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Rectangular Column |
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Rectangular Column |
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This program provides analysis of rectangular concrete columns loaded with axial loads and uniaxial bending moments. The program uses applied loads, reinforcing, material strengths, and slenderness criteria to perform a complete numerical analysis of the column for ACI load combination equations 9-1, 9-2, and 9-3.
Axial loads can be applied at an eccentrically from the column centerline. Concentrated moments can be applied at the top or bottom. Lateral point and distributed loads can also be applied to the column, allowing the column to be analyzed with any degree of bending effects. All applied axial and bending loads can have dead, live, and short term components.
The evaluation of slenderness effects is provided by entering sidesway bracing status, effective length factors, and column heights. ACI section 10.11 is used to determine a moment magnification factor which will be applied to the resultant eccentricity. Up to five layers of reinforcing can be specified in the column, each with its own d distance. This provides the ability to analyze columns with different bar layouts (see below).
Basic Usage
| • | Design Data Section allows you to enter material strengths, data effecting short-term load combinations, and slenderness data. Seismic Zone can be set to 0" to indicate wind, or 1 thru 4 to indicate seismic zones. For seismic zone 3 and 4, the additional load factoring provision of UBC section 2625 will be used. You can also specify whether the live load should be used in short-term load combinations. |
| • | Slenderness factors describing the column bracing and effective length factors effect the actual length used to calculate column slenderness. |
| • | In the Column Data section, enter the depth, width, and height of your column. The height will be used to calculate transverse column moments, due to applied distributed and point loads, and also for actual column length used for slenderness. |
| • | You can enter up to five rebar sets for each column. These can be used to place reinforcing in different locations within the column. All sets allow you to specify bar #, count, and a d distance from the compression face (which, for reference of axial eccentricity and applied loads, is the right side of the column in the examples). |
| • | Applied Loads include the axial and bending loads applied to the column. Because of the variety of loads available, sign convention is important. |
| • | Axial Eccentricity: Positive eccentricity indicates the load is placed to the left of the column, applying a clockwise moment to the top of the column. |
| • | Applied Moments: Positive moment applies a clockwise moment to the column. |
| • | Applied Loads : Positive point and distributed loads force column's mid-span to deflect to the left, inducing tension on the left-hand fibers. (The deflection signs are created by applying the axial load at a positive eccentricity.) |
Unique Features
| • | Evaluates column slenderness and strength for all three ACI load factor combinations. |
| • | Up to five layers of reinforcing are allowed at various depths in the column. |
| • | Complete slenderness analysis is provided. |
| • | Ability to combine beam type loads with eccentric axial loads. |
Assumptions & Limitations
| • | Columns may be COMPRESSION ONLY....no tension. |
| • | Determines an equivalent axial load and eccentricity for use in analyzing the concrete section. Applied moments, distributed loads, and point loads are combined with moments induced from axial load eccentricity, to form a moment diagram at 250th points along the column. The maximum moment is obtained from this analysis and used to calculate an equivalent eccentricity of the axial load. This eccentricity is then factored according to slenderness effects and used to calculate column load capacity. |
| • | The program compares actual load and eccentricity with Ecc. Balanced to determine whether a compression or tension failure mode will control design strength. |
| • | Column capacity assumes the code required lateral ties are provided. |
Example
The data entry for this example is shown in the screen captures that accompany the Data Entry Tabs and Results & Graphics Tabs sections to follow. Here is a sketch of the example:
Data Entry Tabs
This set of tabs provides entries for all input in this calculation. While you are entering data and switching between these tabs you can view the desired resulting information on the tabs on the right-hand side of the screen (calculated values, sketches, diagrams, etc.). A recalculation is performed after any entry data is changed. After each data entry you can view the results on the right-hand set of tabs.
General Tab
Column Width
Enter the column width here.
Column Depth
Enter the column depth. Moments applied to the column will create tension and compression on the outer edges of the "depth" dimension.
