Chapter 3 Prestressing with Post-Tensioning
7.3 Strength Limit Verification—Flexure
7.3.2.2 Material Stresses and Internal Forces
Figure 7.20 also shows the stress and resultant forces developed in the cross section at nominal resistance. The compressive force in the concrete is found by integrating concrete stresses from the neutral axis to the extreme compression fiber. Forces in the prestressing steel are found by multiplying the stress in the steel times the area of the reinforcing steel.
Material stress-strain relationships, such as those presented in chapter 2 of this manual, for both concrete and reinforcing steel are required to compute these internal forces.
To simplify the computation of the resultant compressive force in the concrete, the LRFD Specifications permit the use of an equivalent rectangular stress block (Whitney’s Stress Block).
Details of the rectangular stress block as defined in LRFD Article 5.7.2.2 are shown in figure 7.21.
Figure 7.21 – Rectangular Stress Block to Represent Concrete Compression
The actual distribution of concrete stress is replaced by a constant stress equal to 85 percent of the 28-day concrete strength. The depth of the block (a), measured from the extreme compression fiber, is taken as a percentage of the neutral axis depth (c):
(Eqn. 7.29)
Where, εps = strain in prestressing steel (in/in)
εpe = effective strain in prestressing at time of loading (in/in) Δεps = change in strain in prestressing as a result of loading (in/in)
Chapter 7 – Longitudinal Analysis & Design 151 of 369 The relationship between the depth of the stress block and the neutral axis depth is a function of concrete strength. The parameter β1 is taken as 0.85 for concrete strengths equal to, or less than, 4.0 ksi. For greater strengths the parameter is reduced at a rate of 0.05 per 1.0 ksi increase in concrete strength above 4.0 ksi, with a lower bound of 0.65. In equation form this is expressed as:
(Eqn. 7.30)
The computation of the resultant tensile force in the prestressing steel also relies on a representation of the material’s stress-strain relationship. Figure 7.22 shows the comparison of typical stress strain relationships for mild reinforcing and prestressing strand presented in figure 2.10 of this manual.
Figure 7.22 – Comparison of Typical Stress-Strain Relationships for Prestressing Strand and Mild Reinforcing
Mild reinforcing exhibits a bilinear relationship with a well-defined yield strain. The relationship of strains and stresses in the prestressing steel are assumed linear for stresses up to 90
percent of the ultimate strength of the strand. Beyond this level of stress the relationship is highly nonlinear.
LRFD Article 5.7.3.1 provides an equation to determine the stress in the prestressing steel at nominal resistance for cross sections where it is appropriate to lump the prestressing steel into a single level (as shown in figure 7.20) and where the effective stress in prestressing steel is not less than 50 percent of specified tensile strength of the prestressing steel. The LRFD expression for stress in the prestressing steel under these conditions is:
(Eqn. 7.31)
Where, fps = average stress in prestressing steel at nominal resistance (ksi) fps = specified tensile strength of prestressing steel (ksi)
c = depth from extreme compression fiber to the neutral axis (in)
dp = depth from extreme compression fiber to the centroid of the steel (in)
The parameter k in equation 7.31 relates the stress in the prestressing steel to the type of prestressing steel that is being used. This variation is related to the ratio of the yield stress of the prestressing steel to its specified tensile strength:
(Eqn. 7.32)
LRFD Table C5.7.3.1.1-1 provides values for the ratio of stresses in equation 7.32. The great majority of prestressing steel used for cast-in-place concrete box girder construction is low relaxation steel with a specified tensile strength of 270 ksi. The corresponding values from Table C5.7.3.1.1-1 are:
(Eqns. 7.33)
So that for 270 ksi, low relaxation steel,
(Eqn. 7.34)
When the arrangement of prestressing steel is such that the steel cannot be lumped into a single layer, another relationship between prestressing steel strain and stress must be used.
An often used source for estimating the stress in the prestressing steel is found in the Precast Concrete Institute Design Manual. Figure 7.23 shows a reproduction of the stress-strain relationship presented in Design Aid 11.2.5 of the PCI Design Handbook, 6th Edition.
Chapter 7 – Longitudinal Analysis & Design 152 of 369
Chapter 7 – Longitudinal Analysis & Design 153 of 369 Up to a strain of 0.0086, the stress in the prestressing steel varies linearly as a function of the modulus of elasticity of the steel—in this case 7-wire strand. This level of strain corresponds to a stress of approximately 90 percent of the specified strength of the strand for the case of 270 ksi steel. When the strain in 270 ksi steel is greater than 0.0086 the relationship between stress and strain is:
(Eqn. 7.35)
Figure 7.23 –Stress-Strain Relationships for Prestressing Strand (PCI Design Handbook, 6th Ed.)