Eriksson Beam calculates deflections using the bi-linear method in addition to the deflection mutlipliers present in the PCI Design Handbook. In general, four items needed for accurate camber/deflection prediction:
Accurate knowledge of material properties, namely the elasticity of the concrete, preferably on an hourly basis, daily basis is acceptable
Accurate knowledge of the prestress losses, again preferably on an hourly basis, again daily is acceptable, use of a time-dependent method for prestress losses is required (not ‘lump sum’)
Changes in loading and support conditions, again at least daily
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
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
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Stage
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Loads
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Material Properties
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Section Properties
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1
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Self Weight + Prestress
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Initial
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Gross
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2
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Non-Composite DL
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Initial
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Gross
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3
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Topping Weight
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Final
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Gross
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4
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Composite DL
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Final
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Composite
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5
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All Others
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Final
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Composite
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:
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The above procedure is done on a point by point basis. This results in splitting the moment curve at the cracking moment and applying part of it to the gross section and part of it to the cracked section, as shown.
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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:
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where A is the old deflection multiplier and B is the reduced multiplier.
Mild Deflection Multipliers
Mild deflection multipliers used ACI-318 24.2.4.1.1. For this method each load combination computes it’s own reinforcement ratio and the time factor is set to 2 for final loading, and all other other stages.
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This deflection multiplier is applied to all dead loads on the members.
Shortening
The shortening of members is caused by prestressing, shrinkage, and from flexure. The shortening at the centroid of the member is calculated using
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The modifiers are calculated based on information from ACI 209. For this calculation, the elastic and inelastic shortening are added together. The inelastic shortening of the concrete is a combination of the creep of the concrete caused by the continued application of the prestressed force, along with the shrinkage of the concrete. This value is the shortening of the member that occurs between casting and erection of the precast member and can be assumed to be constant over the height of the cross section.
Creep Modifier
The creep modifier is a multiplier applied to the elastic shortening from first principals.
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The creep coefficient, Cu, is defined as the ratio of the creep strain to the initial strain. A typical value for hardened concrete, according to ACI-209, is 2.35. This value is further modified by the volume to surface ratio (V/S) of the member and the relative humidity (RH) as follows:
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Shrinkage Modifier
The shrinkage modifier is used to account for the member shrinking. This is produced using the ultimate strain in the concrete produced by shrinkage of the concrete. A typical value, according to ACI-209, is 780x10-6. This strain is used to estimate the total shortening as follows:
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The shrinkage strain is further modified by the volume to surface ratio (V/S) of the member and the relative humidity (RH) as follows:
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Flexure Effects
Flexure also effects the shortening at the ends of the member. For a member cambering up, this will make the top corners move further apart and the bottom corners moving closing together. This can be computed using some simple trigonometry as follows:
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Hand Calculation
Below is a hand calculation and the corresponding Eriksson Beam file showing the procedure Eriksson Beam using to calculation deflections, cambers, and shortening. Not Note that in the hand calculation, it 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 conservative results. Following this approach you can see the total moment being applied to the cracked section, shown below as shaded in red, is much larger than doing it on a point by point basis as shown above.
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