LIMIT STATE AnD PROBABILISTIC FORMATS In THE AnALYSIS OF BRACInG PIERS OF AnnULAR CROSS-SECTIOnS

The expediency and efficiency of concrete bridge piers of annular cross-sections reinforced by steel bars uniformly distributed throughout their perimeter are considered. Modelling of permanent and live load effects and bearing capacity of bracing tubular piers is presented. The features of unsophisticated probability-based design formats are analysed. A simplified but fairly exact analysis of eccentrically loaded piers by limit state and probabilistic approaches is provided. The design practice of bracing tubular piers using limit state and probabilistic approaches is illustrated by a numerical example.


Introduction
Any bridge system and its components must be economically reasonable, structuraly appropriate and aesthetically satisfactory. The concrete piers of annular cross-sections satisfy these requirements and provide the most appropriate moments of inertia of cross-sections in two main directions. Bracing piers of bridges are fixed at their base and top in order to provide the load from decks to foundations. Therefore, they can transmit to the foundations all effects caused by vertical and horizontal random forces and are assumed to contribute to the overall horizontal stability of a bridge.
The reliability verification and rationality assessment of structures may be made by using partial safety factors (in Europe) and load and resistance factors (in the USA and other countries) based on probability criteria or by applying probability-based approaches. The choice of the level of reliability of structural members and their systems should take into account the predicted consequences of failure in terms of risk in life and potential economic losses. However, the reliability level of columns designed by semi-probabilistic methods may be differ considerably (Diniz 2005). On the other hand, reliability verification formats based on probabilistic concepts help us evaluate objectively all uncertainties of design models and ascertain effective solutions of structures.
The results of this study involve the formulations of limit state and safety margin criteria, as well as the perfections of load effects, bearing capacity and survival probability formats for piers of annular cross-sections reinforced by steel bars uniformly distributed throughout their perimeters. They encourage designers and highway engineers to use the presented unsophisticated semi-probabilistic and probabilistic approaches in design practice of tubular piers of bridges.

Compressive forces and bending moments
The permanent gravity forces of piers N G = N G1 + N G2 are caused by self-weight of structures, G 1 , and roadway surfacing weight, G 2 . The value N G1 also depends on propped and unpropped members of continuous beams (Kudzys et al. 2007). The coefficients of variation of these loads are δG 1 = 0.10 and δG 2 = 0.25 (Czarnecki, Nowak 2008;Eamon, Nowak 2004). The surfacing weight, G 2 , may be determined taking into account possible additional roadway topping.
For the general verifications of bridge piers, the transient live load models must cover most of the traffic effects. The representing variable live gravity and horizontal longitudinal forces N Q and Q l are caused by heavily-loaded trucks, cars and special vehicles. The horizontal force Q l consists of braking, temperature and wind components spreading out over the entire pier cap. For the 75-year of girder bridges service, the coefficient of variation of static live loads δQ st can vary from 0.14 to 0.18 (Czarnecki, Nowak 2008;Eamon, Nowak 2004;Szerszen et al. 2005). For two heavily loaded trucks travelling side-byside, the mean value of a dynamic load factor Q din /Q st may be taken as 0.10 with the coefficient of variation δQ din = 0.80 (Eamon, Nowak 2004). Thus, the coefficient of variation of bridge live loads may be expressed as: (1) It is equal to 0.22-0.26 and may be taken as δQ = 0.25.
Low bracing piers (h < 15 m) may be considered as eccentrically loaded columns. According to EN 1992EN -2:2005 Eurocode 2: Design of Concrete Structures -Concrete Bridges -Design and Detailing Rules, the first order eccentricity of force N of bracing piers (Fig. 1a) where -the moment of inertia of a cross-section.
The 2 nd order eccentricity of the applied compressive force N E = N G + N Q is defined as: From Eq (3), the mean and variance of a flexural stiffness of pier shafts may be expressed as: where the variances of the modulus of elasticity E c and the moment of inertia I of considered pier shafts are calculated using the coefficients of variation δE c = 0.15 (JCSS 2000) and δI = δA c = 1 2 The total destroying moment of bracing piers is expressed as: The means and variances of its components are: When the design eccentrity e d = e o + e Qld + e Nd , the design value, M Ed , of the total bending moment by (9) is equal to γ F M Ek , where γ F = 1.35 (ENV 1991(ENV -3:1995. Eurocode 1 -Part 3. Basis of Design and Actions on Structures. Traffic Loads on Bridges) is the partial safety factor for actions and M Ek = Q lk h + N Ek e d is its characteristic value.

