|
Torsional Analysis of Rigid Diaphragm |
|
|
|
Torsional Analysis of Rigid Diaphragm |
|
|
This module provides horizontal force distribution analysis for a rigid diaphragm laterally supported by up to 160 resisting elements (walls, columns or generic resisting elements). An applied shear (and an optional orthogonal force) is applied to a center of mass location, and that force is distributed to all elements after the rotational stiffness analysis has been completed.
All lateral forces are distributed to each element on the basis of relative rigidities and resisting element locations. Lateral shear forces, induced torsional forces, and minimum eccentricity are considered after determining the location of the center of rigidity.
The module provides analysis for one level only. For structures where elements are symmetrically placed on many levels, a calculation may be performed for each level and results added to determine shears and overturning moments for each element. When determining center of mass (where the lateral force is applied) on successively lower levels when elements are NOT all aligned vertically, a new center of mass position should be calculated based upon element forces acting from the diaphragm from the level above and combined with the force at that level.
A very unique capability of this module is to have the applied lateral load applied at angular increments for a full 360 degree rotation. The prior version of this module in our Version 5.8 software only applied the lateral load at 90 degree increments. Because seismic or wind loads can occur at any angle, we provide the ability for the user to define the angles at which the lateral load is applied to the rigid diaphragm for distribution to the resisting elements.
Another feature is a more advanced application of minimum force eccentricity. The code specifies that a minimum eccentricity must be used to increase the torsional effect created by the moment arm between the point of load application and the center of rigidity. The minimum eccentricity is specified as 5% of the building dimension measured perpendicular to the direction of load application. This module creates an ellipse (measuring 5% of the building dimension on each axis) along which the lateral load is applied.
So to recap.....the applied lateral load is applied at the angular increments you specify for a full 360 degrees, and this is performed for the number of angular locations you specify around the minimum eccentricity ellipse. This means if you use 15 degree angular increments for load direction and 15 degree increments for accidental eccentricity, then the lateral load is actually applied in (360/15+1) * (360/15+1) = 625 locations. This can provide a very accurate calculation of applied torsions and direct shears to all resisting elements connected to a rigid diaphragm.
Basic Usage
| • | The most important step for successful use of this module is to properly enter the X and Y location of the center of rigidity of each resisting element and its angle in degrees counterclockwise from a normal Cartesian "0" degree orientation. |
| • | For each resisting element, its center of rigidity will be at the centroid of the wall or column section. |
| • | The angle is measured with respect to the centerline of the length measurement (long dimension). 0 degrees and 180 degrees defines a wall's angle as parallel to the X-axis. 90 degrees and 270 degrees defines a wall as being parallel to the Y-axis. The angle increases positively in a counterclockwise direction. |
| • | Lateral shears are typically the force at the diaphragm level due to wind or seismic forces at that level. Distance to Center of Mass specifies the X-Y location where the lateral shears act. If lateral forces must be added to the diaphragm from the level above or below, you must combine all forces to calculate an adjusted mass application point. Maximum Dimensions are used to calculate the minimum additional eccentricity that will be added to and subtracted from the inherent eccentricity to calculate governing forces for each resisting element. |
| • | When defining walls as resisting elements, the thickness, length, and height are required for each wall providing lateral support to the diaphragm. These values are used with the elastic modulus to establish the relative stiffness of each wall. For other resisting elements you can enter the section information or just enter the resisting element deflection under the same load for all elements. |
| • | The Elastic Modulus does not have to be an exact value if all of the elements are of identical construction. In this situation, it may be simpler to just use a value of 1. |
| • | X & Y Distances for each resisting element define the location of the center of stiffness of each element in plan view. This location will be used when combining all stiffnesses and calculating the overall center of rigidity for all elements acting as a system. |
| • | When using walls or columns as resisting elements, an inclination angle can be entered for each element. For walls, this angle describes the orientation of the long axis of the wall, measured in degrees counterclockwise from a normal Cartesian "0" degree orientation. For columns, this angle describes the orientation of the X axis of the column, measured in degrees counterclockwise from a normal Cartesian "0" degree orientation. Enter all angles as positive. |
| • | Enter the fixity condition that best describes the element's top and bottom restraint against rotation about the longitudinal and transverse axes. FP (Fix/Pin) would be appropriate for an inverted pendulum condition (where walls or columns cantilever up from a fixed base condition, but are free to rotate at their tops). FP would also be appropriate for a moment frame structure with pinned column bases (a structure that behaves like a table). The FF (Fix/Fix) would be appropriate for conditions where both the tops and the bottoms of the columns and/or walls are fixed against rotation about their longitudinal and transverse axes. This setting results in double curvature in the vertical lateral force resisting elements. |
Unique Features
This module uses a numerical approach to determine center of rigidity location and to distribute lateral forces to each resisting element. Because resisting elements may be located at any angle, a rigorous stiffness analysis is performed, calculating each element's stiffness about both axes and combining the stiffnesses of all the elements to determine a center of rigidity location.
