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- Structural Reliability Analysis and Design R. Ranganathan
- ISBN 13: 9788172248512
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Structural Reliability Analysis and Design R. Ranganathan
A methodology has been developed for RBOD of reinforced concrete frames which considers the failure of structure, both at the component and system level simultaneously, under two different live load and wind load combinations. System failure probability is evaluated based on the generated stochastically dominant failure modes which consist of collapse mechanism failure modes and rotational failure modes arising due to critical sections reaching their permissible inelastic rotation capacities.
Optimal values of the moment capacities of critical sections are obtained based on the minimization of the total structural cost of the frame using the cost functions for member sections as a function of their moment capacities, which are derived by optimizing the member sections. It is observed that designs obtained using RBOD procedure give cost reduction for the chosen target reliability level, equal to that inherent reliability in the designs obtained as per the present Indian standard specifications.
Optimal designs based on structural reliability theory aim to minimize the weight [l] with failure probability constraint for each failure mode and for the system, or the material cost with failure probability constraint for maximum safety based on fixed weight or total expected cost .
Surahman and Rojiani  have presented the optimization problem as minimization of total cost sum of initial cost and cost of failure for reinforced concrete RC frames which is carried out in two steps-minimization of initial structural cost for a given risk level expressed in terms of probability of failure followed by minimization of the sum of the initial cost and cost of failure.
Multicriteria reliability based optimization procedures for structural systems subject to probability constraints imposed at both serviceability and ultimate limit states have been studied by Frangopol[3,4]. Sensitivity analysis has been carried out by him [5, 61 to establish response sensitivities of the optimum structures to changes in parameters.
Parimi and Cohn  have presented an optimality criterion based on a general formulation of the multicriteria probabilistic structural design which includes an economy criterion minimum structural cost , behavioural criterion maximum reliability and a combination of the two. Murotsu et al. It is observed that very little work has been done on reliability based optimal design RBOD of RC frames from the point of view of both system reliability and component reliability and hence a formulation and methodology for optimal design of RC frames with constraints on both component reliability and system reliability is presented.
Based on this idealization, the load-deformation characteristics of a structure are predicted to follow the piecewise-linear-elasticplastic PWLEP curve, Fig.
It is also assumed that structural members have bilinear momentcurvature relationships Fig. Applied loads are approximated as equivalent concentrated forces acting at nodes. Nonlinear effects are neglected and effects of geometrical changes are not significant. The nonlinear stress A. SriVidya and R. Ranganathan strain relations for concrete and steel are idealized as shown in Fig.
The elastic limit L, corresponds to yielding of steel or concrete. The elastic limit L, in Fig. Moment of resistance of section corresponding to limit L, in Fig. The plastic limit L, corresponds to the failure of either concrete or steel. In Fig. Moment of resistance of section corresponding to limit L?
Moment capacities of critical sections and applied loads are assumed as statistically independent random variables. Modulus of elasticity of materials and geometrical dimensions are considered as deterministic. This function is based on the optimal design of critical sections using the classical approach. For beam-sections, the expressions for optimal section size and reinforcement, as a function of moment capacity, derived by Sahasrabuddhe[lo], have been modified and used after eliminating the partial safety factors.
Further, for beam-sections, cost functions are derived as a function of moment capacity for the two cases: i when depth of beam is fixed; and ii when percentage reinforcement is fixed. Similarly, for column-sections, expressions for optimal depth and optimal reinforcement are derived and cost function is obtained as a function of moment capacity.
Further, cost functions are derived for the two cases. The cost functions for the rectangular beam-sections and columnsections are derived based on the assumptions for RC sections listed in Section 2. The width of section b is assumed and expressions for depth d of section and reinforcement A,, are obtained in terms of moment capacity, M. The expressions of depth, reinforcement and cost for different cases are given below [I I]. Beam-section: optimal depth and optimal reinforcementThe cost function of beam-section in terms of the ultimate moment capacity, M.
The optimal cost of beamsection per unit length as a function of moment A. Ranganathan capacity, M, is derived using the expression for M, given as,where f,,,, is the maximum ordinate of stress-strain curve for concrete; f,, is the stress in tensile steel; k, is the ratio of average stress in concrete to maximum stress in concrete; and k, is the ratio of depth of resultant compressive force to depth of neutral axis.
Expressions for optimal values of depth, reinforcement and cost are developed following the procedure given below:. Assuming equal areas of compressive and tensile steel i. Wa k; is the ratio of average stress in concrete to maximum stress in concrete. Since percentage of reinforcement is fixed, solving the quadratic eqn 47 [ , is considered in evaluating the statistics of moment capacity.
