ME 160 Introduction to Finite Element Method Instructor:

Tai-Ran Hsu, Professor

Department of Mechanical Engineering San Jose State University

Spring, 2016

Textbook: “The Finite Element Method in Thermomechanics,” by T.R.Hsu, Allen & Unwin, Boston, 1986. ISBN 0-04-620013-4 (SJSU Library: TA418.58 H78 1986 at 8th Floor with one reserved copy at 1st Floor) 15 copies available at the Spartan book store at a Shelve price of $63.50 E-version available @ http://www.springer.com/gp/book/9789401/60001 at a price of 67.82€

Principal References: [1] “Computer-Aided Design: an integrated approach” by Tai-Ran Hsu and Dipendra Sinha. West Publishing Company, St. Paul, Minnesota, 1992. ISBN: 0-314-80781-0. (SJSU Library: TA174.H77 1992 at 8th Floor with one reserved copy at 1st Floor) [2] “The Finite Element Method in Mechanical Design,” Charles Knight, PWS-Kent Co., 1993 [3] “Applied Finite Element Analysis” L. J. Segerlind, John Wiley & Sons, 1976

Adopted textbook for the class Recommended Principal Reference Book

Chapters 6 and 7

The Fundamental Mathematics Skill and ME Discipline Knowledge

A. Mathematics Skill: • Ordinary and partial differential equations relating to ME disciplines of deformable mechanics and heat conductions • Matrices: • Square, rectangular row and column matrices • Multiplication of matrices • Transposition of matrices: row matrices, column matrices, rectangular matrices • Differentiations and integration of matrices • Inverse of matrices • Interchange of rows and columns for matrices • Solution of simultaneous equations in matrix forms

B. ME discipline knowledge: • Static equilibrium of deformable solid structures ● Stresses and strains in deformable solids • Displacements of deformed solids • Relationships in deformed solids in “strength of materials” and “theory of elasticity:” Such as: Strains vs. displacements Stresses vs. strains – classical Hooke’s law and generalized Hooke’s law Strain energy, work by forces and potential energy • Equation for heat conduction in solids and formulations of boundary conditions • Principles of thermal expansion and stresses

The Fundamental Mathematics Skill and ME Discipline Knowledge – Cont’d

Fundamentals of Finite Element Method

Learning objectives: • The fundamental mathematics skill and ME discipline knowledge ● What is finite element method (FEM) ● Why FEM? ● Why FEM can solve engineering analysis problems that many traditional analytical methods cannot? ● FEM offers APPROXIMATE solutions to problems. But it is the only viable solution methods for most realistic engineering problems ● Knowledge and experience in the fundamentals of FEM are essential for obtaining better results. ● Broad applications of FEM in solving science and engineering problems. ● So, it will be great benefits for you to learn this powerful method.

WHAT TAKES TO LEARN THIS POWERFUL METHOD?

● Obviously, there are a lot of mathematics in the subject. Especially in Matrix algebra and calculus ● There is a lot of physics involved too. For mechanical engineers, the “Mechanics of materials” and Introductory Theory of elasticity and Heat conduction ● There are several good commercial FE codes in the marketplace and some are adopted by major industry. ● The use of these codes is not hard (a “cookbook” practices), and technicians can managed. Intelligent and effective use of these codes is something else. Only engineers with knowledge and experience in the theory and principles of FEM can do, and this is the purpose of this course.

What is Finite Element Method (FEM)? ● It is a powerful numerical analysis method used in many fields of science and engineering ● It is the only viable method to handle problems involving complex: ● geometry ● boundary (or support) conditions, and ● loading conditions These are the problems that cannot be solved by classical methods (e.g., the classical mathematical methods such as differential equations offered in ME 130 and ME 230) ● The principle of FEM is to discretize the “real” solid continua or fluid media into a finite number of sub-divisions of specific geometry – the elements, interconnected at specific points of the element –called nodes. ● Mathematical formulations and solutions are then derived for the “elements” of specific geometry ONLY– called the “element equations.” ● Assemble all “element equations” to establish the “overall structure equations” from which one obtains the required solutions at every nodes and elements – but not everywhere in the solid or fluid. ● So, FEM will give us “APPROXIMATE” solutions, but not the “Exact solutions” as would be by the classical methods. Because the solution obtained by FEM is on a “discretized” continuum, but not on the “real continuum.”

