1. Introduction
This article presents the joint computational implementation of theories related to structural reliability and the advanced analysis of structures used by ^{Agostini (2014)}. This research is the first step in an attempt to evaluate the structural safety of steel portal frames designed with advanced analysis.
^{Agostini (2014)} aimed at developing a computational program that could provide an analysis of the structural reliability integrated into the structural analysis of steel portal frames. Reliability analysis was carried out with the First Order Reliability Method (FORM) method. The structural advanced analysis was carried out with the program Computational System for Advanced Structural Analysis (CS-ASA) (^{Silva, 2009}) considering the nonlinear geometric behavior and the flexibility of beam-to-column connections.
2. Computational development and implementation
To create the Structural Reliability Module (SRM) computational program, researchers used the Fortran 95 programming language in the Microsoft Visual Studio 2010 environment (^{Microsoft Corporation, 2010}). SRM was designed to be integrated with the mathematical and statistical routines of the numeric library for Fortran, the IMSL™ version 6.0 (Visual Numerics^{®}, 2006), as well as the CS-ASA structural analysis program version developed by ^{Silva (2009)}.
SRM was developed to work on an implicit performance function (limit state equation) allowing the reliability analysis of different types of portal frames. The condition for not fulfilling a certain service limit state is linked to a specific displacement.
The FORM method, present in SRM, requires the calculation of gradients of the performance function for all random variables. SRM uses a central finite difference process for calculating gradients of the performance function. This process consists of changing the value of each random variable by a small increase and decrease. For each increase and decrease, a call to the CS-ASA finite element program for the displacement supply occurs. With the calculated displacement (cd) for the node and direction under analysis as well as the fixed displacement limit (dl), the performance function G(u)=1-(cd/dl) is evaluated. In the SRM, the performance function was allocated in a specific subroutine. The two results of the evaluation of the performance function, generated from the increase and decrease in the value of the random variable, will enable the calculation of the performance function gradient for the random variable in question. This entire process is performed for all the random variables present in the reliability analysis to compose the gradient vector of each iteration of the FORM method.
The proposed implementation allows one to consider, as random variables, one elastic modulus, two transverse cross-section areas, two moments of inertia, three dead loads, and five live loads. To model these variables, one implements the following probability distributions: normal, lognormal, and type I extreme (largest value). For the latter two, two subroutines were created with a specific link to the reliability module and aim at calculating the mean and the standard deviation equivalent to a normal distribution.
To transform the variables, a Nataf transformation (^{Melchers, 1999}) was used and, to determine the design point in reduced space, the HLRF algorithm was used (^{Hasofer and Lind, 1974}; ^{Rackwitz and Fissler, 1978}). Equations to determine the sensitivity indices and the importance factors, as shown by ^{Sagrilo (2004)}, were also implemented in the SRM. To integrate the SRM with the IMSL™ library (Visual Numerics^{®}, 2006), it was necessary to make adjustments in the project developed on the Microsoft Visual Studio 2010 environment (^{Microsoft Corporation, 2010}), following the orientations available on the Intel^{®} website (Intel^{®} Developer Zone, 2011).
The process of reliability analysis follows the original structure from the CS-ASA program. This program works with three input files. The first input file contains general information such as the type of analysis, coordinates, loads, properties of materials and sections. The second contains the parameters related to the nonlinear solution strategy. And, the third, refers to the dynamic analysis, which is not approached herein. A fourth input file was created to carry out the reliability analysis. Readings of this file can only be carried out by SRM. The first parameters that make up the fourth input file are as follows: reliability analysis method, number of iterations for FORM, tolerance for interruption of the iterative process involving the FORM method, displacement limit for the node in analysis, numerical parameter of finite differences for carrying out an increase and a decrease on the value of each random variable, and the indicator of correlated variables. In explanation, for the actual version of SRM, it is not possible to consider the correlation between variables. After those parameters and in the same line for each random variable: type of random variable, mean, and standard deviation values. Integrating the CS-ASA output files, there was created a specific output file that shows both the input file parameters related to the reliability analysis and the results of this analysis.
