General
A general second-order elastic analysis requires consideration of effects caused by displacements between brace points (P-delta) and effects caused by displacements of brace points (P-Delta). Ram Frame uses a geometric stiffness method to approximate P-Delta effects. P-Delta can be turned on/off in Ram Frame – Criteria – General. This method does not capture P-delta effects in the analysis. P-delta effects can be accounted for in the steel post processor by using the B1 factor in the AISC 360 codes. Refer to section 6.13 in the Ram Frame Analysis documentation for more information on P-Delta.
Geometric Stiffness Method
The geometric stiffness method implemented in Ram Frame is a non-iterative approach. A single modification to the stiffness matrix is made and applied to all load cases. This is advantageous over approaches that use an iterative non-linear solution, because it reduces analysis time and can be used with dynamic load cases that require modal combination of results.
The program calculates a global stiffness matrix (assembled from each finite element). Then, based on diaphragm masses or gravity loads, it calculates a global geometric stiffness matrix, which is subtracted from the global stiffness matrix. The global stiffness matrix becomes less stiff, which increases calculated displacements when considering P-Delta effects. All of these computations take place at the global level and not at any individual element level. Based on computed displacements, the program calculates member forces using the individual element stiffness matrices. Since these matrices are not modified, the member forces are larger when P-Delta is considered. Base shears are calculated based on member forces. Hence, we see a difference between the sum of the applied loads and sum of the base shear when P-Delta is considered. In a theoretical exact approach, the total shear reaction would match the applied load.
Scale Factor
P-Delta effects in Ram Frame are a function of the mass or gravity loads on the diaphragm and the stiffness of the members attached to the diaphragm. Codes generally require that second order effects be calculated for load combinations. Ram Frame analyzes load cases and superimposes the results for load combinations. Therefore, it is necessary to enter an appropriate scale factor so that superposition of the load cases produces the proper load combination results with P-Delta. The scale factor should represent the average factor on the gravity load cases in the controlling load combination. Since this may vary or each member, enter a conservative scale factor that covers all load combinations. When the gravity load option is used in Ram Frame, this is fairly straight forward because you can enter separate factors for dead load and live load. If the mass option is used, the scale factor must also consider the magnitude of the mass that is modeled relative to the gravity loads. For example, assume the modeled mass is 20% higher than the dead load and the live load is about 50% of the dead load. If the appropriate scale for mass is based on the load combination factors 1.2DL and 1.6LL, then the scale factor for mass is (1.2 + 0.5 * 1.6) / 1.2 = 1.67.
Generally the most preferred option is the Use Gravity Loads option. The scale factors should be those associated with the load combination most likely to govern for the lateral columns. For example, since the seismic or wind loads are likely to control the designs, the strength design combinations 4 or 5 of ASCE 7 Section 2.3.2 are likely to control. In those combinations, the factor on Dead Load is 1.2 and the factor on Live Load is either 0.5 or 1.0, as specified by Exception 1. Conservatively, the factors of 1.2 and 1.6 per combination 2 could be used, guaranteeing that the worst P-Delta condition is covered for all combinations. Note that these should be ultimate factors even if ASD will be used in design of the members so that the P-Delta analysis will be performed at an ultimate level, which is necessary for the principle of superposition of load cases to be valid. Also, note that these are not the factors that will be used in the load combinations for design, these are merely the factors used to calculate the ultimate gravity loads used in the P-Delta analysis method.
Limitations
The geometric stiffness method is only utilized with diaphragms are rigid or semirigid. If diaphragms are flexible or pseudo-flexible, the AISC 360-05 code in the steel post processor has an option to include the B2 factor. This moment magnification factor is conceptually the same thing as using P-Delta in the analysis.
The Eigen solution can be very sensitive to P-Delta. The Eigen solution is required for any dynamic load case and static load cases that use calculated periods/frequencies. If you are encountering an instability running a dynamic load case with P-Delta, first run the analysis without P-Delta to make sure the displacements are reasonable. If the displacements look fine, try creating an Eigen solution load case to reduce the number of modes the program is using. If you are encountering an instability running a static wind or seismic case, consider explicitly defining the periods/frequencies to avoid the Eigen solution. The Eigenvalue Error technote explains further.
Some building configurations are problematic for the geometric stiffness method implemented in Ram Frame. In particular, models with many disconnected nodes, short story heights, and offset diaphragms (e.g. mezzanines and low roofs) can be problematic. Generally, the limiting factor is a particular diaphragm and the problem presents itself as an RZ instability. If you encounter this error, run the model without P Delta and review the displacements. If the displacements are reasonable and the Eigen solution is not causing the issue, then you are experiencing a limitation with P-Delta. If you must run P-Delta in the analysis, your only option is to combine the mass on the problem diaphragm with another diaphragm in Ram Frame – Loads - Masses. However, this will impact any dynamic analysis and the distribution of static seismic loads.
Knee braces can produce unconservative P-Delta effects. Nodes are defined where the braces meet the columns necessitating the evaluation of lateral stiffness coefficients at a plane containing all such nodes. Since there is no diaphragm at this level, there will not be any floor lateral stiffness to be corrected.
AISC 360 Direct Analysis Method in RAM Structural System
RAMSS Eigenvalue Error
RAM Instability In Finite Element Analysis