{ FlexPDE REL. 5.0.4, VERIF. St2LM_1.pde } { ***************************************************** Structural analysis, theory of elasticity (St) Two-dimensional problem (2) Linear problem (L) Modal analysis (M) Plane stress Compound Double-T Region (steel and copper) Bottom boundary is fixed ***************************************************** } title "Modal Analysis of a Double-T Region" select modes =4 {number of eigenvalues and modes} errlim=0.5e-2 Variables U { U and V are the variables } V definitions {geometrical parameters in meters} h=2e-2 {thickness} b=16e-2 {half-height} l=5e-2 {half-width of cleat} nu = 0.29 { Poisson's ratio for steel } E = 2e11 {Youngs modulus in Pascals for steel} rho = 7.86e3 {density in kg/m^3 for steel } mu = E/2/(1+nu) {shear module} lambdaL = 2*mu*nu/(1-2*nu) {Lame's parameter} {Lame's parameters for plane stress } lambdaz = 2*mu*lambdaL/(lambdaL+2*mu) Sxx = dx(U) {longitudinal strain Sxx} Syy = dy(V) {longitudinal strain Syy} Sxy = (dy(U)+dx(V))/2 {shear strain Sxy} Txx= lambdaz*(Sxx+Syy)+2*mu*Sxx {stress Txx} Tyy= lambdaz*(Sxx+Syy)+2*mu*Syy {stress Tóó} Txy=2*mu*Sxy {stress Txy} Scal=0.2*globalmax(magnitude(x,y))/globalmax(magnitude(u,v)) Freq=sqrt(lambda)/(2*pi) {Frequency "f" in Hz} equations {The displacement equations for modal analysis} U: dx(Txx) + dy(Txy) + lambda*rho*U = 0 V: dx(Txy) + dy(Tyy) + lambda*rho*V = 0 boundaries { regions and boundary conditions } Region 1 {upper region 1 from steel} start (h/2,0) natural(U)=0 natural(V)=0 {free boundary} line to (h/2,b-h) to (l,b-h) to (l,b) to (-l,b) line to (-l,b-h) to (-h/2,b-h) to (-h/2,0) to close Region 2 {lower region 2 with another material properties} {material properties for copper} nu=0.33 E=1.2e11 rho=8.9e3 start (h/2,0) natural(U)=0 natural(V)=0 {free boundary} line to (h/2,-b+h) to (l,-b+h) to (l,-b) value(U)=0 value(V)=0 {fixed displacements} line to (-l,-b) natural(U)=0 natural(V)=0 line to (-l,-b+h) to (-h/2,-b+h) to (-h/2,0) to close monitors grid(x+Scal*U,y+Scal*V) as "Mode Shape" report Freq as "Frequency in Hz" plots { Resulting Plots } grid(x+Scal*U,y+Scal*V) as "Mode Shape" report Freq as "Frequency in Hz" summary report Freq as "Frequency in Hz" end