SCALED RESPONSE SPECTRA USING SPCD CONSTRAINTS, WITH SUPERELEMENTS, scrspcs.v2001, TAN 4936 Reference: Handbook of Dynamic Analysis, Appendix E 1. BACKGROUND The Scaled Response Spectra capability computes the displacements and internal loads in a structure due to application of a spectrum of enforced motion at its base points. The responses of each natural mode of vibration due to the base motion is estimated, with stresses and displacements for each mode computed. The modal stresses and displacements are then summed, using several methods ranging from optimistic to pessimistic. The present capability is based on the large mass method of enforced motion, described in the Reference. This alter package uses the SPCD method for enforced motion similar to that introduced in V2001, and introduces the superelement capability as well. 2. THEORY The base motion input is applied to the SPC variables, Ys. The subset that has applied motion is named Yx. The motion in the structure uabar is the sum of the fixed base motion ua and the motion due to base motion ux. The equation for enforced motion based on SPCD input only is Zaa Uax ua 0a [ ][ ]=[ ] (1) 0xa Ixx ux Yx Zaa = Kaa + i*omega*Baa - omega*omega*Maa (2) uabar = ua + Uax*ux The static displacements due to base motion Uax are computed from the stiffness matrix and Yxx, made by diagonalizing Yx, Kaa*Uax <= Kax*Yxx (3) The second row of (1) can be used to eliminate ux from the first row, Zaa*ua = -Uax*Yx (4) The undamped modes of the structure with the base fixed are computed from the equation [Kaa - lambda*Maa]PHIaz=0 (5) The mode shapes can be used to reduce (3) to a set of equations with diagonal matrices PHIaz'*Zaa* PHIaz*ua = -PHIaz'*Uax*Yx Zhh*uh = PSIdhx*Yx The matrix product on the right is a form of mode participation factor matrix based on displacement input, PSIdhx = -PHIaz'*Uax The method used in this alter is to re-use modules written of the older form of scaled response spectra, based on the large mass method. The INTERR module performs the interpolation of the input response spectrum to find the input at for each mode. Its matrix inputs are: ZETAH A vector of fraction of critical damping values for each mode FN A vector of natural frequencies for the modes. PSIa The mode participation factor matrix, acceleration input type. The large mass method differs from the SPCD method in that the base points are placed on very large masses rather than placing them on SPCs. As a result, the large mass mode matrix contains rigid body modes and perhaps very low frequency modes whose mode shapes approximate those in Uax. These modes do not exist with the SPCD method. To account for these quasi-rigid body modes they are replaced by the static shapes Uax, the two vectors are expanded with zero values for the generalized coordinates associates with the Uax shapes. That is, the static shapes are undamped, and their equivalent natural frequency is zero, reflecting that they are static shapes. In order to make a PSI matrix that looks like the large mass version the eigenvectors are appended to the static shapes, PHIay = [Uax | PHIaz] The eigenvectors are post-multiplied by the inverse of the natural frequencies squared ([lamdainv]) to convert PSI to acceleration units of measure. The PSI matrix of the acceleration type is then Ixx 0 PSIayx = [ ]PHIay'*Uax 0 lamdainv When compared with a PSId matrix made with the large mass method, the rows associated with flexible modes are usually identical to at least five or more digits, except perhaps for a sign associated with the ambiguity of the signs of eigenvectors. The differences in the last digits are due to the large mass method being an approximate method, while the SPCD method is an exact method. Making the large masses larger will reduce the differences, at the risk of making them so large that they are numerically destabilizing. An advantage of the SPCD method is that you are relieved from the burden of deciding what mass ratio is large enough but not too large. The Uax shapes will be identical to the low frequency modes to a constant, but the constant is quite large, due to the normalization used for the low frequency modes. The large mass method is in SOL 103. The SPCD method has been shifted to SOL 112 because it has many of the components needed for other reasons. The SUPORT entry, which produces the r-set, plays a different role in the SPCD implementation. It is required in the large mass approach, but will not be needed in most SPCD approach cases. When it is present, Uax is computed by the following steps. Compute the forces caused by enforced displacement, Pfx = Kfs*Ysx, Reduce Pfx to the l-set, Poxbar Pfx = [Plxbar] Prxbar Plx = Gol'*Poxbar + Plxbar Kll*ulx <= Plx ulx Uax = [ ] 0rx As is discussed later, Prxbar must be null for proper r-set to s-set relationships. urx must be null for similar reasons. The SPCD method uses SUPORT entries to constrain rigid body modes. A rigid body mode can exist, for example, when shaking a horizontal bar at one end that is unconstrained on the other end. In order to calculate ulx it is necessary that all rigid body modes have been removed from Kll. The SUPORT entry provides this capability. This implies that the SUPORT entries and the SPCs of the model must be orthogonal. That is, the static shapes in Uax must have zero motion in their r-set DOFs. Rigid body modes must be constrained with large masses in the large mass approach. They therefore approximate constrained modes, not rigid body modes of the original structure. 3. INPUT The inputs for the large mass method are discussed in the Reference. To convert a large mass method input file to the SPCD method, 1. Replace SOL 103 with SOL 112 2. Remove all large masses and SUPORT entries. Select DOFs at which spectrums are to be applied for base motion. Place them on SPC1 entries selected with a SPC= command, and on SPCD entries selected by a LOAD= command. DOFs on SPCDs must be in the residual structure. The commands should be above the subcase level. 3. PARAM, SCRSPEC is not needed for the SSSALTER implementation. All other inputs are as described in the Reference. The example problem listed below is similar to the one described in the Reference. 4. OUTPUT Output is identical to output from the large mass method, as described in the Reference. 5. EXAMPLE PROBLEM scrspc3.dat This model is similar to that discussed in the Reference. A small superelement has been added. Entries that are required for the large mass methods are removed by placing a "$" symbol in front of them, converting them to comments. An early version of the SSSALTER is included in the input file. You may replace it with an include statement when you have access to the current SSSALTER.