Download - 9 - Flexural Design for Ultimate Loads
Flexural design for
ultimate loading
Fawad Muzaffar M.Sc. Structures (Stanford University)
Ph.D. Structures (Stanford University)
Civil Engineering
Department
Specified Load Factors
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Load Factors as Specified by ACI-318
Note: Refer to ACI 318-08 for exceptions.
Specified Values of Strength Reduction Factors
β’ ACI 318-08
β’ AASHTO
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The Unified Approach for Calculation of Strength Reduction Factors, π
β’ Nominal moment capacity is reached when ππ = 0.003.
β’ The behavior is tension controlled (or ductile) when ππ‘ = 0.005.
β’ If ππ‘ is small ( as is the case
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for axially loaded members, the behavior of the member will be brittle).
β’ If ππ‘ < ππ¦, the behavior is specified as compression controlled.
β’ Note: ππ‘ values above for ππ = 0.003 can be related to π/ππ ratios. e.g. ππ‘ = 0.005 would result in π ππ = 0.63
The Unified Approach for Calculation of Strength Reduction Factors, π
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β’ Reduction factor as a function of ππ‘
β’ Note: The greater the ππ‘, the more economical will the section be.
Calculation of Nominal Moment-The Strain Compatibility Approach
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β’ Increase in lateral load results in progressive increase of tensile strain, ultimately cracking at bottom. This happens when
β’ After M>Mcr significantly , the member starts to behave like reinforced concrete member.
β’ Before cracking, stresses in concrete at the CGS level have to reduce to zero (i.e. decompression stage)
Calculation of Nominal Moment-The Strain Compatibility Approach
β’ Strain in Prestressing Steel at Different Stages β Due to Prestressing:
β At Decompression:
β At Ultimate Load:
β Consequently:
β The corresponding stress in prestressing steel can then be evaluated.
β’ Compression in Concrete
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Calculation of Nominal Moment-The Strain Compatibility Approach
β’ The Bernoulli Assumptions
β’ Nominal Moment of Rectangular Sections β’ If π΄π
β² = 0, πΆ = π yields
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Calculation of Nominal Moment-The Strain Compatibility Approach
β’ Nominal Moment of Rectangular Sections
β If bonded Reinforcement is to be accounted for, C=T yields
β The internal moment couple can then be written as
β’ Nominal Moment of Flanged Sections
The internal moment couple can then be written as
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Calculation of Nominal Moment-The Strain Compatibility Approach
β’ The Solution Algorithm for Rectangular Sections
β Unknowns: πππ , ππ & c. Goal: Calculate Mn
β Assume c Calculate ππ & πππ Calculate πππ , ππ Evaluate c from
Ξ£πΉπ₯ = 0 Check c Iterate if Necessary
or
β Express: ππ = π(π) & πππ = π(π) Calculate πππ , ππ Constitute
Nonlinear Equation by writing Ξ£πΉπ₯ = 0 Solve the nonlinear equation to evaluate c.
β’ The Solution Algorithm for Flanged Sections β Assume a<tf Check assumption by writing Ξ£πΉπ₯ = 0 Analyze as
Rectangular Section if assumption is true.
β If a>tf Correct the assumption and analyze
section by adopting a T section.
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Calculation of Nominal Moment-The Approximate Approach
β’ Approximate Evaluation is allowed by ACI 318 if
β Bonded Tendons
β Unbonded Tendons
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Design for Ultimate Strength β Reinforcement Limits
β’ Minimum Values of Reinforcement Index β Definition
β If percentage of reinforcement is
too small, section will fail premat-
-urely as soon as Mu>Mcr.
β The total amount of pre-stressed &
non-prestressed reinforcement req-
uired by ACI should result in
ππ’ β₯ 1.2 πππ
Exception: If Mn > 2 Γππ’ and Vn > 2 Γ ππ’
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Design for Ultimate Strength β Reinforcement Limits
β’ Minimum area of bonded non-prestressed reinforcement in beams should be calculated using
β’ Maximum Reinforcement β If percentage of longitudinal reinforcement is large enough, brittle
failure will result.
β To ensure ductile failure, ACI specifies that ππ‘ β₯ 0.005 at ultimate load.
β In prestressed members, it is not always possible to ensure an under-reinforced behavior because
i) Serviceability Requirements ii) Absence of yield plateau of prestressed steel
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Design for Ultimate Strength β Reinforcement Limits
β’ To ensure ductile behavior at ultimate load, the reinforcement index is limited to β For Rectangular Sections with Prestressing Steel only:
β For Rectangular Sections with + and βve bonded Steel:
β For Flange Sections:
where
β Note that for rectangular section
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Design for Ultimate Strength β Reinforcement Limits
β For flange section
β For over-reinforced prestressed beams, section B.18.8.2 of the code specifies
β Use Empirical Relationship to evaluate Mn of Over reinforced Beams
i) Rectangular Section: ii) Flanged Section:
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Evaluation of Nominal Moment of Non-Bonded Prestressed Members
β’ Un-bonded Tendons β Un-grouted Tendons or Asphalt Coated Tendons.
β Stress Concentration do not exist at crack locations of un-bonded Tendons.
β A lesser number of wider cracks exist in un-bonded prestressed members.
β Bonded reinforcement is more important for members with un-bonded tendons as compared to bonded members.
β Bonded reinforcement contributes significantly to increasing the moment strength capacity of the section.
β The bonded reinforcement at the bottom will always be yielding at ultimate moment.
β The expressions presented for bonded prestressed members are equally applicable for evaluation of nominal moment of un-bonded members.
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