fea stress intro.ppt - rice university · • pipe element, pipe elbow, pipe tee, ... stress...

FEA Stress AnalysisFEA Stress AnalysisIs based on the displacements of

parametric solid models

Copyright © 2010 J. E. Akin Rice University, MEMS Dept.

FEA Data Reliability, 1• Geometry: generally most accurate if not• Geometry: generally most accurate, if not

de-featured (sliver faces cause mesh failure)M t i l if d di d d• Material: accurate if standardized or tested. Moduli, Poisson’s ratio, and yield stress are k l 2 3 i ifi fiknown only to 2 or 3 significant figures

• Mesh: requires engineering judgment for element type, sizes, and transition ratios


FEA Data Reliability, 2• Loads: less accurate require assumptions• Loads: less accurate, require assumptions.

Are ‘point’ forces reasonable? Are pressure values known? Are pressure areas reasonablyvalues known? Are pressure areas reasonably located? Are accelerations known for body forces? How are moments approximated? etcforces? How are moments approximated? etc

• Are there industry codes or government i f f l d l ?requirements for safe load values?


FEA Data Reliability, 3• Restraints (prescribed displacements):• Restraints (prescribed displacements):

least accurate, drastically effects results; several reasonable restraint cases should beseveral reasonable restraint cases should be studied. D h l d d i l h i• Does the excluded material, at the restraint, have the strength to supply the supporting

i f ? Ai ld idreaction force? Air seldom provides support forces!!


FEA Result Reliability, 4FEA Result Reliability, 4

• Equation Solver: generally the computedEquation Solver: generally the computed displacements quite accurate (3 or 4 significant figures)significant figures)

• Reactions: the reaction forces (or moments) obtained from the solver at themoments) obtained from the solver, at the restraint points, are similarly accurate, as are the Free Body Diagram (FBD) plotsare the Free Body Diagram (FBD) plots produced from them.



• 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.


FEA Result Reliability, 6FEA Result Reliability, 6• Post-processing: Any temperature

diff bi d i h hdifferences are combined with the coefficient of thermal expansion to define h l i Th h i l & h lthermal strains. The mechanical & thermal

strains are combined with elastic material i f h i lproperties to form mechanical stresses.



P t i Th h i l t• Post-processing: The mechanical stresses or strains are combined with the assumption

f b ittl d til b h i t d fiof brittle or ductile behavior to define a failure value, that is compared to the

t i l i ld t ( d l t d t i lmaterial yield stress (and a related material Factor of Safety, FOS).


FEA Result Reliability, 8FEA Result Reliability, 8• Plotting: displacements are continuous between

l h i (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.


General Approach for Stresses 1General Approach for Stresses, 1

• Select verification tools to independently p ycheck the FEA study– Use analytic, experimental, another FEA

method, etc.– Predict the displacements and stresses at

important locationsimportant locations– When done compare with your prediction– If significantly different, re-access the g y

assumptions used for both solutions– Repeat the prediction and/or FEA study


General Approach for Stresses, 2

• Understand the primary variables (PV) in the ilib i i i h blequilibrium equation governing the problem.

- Displacement vector for stress studies

U d t d b d diti (BC)• Understand boundary conditions (BC)– Essential, or Dirichlet BC specify a displacement

(often zero) at some boundary points (EBC)(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


General Approach for Stresses, 3

• Understand reactions needed to maintain the Essential BC applied at a pointpp p

• 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!


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

M h( )• Mesh(s)– Element type and size, size transition rates– Interface contact or bond conditionInterface contact or bond condition

• Force and Pressure Loading Cases– Types of loads, Factors of safety, Accelerationsyp , y,

• Boundary conditions (Fixtures)– Coordinate system(s) for vector components


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).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– Properties are known only to 3 no 4 significant figures.


Primary Stress Assumptions, 3

• Boundary conditions (Fixtures) are idealizations– Hinge joints, ball joints, roller joints are

standard symbols for approximating real supportssupports

– Springs and elastic foundations approximate real support conditions

– Contacting faces may be fully bonded (default), or non-penetrating, or have friction limits presentpresent.


Primary FEA Solution Costsy

• Assume a sparse, banded, linear algebra system of Eequations, with a half-bandwidth of B. Full system if B = E.– Storage required, S = B * E (Mb)

S l ti C t C B * E2 (ti )– Solution Cost, C α B * E2 (time)– Half symmetry: B’ ← B/2, E’ ← E/2, S’ ← S/4, C’ ← C/8

, answers obtained eight times faster, g f– Quarter symmetry: B’ ← B/4, E’ ← E/4, S’ ← S/16,

C’ ← C/64 , answers 64 times faster– Eighth symmetry, Cyclic symmetry, ...


