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Code References

ACI 318-14

Background

In general, four items needed for accurate camber/deflection prediction: 

1. Accurate knowledge of material properties, namely the elasticity of the concrete, preferably on an hourly basis, daily basis is acceptable. 

2. Accurate knowledge of the prestress losses, again preferably on an hourly basis, again daily is acceptable. Note that the use of a time-dependent method for prestress losses is required (not ‘lump sum’). 

3. Changes in loading and support conditions, again at least daily. 

4. System of equations that ties all of this together. 

From PCA Notes on ACI 318: “Because of the variability of concrete structural deformations, designers must not place undue reliance on computed estimates of deflections.  In most cases, the use of relatively simple procedures for estimating deflections is justified." 

Bi-Linear Deflections

Eriksson Beam calculates deflections using the bi-linear method in addition to the deflection multipliers present in the PCI Design Handbook. When using the bi-linear deflection methods, the member allows the user of the gross section properties up until the point the member cracks. At that point the rest of the applied load goes to the cracked section. Because of this, the loads are applied in stages where each stage is checked for cracking. If at any point the member cracks, the stage is split into a pre-crack and post-crack analysis step where their results are combined using superposition. This methodology can be summarized as follows:

1. Compute I_g for each analysis section.

2. Compute total moment and total stresses for each loading stage for each analysis section.

3. Compute deflections for each loading stage at each analysis section, using either I_g, I_cr, or a combination of the two. E_ci is used for the 1^st loading stage, E_c is used for all others.

   1. If an analysis section cracks during a loading stage, (the applied moment exceeds M_cr) compute the point within the loading stage where the section cracks. Then compute I_cr for this loading stage and for every loading stage after this one.

   2. For the loading stage where an analysis section cracks, two deflections are calculated. One using the portion of the applied load below cracking in conjunction with I_g, and a second deflection using the portion of the load above cracking in conjunction with I_cr, and then sum them.

4. Multiply the initial and erection stage camber/deflections by the creep and shrinkage multipliers in Table 5.9.2 in the PCI Design Handbook. Note that these multipliers can (and probably should) be reduced based on the ratio of the areas of the mild and prestressed reinforcement present at that analysis section (see the section on the Creep and Shrinkage Multipliers for a detailed discussion of these modifiers).

5. Sum the deflections obtained at each loading stage to calculate the total deflection.

Loading Stages

Eriksson Beam uses 5 loading stages

The section properties will change to the cracked section properties if the applied moment exceeds the cracking moment. However, only sections under a positive moment will crack. In example, if Stage 1 cracks along the top of the member from the prestressing, the stage will still use gross section properties at that location.

Cracked Section Analysis

When a given stage cracks, the stage is split into two sub stages, a before cracking stage and a after cracking stage. For these two stages the loads are divided by first computing how much moment can be applied before cracking as follows:

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Beam uses a bi-linear deflection approach (I_g/I_cr) to calculate deflections at sections where the applied M is greater than M_cr during a loading stage.  Only that portion (or %) of M which is greater than M_cr is applied to I_cr, while the remainder of M is applied to I_g.  In the spirit of ACI E24.2.3.7 when the midspan cracks, the program computes I_cr at every section where the member is NOT Class U, using the procedure outlined in Robert Masts’ “Analysis of Cracked Prestressed Concrete Sections: A Practical Approach,”.  The program then calculates the % of load that is applied to I_cr at the first section that cracks (typically at or near midspan) and applies this % of load to every section where I_cr was just calculated.

Creep and Shrinkage Multipliers

For all sections at erection, there are two creep and shrinkage multipliers:

1. Deflection (downward), applied to the elastic deflection due to the self-weight at release of prestress

2. Camber (upward), applied to the elastic camber due to the prestress at the time of release of prestress

For non-composite sections at final, there are three additional creep and shrinkage multipliers:

1. Deflection (downward), applied to the elastic deflection due to the self-weight at release of prestress

2. Camber (upward), applied to the elastic camber due to prestress at the time of release of prestress

3. Deflection (downward), applied to the elastic deflection due to superimposed dead load only 

For composite sections at final, there are four additional creep and shrinkage multipliers:

1. Deflection (downward), applied to the elastic deflection due to the self-weight at release of prestress

2. Camber (upward), applied to the elastic camber due to prestress at the time of release of prestress

3. Deflection (downward), applied to the elastic deflection due to superimposed dead load only

4. Deflection (downward), applied to the elastic deflection caused by the composite topping

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In ACI 318-71 9.5.2.3, the base factor used to modify initial deflections to account for long-term effects was calculated using the following equation:

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Adding 1 to this value (to account for the immediate deflection), gives us 3.00 for this factor.  This factor is applied to the immediate deflection caused by the superimposed dead load.

Modification to Creep and Shrinkage Multipliers

From Shaikh, A. F., and Branson D. E., 1970, “Non-Tensioned Steel in Prestressed Concrete Beams”, the creep and shrinkage deflection multipliers can be reduced by the addition of mild reinforcement. Using this method, a given multiplier is reduced using the following equation:

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               A_ps = area of prestressed reinforcement

Mild Deflection Multipliers

Mild deflection multipliers are calculated using Eq. 24.2.4.1.1 (shown below).  For this method, each load combination computes its’ own reinforcement ratio.

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This deflection multiplier is applied to all sustained loads on the members.

Hand Calculation

References

Martin, L.D., “A Rational Method for Estimation Camber and Deflection of Precast Prestressed Members,” PCI Journal, V. 22, No. 1, January-February 1977, pp. 100-108.

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