Assumptions & Limitations
Because this program performs a very complex stiffness matrix analysis for all resisting elements, the traditional method of listing separate components of direct and torsional shears is not applicable. Also, the module internally adds and subtracts the additional accidental eccentricity (based on both maximum dimensions) about each axis to calculate maximum force for each element. This results in one final force value being displayed for each wall.
Coordinate System
Please note that a STRICT X-Y coordinate system should be used to ensure that the analysis is properly carried out. When setting up an X-Y coordinate axis, please follow the standard Cartesian model with the diaphragm located such that X increases to the right and Y increases up the screen. Unless another method is necessary, this will perform very well (but the module can handle variations).
General Input Tab
Applied Lateral Force
This is the main force applied to the rigid diaphragm. The location of application is specified in the input item labeled Location of Shear Application.
Additional Orthogonal Force
This is an optional force that is applied at a 90 degree angle to the main force. Some codes specify that this force must be applied along with the main force.
Maximum Load Used for Analysis
The final force applied to the diaphragm is sqrt(Main2 + Orthogonal2).
Load Angular Increment
This module allows the force to be applied in almost all angular directions to the rigid diaphragm.
According to the entry for angular increment, the module will apply the load to the diaphragm at multiple angular increments. For example, if you select "15 deg", the module will apply the "Maximum Force Used for Analysis" at 0, 15, 30, .... 360 degree angles. When the Load Angular Increment is set to smaller values, it will result in slightly longer calculation times, but it wall also allow the module to "zero in" more accurately on the actual maximum shear forces in all of the resisting elements.
Note that there is also an option named "Specify". This allows you to very precisely specify a singular direction for the application of load.
Accidental Eccentricity Angular Increment
Most building codes require the consideration of an "accidental eccentricity". This is a prescribed additional amount of moment arm that must be compounded with the inherent eccentricity that already exists in the system; i.e. the distance between the center of rigidity and the center of mass for seismic loads or the distance between the center of rigidity and the center of exposure for wind loads. This additional eccentricity accounts for the variability of the exact location of applied load in normal as-built conditions.
Normally an "X direction" and a "Y direction" accidental eccentricity would be determined as a function (typically 5%) of the overall building dimension in each direction. Then, the X directed force would be applied at two locations:
| • | center of mass PLUS "Y direction" eccentricity, and |
| • | center of mass MINUS "Y direction" eccentricity. |
And the Y directed force would be applied at two locations:
| • | center of mass PLUS "X direction" eccentricity, and |
| • | center of mass MINUS "X direction" eccentricity. |
However, in this module the "X direction" and "Y direction" eccentricities are used to specify the dimensions of an ellipse that encircles the center of mass. This ellipse creates a continuous path that smoothly incorporates the "X direction" and "Y direction" eccentricities. In this way, it defines all possible locations where the load should be applied to account for all possible accidental eccentricity locations.
The entry for Accidental Eccentricity Angular Increment specifies the angular increment that will be used to subdivide the ellipse into a number of locations where the force will be applied to the diaphragm.
Summary of Angular Increment & Accidental Eccentricity Angular Increment
The module applies the "Maximum Load" at the "Load Angular Increments" at each location of "Accidental Eccentricity Angular Increment" to generate an extensive set of results from which the maximum force values for each resisting element may be inspected.
For example, setting both "Load Angular Increment" and "Accidental Eccentricity Angular Increment" to 15 degrees tells the module to run (360/15 + 1) * (360/15 + 1) = 625 separate analyses of force distributions to the resisting elements. This is not the highest degree of detail that the program can provide, but it may offer a good balance of accuracy versus analysis time.
Location of Shear Application
This specifies the X and Y location of the center of mass. The Accidental Eccentricity ellipse will be circumscribed around this location.
Accidental Torsion Values
Accidental torsion is defined as a percentage of overall constructed diaphragm dimension in each of two orthogonal directions. Therefore enter the necessary eccentricity percentage and both maximum diaphragm dimensions here.