The reliability analysis starts, from the CS-ASA program, with readings on the first input file. The SRM is activated as soon as a reliability analysis option is found in this file. If the reliability analysis is not defined, reading of file with general information goes on normally to carry out one of the types of analysis that make up CS-ASA. Managing the analysis process, SRM calls upon two subroutines which belong to the CS-ASA program. Firstly, the subroutine responsible for assembling the vector of nodal loads. Secondly, the subroutine that is responsible for producing the linear or nonlinear solution of the structure under analysis, providing the structural response in terms of displacements.
Besides taking the geometric nonlinearity into account, researchers enhanced the SRM with a functionality that enables it to carry out reliability analysis in structures with semi-rigid connections that display linear or nonlinear moment-rotation behavior. To do this, this study adjusted the ^{Richard and Abbott, (1975)} model, already implemented in the CS-ASA program, to carry out reliability analyses of structures that show a nonlinear moment-rotation behavior. The choice to adjust this model is due to the availability of results by ^{Haldar and Mahadevan, (2000)} related to the ^{Richard and Abbott, (1975)} model. Also, as affirmed by ^{Pinheiro (2003)}, this model always offers a positive rigidity, is computationally effective, and is one of the most used to represent semi-rigid connections. To carry out the reliability analysis considering the connections present in the structure as semi-rigid, but with linear moment-rotation behavior, the input file with general information is edited so as to indicate the initial rigidity (Sc) of the connection.
The flow chart in Figure 1 indicates CS-ASA with new functionality in the dotted box, reliability analysis. Interaction between static reliability analyses, denoted in Figure 1 by a double-headed arrow, occurs with a command from the SRM and aims at providing a structural response in terms of displacement. The dotted boxes in Figure 1 - labeled “geometric nonlinearity” and “rigidity of connections” - indicate the possibility of considering these effects when calculating displacement. Figure 2 presents the flow chart for SRM.
3. Results
Two examples analyzed by ^{Haldar and Mahadevan, (2000)} were used to compare with the methodology developed in the previous section. Among the main considerations regarding the geometric nonlinear analysis are the following: adoption of second order formulation SOF-2 (^{Silva, 2009}) defined in an updated Lagrangian referential and based on the ^{Yang and Kuo, (1994)} formulation and Euler-Bernoulli theory, use of a constant load increment strategy, use of constant load iteration strategy, determination of the initial increment in load parameter at 10% of the total load, division of loading into 10 increments, and adoption of Newton-Raphson standard method. It should be noted that the validation of the SRM was done by comparing it with data provided by ^{Agostini and Freitas, (2011)} and ^{Haldar and Mahadevan, (2000)}.
Example 1: Plane frame with three elements
The first structure that was analyzed is represented in Figure 3. Statistical properties of random variables considered in the analysis of that structure are described in Table 1. As regards the present example, structural reliability analysis was carried out considering the geometric nonlinearity and the presence of semi-rigid connections in the joints between columns and beam. The limit state in question refers to the horizontal displacement of Node 2, limited to 0.0114m in service.
Variables | Units | Nominal values | Mean /Nominal value | Coefficient of variation | Type of distribution |
---|---|---|---|---|---|
E | MPa | 199,948.04 | 1.00 | 0.06 | Lognormal |
A | m^{2} | 0.01271 | 1.00 | 0.05 | Lognormal |
I | m^{4} | 0.0003970848 | 1.00 | 0.05 | Lognormal |
D (Dead load) | kN/m | 43.78 | 1.05 | 0.10 | Lognormal |
L (Live load) | kN/m | 16.05 | 1.00 | 0.25 | Extreme Type I |
W (Wind load) | kN | 28.91 | 0.78 | 0.37 | Extreme Type I |
As for the portal frame in Example 1, represented by the structure shown in Figure 3, three different connecting behaviors were simultaneously analyzed for Nodes 2 and 3. These analyses took into account the nonlinear behavior of the connection using the ^{Richard and Abbott, (1975)} mathematical model. The model comprises the following parameters: initial stiffness (k), strain-hardening stiffness (k_{p}), reference moment (M_{0}), and a parameter defining the shape of the curve (n). Figure 4 shows the values of the four parameters for all three analyzed cases; the values were inserted into the analysis as non-random. ^{Haldar and Mahadevan, (2000)} analyzed the reliability considering also the random behavior of the parameters that make up the ^{Richard and Abbott, (1975)} model. In order to better visualize the behavior of the connections under analysis, moment (M)-rotation (φ_{c}) curves 1, 2, and 3 - indicated in Figure 4 - were generated using the equation presented in the figure.
