Deflections
This section explains the assumptions and methodology of the Concrete Beam Deflection calculations. Every effort has been made to include a discussion of significant decisions and assumptions made by the program.
Sign Convention
All upward deflections are assigned a negative sign convention and all downwards acting beam deflections have positive sign. Only local member deflections are calculated as described in the Technical Section of the Concrete Analysis Chapter in this addendum.
Effective Moment of Inertia (Ieff)
| (9-7) |
| = | ||
| = |
The cracked moment of inertia (Icr) is calculated based on the procedure outlined in Notes on ACI 318-99, Portland Cement Association. Icr is calculated considering the developed strength of both compression and tension reinforcement. Where a bar set is not fully developed the area is reduced to account for the undeveloped length. Note that for T-Sections where the top flange is in tension Icr is calculated assuming a rectangular section (i.e. it is assumed the neutral axis is located below the flange depth in the beam web). Ma is the applied moment at the cross-section under consideration. Note that when Mcr > Ma, the gross moment of inertia (Ig) will be used.
Continuous Beam and Cantilevers
A beam's Ieff is calculated at mid-span and both supports (at face of supports). The overall beam effective moment of inertia (Ieff ) is based on the average of the end and mid-span calculated values as defined below and in "Notes on ACI318-99", PCA.
| Ieff = 0.15 Ieff-support1 + 0.70 Ieff-mid-span + 0.15 Ieff-support2 |
| Ieff = 0.15 Ieff-fixed support + 0.85 Ieff-mid-span |
| Ieff = 1.0 Ieff-mid-span |
| Ieff = 1.0 Ieff-support |
Upward and Downward Deflections
When a structure is skip-loaded there is a likelihood of both upward and downward deflections existing on a single beam. This is as described in the section on Design Deflection Curves in the Concrete Analysis Technical Section of this addendum. The upwards and downwards deflections are considered separately when checking deflections. For the calculation of Ieff the program adds the moments (at mid-span and support) for all skip load conditions which contribute to the upward and downward deflection of the beam respectively. The illustration and table below show the values calculated for the second span (bold) from the four skip load cases (Mi1 = moment at left support of span for load case 1, etc)
| Ma (Up Deflection) | Ma (Down Deflection) | |
|---|---|---|
| Left Support | Mi1 + Mi3 | Mi2 + Mi4 |
| Mid Span | Mmid1 + Mmid3 | Mmid2 + Mmid4 |
| Right Support | Mj1 + Mj3 | Mj2 + Mj4 |
The following is in reference to the above illustration. To calculate Ieff at the left support of span 2 for the upward deflection the program adds the left support moments from load cases 1 and 3 (Ma = Mi1 + Mi3). Depending on the sign of Ma at this support the program will calculate Mcr and Icr for the bottom of the section in tension (Ma > 0) or top of section in tension (Ma < 0 ). That is, Ma is calculated based on the direction of the mid-span deflection for each load case. Mcr and Icr are then respectively based on the direction of the Ma. This calculation of Ma, Mcr, and Icr is performed for the supports and mid-span and the beam Ieff is calculated from these data per ACI Eq. 9-7 described above.
Analysis
It is important to note that the analysis is not rerun after Ieff is calculated for each deflection check. That is, the program assumes that the relative stiffness of the beams used in the actual gravity analysis is commensurate with their relative stiffness after the actual Ieff are calculated (so that there would not be significant redistribution of forces if the actual Ieff values were used in the analysis). If this assumption is not adequate the engineer is encouraged to run the analysis using relative beam stiffness that more closely depict the cracked section behavior and the true force distribution.
| Setting | Description | |||||
|---|---|---|---|---|---|---|
| Final Deflections | For all deflection checks the calculated deflection from the analysis is modified to take into account the calculated Ieff and the applied loads. In general (described in more detail with each check below) the design deflection is computed as the deflection calculated in the analysis times the ratio of Ieff to Ianalysis. Ieffwill only differ from the moment of inertia used in the analysis if the user selects to calculate Ieff per ACI in the Criteria-Deflection menu item. For each beam with cantilevers the deflections are computed on each span independently and the span with the larger deflection ratio controls. | |||||
| Deflection Ratio | For each deflection check (Dead Load, Live Load, Long-term, and Net) the user can specify allowable limits in the Criteria-Deflection dialog. For all those deflection checks where appropriate criteria are defined (i.e., there is an allowable absolute deflection (delta) limit or span-to-depth ratio limit) the program will calculate a deflection ratio. The deflection ratio is calculated as the larger of the calculated deflection to the absolute limit specified and the calculated span-to-deflection ratio over the allowable span-to-deflection ratio. For cantilevers the span length is doubled when calculating the span-to-deflection ratio. | |||||
| Dead Load Deflection | The dead load deflection on a member is calculated as described in the Final deflections section above. Ieff for dead load is based only on the moments on the span due to dead load. The dead load deflection check is performed irrespective of the direction (up or down) of the deflection. Note that increasing the quantity of tension reinforcing or the member dimensions will reduce the magnitude of the deflection. | |||||
| Live Load Deflection | The live load deflection on a member is calculated as the deflection due to all Dead Load and Live Load applied, less that due to only Dead Load (Live Load Deflection is not simply the deflection due to Live Load alone). This ensures that the correct Ieff is used when calculating the deflection from Dead Load and Live Load together and the deflection due to Dead Load alone is subtracted from this quantity. Ieff for dead load and live load is based on the moments on the span due to both dead load and live load. For skip load cases the deflection check is performed once for upwards deflection and again for downwards deflection. The controlling case is reported. Note that increasing the quantity of tension reinforcing or the member dimensions will reduce the magnitude of the deflection. |
|||||
| Long-term + Live Load Deflection |
Due to creep and shrinkage the deflection of concrete members continues over the life of the structure. According to the ACI long-term deflection is calculated as the deflection due to dead and some portion of live load times a deflection factor (λ) defined below.