Total Height
This is the total height of the column and is only needed if you are entering applied lateral loads to the column. If you are this length is needed to perform the beam analysis to determine the moments along the column's length.
Unbraced Length
Enter the unbraced length of the column that should be use in the calculation of column slenderness.
f'c
Specified 28 day compressive strength of concrete to be used.
Fy
Yield strength of reinforcing steel to be used.
Seismic Zone
This item specifies whether the short term loads are due to wind or seismic forces. Entering a 0" indicates that short-term loads are due to wind. Entering 1" to 4" specifies that the short-term loads come from UBC seismic zone loads. Additionally, when zone 3 and 4 are used, the special load factoring for that zone required by the UBC are used.
Include Live Load w/ Short Term Loads
This entry controls whether or not live loads are used when ACI load combination C-2 calculations are executed. When short-term loads are due to wind forces, live loads are typically used, whereas seismic forces usually ignore live load effects (except high storage loads).
Reinforcing Specification
Enter your reinforcing selection in this area. You have the option of entering up to five layers of rebar, each with its own size, quantity, and depth. The concrete section analysis procedures will use the total area and d distance to each layer to determine internal strains and forces.
Bar Size #
Enter the bar size in each layer. The size # and count will determine the reinforcing area in each layer.
Bar Count
Enter the number of bars in each layer.
d - Depth to Rebar
Enter the distance from extreme compression/tension fiber to the centerline of reinforcing layer.
Effective Length Factor
This factor is applied to the unbraced length to determine Lu. The common reference for k factors can be found in the AISC handbook, Figure C1.8.1
Unbraced Column ?
Leave this box unchecked if the column is braced against sidesway. Check this box , if it's free to deflect laterally when loaded.
When the column is unbraced, the effective length used to calculated slenderness will be adjusted. For unbraced columns, a factor must also be entered to magnify eccentricity further. Please see the explanation for M2S * DeltaS.
For braced columns, slenderness is neglected if:
k LU/r < ( 34 - 12(M1/M2) Where M1>M2
For unbraced columns, slenderness is neglected if:
k LU/r < 22.
Delta:s
For unbraced columns, ACI equation 10-6 requires that a magnification factor to account for the entire frame systems potential to drift be specified. Enter that factor here, and it will be added to the magnification factor calculated in the program, before application to the design eccentricity to calculate final eccentricity.
Load Entry Tab
This section allows you to enter loads on the column to be included in the analysis. Using the applied moments, point loads, distributed loads, and moment induced by axial load eccentricity, a moment diagram for the column is internally constructed. The maximum moment in the column is then used to determine an equivalent eccentricity of the applied axial load. This eccentricity is used for all further analysis procedures to determine neutral axis location, internal strains, and compressive capacity. All applied loads may have dead, live, and short term components. Load factoring of short-term components is governed by the Seismic Zone entry.
Axial
Enter the maximum axial dead, live, and short-term loads and moments the column will be subjected to. Do not apply load factors to these loads.
Applied Loads
Eccentricity For the previously entered axial loads, you may specify an eccentricity to the column Depth centerline. This induced top moment will be combined with the other moments to determine final moment and load eccentricity. A positive eccentricity applies a clockwise moment to the top of the column.
Moments
You may enter a concentrated moment applied to the top or bottom of the column. Positive moments apply a clockwise torque to the column.
Uniform
Up to two full or partial length uniform loads can be applied to create bending moments between supports. Positive uniform loads act to deflect the column to the left (side of negative axial eccentricity).
Point
Up to two concentrated loads can be applied to create bending moments between supports. Positive uniform loads act to deflect the column to the left (side of negative axial eccentricity).
ACI Factors Tab
This tab specifies the load factors to be used by the program when calculation the factored dead, live, and short term loads to be used in the internal load combinations for determining Pu and Mu for the column.
