FEEG6010 Advanced Finite Element Analysis SEMESTER 2 FINAL ASSESSMENTS 2020/21

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FEEG6010W1

SEMESTER 2 FINAL ASSESSMENTS 2020/21

Advanced Finite Element Analysis

Section A (all students)

A.1

(i) The plate of Figure A.1.1 is fully fixed on side-1 and simply supported on side-2, while side-3 and side-4 are free. Write the boundary conditions applicable to side-1 and side-2 according to:

(a) the Kirchhoff plate theory. [3 marks]

(b) the Reissner-Mindlin plate theory. [2 marks]

(ii) Assume the Timoshenko beam finite element of Figure A.1.2, which is quadratic in the deflection u and linear in the rotation e .

(a) Derive the shape functions of this element for the

deflection N1(u)(x), N2(u)(x) and N3(u)(x), and the rotation N1(e)(x) and N2(e)(x) . [5 marks]

(b) All parts of the stiffness matrix will be integrated using the

same  Gauss  quadrature  scheme.  How  many  Gauss points are needed for full integration? [5 marks]

(c) Do you expect this element to suffer from locking at the “slender limit” or not, and why? [5 marks]

(iii) Figure A.1.3 shows the three-dimensional  model of a  beam discretised with constant strain tetrahedral elements. What issues do you expect regarding the predicted flexural stress? Use simple sketches to support your arguments, if necessary. [5 Marks]

[TOTAL 25 MARKS]


Figure A.1.1: A plate fully fixed on side- 1, simply supported on side-2, and

free on side-3 and side-4. The circled numbers are labels indicating the

number of each side.

Figure A.1.2: A Timoshenko beam element with quadratic deflection and

linear rotation.

Figure A.1.3: Three-dimensional model of a beam discretised with

constant strain tetrahedral elements.


A.2

(i) Figure A.2.1 shows a cantilever beam with a transverse tip load, for which the four different finite element models of Figure A.2.2 are considered. Which of the four models is expected to give better results and why? [5 Marks]

(ii) An elastoplastic material has elastic constants Ε and v and the following yield function:

F = 5(G11  + G22) − S (p ) = 0

where  S (p ) = 500 + 1000p    and p  = 1(p)1  + 2(p)2 .  Plastic  flow  is associated. Conditions of plane stress are assumed, where all out of plane stresses and strains are zero.

(a) Write the expressions linking stress increments Ġ11   and

Ġ22 to the total strain increments ̇11 and ̇22 and the plastic multiplier 入̇ . [10 Marks]

(b) Write the consistency condition, introduce the expressions

derived above, and thus solve for the plastic multiplier 入̇ as a function of the total strain increments ̇11  and ̇22 . [10 Marks]

[TOTAL 25 MARKS]


Figure A.2.1: A cantilever beam with a transverse load at the tip.

(a)                                                                (b)


(c)                                                                 (d)


Figure  A.2.2:.Four  finite  element  models  of  a  cantilever  beam: (a)   a   single   Bernoulli-Euler   beam   element,   (b)   four   linear Timoshenko beam elements, (c) a single plane-stress biquadratic quadrilateral  with  reduced  integration,  and  (d)  two  plane-stress bilinear quadrilaterals with full integration.

A.3

A three-dimensional continuum body undergoes homogeneous           deformations such that the Lagrangian displacement of every material particle of this body at any given time t is given by:

3X2

1

2X2

U3

where the position of each material particle in an orthornormal basis is labelled as:

X

X

X3

(i) Derive  the  explicit  expression  of  the  Lagrangian  displacement gradient tensor in matrix form. [3 marks]

(ii) Derive the explicit expression of the deformation gradient in matrix form. [3 marks]

(iii) Derive   the   explicit   expression   of   the   right   Cauchy-Green deformation tensor in matrix form. [3 marks]

(iv) Derive the explicit expression of the left stretch tensor in matrix form. [3 marks]

(v) Derive the explicit expression of the Cauchy strain tensor in matrix form. [3 marks]

(vi) Derive the explicit expression of the Green-Lagrange strain tensor in matrix form. [3 marks]

(vii) Derive the explicit expression of the rate of Green-Lagrange strain tensor in matrix form. [2 marks]

(viii) Determine the position of the material particle which does sustain any strain. [1 mark]

(ix) Establish an analytical relationship between t, X1 , X2  and X3  that

ensures that the deformation of the continuum body is volume-preserving (i.e. the material is incompressible). [4 marks]

[TOTAL 25 MARKS]

Section B (Aero/Mech/Ship only)

B.1

The strain energy function of a polymeric material is given by the

following  hyperelastic  potential where c1  , c2     and c3     are

material parameters, and I1 , I2   and I3   are respectively the first, second  and  third  principal  invariants  of the  right  Cauchy-Green deformation  tensor C (with  components Cij in  a  Cartesian

coordinate system, i,j 1,2,3). Exp and Log are respectively the

exponential and logarithm functions.

c3   log

(i) Give the explicit expression of the first principal invariant I1  as a function of the components of C [2 marks]

(ii) Define the second Piola-Kirchhoff stress tensor arising from

the hyperelastic potential as a function of and C . [2 marks]


(iii) Provide the explicit expression of the second Piola-Kirchhoff

stress  tensor  arising  from  the  hyperelastic  potential , in

compact form (tensor). You will use the following formulas for

the derivative of the determinant  det(A)  of a second-order tensor A :

det(A)

A

and the derivative of the trace of the square of a second-order tensor A :

trace 2

A

A

[15 marks]

(iv) Provide the explicit expression of the second Piola-Kirchhoff

stress  tensor  arising  from  the  hyperelastic  potential , in

matrix form (assuming a 3D problem). You will denote the

components of C 1   as Cij 1 . To lighten notations replace the

/ I3   by

respectively the letters A, B and K.

[6 marks]

[TOTAL 25 MARKS]


Section C (Civil only)

C.1

(i)   You  are  tasked  with  producing  a  finite  element  model  to estimate the bearing capacity of a pile with square cross-section in undrained clay. Using sketches where necessary to support your arguments, answer the following questions:

(a) What dimensionality will the model have and why?

(b) How will you decide the extent of the domain to model?

(c) What boundary conditions will you apply, where and why?

(d) What initial conditions will you apply, if any, and why?

(e) How will you apply the load, where and why?

(f) What material model will you use and why?

(g) What type of element(s) will you use and why?

(h) Discuss your strategy for meshing the domain.

Note: Your sketches can be drawn using software or by hand. [20 marks]

(ii) You are asked to  model groundwater flow with a four-node bilinear quadrilateral element interpolating the pore pressure only. Is locking expected to be an issue with this element or not, and why? [5 marks]

[TOTAL 25 MARKS]

2023-05-22