Example on why FEM can solve stress analysis of a plate but not by classical method:

The case is related to determine the induced stress in a solid plate by uniform force F applied at the two edges in the direction of the x-coordinate:

The induced stresses: Stress along the x-direction = σxx = S = F/(DW) Stress along the y-direction = σyy = 0 Shearing stress = σxy = 0

The above solution may sound easy and straightforward for a solid plate, but how will you calculate the induced stresses in the same plate if a small hole of radius d is introduced at the center of the plate???

Stress analysis of a perforated plate:

How would the hole affect the stresses in the plate subjected to the same load at the two edges? The problems is complicated by the following facts: ● there is no material in the plate where the hole is – So, there is no material to share the load F inside the hole ● this alteration of the “geometry” of the solid plate will obviously affect the stress variation in the plate ● so, the stresses will no longer be uniform in the plate ● the stresses near the hole will not be the same as those away from the hole ● the question is “How much difference in the induced stresses in the plate near the hole” will be??

Stresses in perforated plates by classical theory of elasticity: Fortunately, we have the solution for this particular problem by using the classical theory of elasticity as shown below:

θσ 2cos4312

12 2

2

4

4

2

2

−++

−=

ra

raS

raS

r θσθ 2cos312

12 4

4

2

2

+−

+=

raS

raS

θτ θ 2sin2312 2

2

4

4

+−−=

ra

raS

r

σr = 0, σθ = 3S and τrθ = 0 We will find that at r = a and θ = π/2 – at Point A with S = stress away from the hole

where S = F/(DW)

The above results indicates that σxx = 3S, meaning it is 3 times higher at the rim of the hole than those at points away from the hole – It is the well-known stress concentration factor of a perforated plate.

Unloaded plate Perforated plate under lateral forces

Coordinates for induced stresses Induced stresses by theory of elasticity

What will happen to the induced stresses in plates with further complication in the geometry? A perforate plate with slant or curved edges:

The formula for computing induced stresses derived by using the classical theory of elasticity ceases to be valid!!

What can we do to compute the induced stress distribution in the plate?

FEM appears to be the only viable solution method to solve this problem.

Why FEM?

Converting the real CONTINUUM solid plate

TO An assembly of finite number of individual sub-divisions (elements) inter-connected at nodes

with elements of particular plate geometry of: 3-side Triangular and 4-sides Quadrilateral geometry

One needs only to deal with these two plate geometry for the solutions – much easier than to handle the whole, undivided continuum solid plate.

Benefits of discretizing the solid plate:

Typical Element Geometry

History and Application of FEM This powerful method was initiated for stress analysis of aircraft structures in 1959 with attribution to two Boeing Company engineers and one UC-Berkeley professor. It gained instant acceptance by engineering and science communities. Major application of FEM include the following:

1) Industrial applications:

□ Aerospace: for stress analysis of aircraft structures and engine components; aerodynamic and performance analyses.

□ Automobile design and manufacturing: stress analysis of vehicle structures induced by dynamic and impact loads with simulations of shape changes under these loads.

□ Shipbuilding: stress analysis of structures for components and assembled products; hydrodynamic performance analyses of ocean-bound vessels.

□ Nuclear power: thermomechanical and thermohydraulic analyses of reactors, nuclear power plant equipment and pipelines; analysis and simulation of normal operating and accident conditions.

□ Steel and metal processing: stress and thermal analyses of process equipment; prediction of residual stress and distortions in the finished products.

1) Industrial applications – (cont’d):

□ Construction: stress analysis of building and bridge structures, concrete foundations, underground tunnels and structures.

□ Resource and mining: stress analysis of excavation equipment, geotechnical materials; analysis of response of geological materials to static and dynamics loads; mining safety design and analyses.

□ Computer and electronics: design analyses of computer and electronics systems; thermal management with proper cooling, and induced thermomechanical analysis of delicate components, antenna design.

□ Renewable energies: stress analysis of wind turbogenerators subject to static and dynamic loads; solar panels structures, thermal performance analysis

2) Major scientific and engineering applications:

□ Elastic-plastic stress analysis of solid structures [Owen and Hinton1980]

□ Thermomechanical stress analysis with finite element formulations for heat transfer, thermoelestic, thermoelastic-plastic-creep involving material and geometric nonlinearities, and multi-dimensional fracture mechanics. [Hsu 1986]

□ Thermoelestic-plastic-creep of micro structures design and packaging [Hsu 2008]

□ Heat and mass transfer [Lewis et. al. 1981, Foulser 1984]

□ Dynamics and vibration [Bathe and Wilson 1976]

□ Fluid mechanics [Gallagher et.al. 1975]

□ Diffusion and mass transport [Abdel-Hadi et.al 1985]

□ Soil mechanics [Aalto 1984]

□ Geomechanics [Naylor and Pande 1981]

□ Biomechanics and bioengineering [Huiskes and Chao1983, Furlong and Palazotto 1983, Lee et.al, 1983]

□ Material science [Basombrio 1984]

□ Physical science [Sigh and Lai 1984]

The above list of application of FEM, of course, is by no means complete. New applications are reported at all times.

