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

• ACI 318-08

• AASHTO

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The Unified Approach for Calculation of Strength Reduction Factors, 𝜙

• Nominal moment capacity is reached when 𝜀𝑐 = 0.003.

• The behavior is tension controlled (or ductile) when 𝜀𝑡 = 0.005.

• If 𝜀𝑡 is small ( as is the case

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for axially loaded members, the behavior of the member will be brittle).

• If 𝜀𝑡 < 𝜀𝑦, the behavior is specified as compression controlled.

• Note: 𝜀𝑡 values above for 𝜀𝑐 = 0.003 can be related to 𝜌/𝜌𝑏 ratios. e.g. 𝜀𝑡 = 0.005 would result in 𝜌 𝜌𝑏 = 0.63

The Unified Approach for Calculation of Strength Reduction Factors, 𝜙

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• Reduction factor as a function of 𝜀𝑡

• Note: The greater the 𝜀𝑡, the more economical will the section be.

Calculation of Nominal Moment-The Strain Compatibility Approach

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• Increase in lateral load results in progressive increase of tensile strain, ultimately cracking at bottom. This happens when

• After M>Mcr significantly , the member starts to behave like reinforced concrete member.

• 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

• Strain in Prestressing Steel at Different Stages – Due to Prestressing:

– At Decompression:

– At Ultimate Load:

– Consequently:

– The corresponding stress in prestressing steel can then be evaluated.

• Compression in Concrete

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Calculation of Nominal Moment-The Strain Compatibility Approach

• The Bernoulli Assumptions

• Nominal Moment of Rectangular Sections • If 𝐴𝑠

′ = 0, 𝐶 = 𝑇 yields

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Calculation of Nominal Moment-The Strain Compatibility Approach

• Nominal Moment of Rectangular Sections

– If bonded Reinforcement is to be accounted for, C=T yields

– The internal moment couple can then be written as

• 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

• The Solution Algorithm for Rectangular Sections

– Unknowns: 𝑓𝑝𝑠, 𝑓𝑠 & c. Goal: Calculate Mn

– Assume c Calculate 𝜀𝑠 & 𝜀𝑝𝑠 Calculate 𝑓𝑝𝑠, 𝑓𝑠 Evaluate c from

Σ𝐹𝑥 = 0 Check c Iterate if Necessary

or

– Express: 𝜀𝑠 = 𝑓(𝑐) & 𝜀𝑝𝑠 = 𝑓(𝑐) Calculate 𝑓𝑝𝑠, 𝑓𝑠 Constitute

Nonlinear Equation by writing Σ𝐹𝑥 = 0 Solve the nonlinear equation to evaluate c.

• The Solution Algorithm for Flanged Sections – Assume a<tf Check assumption by writing Σ𝐹𝑥 = 0 Analyze as

Rectangular Section if assumption is true.

– 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

• Approximate Evaluation is allowed by ACI 318 if

– Bonded Tendons

– Unbonded Tendons

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Design for Ultimate Strength – Reinforcement Limits

• Minimum Values of Reinforcement Index – Definition

– If percentage of reinforcement is

too small, section will fail premat-

-urely as soon as Mu>Mcr.

– The total amount of pre-stressed &

non-prestressed reinforcement req-

uired by ACI should result in

𝑀𝑢 ≥ 1.2 𝑀𝑐𝑟

Exception: If Mn > 2 ×𝑀𝑢 and Vn > 2 × 𝑉𝑢

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Design for Ultimate Strength – Reinforcement Limits

• Minimum area of bonded non-prestressed reinforcement in beams should be calculated using

• Maximum Reinforcement – If percentage of longitudinal reinforcement is large enough, brittle

failure will result.

– To ensure ductile failure, ACI specifies that 𝜀𝑡 ≥ 0.005 at ultimate load.

– 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 – Reinforcement Limits

• To ensure ductile behavior at ultimate load, the reinforcement index is limited to – For Rectangular Sections with Prestressing Steel only:

– For Rectangular Sections with + and –ve bonded Steel:

– For Flange Sections:

where

– Note that for rectangular section

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Design for Ultimate Strength – Reinforcement Limits

– For flange section

– For over-reinforced prestressed beams, section B.18.8.2 of the code specifies

– 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

• Un-bonded Tendons – Un-grouted Tendons or Asphalt Coated Tendons.

– Stress Concentration do not exist at crack locations of un-bonded Tendons.

– A lesser number of wider cracks exist in un-bonded prestressed members.

– Bonded reinforcement is more important for members with un-bonded tendons as compared to bonded members.

– Bonded reinforcement contributes significantly to increasing the moment strength capacity of the section.

– The bonded reinforcement at the bottom will always be yielding at ultimate moment.

– The expressions presented for bonded prestressed members are equally applicable for evaluation of nominal moment of un-bonded members.

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