Table 2 presents, for curves 1 to 3, the results from ^{Agostini (2014)} on the reliability analysis carried out considering the presence of semi-rigid connections in the structure. Table 3 presents the comparison between the results of the reliability index β given by ^{Agostini (2014)} and those given by ^{Haldar and Mahadevan, (2000)} considering the presence of semi-rigid connections.
Variable | Curve 1 | Curve 2 | Curve 3 | ||||||
---|---|---|---|---|---|---|---|---|---|
Sensitivity index | Initial value of variable | Final value of variable | Sensitivity index | Initial value of variable | Final value of variable | Sensitivity index | Initial value of variable | Final value of variable | |
E | -0.1960 | 199,948.04 | 188,354.23 | -0.1792 | 199,948.04 | 189,840.12 | -0.1630 | 199,948.04 | 191,314.86 |
A | -0.0047417 | 0.01271 | 0.01268 | -0.0042829 | 0.01271 | 0.01268 | -0.003646 | 0.01271 | 0.01268 |
I | -0.1587 | 0.0003970848 | 0,0003813867 | -0.1451 | 0.0003970848 | 0.0003834094 | -0.1322 | 0.0003970848 | 0.0003853952 |
D | 0.009346 | 45.97 | 45.95 | 0.012188 | 45.97 | 46.00 | 0.011894 | 45.97 | 45.98 |
L | 0.00814 | 16.05 | 15.54 | 0.010107 | 16.05 | 15.56 | 0.009421 | 16.05 | 15.54 |
W | 0.9676 | 22.55 | 109.21 | 0.9729 | 22.55 | 101.81 | 0.9776 | 22.55 | 92.84 |
Performance function | - | 0.7987 | -0.000115 | - | 0.7829 | -0.0000456 | - | 0.7618 | -0.0000155 |
Reliability index (β) | - | 12.14 | 4.93 | - | 10.94 | 4.66 | - | 9.60 | 4.33 |
Probability of failure | - | - | 0.4114e-6 | - | - | 1.5657e-6 | - | - | 7.3658e-6 |
Number of iterations | - | - | 4 | - | - | 4 | - | - | 4 |
β Rigid connection (nonlinear analysis) | β_{1} (Curve 1) | β_{2} (Curve 2) | β_{3} (Curve 3) | |||||
---|---|---|---|---|---|---|---|---|
Haldar and Mahadevan, (2000) | Present work | Haldar and Mahadevan, (2000) | Present work | Haldar and Mahadevan, (2000) | Present work | Haldar and Mahadevan, (2000) | Present work | |
β (Node 2 horizontal displacement) | 5.47 | 5.03 | 5.17 | 4.93 | 4.47 | 4.66 | 4.06 | 4.33 |
Example 2: Plane frame with eleven elements
The second example, shown in Figure 5, consists of a frame with an asymmetric arrangement of its elements. For the actual example, the presence of semi-rigid connections was not taken into account. Statistical properties of the variables related to the structure in question are indicated in Table 4, in which the sub-indices “b” and “c”, alongside the area and moment of inertia variables, stand for beam and column. The horizontal displacement of Node 1 was limited to 0.0254m, thus representing the service limit state. Table 5 presents the results of the reliability analysis carried out for the structure in Example 2, based both on linear and nonlinear analysis. Table 6 presents a summary of the reliability indexes β determined by linear and geometric nonlinear analyses, including values obtained by ^{Haldar and Mahadevan, (2000)}.