Input/output in Finite Element Stress Analysis of

Solid Structures

Input to FE for Stress Analysis

Group 1 General information: ● Profile of the structure geometry (may be transmitted from a CAD package) ● Establish the coordinates: ● x-y for plane ● r-z for asymmetrical ● x-y-z for 3-dimensional geometry ● Often, the FE model (or Mesh) is by “automatic mesh” option by commercial code with geometry transferred from a CAD package, as illustrated below:

Solid model of a cam-shaft assembly from a CAD software FE model by automatic mesh generation

Group 2 Develop and establish FE mesh (model): ● For automatic mesh generation: Specify the desirable densities of nodes and elements in specific regions ● General information includes: ● Node number, nodal coordinates, nodal conditions (e.g., constraints, applied forces, temperatures). ● element number, element description (e.g., element designations and involved nodes)

F

Example on FE mesh for a tapered bar:

● ● ●

● ● ● ●

●

●

● ● ● ●

● ● ● ● ●

●

●

●

ο ο

ο

ο X 1 2 3 4 5 6 7

8 9 10 11 12 13 14

15 16 17 18 19 20 21

22 23 24 25

1 2 3 4 5 6

7 8 9 10 11 12

13 14

15 16 17

18 F

F

F

FE Mesh

L

H

L1 r

Intelligent FEA should follow the following practices: 1) Always place high density of elements in regions in the structure with DRASTIC variation of stresses, such as in the rim of the holes and the fillet of tapered bars and gear teeth:

F L

H

L1 r

Input to FEA – cont’d

2) Avoid using element with high aspect ratios (the ratio of longest edge of the element to the shortest edge in the same element) for numerical accuracies and stability

Group 3 Material properties: ● Young’s modulus(E) ● Poisson’s ratio (γ) ●shear modulus of elasticity (μ) ●yield strength (σy) ●ultimate strength (σu) Group 4 boundary and loading conditions: ● Nodes with constrained displacements (e.g., in x-, y- and z-directions) ● Concentrated forces at specific nodes ( pressures are converted to concentrated forces at the nodes and are involved in the area of application)

Input to FEA – cont’d

Output from FEA

Group 1 Textural output: ● Nodal and element information ● Displacements at nodes ● Stress and strains in each element: ● Normal stress components in x- y- and z-directions ● Shear stress components on xy, xz and yz planes ● Normal and shear stress components ● Maximum and minimum principal stress components ●The von-Mises stress defined as: von-Mises stress is used for the situations with structures subjected to multi-axial load applications. This stress represents the induced stress in the elements in the FE model.

( ) ( ) ( ) ( )222222 62

1xzyzxyzzyyzzxxyyxx σσσσσσσσσσ +++−+−+−=

Output from FEA – cont’d

Group 2 graphic output: ● A solid model such as the cam-shaft assembly ● User input discretized FE mesh from the solid model ● A unreformed FE mesh superimposed with the deformed FE mesh subject to the loading for visual display of the “shape change” of structures ● Color coded zones to indicate the distributions of stresses, displacement, etc. ● color coded zones to indicate other required output such as temperature, etc. ● Animated movements of deformation of of structures under varying loading

SUMMARY

1. The principle of FEM is “Divide and Conquer”; i.e., to solve the problems with continua of complex geometry with complicated loading/boundary conditions by dividing the continua by a finite number of elements interconnected at nodes. This form of division of a continuum is called “Discretization.” 2. The versatility of the FEM has made its applications wide-spread in science and engineering fields. FEA is now a common practice by all disciplines of engineering and science. 3. Solution obtained by FEM is of approximate nature. The closer the discretized continuum to the real situation, the better results is obtained. So, intelligent users apply the fundamental knowledge of the FEM to come up with better discretization, and thereby better solutions. 4. The are several “general purpose” commercial FE codes available in the marketplace. ANSYS code appears more commonly used by industry than all other codes. Students in this class will have opportunity of learning the use of this code from experts of the local company in solving problems relevant to mechanical engineering applications.