Resisting moment
The modelling of stress-strain state and bearing capacity of eccentrically loaded concrete piers of annular crosssections should take into consideration their mechanical features. According to Вадлуга (1979, 1984), the ultimate bending moment M R (Fig. 1b) of columns reinforced by hot-rolled steel bars could be expressed as: The characteristic values of reinforcement strength in tension, f stk , and compression, f sck , should be not more as f yk and 500 MPa, respectively. The coefficients of variation of reinforcement strengths f st and f sc in tension and compression zones of cross-sections may be expressed as δf s = (δ 2 f s1 + δf s2 ) 1/2 ≈ 0.10, where the components δf s1 ≈ 0.06 (JCSS 2000) and δf s2 ≈ 0.08 (Вадлуга 1979) define their statistical deviations and the errors of right-angled epures of stresses.
According to JCSS (2000), Holicky and Markova (2007), the mean strength of hot rolled bars in tension and compression f stm = f scm = 560 MPa. By Kudzys and Kliukas (2008), when high-strength coldworked steel is used, its conventional mean strength in tension and compression may be defined as f stm = 500 MPa and f scm = 600 MPa.
The compressive strength of concrete in tubular piers is presented as: where f ck is its cylinder strength, MPa; and are the factors of the sustained load and stress block, respectively. According to Kudzys and Kliukas (2008), the coefficient of variation of concrete strength is δf c = 0.089 + 3 × (70 -f ck ) 2 × 10 -5 . For design practice, Eq (14) may be rewritten in the form: where The statistics of resistance R = M R of pier shafts under bending with concentrical compressive force are given by: where the variances of random variables are calculated by the expressions: Usually, the reliability verification of bridge structures is based on the limit state concept used in conjunction with partial factor methods (ENV 1991(ENV -3:1995 Bridges). The multiplication factor, K F , should be applied to unfavourable actions using its value equal to 1.0 or 1.1, where consequences of failure are medium and high, respectively (EN 1990(EN :2002 Eurocode: Basis of Structural Design). However, the indispensable reliability level may be not achieved for slightly and strongly reinforced columns (Holicky, Markova 2007). The appropriate level of structural quality and reliability of piers and other members of bridges may be obtained only using probabilistic design approaches.

