Is based on the displacements of parametric solid models. J. E. Akin Rice University, MEMS Dept.

Transcription

1 FEA Stress Analysis Is based on the displacements of parametric solid models

2 FEA Data Reliability, 1 Geometry: generally most accurate, if not de-featured (sliver faces cause mesh failure) Material: accurate if standardized di d or tested. Moduli, Poisson s ratio, and yield stress are known only to 2 or 3 significant ifi figures Mesh: requires engineering judgment for element type, sizes, and transition ratios

3 FEA Data Reliability, 2 Loads: less accurate, require assumptions. Are point forces reasonable? Are pressure values known? Are pressure areas reasonably located? Are accelerations known for body forces? How are moments approximated? etc Are there industry codes or government requirements for safe load values?

4 FEA Data Reliability, 3 Restraints (prescribed displacements): least accurate, drastically effects results; several reasonable restraint cases should be studied. Does the excluded d material, at the restraint, have the strength to supply the supporting reaction force? Air seldom provides support forces!!

5 FEA Result Reliability, 4 Equation Solver: generally the computed displacements quite accurate (3 or 4 significant figures) Reactions: the reaction forces (or moments) obtained from the solver, at the restraint points, are similarly accurate, as are the Free Body Diagram (FBD) plots produced from them.

6 FEA Result Reliability, 5 Post-processing: calculates the gradient of the displacement vector. The gradient of an approximate solution is always less accurate than the approximate solution. The gradient vector components are combined to define mechanical strains.

7 FEA Result Reliability, 6 Post-processing: Any temperature differences are combined with the coefficient of thermal expansion to define thermal strains. The mechanical & thermal strains are combined with elastic material properties to form mechanical stresses.

8 FEA Result Reliability, 7 Post-processing: The mechanical stresses or strains are combined with the assumption of brittle or ductile behavior to define a failure value, that is compared to the material yield stress (and a related material Factor of Safety, FOS).

9 FEA Result Reliability, 8 Plotting: displacements are continuous between elements, so their contours are accurate (may display too many significant figures) Strains and stresses are discontinuous between elements, but are averaged to look smooth. Two or three significant figures might be accurate, depending on mesh fineness.

10 General Approach for Stresses, 1 Select verification tools to independently check the FEA study Use analytic, experimental, another FEA method, etc. Predict the displacements and stresses at important locations When done compare with your prediction If significantly different, re-access the assumptions used for both solutions Repeat the prediction and/or FEA study

11 General Approach for Stresses, 2 Understand the primary variables (PV) in the equilibrium i equation governing the problem. - Displacement vector for stress studies Understand d boundary conditions (BC) Essential, or Dirichlet BC specify a displacement (often zero) at some boundary points (EBC) Natural (stress free), or Neumann (known stress) BC apply at other boundary points (NBC) One or the other acts at a boundary point, never both conditions

12 General Approach for Stresses, 3 Understand reactions needed to maintain the Essential BC applied at a point Force at a given displacement Moment at a given rotation (if active) Can the omitted material at the reaction supply the necessary force? DO NOT use air to impose displacements!

13 Primary Stress Assumptions, 1 Model geometry (? De-featured, neglected regions?) Material Properties for stresses Elastic moduli, Poisson s ratio, Yield stress, Coefficient of thermal expansion Mesh(s) Element type and size, size transition rates Interface contact or bond condition Force and Pressure Loading Cases Types of loads, Factors of safety, Accelerations Boundary conditions (Fixtures) Coordinate system(s) for vector components

14 Primary Stress Assumptions, 2 Material Properties are tabulated for common materials Elastic modulus, E, Poisson s ratio, μ, and Shear Modulus, G, define the stiffness contributions of an element (terms in the square matrix of the algebraic system). The coefficient of thermal expansion, α, defines strains due only to temperature differences. Yield stress or yield strain or ultimate stress values are used to estimate material failure in each element. Properties are known only to 3 no 4 significant figures.

15 Primary Stress Assumptions, 3 Boundary conditions (Fixtures) are idealizations Hinge joints, ball joints, roller joints are standard symbols for approximating real supports Springs and elastic foundations approximate real support conditions Contacting faces may be fully bonded (default), or non-penetrating, or have friction limits present.

16 Primary FEA Solution Costs Assume a sparse, banded, linear algebra system of E equations, with a half-bandwidth of B. Full system if B = E. Storage required, S = B * E (Mb) Slti Solution Cost, C α B*E 2 (time) Half symmetry: B B/2, E E/2, S S/4, C C/8, answers obtained eight times faster Quarter symmetry: B B/4, E E/4, S S/16, C C/64, answers 64 times faster Eighth symmetry, Cyclic symmetry,...

