THE FINITE ELEMENT METHOD 2017 Dept. of Solid Mechanics

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Jasmin Miles
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1 THE FINITE ELEMENT METHOD 07 Dept. of Solid Mechanics EXAMINATION: A maximum of 60 points can be achieved in this examination. To pass at least 0 points are required. Permitted aid: Pocket calculator. Anonymkod / Anonymous Code (XXX-NNNN)... Kurskod / Course Code (FHLXXX)... Personlig identifierare / Personal Identifier... Hand in the exam!!!!! Problem : (p) Consider the four node element used to model an elastic boundary value problem. The nodes --4 are located along the boundary and the traction along -4 is sketcked below. Moreover, in point (0, 4.5) a load F of magnitude 000N/m is applied. The direction of the load F is n = / [ ]. Calculate the contribution from the traction and the load F to the boundary load vector, f b = L 4 N T ttdl. The thickness of the element is 5mm. 00N/mm 00N/mm y node 4 ( 0,) 00N/mm node (0,) F x node ( 0, 7) node (0, 7) Figure : Four node element. All coordinates are in mm.

2 Problem : (6p) In the element routine plani4e.m the stiffness matrix k e = A e B et DB e tda is calculated for an isoparametric plane four node element. One part of the code has been erased. State the code that has been erased! Figure : Part of the routine plani4e.m Hint: The strain components in a plane analysis is given by are ǫ T = [ǫ xx ǫ yy γ xy ] and ǫ xx = ux x ǫ yy = uy y, γ xy = ux y + uy x

3 Problem : (8p) The element displacement vector for the element below is a e = [ ] T u 8 u 6 u 7 y u 5 (, ) (,) 4 (, ) (, ) u u 4 u u x Use one integration point to calculate the strain energy, A tǫt σda = A tǫt DǫdA where D is a constant constitutive matrix. Motivate your answer. Hint: The strain components in a plane analysis is given by are ǫ T = [ǫ xx ǫ yy γ xy ] and ǫ xx = ux x ǫ yy = uy y, γ xy = ux y + uy x. Theelasticity matrixd isgivenbyd = E ν material parameters. ν 0 ν ( ν) wheree andν areconstant Problem 4: (p) The governing equation for heat conduction in a one-dimensional rod subjected to convection along the constant perimeter, P, is given by ( d Ak dt ) αp(t T ) = 0 0 < x < L dx dx The boundary conditions are given by q(x = 0) = αplt /A and T(x = L) = T. The D Fourier law reads q = k dt dx. Derive the weak form and finite element formulation for the problem. Use two elements of equal size to determine the temperature distribution. The convection coefficient is given by α = 6kA/(PL ). All geometrical and material parameters are constant along the rod.

4 Problem 5: (0p) The finite element formulation for two dimensional linear elastic problem takes the form Ka = f b +f l where K is the stiffness matrix and a the vector of unknown node displacements. The right hand side vectors f b stem from the line load q and f l from gravitation b which acts on all elements x y q b The above depicted mesh consists of five linear triangular elements. For the indicated boundary and load conditions mark in the system of equations below with x components that are known and different from zero? components that are unknown and different from zero all blank positions are interpreted as zero. K a = f b + f l 4

5 Problem 6: (p) (Only Pi and I (FHLF0)) A beam of length L and stiffness EI is supported and loaded according to the figure below. The moment, M, is applied at a distance L from the left support. Determine the slope of the beam at the left support. M L L L Problem 6: (p) (Only F (FHLF0)) The finite element formulation for heat conduction (div(q) Q = 0) takes the form: B T DBdV V } {{ } K αn T NdS S } c {{} a = N T αt ds S } c {{} K f c + N T q n ds N T hds S g S }{{ h } f b + N T QdV V } {{ } f l where S c is the boundary where convection applies, S h is the boundary where the heat flow is prescribed and S g is the boundary where the temperature is prescribed. a) Show that the matrix K + K is positive definite for α > 0. The constitutive matrix D is symmetric and positive definite. b) Show that T [ S NT q n ds + V NT QdV ] = 0 where V and S repesents the total volume and surface, respecively. Moreover, is a vector with all entries being equal to one, i.e. the scalar product can be identified as the sum over all components. 5

6 Some hints that might be helpful Hint: Green-Gauss s theorem states: φdivqda = Hint: Some trigonometric relations: A L φq T ndl ( φ) T qda A sin(α) +cos(α) = sin(α+β) = sin(α)cos(β)+cos(α)sin(β), cos(α+β) = cos(α)cos(β) sin(α)sin(β) sin (α) = cos(α), cos (α) = +cos(α) sin(α) = sin(α)cos(α), tan(α) = tan(α) tan (α) Hint: A quadratic matrix is positive semidefinite if a T Ka 0, a, and a T Ka = 0 for some a 0 Hint: The interpolation formula of Lagrange is given by l n k = (x x )...(x x k )(x x k+ )...(x x n ) (x k x )...(x k x k )(x k x k+ )...(x k x n ) Hint: Fourier s law is given by q = D T h V= h A Hint: The position, ξ i, of the integration points and weights, H i, for n number of integration points can be found from A n ξ i H i 0 ±/ where integration from to is assumed. Hint: 0 0 a b c d = ad bc (ad bc) 0 0 d+b d b c a c a 6

ME FINITE ELEMENT ANALYSIS FORMULAS

ME FINITE ELEMENT ANALYSIS FORMULAS ME 2353 - FINITE ELEMENT ANALYSIS FORMULAS UNIT I FINITE ELEMENT FORMULATION OF BOUNDARY VALUE PROBLEMS 01. Global Equation for Force Vector, {F} = [K] {u} {F} = Global Force Vector [K] = Global Stiffness