Considerations for the Design and Construction of Bonded and Unbonded Post-Tensioned Concrete Bridge Elements (2025)
Chapter: References and Bibliography
References and Bibliography
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Table R.1. Definitions of symbols used in equations from Table 2.2.
| Reference | Variable symbols | Variable definitions |
|---|---|---|
| AASHTO (2020) based on MacGregor (1989) | Aps | area of prestressing steel (in.2) |
| As | area of nonprestressed tension reinforcement (in.2) | |
| A’s | area of compression reinforcement (in.2) | |
| b | width of the compression face of the member (in.) | |
| bw | web width (in.) | |
| c | distance from the extreme compression fiber to the neutral axis (in.) | |
| dp | distance from extreme compression fiber to the centroid of the prestressing steel (in.) | |
| f′c | specified compressive strength of concrete (ksi) | |
| fps | average stress in prestressing steel at the time for which the nominal resistance of member is required (ksi) | |
| fy | specified minimum yield strength of reinforcement (ksi) | |
| hf | compression flange depth (in.) | |
| le | effective strand length (in.) | |
| li | length of strand between anchorages (in.) | |
| Ns | number of support hinges crossed by the strand between anchorages or discretely bonded points | |
| β1 | stress block factor taken as the ratio of the depth of the equivalent uniformly stressed compression zone assumed in the strength limit state to the depth of the actual compression zone | |
| ACI 318-19 (2019) | h | overall height or depth of member (in.) |
| ln | length of clear span measured face-to-face of supports (in.) | |
| µ | factor dependent on span-to-depth ratio | |
| ρp | ratio of Aps to bdp | |
| Warwaruk et al. (1962) | fpe | effective stress in prestressing steel after losses (ksi) |
| fu | tensile strength of prestressing steel (ksi) | |
| Pannell (1969) | f′c | compressive strength of concrete cube specimen (ksi) |
| lp | plastic hinge length (in.) | |
| n | ratio of depth of neutral axis at failure to the distance from extreme compression fiber to the centroid of the prestressing steel | |
| α | stress block factor taken as the ratio of equivalent rectangular concrete compressive stress block intensity to the compressive strength of concrete | |
| ϵu | limiting strain at which the concrete in the member crushes (in./in.) | |
| ψ | scaled plastic hinge length | |
| a | factor for compressive force | |
| Tam and Pannell (1976) | C | total compressive force at ultimate (kip) |
| Es | modulus of elasticity of steel (ksi) | |
| fcu | compressive strength of concrete cube specimen (ksi) | |
| L | length of prestressing strand between anchorages (in.) | |
| qe | ratio of fpeAps to fcubdp | |
| qs | ratio of fyAs to fcubdp | |
| qu | ratio of fpsAps to fcubdp | |
| λ | parameter for unbonded strand |
| Reference | Variable symbols | Variable definitions |
|---|---|---|
| Du and Tao (1985) | ds | distance from extreme compression fiber to the centroid of the nonprestressed tensile reinforcement (in.) |
| fy | yield strength of nonprestressed reinforcement (ksi) | |
| q0 | reinforcement index | |
| Δfps | stress increment in unbonded strands at failure (ksi) | |
| Kordina and Hegger (1987) | Ab | cross-sectional area of beam (in.2) |
| Eps | modulus of elasticity of prestressing steel (ksi) | |
| i | number of plastic hinges | |
| kbi | parameter dependent on concrete strength | |
| kvi | parameter dependent on percentage of prestressed reinforcement | |
| kfi | parameter dependent on cross-sectional shape | |
| lGi | equivalent hinge length dependent on type and distribution of loading (in.) | |
| l | length of span between supports (in.) | |
| Naaman and Alkhairi (1991) and Naaman et al. (2002) | l | length of span for which computation is carried out (in.) |
| ls | sum of lengths of loaded spans containing tendon(s) considered (in.) | |
| εcu | failure strain of concrete in compression | |
| Ωu | factor dependent on the number of loading points across a span | |
| f | factor dependent on the number of loading points across a span | |
| l | span length between end anchorages (in.) | |
| ρp | ratio of Aps to bdp | |
| Au and Du (2004) | cpe | distance from the extreme compression fiber to the neutral axis at service state (in.) |
| Ozkul et al. (2008) | e | eccentricity of load parallel to axis measured from centroid of section |
| fpu | tensile strength of prestressing steel (ksi) | |
| k1 | factor dependent on member continuity and the number of loading points across a span | |
| l | span length (in.) | |
| lh | length from support to the plastic hinge (in.) | |
| ll | distance from support to the applied load (in.) | |
| lp | equivalent plastic hinge length (in.) | |
| He and Liu (2010) | em | strand eccentricity at beam midspan (in.) |
| Rs | stress increment reduction factor for considering the second-order effect in external strands | |
| γ | deflection reduction factor | |
| δmid,u | beam midspan deflection (in.) | |
| η | parameter determined based on load type and strand profile | |
| φ | parameter determined based on load type and strand profile | |
| Ωe | bond reduction coefficient in the elastic state | |
| Harajli (2011) | f | ratio of span length to distance between two point loads |
| Np | continuity parameter | |
| np- | number of negative plastic hinges | |
| np+ | number of positive plastic hinges | |
| Np | continuity parameter | |
| φps | stress reduction factor |
Note: Symbols that are used in the same way for different equations are defined once in this table.
Table R.2. Definitions of symbols used in equations from Table 2.4.
| Reference | Variable symbols | Variable definitions |
|---|---|---|
| Ng and Tan (2006a) | c | distance from the extreme compression fiber to the neutral axis (in.) |
| dps0 | initial effective strand depth (in.) | |
| Eps | modulus of elasticity of prestressing steel (ksi) | |
| fpy | yield strength of prestressing steel (ksi) | |
| h | overall depth of member (in.) | |
| ks | factor for considering the second-order effects | |
| l | span length (in.) | |
| ll | distance from the beam support to loading point | |
| Sd | distance between two deviators placed symmetrically with respect to midspan of beam | |
| Ωu | bond reduction coefficient | |
| He and Liu (2010) | em | strand eccentricity at beam midspan (in.) |
| Rs | stress increment reduction factor for considering the second-order effect in external strands | |
| γ | deflection reduction factor | |
| δmid,u | beam midspan deflection (in.) | |
| εcu | failure strain of concrete in compression | |
| η | parameter determined based on load type and strand profile | |
| φ | parameter determined based on load type and strand profile | |
| Ωe | bond reduction coefficient in the elastic state |
Note: Symbols that are used in the same way for different equations are defined once in this table.