17 Stress Result Accuracy Displacements are most accurate at the mesh nodes. Stresses are least accurate at the mesh nodes, most accurate at element center. Stresses are discontinuous at element interfaces Stresses can be post-processedprocessed for accurate averaged nodal values

18 Local Stress Singularities All stress analysis problems have local radial gradient singularities near re-entrant corners in the domain. The stress there is theoretically infinite, but not in practice. Mesh refinement never helps there. Radius, r Strength, p = π/c Re-entrant, C u = r p f(θ) u/ r = r (p-1) f(θ) Corner: p = 2/3, weak Crack: p = 1/2, strong u/ r as r 0

19 Local Stress Error, 1 The stress error at a (non-singular) point is the product of the element size, h, the stress gradient, and a constant dependent on the domain shape and dboundary conditions. Large stress gradient points need small element sizes, h Small stress gradient regions can have large element sizes

20 Local Stress Error, 2 Plan local mesh size with engineering judgment based on estimated stress gradients (stress concentrations). Always utilize the SolidWorks Mesh Control Option Revise the mesh where you see (nonsingular) stress concentration results.

21 Stress Mesh Considerations Crude meshes that look like a part are ok for images and mass properties but not for FEA stress analysis. Local stress error is proportional to product of the local lmesh size (h) and the stress gradient. Displacement components are piecewise i continuous polynomials of degree p, while the stresses are piecewise discontinuous polynomials of degree (p-1). In SW p=2.

22 FEA Stress Models 3-D Solid: has 3 displacements (no rotations), and d6 stresses (3 normal l&3 shear stresses) 2-D Approximations Plane Stress (σ zz = 0): has 2 displacements, and 3 stresses (2 normal & 1 shear) Plane Strain (ε zz = 0) : has 2 displacements, and 3 stresses (and σ zz from Poisson s ratio) Axisymmetric ( / θ = 0): has 2 displacements, and 4 stresses (3 normal & 1 shear)

23 FEA Stress Models, 2 2-D Approximations Thick Shells: 3 general displacements (no rotations), and 5 (or 6) stresses Thin Shells: 3 displacements and 3 rotations, and 5 stresses (each at top, middle, and bottom surfaces) Plate bending: 1 normal displacement, 2 inplane rotations, and 3 stresses (each at top, middle, and bottom surfaces)

24 FEA Stress Models, 3 1-D Approximations Bars (Trusses): 3 displacements (1 local axial displacement), and 1 axial stress Torsion member: 3 rotations (1 local axial rotation), and 1 torsional stress Beams (Frames): 3displacements, and 3 rotations, with axial, bending, & shear stress Thick beam, thin beam, curved beam Pipe element, pipe elbow, pipe tee, etc

25 Symmetry y& Anti-symmetry y Planes Use symmetry planes for the maximum accuracy at the least cost in stress problems. pobe Cut the object with symmetry planes and apply new boundary conditions (EBC or NBC) to account for the removed material.

26 Symmetry (Anti-symmetry) Planes Requires symmetry of the geometry and material properties. p Requires symmetry (anti-symmetry) of the force terms. Requires symmetry (anti-symmetry) of the imposed displacements or rotations.

27 Symmetry, Anti-symmetry Symmetry Displacement EBC Zero displacement normal to surface Zero rotation ti vector tangent tto surface Anti-symmetry Zero displacement vector tangent to surface Zero rotation normal to surface

28 Stress Analysis Verification, 1 Prepare initial estimates of deflections, reactions and stresses at important points. Eyeball check the deflected shape and the principal stress vectors. Eyeball check the stress contour lines for wiggles. (OK in low stress regions.)

29 Stress Analysis Verification, 2 The stresses often depend only on the shape of the part and are independent of the material properties. You must also verify the displacements which almost always depend on the material properties. p

30 Stress Analysis Verification, 3 The reaction resultant forces and/or moments are equal and opposite to the actual applied loading. SolidWorks can draw Free Body Diagrams (FBD) of support regions as an optional verification of the loadings.

31 Stress Analysis Verification, 4 For pressures or tractions remember to compare their integral (resultant) to the solution reactions. The resultants can be obtained via the List Selected feature in SW Reactions can be obtained at elements too.

32 Stress Analysis Verification, 5 Compare displacements, reactions and stresses to initial estimates. Investigate any differences. Then, re-run the study Ask what if questions about the loads and supports