PRECAST SPUn COnCRETE PIERS In ROAD BRIDGES AnD fOOTBRIDGES

The usability of precast spun concrete members of annular cross-sections as pier shafts for road bridges and footbridges is discussed. The probability distribution of traffic loads is considered, their coefficient of variation is specified. First and second-order load effects for shafts of braced and bracing piers are analysed. The modeling of resisting compressive forces and bending moments of eccentrically loaded spun concrete shafts is considered. The features of mechanical properties of compressed spun concrete specimens reinforced by cold worked high-strength steel bars are presented. A simplified but fairly exact analysis of pier shafts under persistent situations by limit state and probabilitybased approaches is provided. A design of tubular shafts of braced piers using semi-probabilistic and probabilistic reliability verifications is illustrated by a numerical example.


Introduction
Precast spun (centrifugally cast) concrete shafts of annular cross-sections reinforced by steel bars uniformly distributed throughout their parameters and having high microcracking and low creep parameters satisfy economical, constructive and aesthetical requirements for bridges. The economically and structurally effective spun concrete shafts allow engineers to design and erect prefabricated piers subjected to vertical and horizontal forces. Therefore, they may be successfully used in construction practice of piers of short-span road bridges and footbridges. In some cases, it is expedient to use high-strength reinforcing steel bars increasing a bearing capacity of slender shafts exposed to compression with a small bending moment (Kudzys et al. 1993).
Multiple-span bridges may consist of one or few bracing piers and a large number of braced piers. The braced and bracing piers with tubular shafts may be treated as being isolated members fixed at their foundation which must also be fixed in the ground. The braced piers with movable bearings are almost free at their top and are assumed not to contribute to the overall horizontal stability of multiplespan bridges. The precast spun concrete shafts may be successfully used as bracing piers subjected to bending moments caused by vertical and horizontal forces. Therefore, the bracing piers should be fixed at their top.
The recommendations and directions presented in codes and standards for design and detailing rules of concrete structures EN 1992EN -1:2004 Eurocode 2: Design of Concrete Structures -Part 1: General Rules and Rules for Buildings and ACI 318-99:1999 Building Code Requirements for Structural Concrete are not fully formulated. In some cases, it hampers the development of analysis methods of spun concrete structures. Undoubtedly, the analysis of bearing capacity and structural safety of eccentrically loaded spun concrete piers under compression with a bending moment or bending with a compressive force has some characteristic features. Therefore, their design in a simple and easily perceptible manner is desirable by design engineers.
Contemporary design codes for bridge structures prescribe reliability verification methods exposed in limit state concepts. However, the reliability level of spun concrete piers designed by these concepts may differ considerably. The actual reliability level of piers may be defined only by probability-based concepts and approaches. However, for practical sake, the methodological and mathematical features of probabilistic approaches should be unsophisticated.
The object of this paper is stimulating the highway and structural engineers to use effective precast spun concrete shafts in bridge engineering and simplified but fairly exact semi-probabilistic and probabilistic approaches in their design practice.

first order effects
The destroying compressive forces and bending moments of pier shafts are caused by permanent and transient loads of persistent situations. They may be grouped into vertical (gravity) and horizontal actions (Fig. 1b, c).
The permanent gravity forces are caused by self-weight of structures, G 1 , and roadway surfacing weight, G 2 . The coefficients of variation of permanent loads are: δG 1 ≈ 0.10 and δG 2 ≈ 0.25 (Charnecki, Nowak 2008;Eamon, Nowak 2004). The value N G1 also depends on propping of precast members of continuous beams (Kudzys et al. 2007). The variable gravity, N Q , and horizontal braking, Q l , live forces are caused by heavily-loaded trucks, cars and special vehicles. The forces Q l may include temperature, concrete shrinkage and wind components. It is assumed that these horizontal forces act in longitudinal direction and are spread out over the entire pier cap of bracing piers.
The static, Q st , and dynamic, Q din , live loads of road girder bridges were investigated by Eamon and Nowak (2004) and Charnecki and Nowak (2008). It was deter-mined that the coefficient of variation of static live loads, δQ st , varies from 0.11 to 0.14-0.18 for a single and two heavily loaded trucks and the mean value of a dynamic factor may be taken as For new bridges, the dynamic factor may be equal to 0.25 or 0.15. Thus, δQ by (1) can be introduced from 0.23 to 0.26 and may be taken equal to 0.25 or 0.30, when consequences of failure can be medium or high, respectively.
According to European Standards, the reliability verification of bridge structures is based on the limit state concept used in conjunction with partial factor methods. Using these design approaches, the multiplication factor, K F1 , should be applied to unfavourable actions using its value equal to 1.0 or 1.1, when consequences of failure may be medium or high, respectively, as it is recommended by EN 1990by EN :2002   is the moment of inertia of a cross-section, where r 2 and r 1 -the radii of annular cross-section circles (Fig. 1a). The mean value of the modulus of elasticity of a spun concrete may be defined by: The mean and variance of a flexural stiffness may be presented as: where the component statistics are: σ 2 I = (δI × I m ) 2 with δI by Eq (5).
According to EN 1992-2:2005 directions, the 1 st order eccentricity of a compressive force N E = N G + N Q of piers ( Fig. 1) may be given by: is the shift of a bearing due to its movement or execution imperfections.