Variable | Unit | Mean | Coefficient of variation | Type of distribution |
---|---|---|---|---|
E | MPa | 199,948.04 | 0.06 | Lognormal |
Ab | m^{2} | 0.007613 | 0.05 | Lognormal |
Ac | m^{2} | 0.011419 | 0.05 | Lognormal |
Ib | m^{4} | 0.0002151916 | 0.05 | Lognormal |
Ic | m^{4} | 0.0001431836 | 0.05 | Lognormal |
P1(Dead load) | kN | 44.48 | 0.10 | Lognormal |
P2(Dead load) | kN | 88.96 | 0.10 | Lognormal |
P3(Dead load) | kN | 88.96 | 0.10 | Lognormal |
P4(Wind load) | kN | 44.48 | 0.37 | Extreme Type I |
P5(Wind load) | kN | 22.24 | 0.37 | Extreme Type I |
Variable | Sensitivity index | Initial value of the variable | Final value of the variable | |||
---|---|---|---|---|---|---|
Linear | Nonlinear | Linear | Nonlinear | Linear | Nonlinear | |
E | -0.182986 | -0.183625 | 199,948.04 | 199,948.04 | 194,716.96 | 194,731.43 |
Ab | -0.001573 | -0.001566 | 0.007613 | 0.007613 | 0.007602 | 0.07602 |
Ac | -0.001099 | -0.000923 | 0.011419 | 0.011419 | 0.011403 | 0.011404 |
Ib | -0.065100 | -0.065437 | 0.0002151916 | 0.0002151916 | 0.0002133527 | 0.0002133548 |
Ic | -0.084845 | -0.085225 | 0.0001431836 | 0.0001431836 | 0.0001416446 | 0.0001416474 |
P1 | -0.001382 | -0.000862 | 44.48 | 44.48 | 44.25 | 44.25 |
P2 | 0.008176 | 0.008469 | 88.96 | 88.96 | 88.68 | 88.69 |
P3 | 0.003952 | 0.004246 | 88.96 | 88.96 | 88.60 | 88.60 |
P4 | 0.973712 | 0.973498 | 44.48 | 44.48 | 91.69 | 91.21 |
P5 | 0.082917 | 0.083330 | 22.24 | 22.24 | 22.32 | 22.31 |
Performance function | - | - | 0.4763 | 0.4743 | -0.0000002 | -0.0000002 |
Reliability index (β) | - | - | 3.289 | 3.262 | 2.254 | 2.240 |
Probability of failure | - | - | - | - | 1.208e-2 | 1.255e-2 |
Number of iterations | - | - | - | - | 4 | 4 |
β (linear analysis) | β (geometric nonlinear analysis) | |||
---|---|---|---|---|
Haldar and Mahadevan, (2000) | Present work | Haldar and Mahadevan, (2000) | Present work | |
β (Node 1 horizontal displacement) | 2.283 | 2.254 | 2.274 | 2.240 |
4. Discussion
Comparing the results of Example 1 - presented in Tables 2 and 3 - it can be seen that the presence of semi-rigid connections resulted in a decrease in the reliability index, when the behavior of semi-rigid connections of nodes 2 and 3 goes from more stiff to less stiff. This decrease in the value of the reliability index β were also detected by ^{Haldar and Mahadevan, (2000)}. These authors considered the idealization of connections as perfectly pinned or stiff as inadequate and suggested a careful investigation, even for the serviceability limit state, in frames modelled with semi-rigid connections. ^{Hadianfard and Razani’s, (2003)} reliability analysis also showed results that indicated substantial differences in behavior of structures modelled with semi-rigid connections compared to those modelled with ideally pinned or rigid connections. ^{Hadianfard and Razani, (2003)} stated that to achieve more reliable results, it is necessary to consider the semi-rigid behavior of connections in the reliability analysis.
By comparing the results in Example 2, shown in Table 6, with those determined by ^{Haldar and Mahadevan, (2000)}, one is able to notice an agreement between values. Results shown in Tables 5 and 6 indicate mainly that the consideration of the geometric nonlinearity did not result in great differences in the reliability index, sensitivity indices or final values of the variables. However, a small decrease in the reliability index β can be noticed. One noted that, for the incident loading, the properties of the material and the geometric characteristics of the structure in Example 2 show only small manifestations of the effects of geometric nonlinearity in the limit state under study.
5. Conclusions
This study was carried out to understand the influence of the effects of geometric nonlinearity and of semi-rigid connections on the reliability of steel plane frames. The results indicate the efficiency of the proposed implementation. The reliability analyses involved the service limit state and provided evidence of a slight decrease in reliability index β when including the effects of geometric nonlinearity. When semi-rigid connections were considered, the result was a considerable decrease in the value of reliability index β as the stiffness of connections decreased. These results suggest the need to take the semi-rigid behavior of connections into account when carrying out reliability analyses of steel plane frames.