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Glossary of Notations 108 GLOSSARY OF NOTATIONS A = Earthquake peak ground acceleration. IρM = Soil influence coefficient for moment. = A0 Cross-sectional area of the stream. K1, K2, = aB Barge bow damage depth. K3, and K4 = Scour coefficients that account for the nose AF = Annual failure rate. shape of the pier, the angle between the direction b = River channel width. of the flow and the direction of the pier, the BR = Vehicular braking force. streambed conditions, and the bed material size. = BRa Aberrancy base rate. Kp = Rankine coefficient. = = bx Bias of ¯x x/xn. KR = Pile flexibility factor, which gives the relative c = Wind analysis constant. stiffness of the pile and soil. C′=Response spectrum modeling parameter. L = Foundation depth. = CE Vehicular centrifugal force. Le = Effective depth of foundation (distance from = CF Cost of failure. ground level to point of fixity). = CH Hydrodynamic coefficient that accounts for the effect LL = Vehicular live load. of surrounding water on vessel collision forces. LOA = Overall length of vessel. = CI Initial cost for building bridge structure. LS = Live load surcharge. = Cp Wind pressure coefficient. max(x) = Maximum of all possible x values. = CR Creep. M = Moment capacity. = cap CT Expected total cost of building bridge structure. M = Moment capacity of column. = col CT Vehicular collision force. M = Design moment. = design CV Vessel collision force. n = Manning roughness coefficient. = D Diameter of pile or column. N = Number of vessels (or flotillas) of type i. = i DC Dead load of structural components and nonstructural PA = Probability of aberrancy. attachments. P = Nominal design force for ship collisions. DD = Downdrag. B P = Base wind pressure. D = K–S maximum difference between measured B0 n P = Axial force capacity. cumulative probability and expected probability. cap PC = Probability that bridge will collapse given that DW = Dead load of wearing surfaces and utilities. i,k a vessel of type i has collided with bridge e = Height of column above ground level. member k. EH = Horizontal earth pressure load earth surcharge load. P = Axial capacity of column. E = Modulus of elasticity for pile. col p P = Design axial force. EQ = Earthquake. design P = Probability of failure. EV = Vertical pressure from dead load of earth fill. f PG = Geometric probability. E = Wind exposure coefficient. z = f = Location of maximum bending moment in pile shaft PL Pedestrian live load. = below the soil surface. Pp Passive resultant resisting force of the soil. Psoil = Soil bearing capacity. F0 = Froude number. Q = River flow discharge rate. Fa = Site soil coefficients for short periods. R = Resistance or member capacity. Fapl = Equivalent applied force. RB = Correction factor for impacting barge width. Fi = Equivalent inertial force. RBa = Correction factor for bridge location for vessel FK = Applied force for load type K. FR = Friction. aberrancy. R = Correction factor for current acting parallel to Fv = Site soil coefficient for system with natural period c T = 1 sec. vessel path. R = Correction factor for vessel traffic density. FY(Y*) = Cumulative probability = the probability that the D = variable Y takes a value less than or equal to Y*. RH Hydraulic radius. = G = Wind gust factor. Rm Response modification factor. = g = Acceleration caused by gravity. Rxc Correction factor for crosscurrents acting H = Moment arm of applied force. perpendicular to vessel path. HL-93 = AASHTO LRFD design live load. S = Load effect. = IC = Ice load. S0 Slope of the river bed stream. = IIM = Dynamic amplification for live load. Sa Spectral acceleration. = ILL = Live load intensity in terms of HL-93. SC Scour. IM = Vehicular dynamic load allowance. SD1 = Spectral acceleration for a natural period T1 = 1 Ip = Moment of inertia of pile. sec. IρH = Soil influence coefficient for lateral force. SDs = Spectral acceleration for short period Ts = 0.2 sec. 109 SE = Settlement. y0 = Depth of river flow just upstream of bridge pier SH = Shrinkage. excluding local scour. T = Natural period of the system. ymax = Maximum depth of scour. = t′=Natural period modeling factor. yscour Scour depth. = = T0 = Natural period at which the maximum spectral Z Safety margin R–S. = acceleration is reached. Z0 Friction length for wind. β = TG = Temperature gradient. target Target reliability index used for calibration. β= T = Natural period at which the spectral acceleration begins Reliability index. s ε= to decrease. Standard error in regression equation. φ= TU = Uniform temperature. Resistance factor. φ = V = Velocity (for wind speed, vessels at impact, or river s Angle of friction for sand. Φ=Cumulative standard normal distribution function. flow). Φ = Unit adjustment parameter = 1.486 for U.S. units V = Wind friction velocity. 