Second-order effects for braced piers
The buckling load of a shaft of braced piers may be expressed in terms of EI by Eq (2) and given by: Its design value, mean an variance follow from: Due to geometric imperfections, a buckling failure of a shaft under perfectly concentric compression is not a relevant limit state of piers. According to EN 1992-1:2004, a buckling load can be used as a parameter in their 2 nd order analysis assuming that the 2 nd order bending moment has a sine-shaped distribution. For braced piers (Fig. 1b), the 2 nd order eccentricity of the applied compressive force N E is defined as: where N B -by Eq (13); c 0 = 8 for a constant 1 st order bending moment. The design value, mean and variance of a 2 nd order eccentricity may be written as follows: where N Bd , N Bm and σ 2 N B -given by Eqs (14)- (16):

Second-order effects for bracing piers
The second-order eccentricity of the applied compressive force N (Fig. 1c) For bracing piers, the total destroying moment and its design value are expressed as: where

Compression with a bending moment
The compressive strength of spun concrete in bridge piers may have the form where the sustained load factor is expressed as: and the strength factor is defined by: where ρ = A A s c is the reinforcement ratio (Fig. 1a); f ck -characteristic cylinder strength, MPa. The coefficient of variation of this strength may be expressed by the Eq where f ck is in MPa . The ultimate compressive stresses in reinforcing steel bars of concentrically and eccentrically loaded shafts of braced and bracing piers, respectively, may be defined as: According to test data, these values were not more as yield strength f y and 800 MPa for hot rolled and cold worked steel bars, respectively, and were close to the stresses calculated by Hussaini et al. (1993) recommendations. The standardized second central moment of ultimate compressive stresses may be expressed as The shafts of braced piers, usually, are under compression with a small bending moment. The modelling of strain and stress distribution in concrete and high-strength steel of eccentrically loaded shafts may be based on a plane crosssection hypothesis and bi-linear concrete strain-stress relation when the conventional strain of concrete at its peak stress is equal to 0.5ε cu (Fig. 2).
When the eccentricity ratio e r s ≤ 1, the response factors, characterizing an extent of the intelligent use of the compressive resistance of concrete and reinforcement cross-sections, may be calculated by the following Eqs: where f cc -given by Eq (27); ′ σ sc -by Eq (31). Using the response factors k c and k s , given by Eqs (33) and (34), the resisting compressive force N R or the resistance R N of eccentrically loaded shafts of annular cross-sections of braced piers may be presented in the form: where f cc -given by Eq (27), ′ σ sc -given by Eq (31), e -defined by Eq (17).
According to the partial safety factors design (PSFD), the design value of this resistance can be considered as: where the variances of random variables of this resistance are: where δA -defined by Eq (5);

Bending with a compressive force
According to Vadluga (Вадлуга 1985), the ultimate bending moment M R of shafts of annular cross-sections (Fig. 3) reinforced by hot-rolled steel bars could be expressed as follows: The characteristic strength of reinforcement in tension, f stk , and compression, f sck , should be not more as f yk and 500 MPa, respectively. The compressive strength of concrete, f cc , is given by Eq (27). When a high-strength cold worked steel is used, its mechanical parameters may be defined as: For design practice, Eq (39) may be rewritten in the form: where The design value of resisting moment, M Rd , may be calculated by Eq (40) with the parameters with the following variances of variables: The coefficients of variation of steel strengths f st and f sc may be modelled as: where the components δf s1 ≈ 0.06 and δf s2 ≈ 0.08 define their statistical uncertainties and the errors of rightangled epures. The indispensable reliability level may not be achieved for strongly reinforced members under compression (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.

