This work presents reliability analysis of optimally designed reinforced concrete frames in accordance with Eurocode 2 (2004). A computer program in MATLAB (2004) was developed to analyze the frames based on direct stiffness method. The mathematical models of the design optimization problem were formulated and computer programs developed to obtain economical cross sectional dimensions and the corresponding amount of reinforcing steel using optimization process. MATLAB optimization tool box,was employed to perform the optimization. Reliability analysis based on the First-Order Reliability Method of analysis as in FORM5 (Gollwitzer, et al., 1988) was employed to determine reliability levels of optimized frame elements. The results showedthat optimum design resulted in cost savings of 5.1% in beams and 3.2% in columns.The reliability indices of the critical beam in flexural failure mode were3.96 and 3.43 for deterministic and optimum designs respectively. The results of column axial failure mode gave reliability indices of 5.83 and 5.04 for deterministic and optimum designs respectively.The results of system reliability analysis gave system reliability index value of 3.33 as against the minimum component reliability index of 3.43 for the critical member. The optimally designed reinforced concrete frames resulted in cost savings but the component and system reliability indices were both lower than the reliability indices of the deterministic design.
TABLE OF CONTENTS
Title Page i
Table of Contents viii
List of Figures xii
List of Tables xiv
List of Appendices xv
CHAPTER ONE: INTRODUCTION 1
1.1 Preamble 1
1.2 Statement ofResearch Problem 3
1.3 Justification of the Study 4
1.4 Aim and Objectives 5
1.4.1 Aim 5
1.4.2 Objectives 5
1.5 Scope and Limitations 6
1.5.1 Scope 6
1.5.2 Limitations 6
CHAPTER TWO: LITERATURE REVIEW 7
2.1 Analysis of Frames 7
2.1.1 Force or Flexibility method 11
2.1.2 Displacement or Stiffness Method 11
2.2 Structural Optimization 21
2.2.1 Design oriented structural analysis approach 24
2.2.2 The approximation concept approach 25
2.2.3 Optimality Criteria methods 25
2.2.4 Works on Optimization of Reinforced Concrete Structures 27
2.3 Reliability Analysis 32 2.3.1 Target Reliability 362.3.2 Component Reliability Analysis 37
2.3.3 System Reliability Analysis 37
2.4 Works on Reliability Analysis of Reinforced Concrete Structures 41
CHAPTER THREE: MATERIALS AND METHODS 48
3.1 Analysis of the Frame 48
3.2 Formulation of Optimum Design Functions 49
3.2.1 Objective function 49
3.2.2 Design Constraints 57
3.3 Optimum Design Input Parameters 63
3.4 Formulation of Explicit Design Functions 64
3.5 Reliability analysis 67
3.5.1 Beam Failure Modes and Limit States 68 126.96.36.199 Flexure Failure Mode 68 188.8.131.52 Shear Failure Mode 71
184.108.40.206 Deflection Control 71
3.5.2 Column Failure Modes and Limit States 73 220.127.116.11 Flexure Failure Mode 73 18.104.22.168 Axial Failure Mode 76 22.214.171.124 Combine Flexure and Axial Failure Mode 77
3.6 System Reliability Analysis of the Frame 77
CHAPTER FOUR: RESULTS AND DISCUSSION 80
4.1 Frame Analysis 80
4.2 Deterministic and Optimum Designs of Beams 814.3 Formulation of Expressions for Optimum Design of Beams 84 4.3.1Expression for Optimum Cost of Beams 85
4.3.2Expression for Beam Span to Effective Depth Ratios 88
4.3.3 Expression for Beam Reinforcement Ratios 89
4.3.4 Beam Design chart 92
4.4 Deterministic and Optimum Design of Critical Column 94
4.5Reliability analysis 95 4.5. 1 Beam Reliability Analysis 95 4.5.2 Column Reliability Analysis 98 4.5.3 System Reliability Analysis 101 CHAPTER FIVE: CONCLUSION AND RECOMENDATIONS 103
5.2 Conclusion 103
5.3 Recommendations 105
List of Figures
2.1: Statically Indeterminate Structures 7
2.2: External Statically indeterminate System 8
2.3: Frame built with intermediate hinges 8
2.4: Internal Statically indeterminate System 9
2.5: Member stiffness Influence Coefficients 14
2.6: Transformation to nodal loads 16
2.7: Coordinate transformation 17
2:8:Probability densities for typical resistance (R) and load (Q) 34
2:9:(a)Probability density and (b) cumulativeprobability for margin (M) 34
3.1: The basic frame 48
3.2: Beam details for singly reinforced section 51
3.3: Stirrup details 53
3.4:Column Details 55
3.5:Length of bent up bars in columns 56
3.6: Rectangular stress – block for singly reinforced section58