Results & Graphics Tabs
This set of tabs provides the calculated values resulting from your input on the "Data Entry Tabs". Because a recalculation is performed with each data entry, the information on these tabs always reflects the accurate and current results, problem sketch, or stress/deflection diagram.
Results Tab
The program performs a detailed analysis of the column for ACI load combinations C-1, C-2, and C-3. Using the applied loads, an equivalent eccentricity is calculated and then factored by the Moment Magnification Factor which was evaluated considering slenderness effects. This eccentricity is used to calculate internal strains and forces. These forces are resolved into allowable axial capacities. A satisfactory design results when Pu <= Pn * Phi for all three ACI load combinations.
Pu : Max Factored Axial Load
Maximum factored axial load for each ACI equation.
Pn * Phi @ Design Eccentricity
Maximum compressive load capacity of the column considering size, reinforcing, and slenderness. Phi equals 0.70 since this program deals with rectangular tied columns.
Final Eccentricity
This eccentricity is obtained by calculating an equivalent eccentricity at which the axial loads would have to be applied to create the same maximum moment as created by all applied bending loads. Minimum eccentricity is 0.6 + 0.03*Diameter.
Magnification Factor
This factor is determined by slenderness effects (see next section).
Design Eccentricity
This is the eccentricity used for determining column capacity, and is equal to Final Eccentricity * Magnification Factor.
Po * .80
This is the column capacity if no slenderness or bending were effecting the column.
P : Bal & Ecc. Bal.
Balanced load and eccentricity are calculated to verify whether the column is working in a Tension Failure or Compression Failure mode. This load, when applied at the Balanced Eccentricity , produces simultaneous ultimate strains in concrete and steel. This value is given only for reference.
Slenderness Effects
Actual k*Lu/r
This value will be used for the column slenderness calculations. k and Lu have already been entered. r, the radius of gyration, is equal to 0.288*Depth.
Beta
This is the typical concrete factor: Beta=0.85 - 0.05(f'c-4000)/1000
Elastic Modulus
This is calculated as: 57,000 * (f'c)½ Because the neutral axis distance varies according to the Design Eccentricity, the slenderness analysis is completed for all ACI load combinations.
Neutral Axis Dist.
The neutral axis distance represents the point at which compression and tension forces are balanced for the section and design eccentricity.
Phi
The Phi factor is calculated by balancing the effects of axial and bending loads in accordance with ACI section 9.3.2.2.
Max. K Lu/r to Neglect
The maximum allowable slenderness ratio is calculated in accordance with ACI 10.11.4, considering unbraced length, effective length factor, radius of gyration, and bracing status.
Beta
Ratio of factored dead load moment to factored total load moment.
Cm
CM is determined in accordance with ACI 10.11.5.3.
EI
This represents the minimum value of ACI equation 10-10 and 10-11. (ACI states that either value can be used, so both are calculated and the minimum taken to be conservative).
Pc
Critical buckling load, calculated using equation 10-9.
Alpha
This value will be used to calculate the actual magnification factor, and is the ratio of Pu/(Phi *Pc) from ACI equation 10-7.
Delta
Actual moment magnification factor calculated using ACI equation 10-7.
Eccentricity
Please see Final Eccentricity in the section titled Summary.
Design Ecc.
Please see Design Eccentricity in the section titled Summary.
Sketch Tab
This tab provides a sketch of the beam with loads and resulting values shown. Using the [Print Sketch] button will print the sketch in large scale on a single sheet of paper.
Diagrams Tab
This displays a moment, shear, and deflection diagram for the beam with the applied loads and end conditions. Note the two tabs...."Graphic Diagram" and "Data Table". The Data Table tab provides the entire internal analysis at the 1/500th points within the beam.
Printing Tab
This tab allows you to control which areas of the calculation to print. Checking a box will signal that the information described by the item will be printed. However, if there is no information in for a particular selection it will not be printed. So these checkboxes are best described as "If this particular area of the calculations contains data then print it".
Sample Printout
