Moment Capacity by Strain Compatibility
The algorithm used in Precast/Prestressed Girder for computing moment capacity by Strain Compatibility method is as follows:
- Assume that the concrete at the extreme compression fiber has attained the ultimate concrete strain, εcu. By default, Precast/Prestressed Girder would assume the value of εcu as 0.003. It should be noted here that for a composite section, a unique value of εcu is assumed for both deck concrete and beam concrete.
- Assume a neutral axis depth, c from the extreme compression fiber and compute the depth of equivalent compression block, a = β1c. If there is a deck, β1 is taken as that corresponding to deck concrete. If deck is absent, β1 is taken as that corresponding to beam concrete.
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Assume that each strand is stressed to the same jacking stress and goes through same amount of final loss. So, each strand should have same stress, fpe when the beam is under the action of prestressing force alone. At this stage, the strain in each strand is given by,
where, Ep is the modulus of elasticity for the strand and n is the strand row number starting from the extreme tension side (row 1 is the strand row with highest tensile strain). This is the first component of strain in strand row n.
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As the beam concrete surrounding the strand goes through decompression due to the action of superimposed load, the strain in strand would increase. This increment in strand strain when the surrounding concrete is fully decompressed (attain zero strain) is equal to the strain in surrounding concrete with Pe acting alone (Pe is the total prestressing force in the beam after final loss). That is the second component of strain in strand and is given by (Design of Prestressed Concrete, by Arthur Nilson),
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Find the third component of strains in each strand rows using the formula,
where, dpn is the depth of nth strand row from extreme compression fiber. This component of strain is due to overloading to failure stage.
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Find the total strain in strand as,
- Find the stress fpsn corresponding to epsn using the stress strain relation for strand given in Reference 9, PCI Bridge Design Manual, as follows:
- Find total force in nth strand row, Fpn given by, Fpn = fpsn Apn where, Apn is the total area of strands in nth strand row
- Find the moment contributed by nth strand row, Mpn Mpn = Fpn dpn
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Repeat steps 3 through 9 for all strand rows and find total Fp as the sum of Fpn and total Mp as the sum of Mpn.
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Find the first part of compressive force in concrete C1 and moment MC1. The deck of section contributes this part. If deck is absent, then C1 and MC1 would be zero.
where, f`cs is the final compressive strength of deck concrete.
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Find the second part of compressive force in concrete C2 and moment MC2. The top flange of section contributes this part. If the depth of compression block is less than dPCTop, then C2 and MC2 would be zero. dPCTop is the distance from top of deck to top of precast section and hence it includes the thickness of haunch.
It should be noted here that although thickness of haunch is included in MC2, its contribution to C2 is ignored. f`cb is the final compressive strength of beam concrete.
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Find the third part of compressive force in concrete C3 and moment MC3. The web of the section contributes this part. If the depth of compression block is less than dPCTop + ttf, then C3 and MC3 would be zero.
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Find total compression C and total moment contributed by concrete MC in the section.
C = C1 + C2 + C3
MC = MC1 + MC2 + MC3
- Compare Fp with C. If Fp is almost equal to C within a certain tolerance, then the assumed depth of neutral axis, c is correct and go to Step 16 to compute moment. If Fp is greater than C, then increase the value of c and go back to Step 2. If Fp is less than C, then decrease the value of c and go back to Step 2.
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Find the moment of resistance as
where, ΦM is the resistance factor for moment.