3.7: Rectangular stress – block for singly reinforced section 59
3.8: Shears and moments in a continuous beam 69
3.9: Substitute frame 74
3.10: Fault-tree of the frame 78
4.1: Variation of optimized cost with live loads 81
4.2a: Variation of best values of design variables forqk of 3kN/m2 83
4.2b: Variation of best values of design variables forqk of 4kN/m2 83
4.2c: Variation of best values of design variables forqk of 5kN/m2 83
4.2d: Variation of best values of design variables forqk of 6kN/m2 84
4.2e: Variation of best values of design variables forqk of 7kN/m2 84
4.3: Variation of cost with the change of beam span 86
4.4: Constants C1 for cost of beam expression 87
4.5: Constants C2 for cost of beam expression 88
4.6 Variation of span – effective depth ratio with span 89
4.7: Variation of reinforcement ratios with the change of span 90
4.8: Constants C1 for reinforcement ratio expression 91
4.9: Constants C2 for reinforcement ratio expression 91
4.10: Typical optimum design chart for tension reinforcement 93
4.11: Typical optimum design chart for compression reinforcement 93
4.12: Beam flexure failure reliability index versusload ratio 97 4.13: Beam shear failure reliability index versus load ratio 97 4.14: Span – effective depth reliability index versus load ratio 97
4.15: Column flexure failure reliability index versusload ratio 100
4.16: Column axial failure reliability index versus load ratio 100
4.17: Column combined flexure and axial failure reliability index versus load ratio 101
Reinforced concrete (RC) is one of the most widely used modern building materials. The wide use of RC in construction stems from the wide availability of reinforcing steel as well as concrete ingredients (cement, sand, gravel and water). It has the advantage that any shape can be formed (MacGregor and Wight, 2009).
RC Frames consist of horizontal beam elements and vertical column elements connected by rigid joints. These structures are cast monolithically, that is, the beams and columns are cast in a single operation in order to act in unison. Design applications range from low – rise buildings that are few stories to high-rise buildings with many floors.They provide resistance to both gravity and lateral loads through bending in the beams andcolumns(Nawy, 1996).
The design of structural components in a building frame is based on bending moment, shearforce and other load effects at each critical section in beams and columns. The critical sections for bending in columns are located at the top and bottomarears near the joints. For beams, the critical sections for positive and negative moments are assumed to be located at mid-spans and near the joints, respectively.
The trend in structural design has been towards improving the final design to the maximum degree possible without impairing the functional purposes the structure is supposed to serve
(Camp, et al., 2003). This has led to the emergence of the concept of structural optimization. An optimum structure is that which satisfies the performance requirements related to safety and serviceability while economy (minimum cost or material consumption) is maximized (Moharrami, 1993). The objective of reinforced concrete optimization is usually to find the concrete cross-sectional dimensions and the corresponding amounts of reinforcing steel. The objective function is minimized under a set of conditions called constraints. The most important of these are the design constraints such as the Eurocode 2 (2004) requirements. These include design variables such as the area of steel and the cross sectional dimension of the members. The design constraints on dimensions, strength capacities and areas of reinforcement were based on the specifications of Eurocode 2 (2004).
Reliability – based structural design is necessary if uncertainties exist in loads, material and geometric properties and /or mathematical models (Agrawal and Bhattacharya, 2010).
Structural reliability is the probability of not attaining the acceptable limits for safety and serviceability requirements of the structure before failure occurs.Structural analysis deals with the relation between the loads a structure must carry and its ability to carry the loads. It is usually measured by the reliability index, β which is a relative measure of the reliability or confidence in the ability of a structure to perform its function in a satisfactory manner.