Probability-based analysis
The safety margin of bridge piers may be defined as their time-dependent performance of the form: where the additional random variables θ i , as professional factors, represent the uncertainties of design models in transformation of basic variables into resisting and destroying moments (JCSS 2000;Melchers 1999). Their means and standard deviations may be presented as θ Mm ≈ 1.0, σθ M ≈ 0.10 (JCSS 2000; Holicky, Markova 2007) and θ Rm ≈ 1.02, σθ R = 0.08 (Вадлуга 1979). The values 1.02 and 0.06-0.08 also refer to the resistance of composite bridge girders (Eamon, Nowak 2004) and flexural reinforced concrete members . The statistics of random bending moments may have the forms: where M Gm , σ 2 M G , M Qm , σ 2 M Q are calculated from Eqs (10)-(13). The statistics of the pier resistance are defined as follows: The probability distribution of permanent bending moment, M G , and concrete pier resistance, R, is close to a normal distribution (Ellingwood 1981; JCSS 2000) (ENV 1991(ENV -3:1995 ;). The probability distribution of bending moment M Q (t) caused by random live loads within 75-year reference time may be treated as lognormal one (Eamon, Nowak 2004) (ISO 2394(ISO :1998. General Principles on Reliability for Structures). An application of the lognormal distribution is convenient for loads due to the road traffic consisting of the sum of a number of identically distributed independent lorries, cars and special vehicles. A stationary lognormal process is used in the analysis of bridge extreme live load distribution (Bhattacharya 2008).
Therefore, for the sake of simplified but fairly exact probabilistic analysis of pier shafts, it is expedient to present Eq (24) in the form: are the conventional resistance and bending moment of pier shafts, respectively. Their statistics are defined as: The parameters R c and M c are statistically independent variables. Therefore, the survival probability of piers may be calculated by the formula where f R (x) is the density function of R c by (32) and F M -the cumulative distribution function of M c by (33).
The survival probability of piers may be introduced by the reliability index where Φ -1 (P S ) -the inverse of the standard normal distribution. For columns designed by limit state methods, this index is between 3.3 and 5.0. It increases significantly if a reinforcement ratio increases and a concrete compressive strength decreases (Diniz 2005). According to EN 1990to EN :2002 Eurocode: Basis of Structural Design, the target reliability index, β T , of structural members may change from 3.3 to 4.3, depending on their failure consequence classes. For eccentrically loaded reinforced concrete columns and piers, the index β T must be not less 3.5 and may be selected equal to 4.0 because their failure can be more brittle comparing to a failure of bending members (Szerszen et al. 2005;.

The parameters of analysis
The bracing piers (Fig. 2) of annular cross-sections of the cross-street bridge in Kaunas are subjected to permanent and variable vertical forces N G = N G1 + N G2 and N Q , as well as horizontal braking force Q l . Their characteristic values are presented as: The statistics θ Mm = 1.0, σθ M = 0.1, θ Rm = 1.02, σθ R = 0.08.

Limit state analysis
The design value of flexural stiffness by (3) for pier shafts is defined as: Thus, the piers are suitable for the considered bridge.

Probability-based analysis
The mean values and variances of forces are as follows: The factor . Therefore, the statistics of flexural stiffness of piers by (7) and (8)   According to Eqs (38) and (39), the survival probability and reliability index of pier shafts are as follows: P s = 0.999958 and β = 3.93 ≈ β T = 4.0. Thus, their structural safety is sufficient and the constructive solution is effective.
The dimensions and the reinforcement of considered bracing pier satisfy reliability requirements of the current limit state and suggested probability-based design methods. The cross-sectional areas of concrete and reinforcing bars of solid and tubular pier shafts (Fig. 2) respectively are of the size: A c = 4.73 m 2 , A s = 0.0364 m 2 and 2A c1 = 1.304 m 2 << 4.73 m 2 , 2A s1 = 0.045 m 2 > 0.0364 m 2 . However, the solid concrete pier is in need of a great amount of the transient reinforcement. There seems to be a clear rational use of concrete pier shafts of annular cross-sections.

Conclusions
The concrete bracing piers of annular cross-sections reinforced by steel bars uniformly distributed throughout their perimeter may be treated as economically reasonable and constructively appropriate structures for bridges. The analysis of bearing capacity and reliability of fairly complicated eccentrically loaded shafts of piers may be successfully based on the semi-probabilistic and full probabilistic concepts and approaches, presented in this paper. In both cases, the destroying and resisting bending moments of bracing piers may be calculated in a simple engineering manner by Eqs (9) and (14). The statistics of a shaft resistance may be expressed by Eqs (22) and (23).
For road bridges, the live load, Q, can be described by a lognormal distribution with a coefficient of variation δQ = 0.25. The quantitative reliability index β by Eq (39) is indispensable in the objective assessment of a structural safety level of bridge piers and for an acceptability of their constructive solutions. The target reliability index β T for pier shafts may be selected equal to 4.0. The presented design methods and their numerical illustration show that the prediction of a survival probability of piers, including their reliability index, may be based on unsophisticated probabilistic approaches. It may stimulate engineers, having a min of appropriate skills, to use full probabilistic approaches in design practice more courageously and effectively.