However, here they are designed as per Indian Standard . Axial rigidity EA and flexural rigidity EI are evaluated. Step 2-the frame is analyzed for the given loading situations. Statistics of moment capacities of beamsections and column-sections are evaluated as explained in the previous section. The axial forces acting in column-sections as obtained during analysis are taken into consideration while evaluating moment capacity of column-sections.
The permissible rotation capacities for all sections are evaluated as suggested in Ref. Step 3-safety equations for component failure under flexural failure mode are obtained. Step stochastically dominant failure modes SDFM for the frame structure are identified using the developed strategy [ Step 5-The optimal cost functions for optimizing member sections are evaluated as discussed above.
Constrained optimization using sequential unconstrained minimization technique with interior penalty function method SUMT is carried out with the objective function given by eqn 1 and constraints, eqns 2 5 , until optimal values of M are obtained. Step 7-section properties EA, EI are evaluated and steps are repeated. The methodology of reliability-based optimal design of RC frame structure for specified reliability level of component and system is illustrated in the following example.
Step 2-the frame is analyzed for both load combinations at the mean values of loads which are fixed using the ratios of mean to nominal values of loads given in Table 2. Step 3-reliability analysis is carried out and safety margin equations of component failure mode under flexure for both load combinations are obtained and are given in Table 4 Table 4.
These equations are used for evaluating the p values of failure modes fiZ given in Table 4 by Cornell's definition, the ratio of mean value to the standard deviation of the safety margin  and further for evaluating the p,. The identified SDFM are synthesized and system failure probability is evaluated which forms the constraints eqn 5. Step 5-the optimal cost functions for the critical sections are evaluated and optimization of the frame using SUMT is carried out.
The results of optimal Table 5. This is the first cycle of RBOD. Step 6-the percentage difference in moment capacities is about Hence RBOD procedure is to be carried out again. Step 7-the optimal solution of cycle-l is taken as the starting point for cycle-2, properties of sections are evaluated and steps are repeated.
A summary of the results of the optimal design of sections and reliability levels of component and system is shown in Table 6. In the third cycle a small increase in cost is observed. This is because, the optimal solution obtained in cycle 2 is governed by the minimum depth of column-section, and on reanalysis of the frame, in the beginning of cycle-3, the system reliability for "load case 1" is 3.
This starting point has further led to the optimal solution of cycle Hence, RBOD has been continued to cycle-4 and cost reduction achieved is again very small.
Total cost reduction achieved is about The starting point for RBOD has been taken as the design obtained, based on the design of frame for the factored load of 1.
Hence the reliability of the frame evaluated at this point gives the inherent reliability of the frame. These values of b under the two load combinations are observed to be high, as seen from Table 6. Sensitivity study for the one-bay one-storey frame Fig. It is observed that, to achieve higher system reliability for the frame. System reliability index 13, Fig. Variation of cost with system reliability index. The constraints considered are the allowable failure probabilities of i component failure under flexure, ii stochastically dominant failure modes.
SDFM and ii system failure. The unique value of system failure probability is taken as the average value of the narrow bounds, evaluated considering all the generated SDFM. Since reliability analysis of RC frames is based on the moment capacities of critical sections, the cost functions for beam and column sections are derived as a function of the moment capacities. The stepwise procedure of RBOD of RC frames has been presented and illustrated by an example of a one-bay one-storey RC frame with constraints on failure probabilities of component failure under flexure, SDFM and system failure.
In order to save computation time in the RBOD process. The starting point for RBOD in this example has been taken as the design of frame based on IS code  for the load combination 1. Hence, from the sensitivity study carried out for this load combination, it is observed that to achieve higher reliability, higher costs are to be incurred, as expected Fig.
It is also observed that the depth of the beam-section increases for higher requirement of fls Table 7. It is also observed from Table 7 that, at the optimal solution, constraint on r! The developed methodology considers the failure of structure. The RBOD procedure is presented in detail considering limit state of collapse in flexure. Reinforced concrete frames designed using the developed methodology of RBOD give cost reduction when designed for the chosen target reliability level equal to that inherent in the designs obtained as per the present code .
ISBN 13: 9788172248512
Reliability and Safety Engineering pp Cite as. In this chapter component reliability and system reliability assessment methods used in structural engineering are discussed. The first-order reliability method FORM , which has evolved from the advanced first-order second-moment method AFOSM is explained, together with the second-order reliability method SORM , which takes into account curvature of failure surface at the design point. Application of these methods in cases of correlated variables is also discussed. Unimodal and bimodal bound methods, the first-order multinormal FOMN and product of conditional margins PCM methods, which are used for system reliability assessment of structures, are also discussed. Unable to display preview. Download preview PDF.
View Structural Reliability Analysis and Design by knutsfordlitfest.orgathan knutsfordlitfest.org from CE MISC at Andhra University. STRUCTURAL RELIABILITY I ANALYSIS.
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