Stress Result AccuracyStress Result Accuracy• Displacements are most accurate at the p

mesh nodes.• Stresses are least accurate at the meshStresses are least accurate at the mesh

nodes, most accurate at element center.– Stresses are discontinuous at elementStresses are discontinuous at element

interfaces– Stresses can be post-processed for accurateStresses can be post processed for accurate

averaged nodal values


Local Stress Singularitiesg• All stress analysis problems have local radial gradient

singularities near re-entrant corners in the domain. gThe stress there is theoretically infinite, but not in practice. Mesh refinement never helps there.

Radius, ru = r p f(θ)

∂u/∂r = r (p-1) f(θ)

Strength, p = π/C Corner: p = 2/3, weak

Crack: p = 1/2 strongRe-entrant, C

Crack: p 1/2, strong

∂u/∂r ⇒∞ as r ⇒ 0


Local Stress Error, 1Local Stress Error, 1• The stress error at a (non-singular) point

i h d f h l i h his the product of the element size, h, the stress gradient, and a constant dependent

h d i h d b don the domain shape and boundary conditions.– Large stress gradient points need small

element sizes, h– Small stress gradient regions can have large

element sizes


Local Stress Error, 2Local Stress Error, 2• Plan local mesh size with engineering

j d b d i djudgment based on estimated stress gradients (stress concentrations).

• Always utilize the SolidWorks Mesh Control Option

• Revise the mesh where you see (non-singular) stress concentration results.g )


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 f h l l h i (h) d hof the local mesh size (h) and the stress

gradient.Di l t t i i• Displacement components are piecewise continuous polynomials of degree p, while the stresses are piecewise discontinuousstresses are piecewise discontinuous polynomials of degree (p-1). In SW p=2.


FEA Stress Models• 3-D Solid: has 3 displacements (no rotations),

d 6 (3 l & 3 h )and 6 stresses (3 normal & 3 shear stresses)• 2-D Approximations

– Plane Stress (σzz = 0): has 2 displacements, and 3 stresses (2 normal & 1 shear)

– Plane Strain (ε zz = 0) : has 2 displacements, and 3 stresses (and σzz from Poisson’s ratio)

– Axisymmetric (∂/∂θ = 0): has 2 displacements, and 4 stresses (3 normal & 1 shear)


FEA Stress Models, 2,

• 2-D Approximations2 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 ( pbottom surfaces)

– Plate bending: 1 normal displacement, 2 in-plane rotations, and 3 stresses (each at top, middle, and bottom surfaces)


FEA Stress Models, 3FEA Stress Models, 3• 1-D Approximationspp

– Bars (Trusses): 3 displacements (1 local axial displacement), and 1 axial stress

– Torsion member: 3 rotations (1 local axial rotation), and 1 torsional stress B (F ) 3 di l t d 3– Beams (Frames): 3 displacements, and 3 rotations, with axial, bending, & shear stress

• Thick beam, thin beam, curved beamThick beam, thin beam, curved beam• Pipe element, pipe elbow, pipe tee, etc


Symmetry & Anti-symmetry Planesy y y y

• Use symmetry planes for the maximumUse symmetry planes for the maximum accuracy at the least cost in stress problems.p ob e s.

• Cut the object with symmetry planes and apply new boundary conditionsand apply new boundary conditions (EBC or NBC) to account for the removed materialremoved material.


Symmetry (Anti-symmetry) Planes

• Requires symmetry of the geometry and material properties.p p

• Requires symmetry (anti-symmetry) of the force termsthe force terms.

• Requires symmetry (anti-symmetry) of the imposed displacements or rotationsthe imposed displacements or rotations.


Symmetry, Anti-symmetryDi l t EBCDisplacement EBC

• Symmetry• Symmetry– Zero displacement normal to surface

Z t ti t t t t f– Zero rotation vector tangent to surface• Anti-symmetry

– Zero displacement vector tangent to surface

– Zero rotation normal to surface


Stress Analysis Verification, 1Stress Analysis Verification, 1

• Prepare initial estimates of deflections, reactions and stresses at important points.

• Eyeball check the deflected shape and y pthe principal stress vectors.

• Eyeball check the stress contour linesEyeball check the stress contour lines for wiggles. (OK in low stress regions.)


St A l i V ifi ti 2Stress 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 y pwhich almost always depend on the material properties.p p


Stress Analysis Verification, 3

• The reaction resultant forces and/or moments are equal and opposite to the q ppactual applied loading.

• SolidWorks can draw Free Body DiagramsSolidWorks can draw Free Body Diagrams (FBD) of support regions as an optional verification of the loadings.verification of the loadings.



• For pressures or tractions remember to compare their integral (resultant) to the p g ( )solution reactions.

• The resultants can be obtained via the ListThe resultants can be obtained via the List Selected feature in SW

• Reactions can be obtained at elements too• Reactions can be obtained at elements too.



• Compare displacements, reactions and stresses to initial estimates. Investigate any differences.

• Then, re-run the study, y• Ask “what if” questions about the loads

and supportsand supports


fea stress intro.ppt - rice university · • pipe element, pipe elbow, pipe tee, ... stress...

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