Reliability index and model uncertainties
The generalized reliability index as a standard reliability measure of bridge members may be defined as: where P s -the survival probability of a member and Φ -1 (•) -the inverse Gaussian distribution. For columns designed by limit state approaches, this index is between 3.3 and 5.0. Its value increases significantly when a reinforcement ratio increases and a compressive strength of concrete decreases (Diniz 2005).
According to EN 1990to EN :2002, the target reliability index, β T , of structural members may be selected from 3.3 to 4.3 depending on their failure consequence classes. For eccentrically loaded reinforced concrete columns (beamcolumns) and piers, the index β T must be not less than 3.5. The failure of a pier can be more brittle compared to the failure of bridge span members and have influence on their structural safety. Therefore, this index may be selected equal to 4.0 Szerszen et al. 2005). This value corresponds to the survival probability of pier shafts, P s , equal to 0.999968.
Usually both the resistance of a pier shaft and its action effects may be treated as stationary random processes. The safety margin of shafts of braced and bracing piers may be defined, respectively, as their performances of the forms: where the components of the vector θ of additional random variables, as professional factors, represent the uncertainties of design models in transformation of the vector X(t) of variables into resisting and destroying action effects (Melchers 1999). According to Ellingwood (1981), Melchers (1999), ENV 1991-3:1995, EN 1990:2002, JCSS 2000and ISO 2394:1998 General Principles on Reliability for Structures, the probability distributions of concrete shaft resistance R N or R M and permanent action effect N G or M G are close to a normal distribution.
An application of a lognormal distribution is conventional 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 by Bhattacharya (2008) in his investigations of bridge extreme loads. Thus, the probability distribution of live load effects may be treated as a lognormal one as it is recommended by ISO 2394:1998, Eamon and Nowak (2004). According to JCSS 2000, Holicky and Markova (2007), the means and standard deviations of the uncertainties of action effects for columns may be defined as: The additional uncertainties of shaft resistances, representing the ratio between their actual and predicted values, may be expressed by its mean, θ Rm , and standard deviation, σθ R . According to test data determined by Vadluga (Вадлуга 1985), , these statistics may be defined as: θ RNm = 0.99, σθ RN = 0.08 and θ RMm = 1.02, σθ RM = 0.08 for the shafts of braced and bracing piers, respectively. The standard deviation σθ R = 0.06 -0.08 is recommended by Szerszen et al. (2005) as statistical parameter for reliability analysis of eccentrically loaded columns.

Survival probabilities
For the sake of simplified but fairly exact probabilistic analysis of pier shafts, it is expedient to present Eqs (47) and (48), respectively in the form: where are the conventional resistances of shafts and are the action effects caused by live loads of road bridges. The parameters R CN and N C , R CM and M C of the safety margins by Eqs (49) are statistically independent variables. The statistics of components of these safety margins may be expressed as: The survival probabilities of pier shafts may be calculated by the Eqs: where f R CN , f R CM are the density functions of normally distributed variables R CN and R CM given by Eq (50) and F N C , F M C are the cumulative distribution functions of lognormally distributed live action effects N C and M C expressed by Eq (51).

The parameters of design
The precast spun concrete shafts of annular cross-sections of braced piers are subjected to permanent, G 1 , sustained, G 2 , and live, Q, loads. The characteristic values of gravity compressive forces are presented as: The statistics of additional random variables are: θ Nm = 1.0, σθ N = 0.10 and θ Rm = 0.99, σθ R = 0.08.

Limit state design
The design value of buckling load for pier shafts by Eq (14) is defined as: ( . (MN) (MN)).

N Ed
It shows that the analysed precast spun concrete shafts are suitable for bridge braced piers. However, when the consequences of failure may be high (K F1 = 1.1), the compressive force N ed = 4.674 MN is inadmissibly more than N Rd = 4.354 MN.
It shows that their structural safety is sufficient and constructive solution is effective, i.e. β N1 ≈ β T (= 4.0).
When the consequences of failure may be high and δQ = 0.30, the reliability index of analysed piers β N1 = 3.79 may be to small.

On acceptability of probability distribution laws
The data given in Tables 1 and 2 show, that all design variables R N , N G and N Q or R M , M G and M Q cannot be adequately modelled only by the normal or lognormal distribution laws. It may lead designers to an overestimation of predicted reliability indices of shafts. As it is shown, case 1 of probability distribution laws for the components of safety margins of considered pier shafts corroborated its acceptability in probability-based analysis of bridge piers.

Conclusions
The design features of economically reasonable precast spun concrete pier shafts depend on constructional solution of braced and bracing piers of road bridges and footbridges. The analysis of load-carrying capacity and structural reliability of pier shafts of annular cross-sections reinforced by steel bars uniformly distributed throughout their perimeter may be determined by unsophisticated semi-probabilistic and probability-based concepts and approaches demonstrated in this paper. They may stimulate engineers having min appropriate skills to use full probabilistic approaches in design practice more courageously.
For shafts of annular cross-sections of braced and bracing piers, the 2 nd order eccentricities of destroying forces may be expressed in a simple and easily perceptible manner. It is expedient to analyse the eccentrically loaded spun concrete shafts of braced and bracing bridge piers as the columns under compression with a bending moment and under bending with a compression force, respectively.
The objective assessment of structural safety level for bridge pier shafts may be introduced only by reliability index β using proper distributions for the components of their safety margins. The data presented in this paper corroborated that a lognormal distribution is to be used for variable live actions and a Gaussian distribution may be used for the joint values of permanent effects and shaft resistances. For spun concrete shafts of piers, the target reliability index β T may be selected equal to 4.0, as it is recommended by the investigators in the USA for eccentrically loaded reinforced concrete columns.