Resisting Element Tab
Resisting Element Type
This module allows you to use three types of resisting elements. In past versions of this module, only walls were allowed. But many users wanted to enter information for braced frames or cantilevered columns for open buildings. So we've expanded this module to allow more general types of lateral resisting elements.
WALL : Click the [Use a Wall] button to define a wall as a resisting element. The wall must be rectangular in plan and must have a non-zero height. The selections for "Fix" and "Pin" will alter the equation used to calculate deflection in BOTH directions of the wall (unless the option is selected to "Force all MINOR AXIS Stiffnesses to ZERO"). Using the entered height, length, thickness, and modulus of elasticity for bending and shear, the module will calculate the bending and shear stiffness of the wall and report the deflection for a unit 1 kip applied load.
COLUMN : Click the [Use a Bending Member] button to define a column as a resisting element. This will be a linear member whose stiffness is specified simply by its X and Y axis moments of inertia. You must also provide a value for the modulus of elasticity of the column for bending. Finally, you must make a fixity selection, which dictates the equation used to calculate deflection in BOTH directions of the column (unless the option is selected to "Force all MINOR AXIS Stiffnesses to ZERO"). Using these settings, the module will calculate the bending stiffness of the column and report the deflection for a unit 1 kip applied load.
GENERIC ELEMENT : Click the [Use Generic Resisting Element] button to specify a generic resisting element whose lateral deflection is known for an applied 1 kip load. This selection is intended for complex resisting elements like braced or moment frames, where another analysis module has determined the unit deflection. Note that this method of defining a resisting element offers and option named "When Stiffness deflections are 0.00, assume completely flexible". This option can be used if your intent is to specify that an element is completely flexible in a certain direction. In this situation, you would need to specify an infinite deflection in that direction. So as a convenience, the system has been configured such that when this option is selected, it will interpret a deflection value of 0.00 as meaning that the element is completely flexible in that direction (i.e. has no ability to resist an applied force in that direction).
Add & Delete Buttons
Use the [Add] and [Delete] buttons to add a new resisting element or delete the one currently highlighted in the list.
Element Data
This area allows you to specify a label and location of the center of resistance for a resisting element.
Resisting Element List
This is the list that you create to define the resisting element locations that give lateral force resistance to the rigid diaphragm.
This table serves to give a summary of the deflections, location and major axis angle for each element. When you click to highlight a line in the table, the information for that resisting element is brought into the variables on the input area.
Summary Maximum Tab
Please note that a STRICT X-Y coordinate system should be used to ensure that the analysis is properly carried out. When setting up an X-Y coordinate axis, please follow the standard Cartesian model with the diaphragm.
Recall that the module calculates the forces to each resisting element by rotating the force about its point of application. That point of application is in increments around an accidental eccentricity ellipse.
This Summary Maximums tab provides the maximum forces for each resisting element along the major and minor axis of the element.
For the wall named "4" in the image below, the module has examined all of the calculated forces and found that the maximum shear force along the MAJOR axis was 142.060 kips, resulting from the "Maximum Load Used for Analysis" being applied in a direction of 315 degrees. The location at which that force was applied was along the perimeter of the accidental eccentricity ellipse and the X,Y coordinates of that location are (94.98, -57.51) feet from the global datum point.
Similarly for the MINOR axis, the maximum shear force was found to occur when the load was applied at 120 degrees.
Force Summary Detail Tab
This tab provides the main table that shows all of the force calculations for each resisting element. It is tree structured, so clicking the [+] sign to the left of each item name will expand the result set for that item.
In the image below we see that the data for the wall labeled "4" is expanded. Below "Label : 4" we see many lines labeled "0 deg". These are the results for the load applied at an orientation of 0 degrees. On each "0 deg" line, observe that the "X Ecc" and "Y Ecc" values are changing. These values are the locations of the applied load as it moves its way around the accidental eccentricity ellipse. The note at the top of the table indicates that the analysis is based on 15-degree "Eccentricity Location" increments. This implies that there will be (360 degrees/15 degrees) = 24 lines of data based on the "0 deg" force orientation. Then, if we scrolled down through the table, we would see that the load application angle has also been set to change in 15-degree increments as well.
Sketch Tab
Analysis Procedure
Please see the following description for the procedure used to calculate the system stiffness matrix and resolve the forces for each resisting element.