0 0 or = 1.0 for SI units. V = Wind velocity above ground level. 10 γ=Specific weight of sand. VB = Base wind velocity = 160 km/h (100 mph). γk = Load factor for load type K. Vcol = Shear capacity of column. λcyc = Variable representing cyclic effects. VDZ = Design wind velocity at design elevation Z. λeq = Modeling factor for the analysis of earthquake loads. Vx = Coefficient of variation (COV) of x = standard λLL = Live load modeling factor. deviation/mean value. λ = Scour modeling variable. = sc W Weight (for vessel or structure). λ = System factor that represents the capacity of the = sys w Vessel weight modeling variable. “system” to continue to carry loads after failure of first WA = Water load and stream pressure. member. = WL Wind on live load. λV = Statistical modeling for estimating wind speed V. = WS Wind load on structure. µcap = Ductility capacity of a concrete column. = x Vessel collision modeling variable. µspecified = Specified ductility capacity. = ¯x Mean value of random variable x. νi = Yearly rate of collisions for each vessel (or flotilla) of Xmax,T = Maximum value of variable X in a period of time T. type i. xn = Nominal value of x as specified by design code. σx = Standard deviation of a random variable x. 106 REFERENCES Aktas, E., Moses, F., and Ghosn, M. (2001). “Cost and Safety Opti- Ditlevsen, O. (1988). “Uncertainty and Structural Reliability: Hocus mization of Structural Design Specifications,” Journal of Relia- Pocus or Objective Modeling?” Report No. 226, Department of bility Engineering and System Safety, Vol. 73, No. 3; pp. 205–212. Civil Engineering, Technical University of Denmark. American Association of State Highway and Transportation Offi- Ellingwood B., Galambos, T.V., MacGregor, J.G., and Comell, cials (1991). Guide Specification and Commentary for Vessel C.A. (1980). Development of a Probability Based Load Criterion Collision Design of Highway Bridges, Washington, DC. for American National Standard A58, National Bureau of Stan- American Association of State Highway and Transportation Offi- dards, Washington, DC. cials (1994). AASHTO LRFD Bridge Design Specifications, Frankel, A., Harmsen, S., Mueller, C., Barnhard, T., Leyendeker, Washington, DC. E.V., Perkins, D., Hanson, S., Dickrnan, N., and Hopper, M. American Association of State Highway and Transportation Officials (1997). “USGS National Seismic Hazard Maps: Uniform Hazard (1996). Standard Specifications for Highway Bridges, Washing- Spectra, De-aggregation, and Uncertainty,” Proceedings of FHWA/ ton, DC. NCEER Workshop on the National Representation of Seismic American Association of State Highway and Transportation Offi- Ground Motion for New and Existing Highway Facilities, NCEER cials (1998). AASHTO LRFD Bridge Design Specifications, 2nd Technical Report 97-0010, SUNY Buffalo, NY; pp. 39–73. edition, Washington, DC. Ghosn, M., and Moses, F. (1998). NCHRP Report 406: Redundancy American Concrete Institute (1995). Building Code Requirements in Highway Bridge Superstructures, Transportation Research for Structural Concrete ACI 318-95, Farmington Hills, MI. American Institute of Steel Construction (1994). Manual of Steel Board of the National Academies, Washington, DC. Construction: Load and Resistance Factor Design, LRFD, 2nd Goble, G.G., Burgess, C., Commander, B., Robson, B., and Schulz, edition, Chicago, IL. J.X. (1991). “Load Prediction and Structural Response, Volume American Society of Civil Engineers (1995). Minimum Design II”; Report to FHWA by Department of Civil and Architectural Loads for Buildings and Other Structures, Washington, DC. Engineering, University of Colorado, Boulder. Applied Technology Council and the Multidisciplinary Center for Haviland, R. (1976). “A Study of Uncertainties in the Fundamental Earthquake Engineering Research (2002). NCHRP Report 472: Translational Periods and Damping Values for Real Buildings,” Comprehensive Specification for the Seismic Design of Bridges, MIT reports, Cambridge, MA. Transportation Research Board of the National Academies, Hwang, H.H.M., Ushiba, H., and Shinozuka, M. (1988). “Reliabil- Washington, DC. ity Analysis of Code-Designed Structures under Natural Haz- Bea, R.G. (1983). “Characterization of the Reliability of Offshore ards,” Report to MCEER, SUNY Buffalo, NY. Piles Subjected to Axial Loadings,” in Proceedings of the ASCE Hydraulic Engineering Center (1986). “Accuracy of Computed Water Structures Congress. Surface Profiles,” U.S. Army Corps of Engineers, Davis, CA. Becker, D.E. (1996). “18th Canadian Geotechnical Colloqium: International Association of Bridge & Structural Engineers (1983). Limit States Design for Foundations. Part II Development for the “Ship Collision with Bridges and Offshore Structures,” IABSE National Building Code of Canada,” Canadian Geotechnical Colloquium, Copenhagen,
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