Oefententamen zonder antwoorden 2023
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Advanced Dynamics (WB2630 T1 S)
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Technische Universiteit Delft
Studiejaar: 2021/2022
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Preview tekst
1 2 3 4
Surname, First name
WB2630 Toets 2 Continuum Mechanics
BRIGHTSPACE
2022/23_Resit
1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0
Instructions
Before working on the exam, read carefully the following instructions:
- The exam lasts for three hours and features four questions. Each question is associated to the
same amount of points. Read all the questions before starting the exam.
The questions are formulated in English. You can choose to answer them in Dutch or in English.
The path to your final answers matters. Provide the steps to reach the final answers. Points are
granted for the intermediate steps and unjustified answers receive no or only partial points.
- After the exam, your copy will be scanned to be graded in ANS. Only write within the answer boxes,
unless asked otherwise! Answers provided outside the boxes cannot be scanned properly and will
not be graded.
The formula sheet can be found at the end of the exam. You are allowed to take it out of the staple.
The use of electronic devices, e., calculators, phones, computers, is not allowed.
Don’t forget to fill in your student number at the top of the exam!
Place your campus card on the right corner of your desk.
If applicable, place your extra-time statement on the other corner of your desk.
Scrap paper, exam questions, and other exam-related documents may not leave the room during
and after the exam and may not be reproduced and/or made public.
- Fraud regulations as stated in the Rules and Guidelines of the Examination Board (RRvE) apply
during this exam.
↷
1 1c We cut the body in two parts and study one of the facets created by the cut with outward normal
n = [1 2 2]
T . Compute the normal stress vector acting on this facet at point P located at xP = [1 0 0]
T .
1p 1d Using the given stress tensor components, compute the forces per unit volume that act on the
structure when it is in equilibrium.
Question 2
Consider a fries cutter as shown in the figure below. The fries cutter can be modeled as a mechanism
made of two rigid components: an L-shaped component and a short straight component connected by
frictionless pins. To describe the mechanism, a reference system [x, y] is used. The L-shaped component
makes an angle θ with the x-axis, while the short component makes an angle φ with the x-axis. The
dimensions of the mechanism are shown in the figure. To cut a potato, a vertical force H is applied on
the L-shaped component. The potato applies a resisting force P on the mechanism. The mechanism is
supported by a clamp at the reference frame origin and a roller where the force P is applied.
1p 2a Derive a relation f [θ, φ] between the virtual rotations δθ and δφ associated to the angles θ and φ,
such that δφ = f [θ, φ]δθ.
1 2b Compute the total external virtual work performed by the forces H and P.
1 2c The fries cutter is in equilibrium, use the principle of virtual work to express the force H as a function
of the force P , the angles θ, φ, and the dimensions L 1 , L 2 , and L 3.
Hint: If you were not able to relate the virtual rotations δθ and δφ, continue with δφ = f [θ, φ]δθ.
↶
Question 3
We consider the truss structure presented below described using a global reference frame [x, y]. The
structure is made of four bars connected by pin joints. Each bar is characterized by a cross-section area
A and a Young’s modulus E. The dimensions of the structure are described in the figure. The structure is
supported at Joints 1 (clamp), 2 (roller), and 3 (roller). A displacement is prescribed in the x-direction at
Joint 2 and a quadratic load is applied on Bars (1) and (4).
A global stiffness matrix K is built for the structure and is given as
K =
EA L
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 +
√ 22
√ 22
− 1 0 0 0 −
√ 22
−
√ 2
√ 2 22
√ 22
0 0 0 0 −
√ 22
−
√ 22
− 1 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0
√ 22
√ 22
−
√ 22
−
√ 22
0 0 0 0
√ 22
√ 22
−
√ 22
−
√ 22
−
√ 22
−
√ 22
0 0 −
√ 22
−
√ 22
√
2
√
2
−
√ 22
−
√ 22
0 0 −
√ 22
−
√ 22
√
2
√
2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
for an associated global displacement vector given as
u = ux 1 uy 1 ux 2 uy 2 ux 3 uy 3 ux 4 uy 4
T .
0 3a If the contribution of one or several bars is not fully included in the proposed global stiffness matrix
K, which one(s) is(are) missing? For each missing bar, provide the associated stiffness matrix
expressed in the global reference system [x, y].
0 3b If one or several missing bars were identified in subquestion a, how many components of the global
stiffness matrix K need to be updated to include the missing bar(s)? Write your answer in the box.
Which components of the global stiffness matrix K need to be updated to include the missing
bar(s)? Circle the components to be updated in the matrix below.
K =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
K 11 K 12 K 13 K 14 K 15 K 16 K 17 K 18
K 21 K 22 K 23 K 24 K 25 K 26 K 27 K 28
K 31 K 32 K 33 K 34 K 35 K 36 K 37 K 38
K 41 K 42 K 43 K 44 K 45 K 46 K 47 K 48
K 51 K 52 K 53 K 54 K 55 K 56 K 57 K 58
K 61 K 62 K 63 K 64 K 65 K 66 K 67 K 68
K 71 K 72 K 73 K 74 K 75 K 76 K 77 K 78
K 81 K 82 K 83 K 84 K 85 K 86 K 87 K 88
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
1 3e Compute the consistent nodal forces at Joints 1, 3, and 4 equivalent to the external distributed load
acting on Bars (1) and (4). The distributed load is quadratic along the bars and takes its maximal
value f at Joint 4. Make use of the finite element method and linear bar elements, no points will be
awarded for solving the problem with statics.
↶
↷
↷
Question 4
Consider the retaining wall shown in the figure below. A reference system [ex, ey, ez ] is used to study the
wall. The wall is modeled within a two-dimensional framework using the finite element method. Bilinear
quadrangle elements with straight edges aligned with the reference system are chosen.
We use four-node quadrangle elements to build the finite element model of the wall and study a particular
quadrangle that is characterized by side lengths a and b.
The nodal displacements for the quadrangle are given as
d = [u 1 w 1 u 2 w 2 u 3 w 3 u 4 w 4 ]
T
where ui and wi with i = 1 , 2 , 3 , 4 are the nodal displacements in the x− and z−direction, respectively.
A bilinear approximation in the physical coordinates x and z is chosen to represent the displacement of
the quadrangle
u[x, z] = α 0 + α 1 x + α 2 z + α 3 xz
w[x, z] = β 0 + β 1 x + β 2 z + β 3 xz,
where αi and βi with i = 0 , 1 , 2 , 3 are the internal parameters.
2 4a Express the internal parameters α 0 , α 1 , α 2 , α 3 and β 0 , β 1 , β 2 , β 3 as functions of the nodal
displacements u 1 , u 2 , u 3 , u 4 and w 1 , w 2 , w 3 , w 4. What does these relations enforce for the
displacement across elements? Build the shape functions N 1 , N 2 , N 3 , N 4 for the quadrangle.
↶
1p 4b Check that the first shape function satisfies the condition
N 1 [xi, zi] = δ 1 i for i = 1 , 2 , 3 , 4 , where δ 1 i is the Kronecker delta.
Hint: If you were not able to find an expression for the first shape function, use
N 1 [x, z] = C 0 + C 1 x + C 2 z + C 3 xz.
1p 4c To describe the retaining wall with a two-dimensional model, an assumption is made, as shown in the
figure. What is this assumption? Explain why this assumption can be made. Using this assumption,
what can be said about the stress and strain tensors?
↶
↷
0 4d Using quadrangles with straight edges parallel to the reference system is a strong restriction. When
modeling the wall, explain the restriction resulting from the use of such elements. What can be done
to overcome this restriction?
