EA Soltion rocedre (demonstrated with a -D bar element problem) MAE - inite Element Analysis Many slides from this set are originally from B.S. Altan, Michigan Technological U. EA rocedre for Static Analysis. repare the E model a. discretize (mesh) the strctre b. prescribe loads c. prescribe spports. erform calclations a. generate stiffness matrix (k) for each element b. combine elements (assemble global stiffness matrix ) c. assemble loads (into global load vector ) d. impose spport conditions e. solve eqations (D=) for displacements. View reslts a. displacements b. reaction forces at restraints c. element strains d. element stresses e. element forces MAE - inite Element Analysis
Given: The -D Bar roblem E = E = Ga ( steel) E = E = E = Ga ( alminm) σ y = σ y = 9 Ma ( steel) σ y = σ y = σ y = 9 Ma ( alminm) MAE - inite Element Analysis MAE - inite Element Analysis The -D Bar roblem A = A = A = mm A = mm A = mm = = m =, N ind: a) displacements (,,, ), b) strains (ε, ε, ε, ε, ε ), c) stresses (σ, σ, σ, σ, σ ), d) internal forces ( i,, i,, i,, i,, i, ), e) safety factor (S). Node nmber = = = m
. repare the E Model The strctre is idealized in terms of nodes and bar elements MAE - inite Element Analysis.a. Generate Elemental Eqations If we fix the left end of a bar (with constant cross section) it s end displacement is given by: AE If the left end is NOT fixed, the relationship between force and displacement is given by: AE AE MAE - inite Element Analysis
MAE - inite Element Analysis Generate Elemental Eqations These two eqations can be conveniently expressed in matrix form as: The different parts are known as: the elemental stiffness matrix the elemental displacement vector the elemental force vector This form allows s to easily combine the eqations from all elements of a strctre. AE AE k d f MAE - inite Element Analysis Generate Elemental Eqations,, A E,,,, A E Element nmber Node nmber,, A E,,,, 8
Generate Elemental Eqations A E,,,, A E,,,, MAE - inite Element Analysis 9.b. Combine Elements Elements are combined by considering the eqilibrim of forces at each node, Node :,,,,, Node :,, MAE - inite Element Analysis,,
Combine Elements Node :,,,, Node :,,,,,, MAE - inite Element Analysis Combine Elements The eqations for each node can then be combined into one vector eqation.,,,,,,,,,, MAE - inite Element Analysis
MAE - inite Element Analysis Combine Elements,,,,,,,,,, Groping forces from each element into individal vectors gives: MAE - inite Element Analysis Combine Elements,, Element : Element :,,
8 MAE - inite Element Analysis Combine Elements,, Element : Element :,, MAE - inite Element Analysis Combine Elements,, Element :
9 MAE - inite Element Analysis Combine Elements MAE - inite Element Analysis Combine Elements The reslting overall matrix eqation is: 8 global stiffness matrix global displacement vector global applied force vector D=
.c. Assemble oads Inserting the load vale(s):. MAE - inite Element Analysis 9.d. Spport Conditions We have one spport condition: =. Spport conditions can be solved two ways: enalty method Mltiply eqation i = i by a large nmber and add to row i. x =. This is eqivalent to adding a stiff spring MAE - inite Element Analysis Stiff spring
MAE - inite Element Analysis Spport Conditions Matrix partitioning method artition global eqation into separate matrix eqations Solve for D first, then solve for : Step : Step :. D D D D D D MAE - inite Element Analysis Spport Conditions Step : Solve for,,. Step : Solve for
.e. Solving for Displacements The global matrices (with applied bondary conditions) are solved for displacements (D): D= - In practice, the compter does not actally calclate -, bt solves for D directly, sing techniqes sch as Gassian Elimination. MAE - inite Element Analysis MAE - inite Element Analysis.a. Solving for Strain Once we know the displacements, we can calclate the strain for each element. In a one dimensional problem, the strain is given by: d dx or the constant cross-section bar,
.b. Stress and Internal orce Below the yield stress, for niaxial stress, the stress is given by: The force in each bar can be calclated by: or by: E A AE( ) MAE - inite Element Analysis.c. Safety actor or a statically loaded member, the Safety actor (S) with respect to yielding failre can be compted as: where s y is the material yield stress, and max is the maximm stress. The S for the entire strctre is eqal to the minimm of the S of each member. MAE - inite Element Analysis
ecommended Safety actors S =. to. for exceptionally reliable materials sed nder controllable conditions, sbject to loads that can be determined with certainty where low weight is important. S =. to for well-known materials, nder reasonably constant environmental conditions, sbject to loads that can be determined readily. S = to. for average materials operated in ordinary environments and sbject to loads that can be determined. S =. to for less tried materials or for brittle materials nder average conditions of environment and load. S = to for ntried materials sed nder average conditions of environment and load. S = to also for better-known materials that are to be sed in ncertain environments or sbject to ncertain loads. Adjst S for repeated loads, impact forces, and brittle materials. Adapted from Jvinall & Marshek, ndamentals of Component Design, th Ed. MAE - inite Element Analysis What abot lanar (-D) roblems? The eqation for a bar element with an arbitrary orientation in planar space is obtained by transforming the local element coordinate system to the global coordinate system. MAE - inite Element Analysis 8
What abot lanar (-D) roblems? Mathematically this is done by mltiplying the elemental stiffness eqation by a rotation matrix: MAE - inite Element Analysis 9 What abot Spatial (D) roblems? Transformation matrices are again sed to transform the eqations in the element coordinate system to the global coordinate system. (This time sing the direction cosines.) MAE - inite Element Analysis