## The bending component of a beam element

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:

⎣

⎢

⎢

⎡

??1??

??1

??2??

??2⎦

⎥

⎥

⎤

= �

??22 ??23 ??25 ??26

??32 ??33 ??35 ??36

??52

??62

??53

??63

??55

??65

??56

??66

� �

??1

??1

??2

??2

�

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 – analyse (a)

• Between 6 and 9 inclusive – analyse (b)

• Between 10 and 12 inclusive – analyse (c)

• Greater than 12 – analyse (d)

Analyse 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 – analyse a

• Between 6 and 9 inclusive – analyse b

• Between 10 and 12 inclusive – analyse c

• Greater than 12 – analyse column d

Two polynomials are given as follows:

(a) (i) ??(??) = ∫ 6??2 + 3?? + 4 1

−1 ???? (ii) ??(??) = ∫ 8?? + 2 1

−1 ????

(b) (i) ??(??) = ∫ 3??2 + 6?? + 5 1

−1 ???? (ii) ??(??) = ∫ 4?? + 7 1

−1 ????

(c) (i) ??(??) = ∫ 9??2 + 3?? + 2 1

−1 ???? (ii) ??(??) = ∫ 12?? + 1 1

−1 ????

(d) (i) ??(??) = ∫ 6??2 + 8?? + 3 1

−1 ???? (ii) ??(??) = ∫ 2?? + 5 1

−1 ????

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 analyse 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 optimise geometric features within a structure. Explain how you

could efficiently optimise the thickness of the fuselage skin based on your choice of element.