Description of ModelIntroduction of Variables and ConstantsAnalyzing the ForceThe General ModelPutting together the piecesBibliographyDescription of Model The General Model BibliographyFinite Element Approach to EarthquakeModelingBenjamin AllenSeptember 17, 2009Description of Model The General Model BibliographyTable of Contents1Description of ModelIntroduction of Variables and ConstantsAnalyzing the Force1The General ModelPutting together the pieces1BibliographyDescription of Model The General Model BibliographyIntroduction of Variablesωkis the damping ratio of the kth node.We shall let m be mass. And M will be the matrix of themasses.K will represent the stiffness matrix that holds the data forthe nodal links.C will show the restoring damping of the system in terms ofthe nodes.We will let x be displacement vector, s will bedisplacement.κ shall be the bulk modulus of a material, G shall be theshear modulus of a material.ωsis the shear stress. ωultis the ultimate shear stress alsocalled the shear strength. sis the sheer strain.Description of Model The General Model BibliographyIntroduction of ConstantsSoil Type Sand Silt Loam Clay Aggregated ClayDensity 1.55 1.15 1.05 1.00Horizontal Damping cxsandcxsiltcxclaycxagrVertical Damping cysandcysiltcyclaycyagrHorizontal Stiffness kxsandkxsiltkxclaykxagrVertical Stiffness kysandkysiltkyclaykyagrShear Modulus GsandGsiltGclayGagrBulk Modulus κsandκsiltκclayκagrDescription of Model The General Model BibliographyStatement of AssumptionsAssumptionsTo create this model we will assume that our earthquake iscaused by a force propagating from outside the model andbelow the bedrock. For our purposes the bedrock will be solid.Our layers will have no edges and be homogeneous and thuswe can ignore the higher stress states and the inconsistenciesof the substances.Description of Model The General Model BibliographyStatement of AssumptionsMore AssumptionsWe will ignore the change of speed in the different groundlayers for the modelDescription of Model The General Model BibliographyDescribing the ForceThe force is assumed to be originating from a distant point andbelow the bedrock. This can then be broken up into horizontaland vertical displacement and analyzed at each node.F =f1xf1yf2xf2y...fnxfnyDescription of Model The General Model BibliographyReactionsNewton’s Second Law is ma = F butif you consider that the ground will compress when force isapplied then there is a stiffness opposing the movement orvelocity like a springand if you consider that it won’t return to it’s previousposition there is a dampingDescription of Model The General Model BibliographyFormulaFrom before we can develop a formula to characterize thereaction.M¨x + K˙x + Cx = F (1)Here the matrices are evaluated at each node as is thedisplacement and its derivative vectors.Description of Model The General Model BibliographyMass MatrixBecause each layer lays on top of the next one in the set themasses are additive creating a lower diagonal matrix.M =m10 0 0 0 0 ... 0 00 m10 0 0 0 ... 0 0m10 m20 0 0 ... 0 00 m10 m20 0 ... 0 0m10 m20 m30 ... 0 00 m10 m20 m3... 0 0... ... ... ... ... ... ... ... ...m10 m20 m30 ... mn00 m10 m20 m3... 0 mnDescription of Model The General Model BibliographyStiffness MatrixK =k1xk1yk2xk2yk3xk3y... k(n−1)xknyk1xk1yk2xk2yk3xk3y... k(n−1)xknyk1xk1yk2xk2yk3xk3y... k(n−1)xknyk1xk1yk2xk2yk3xk3y... k(n−1)xknyk1xk1yk2xk2yk3xk3y... k(n−1)xknyk1xk1yk2xk2yk3xk3y... k(n−1)xkny... ... ... ... ... ... ... ...k1xk1yk2xk2yk3xk3y... k(n−1)xknyk1xk1yk2xk2yk3xk3y... k(n−1)xknyThis can then be evaluated for the reactions that are caused. Thestiffness in the x direction of each level will be the stiffness of thelayer. The stiffness in the y layer will be the inverse of the stiffness ofthat layer plus the inverse of the stiffness of the above layers minusthe stiffness of the layer below.Description of Model The General Model BibliographyStiffness Matrix99K K =k1x0 0 0 0 0 ... 0 001k1y0−1k2y0 0 ... 0 0k1x0 k2x0 0 0 ... 0 001k1y01k2y0−1k3y... 0 0k1x0 k2x0 k3x0 ... 0 001k1y01k2y01k3y... 0 0... ... ... ... ... ... ... ... ...k1x0 k2x0 k3x0 ... knx001k1y01k2y01k3y... 01knyDescription of Model The General Model BibliographyDamping MatrixC =c11c12c13c14c15c16... c1nc21c22c23c24c25c26... c2nc31c32c33c34c35c36... c3nc41c42c43c44c45c46... c4nc51c52c53c54c55c56... c5nc61c62c63c64c65c66... c6n... ... ... ... ... ... ... ...c(n−1)1c(n−1)2c(n−1)3c(n−1)4c(n−1)5c(n−1)6... c(n−1)ncn1cn2cn3cn4cn5cn6... cnnThis can then be evaluated for the reactions that are caused. Thedamping in the x direction of each level will be the damping of thelayer and the damping in the y layer would be the above layer.Description of Model The General Model BibliographyDamping Matrix99K C =c110 0 0 0 0 ... 0 0 00 0 0 0 0 0 ... 0 0 00 0 c220 0 0 ... 0 0 00 c120 0 0 0 ... 0 0 00 0 0 0 c330 ... 0 0 00 c120 c230 0 ... 0 0 0... ... ... ... ... ... ... ... ... ...0 0 0 0 0 0 ... 0 cnn00 c120 c23 0 c34 ... c(n−1)n0 0Description of Model The General Model
View Full Document
Unlocking...