Flexural Analysis of Pre-stressed Concrete

Civil Engineering Department

Flexur Flexural al Analysis Analysis of Prestressed Concrete Members

Fawad Muzaffar M.Sc. Structures (Stanford University) Ph.D. Structures (Stanford University)

Basic Concepts‐Reinforced vs. Prestressed Members •

  Difference Between Reinforced and Pre‐Stressed Concrete  –    Stresses

in Longitudinal Reinforcement

 –    Control

of Cracking of  Cracking and Deflection

 –    Control

of Stiffness of  Stiffness of the of  the Member

 –    Internal

Moment Arm

Internal Stress State of Prestressed of  Prestressed Beam Fawad Muzaffar

Internal Stress State of Ordinary of  Ordinary Reinforced Beam 2

Basic Concepts – Effects of Eccentricity •

The effect of  Eccentricity

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Basic Concepts – Some Terminologies

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Analysis for Flexural Effects – The Basic Concept •

  Methods of Analysis for Flexural Effects  –    Basic

•

Concept Method, C‐Line Method, Load Balancing Method

  Basic Concept Method  –   

Stress at Transfer of Prestress

 –    Stress

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due to Prestress + Self Weight Dead Load

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Analysis for Flexural Effects – The C‐ Line Method  – 

•

At Service Load

The C‐Line Method  – 

Pre‐stressing Force is Treated as External Force

 –    Equilibrium

Equaons (∑M=0; ∑H=0) are used to analyze for force

effects. Fawad Muzaffar

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Analysis for Flexural Effects‐ The C‐Line Method •

  Moment Arm in Prestressed Members

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Analysis for Flexural Effects – The C‐ Line Method •

  Derivation of C‐Line Expressions  –    From ∑M=0; ∑H=0

Or From Figure Plugging value of ‘a’ results in

or e=M/T‐e’ Fawad Muzaffar

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Analysis for Flexural Effects – The C‐ Line Method Derivation •

  From Basic Concept =>

•

  Also: r= (I/A), e = a ‐ e’

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Analysis for Flexural Effects – The C‐ Line Method Derivation •

  From Basic Concept = >

•

  Also: r= (I/A), e = a ‐ e’

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Analysis for Flexural Effects – The C‐ Line Method (Key Takeaway)

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Analysis for Flexural Effects – The Load Balancing Method Using C=T=Pe With

•

The Load Balancing Method  –    Developed  –    Useful  – 

by T.Y. Lin – Applicable to Non‐Straight Tendons Only

in Analysis of Indeterminate Prestressed Structures

The Basic Concept

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Analysis for Flexural Effects – The Cable Equation •

  From Equilibrium

which Indicates that ‐‐Equation 1 •

  Assume  to remain constant for infinitely small fraction of cable

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Analysis for Flexural Effects – The Cable Equation •

  Eliminating T from above equations

•

  Differentiating the above expression results in the following ‐‐Equation 2

•

  Comparing Equations 1 and 2 results in

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Analysis for Flexural Effects‐The Load Balancing Method  –   Load‐Balancing

Method for Parabolic Tendon Profile

Tendon Profile: From Boundary Conditions at x=0 i. e. y=0  and dy/dx=0 

At x=l/2

Differentiating the Profile expression twice, results in

Or From Calculus

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Analysis for Flexural Effects‐The Load Balancing Method  –   Flexural

Stress When Balanced Load is Applied

 –   Flexural

Stresses due to Unbalanced Load

where Fawad Muzaffar

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Variation of Stress in Pre‐stressing Steel with Loading

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Bonded vs. Un‐bonded Tendons, Adoption of Sectional Properties

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Analysis for Flexural Effects‐Analysis Example •

  Example:

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Analysis for Flexural Effects‐Analysis Example •   Example

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Analysis for Flexural Effects‐Analysis Example •

The Basic Concept Approach

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Analysis for Flexural Effects‐Analysis Example

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Analysis for Flexural Effects‐Analysis Example •

The C‐Line Method

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Analysis for Flexural Effects ‐ Analysis Example •

The Load Balancing Method

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Analysis for Flexural Effects ‐ Analysis Example

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