Question 1. 20 marks
You will derive a column of the bending component of a beam element (shown below). Add the last two digits of
your student number (e.g. 12345678 would be 7 + 8 = 15), if the sum is:
• Less than 6 – analyse column 1 [k22 k32 k52 k62]
T
• Between 6 and 9 inclusive – analyse column 2 [k23 k33 k53 k63]
T
• Between 10 and 12 inclusive – analyse column 3 [k25 k35 k55 k65]
T
• Greater than 12 – analyse column 4 [k26 k36 k56 k66]
T
Explain the term Degree of Freedom and discuss the number and type of degrees of freedom for truss and beam
elements and how this relates to the size of the stiffness matrix.
Describe the assumption that governs the relationship between axial and bending deflection for beam elements.
Discuss the validity of this assumption and give an example of when it does not apply.
Derive the stiffness matrix for a beam element in bending with the Unit Displacement method.
External loads for a 2D beam element under bending
The stiffness matrix for beam deflection only is given as:
⎣
⎢
⎢
⎡
𝑓𝑓1𝑦𝑦
𝑀𝑀1
𝑓𝑓2𝑦𝑦
𝑀𝑀2⎦
⎥
⎥
⎤
= �
𝑘𝑘22 𝑘𝑘23 𝑘𝑘25 𝑘𝑘26
𝑘𝑘32 𝑘𝑘33 𝑘𝑘35 𝑘𝑘36
𝑘𝑘52
𝑘𝑘62
𝑘𝑘53
𝑘𝑘63
𝑘𝑘55
𝑘𝑘65
𝑘𝑘56
𝑘𝑘66
� �
𝑣𝑣1
𝜑𝜑1
𝑣𝑣2
𝜑𝜑2
�
Derive a column (given by your student number) of the stiffness matrix for the element shown above using the
Unit Displacement method. Hint: the basic equation for the curvature v′′ of a beam at the distance x from the
left end (beam is lying along x-axis), loaded by the positive moment M1 and the positive force y f1 at the left
end is given by EIv M f x y . = − 1 + 1 ′′
Department of Mechanical and Construction Engineering
Learning and Teaching
Question 2. 20 marks
Four structures involving complex boundary conditions and constraints are shown below. You will analyse one
of them. Add the third and fourth last two digits of your student number (e.g. 12345678 would be 5 + 6 = 11), if
the sum is:
• Less than 6 – analyze (a)
• Between 6 and 9 inclusive – analyze (b)
• Between 10 and 12 inclusive – analyze (c)
• Greater than 12 – analyze (d)
Analyze the structure above according to your student number. Assuming the use of beam elements, explain
how you would set up your model for an efficient, accurate solution.
Department of Mechanical and Construction Engineering
Learning and Teaching
Question 3. 20 marks
You will derive a column of the bending component of a beam element (shown below). Add the last two digits of
your student number (e.g. 12345678 would be 7 + 8 = 15), if the sum is:
• Less than 6 – analyze a
• Between 6 and 9 inclusive – analyze b
• Between 10 and 12 inclusive – analyze c
• Greater than 12 – analyze column d
Two polynomials are given as follows:
(a) (i) 𝐹𝐹(𝜉𝜉) = ∫ 6𝜉𝜉2 + 3𝜉𝜉 + 4 1
−1 𝑑𝑑𝑑𝑑 (ii) 𝐹𝐹(𝜉𝜉) = ∫ 8𝜉𝜉 + 2 1
−1 𝑑𝑑𝑑𝑑
(b) (i) 𝐹𝐹(𝜉𝜉) = ∫ 3𝜉𝜉2 + 6𝜉𝜉 + 5 1
−1 𝑑𝑑𝑑𝑑 (ii) 𝐹𝐹(𝜉𝜉) = ∫ 4𝜉𝜉 + 7 1
−1 𝑑𝑑𝑑𝑑
(c) (i) 𝐹𝐹(𝜉𝜉) = ∫ 9𝜉𝜉2 + 3𝜉𝜉 + 2 1
−1 𝑑𝑑𝑑𝑑 (ii) 𝐹𝐹(𝜉𝜉) = ∫ 12𝜉𝜉 + 1 1
−1 𝑑𝑑𝑑𝑑
(d) (i) 𝐹𝐹(𝜉𝜉) = ∫ 6𝜉𝜉2 + 8𝜉𝜉 + 3 1
−1 𝑑𝑑𝑑𝑑 (ii) 𝐹𝐹(𝜉𝜉) = ∫ 2𝜉𝜉 + 5 1
−1 𝑑𝑑𝑑𝑑
Solve the above polynomials (according to your student number) analytically to obtain the exact solutions.
The Gauss integration rules plane stress elements for 1 and 2 integration points are given by:
1: ( ) 2 (0) 1
1 p = F d ≈ F ∫− ξ ξ
2: ( ) ( 1/ 3) (1/ 3) 1
1 p = F d ≈ F − + F ∫− ξ ξ
Solve the polynomials (i) and (ii) using the Gauss rules for p = 1 and p = 2.
Discuss the accuracy of your answers calculated using Gauss integration in comparison with the analytical
results. Explain any inaccuracies.
How are different Gauss integration rules (e,g, P=1 and P=2) implemented in FEA? Use plane stress elements as
an example and include sketches to aid in your explanation.
What influence does element shape have on the accuracy of your model when using Gauss integration? What
element shape is best in terms of accuracy? Give an example and include sketches to aid in your explanation.
Department of Mechanical and Construction Engineering
Learning and Teaching
Question 4. 40 marks
A schematic diagram of a typical commercial aircraft fuselage is shown below.
Explain how you would set up a finite element model to efficiently analyze this structure. Describe and justify
the element types you would use for the fuselage skin, floor and supports, and landing gear (they can be
different) and explain loads and boundary conditions, and any multiple point constraints used to connect the
components. Sketch, explain and justify your meshing strategy for each component referring to the underlying
element formulation.
Finite element models are commonly used to optimize geometric features within a structure. Explain how you
could efficiently optimize the thickness of the fuselage skin based on your choice of element.

Sample Solution

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