The code defines appropriate values for the time-dependant factor ξ based on the length of time long-term deflection is being calculated for. It is this value (ξ) that the program refers to as the time-dependant factor which the user is required to enter in the dialog box obtained from the Criteria-Deflections menu command. |
Initial and Final Time-Dependant Deflection Factors
The code sets a limit on the acceptable amount of total deflection that should occur after the attachment of non-structural elements (ACI Table 9.5(b)). To facilitate this code requirement the program allows the engineer to determine the long-term deflection (due only to dead load) that occurs prior to the attachment of the non-structural elements and to subtract this amount from the final calculated long-term deflection. Up to half the lifetime long-term dead load deflection can occur in the first three months. Depending on when the non-structural elements are applied the calculated long-term deflection should consider this long-term dead load deflection that has already occurred. The Final Time-Dependant Factor should be measured from day 0 and not from the point in time the initial time-dependant is specified for.
Sustained Load
The sustained load is the live load that is likely to be relatively stable over the life of the building such that it should be considered in the calculated long-term deflection. Ieff fro the long-term load is based on the moments from the Dead Load plus the Sustained Live load. Note that for the sustained loading No skip loading is considered, that is, all the sustained live load is considered to act simultaneously to produce the long-term deflection.
Calculated Deflection
The deflection limits specified by the ACI Table 9.5(b) include not only the long-term deflection but also that deflection due to the immediate application of live load. The deflection calculated for comparison to this limit is as follows:
LT+LL deflection = Final Long-term Deflection - Initial Long-term Deflection + Immediate Live Load Deflection
where
- Final Long-term Deflection considers the dead load and the percentage of sustained live load. It also considers the user specified Final Time-Dependant factor. For skip loaded live loads the upper and lower deflection curves (and associated moments) are summed together. The long-term deflection is therefore based on all the sustained live load being applied continuously (no skip loading is considered).
- Initial Long-term Deflection considers only the dead load and the user specified Initial Time-Dependant factor.
- Immediate Live Load is calculated as described in the standard live load deflection check. Both upwards and downwards deflections are considered separately and added to the long-term deflection for before doing the check. Per Notes on ACI 318-99, Portland Cement Association, the immediate live load portion of the deflection considers all the live load, not just that additional live load over-and-above the sustained percentage of live load.
Net Deflection
According to the last footnote in ACI Table 9.5(b) in some circumstances the engineer may exceed the long-term deflection limit if the total deflection less the camber is within some additional limit. The program defines Total deflection less Camber as the Net Deflection. The Net deflection is calculated as follows where all the deflection magnitudes are as described briefly below:
- Net Deflection = Dead Load Deflection + Live Load Deflection + Long-term Deflection - Camber
| Setting | Description |
|---|---|
| Dead Load + Live Load | Dead and full live load is considered to act concurrently (upwards and downwards live load is considered separately). The two deflection curves are calculated as described the section on Live Load deflection above. However in this check the initial DL deflection is not subtracted from the DL + LL deflection. |
| Long-term Deflection | Long-term deflection is calculated as discussed in the previous section except that the Initial Long-term Deflection is not considered (subtracted from the final long-term deflection). This check considers only the final deflected shape of the section and not the incremental deflection between two points in time. |
| Camber | Where specified and appropriate for the span length the camber is calculated for a beam mid-span only. The camber is taken as the deflection due to dead load times the percentage of dead load to be considered for camber by the user, rounded down to the appropriate increment stipulated by the user. Where the user specified minimum camber is not obtained the program will not call out any camber. Where the maximum allowable camber is exceeded the camber will be set to the maximum user specified limit. No camber is calculated for cantilevers and all camber is assumed to be upward in nature (no downward camber is ever calculated or specified by the program). Camber is also shown on the floor plan DXF output where specified and calculated as being required. |