The partial safety factors in Eurocodes are based on a linear analysis of structures and the satisfaction of ultimateand serviceability limit states using a semi-probabilisticapproach (Castro,et al., 2005).By means of structural reliability methods the design equations, characteristic values, partial safety factors and load combination factors may be chosen such that the level of reliability of all structures designed according to the design codes is homogeneous and independent of the choice of material and the prevailing loading, operational and environmental conditions. The partial factors are calculated so that the reliability of structures is at the predetermined target level (Nowak and Szerszen, 2000).
A computer program was developed to perform the optimization process. The formulationwas programmed using MATLAB‟s intrinsic optimization toolbox function, fmincon (MATLAB, 2004).
1.2 Statement of Research Problem
Codes of practice for example Eurocode 2 (2004) and ACI 318 (2011) are used for the design of RC members. The codes of practice use the concept of the semi-probabilistic method. In this method, the random variables such as the strength of the materials and the loads are treated as deterministic values after obtaining their design values from characteristic values. The transformation of a characteristic value of a random variable into a design value is performed by multiplying or dividing it by a partial factor related to this variable. The resulting deterministic solution is usually associated with a high chance of failure of the member being designed, due to the influence of controllable uncertainties
(e.g. dimensions and material properties), and due to model uncertainties and errors associated with semi-probabilistic design. Reliability design methodologies address these problems (Melchers, 1999; Oberkampf, et al., 2000).
Probability is incorporated in the design of these codes as represented by factors of safety, which results in overdesign of structural elementsas stresses needed to balance or offset the applied loads are very much in excess of what they should be. The effect of overdesign is increase in the cost of the members.Optimum design addresses the problem of overdesign by designing for optimum materials thereby reducing costs.
In the load factor design, when the partial safety factors are high, the reliability of the structure will be also high. Therefore, the reliability of the load factor design is not adequate(Almeida, et al., 2008).Furthermore, deterministic design depends on the experience of the design engineer. For example, if the flexural strength of a beam is inadequate, the decision about whether to increase the section depth or the amount of reinforcement is the choice of the designer. In optimum design, however, this decision is made by the optimization algorithm that guides the modification. The reliability analyses therefore results in efficient and economical use of materials.
1.3 Justification of the Study
The cost of a reinforced concrete structure is influenced by the cost of formwork, concrete and reinforcing bars. Minimizing consumption of these materials therefore leads to cost saving.
Structural optimization is employed to achieve cost saving.A design is optimum if it is proportioned in size and reinforcement such that its cost is minimized while satisfying all design requirements (Fadaee and Grierson, 1996)). The optimization procedure involves minimization of an objective function, which takes into account the structure cost due to concrete, steel reinforcement and form work subject to performance requirements.
Optimum design also has the advantage that it leads to automation, that is, the design variables are chosen by the optimization process and not by the engineer.
Deterministic design is semi-probabilistic. In this method, the random variables such as the strength of the materials and the loads are treated as deterministic values after obtaining their design values from characteristic values. There is no direct link between uncertainties in design parameters and reliability. The resulting deterministic solution is usually associated with a high chance of failure of the member being designed (Babu and Basha, 2008)).
Uncertainty arising from randomness in structural materials and applied loads as well as from errors in behavoural models is inevitable and most be properly accounted to assuresafety and reliability. Reliability – based structural design accounts for such uncertainties (Agrawal and Bhattacharya, 2010).
1.4Aim and Objectives
The aim of this study is to carry out reliability analysis of optimized reinforced concrete frames in accordance with the provision of Eurocode 2 (2004).
The objectives are to:
i. Develop a computer program to analyze plane frames based on stiffness method using MATLAB (2004) and design the frame elements according to Eurocode 2 (2004) design method
ii. Formulate the design optimization problem, develop the mathematical model and a computer program to obtain economical cross sectional dimensions and the corresponding amount of reinforcing steel using optimization process
iii. Develop optimum design charts for design of reinforced concrete elements
iv. Perform reliability analysis and establish the safety levels for the optimum design.
1.5 Scope and Limitations
A computer program in MATLAB was developed to analyze a three bay one storey reinforced concrete plane frame using the direct stiffness method. Optimization functions were formulated and modeled. MATLAB optimization tool box, was employed to perform the optimization.Reliability analysis was also carried out based on the First Order Reliability Analysis Method as in FORM5 (Gollwitzer,et al., 1988).
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