Deflections

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.

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.

Cracking

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:

The above procedure is done on a point by point basis. The moment is split by the maximum percentage that is cracked along the member. IE if midspan moment is 20% above Mcr, 20% of all stage loading will be applied to the cracked section.

Here, only the red part of the moment curve will be applied to the cracked section.

Deflection Multipliers

Deflection multipliers are applied to the immediate deflections caused by sustained loads.

Prestressed Deflection Multipliers

By default, Eriksson Beam uses the deflection and camber multipliers presented in the PCI Design handbook. Based on Shaikh, A. F., and D. E. Branson. 1970. “Non-Tensioned Steel in Prestressed Concrete Beams,” these multipliers can be reduced by the addition of mild reinforcement. Using this method, a given multiplier is reduced using the following procedure:

where A is the old deflection multiplier and B is the reduced multiplier.

Mild Deflection Multipliers

Mild deflection multipliers are calculated using Eq. 24.2.4.1.1 (shown below). For this method, each load combination computes it’s own reinforcement ratio. In addition, the time factor is set to 2 for final loading and all other other stages.

where is the deflection multiplier

             is the time factor, and

             is the ratio of compression reinforcement to the area of concrete.

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

Hand Calculation

Below is a hand calculation and the corresponding Eriksson Beam file showing the procedure the program uses to calculation deflections, cambers, and shortening. Note that the hand calculation assumes that when the member cracks, a percentage of the total load is carried by the cracked section. This is similar to the procedure shown in the PCI Design Handbook and results in a conservative estimation of the deflection of a cracked section. Following this approach, the total moment being applied to the cracked section is shown below as shaded in red. It is important to note that the total moment applied to the cracked section as calculated by hand is larger than the total moment applied to the cracked section as calculated by Eriksson Beam, which calculates the this total moment on a point by point basis as shown above.

Because of this, any loading stage that cracks results in larger deflections in the hand calculations than in Eriksson Beam.

References

PCI Industry Handbook Committee, PCI Design Handbook, 8^th  Ed., PCI, Chicago, 2017.