1 that the mean of each parameter is estimated by means of multiplying the nominal parameter by a certain constant. These constants were reported in [40, 41].
Figure 4 shows the results obtained by the present formulation. As previously mentioned, by using the Monte Carlo method considering both the nominal properties and the statistical parameters in Table 1, a number of 100 different plates were simulated. The variability of each plate corresponds with the specific uncertainty expressed in terms of the coefficient of variation that reports each parameter in Table 1. According to this , Figure 4 shows that the 100 simulated cases presented differences in their behavior due to the fact of considering the uncertainties in mechanical and geometrical properties in which is the deflection at the center point of the plate and is the uniform load. This figure shows the deflection calculated at the center of the plate for 100 simulated cases. Also, it is noticed that the applied load that represents the yield of displacement of the cases varies between 3.9-7.6 MPa while the inelastic behavior due to the effect of the combined large deformations and plasticity are presented between 0.03-0.07 m.
Figure 5 shows the fragility curves obtained by using equation (27), it is noticed that the x-axis is represented in terms of the incremental uniform load, , that the plate is subjected. Moreover, different damage states in terms of displacement, , are selected in order to estimate the probability the structural demand, , is greater than a certain damage state, , for a given load measure, . Figure 5 shows that there is a high probability of exceeding the different damage states, , as the load, , increases. Also, Figure 5 shows that the probability equal to 1 is reached for the damage state equal to 0.005. This implies that there is a certain probability of exceeding the mentioned damage states with a value equal to 1 (safe event). The rest of the selected damage states (i.e., 0.01 to 0.07) present a null probability that the safe event, , can be presented.
6. Conclusions
Fragility curves were estimated using the Boundary Element Method that considers the nonlinear problem of plasticity and large deformations. The classical theory of plasticity and a formulation of initial stresses that allow the formulation of integral boundary equations due to plasticity, were used. For the calculation of the plastic zone, the von Mises criterion was also used.
The obtained fragility curves consider the uncertainties in both geometrical and mechanical characteristics in order to estimate the probability that the structural demand exceeded different damage states, for a given load. The results indicate that there is a probability that certain selected damage states could be exceeding. Moreover, the approach provides information to the structural engineers about the load that produces a certain probability of exceeding each damage states.
Indicial notation is used throughout this work. Indices vary from 1 to 2; indices i, j, and k vary from 1 to 3; index vary from 1 to 2. The following symbology is used throughout the paper:
Abbreviations
| : | Moment stress resultant |
| : | Moment stress resultant nonlinear term |
| : | Membrane stress resultant |
| : | Moment stress resultant derivative |
| : | Membrane stress resultant derivative |
| : | Tension stiffness |
| : | Shear stiffness |
| : | Bending stiffness |
| : | Membrane stress resultant nonlinear term |
| : | Shear stress resultant |
| : | Shear stress resultant derivative |
| : | Generalized displacements in direction 1 |
| : | Kronecker delta function |
| : | Generalized displacements in direction 2 |
| : | Generalized displacements in direction 3 |
| : | Poisson’s rate |
| : | Uniform load |
| : | Jump term |
| : | Two rotations in plate |
| : | Deflection in plate |
| : | Displacements in-plane |
| : | Boundary of the plate |
| : | Generalized tractions |
| : | Domain of the plate |
| : | Weighting functions |
| : | Boundary element influence matrix |
| : | Boundary element influence matrix |
| : | Influence matrices for large deflection |
| : | Influence matrices for plasticity |
| : | Load rate vectors |
| : | System matrix |
| : | Vector of prescribed boundary values |
| : | Shape functions |
| : | Plate bending displacement fundamental solutions for displacement integral equations |
| : | Normal and shear stresses |
| : | Yield stress as a function of a hardening parameter k |
| h: | Thickness of the plate |
| : | Components of fourth order isotropic tensor of elastic constants |
| : | Identity matrix |
| : | Fundamental solution of displacement |
| : | Fundamental solution of traction |
| : | Fundamental solution of strains |
| : | Fundamental solution of displacement on membrane |
| : | Fundamental solution of traction on membrane |
| : | Fundamental solution of strains on membrane |
| : | Probability of failure |
| : | Structural demand |
| : | Damage state |
| : | Load measure |
| : | Standard normal cumulative distribution function |
| : | Mean value of the natural logarithm of the structural demand. |
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research has been supported by CONACYT through the support given under project number 240413. The first and fourth authors thank the Instituto Politécnico Nacional for the economic support; the second author thanks CONACYT for the economic support during his PhD studies; and the third author thanks Universidad Autónoma Metropolitana for the economic support.