代做DEN401/DENM004 Computational Engineering代做Python语言

DEN401/DENM004

Computational Engineering

Assessment 2022/2023

Question 1

A chimney is designed to vent gases over the top of an industrial building. For the design purposes chimney can be considered as a uniform hollow circular tube with a mean diameter D, a wall thickness t and the height H. The design requirements are that the chimney must not fail under the wind loads, and that the amount of material it is made of is the lowest possible. For the design purposes the chimney can be considered as a cantilever beam that is subjected to a uniform. lateral wind load q (N/mm), as shown in Figure 1. The following requirements are to be satisfied:

1. the chimney must not fail in either bending or shear,

2. the deflection at the top should not exceed 127 mm.

3. the ratio of the mean diameter to the wall thickness must not exceed 60.

All relevant information and equations are given in the Table of figure 2. According to the design requirements the ranges of values are D and t are: 150 ≤ D ≤ 650 mm and 2 ≤ t ≤ 15 mm.

Figure 1: Chimney showing loading and dimensions

a) Formulate the optimum design problem. [25 marks]

b) Solve the optimization problem using the graphical method using only the following 11 values

of the mean diameter: (150, 200, 250, 300, 350, 400, 450, 500, 550, 600, 650) [25 marks]

Figure 2: Material properties and relevant equations for the problem shown in Figure 1

Question 2

In the design of a structural component the criteria F1   and F2   are to be minimised; where F1 describes the maximum stress in MPa and F2  describes the cost of the component in pounds £ . These criteria depend on one design variable x in the following way:

F1 (x) = 200(x2 - 4x + 5)

F2 (x) = 25(48 - 9x)

The design variable x can take any values in the range between 0 and 4.

a) Plot the Pareto optimum set by identifying several points in the space of the two criteria F1  and F2 . [10 marks]

b) Assuming that the desirable value of the criterion F1  is 250 MPa and that of the criterion F2  is £500, formulate and solve the corresponding minmax problem. Identify the obtained design as a point of the Pareto optimum set. [8 marks]

c) Describe briefly how this minmax problem can be re-formulated using the bounded objective function formulation.  State (i) the benefits and (ii) any setbacks this formulation brings as compared to the original formulation of the multiobjective problem. [8 marks]

Consider the two-beam problem shown in Figure 3. The beams are of length L = 1m and L = 0.25m, have moment of inertia I=2 × 10-4  m4  and modulus of elasticity E=200 GPa. Some force is applied to the left side of the beam causing a vertical displacement of 0.001 m.

Figure 3: Beam bending problem domain.

d) Using two elements to resolve the two-beam sections, solve the displacement problem and find the rotations and displacements at the three points of the two beams. You may use any node and element numbering as you wish. [17 marks]

e) What force was used to cause the displacement on the left side? [8 marks]

Question 3

The plane stress structure, of thickness t = 0.01 m, shown in Figure 4 is represented by a single triangular element. The node numbers of the elements are shown in red, their positions are stated in metres.

Figure 4: Diagram of 2D Plane Stress Structure

a) Using the interpolating properties of the FEM basis functions, write an expression for the 3

linear basis functions at the nodes of the element. [12 marks]

b) Form. the force vector of the element’s stifness equations for the surface pressure applied to the bottom edge. Do this by evaluating the integral along element’s surface using an appropriate numerical integration rule. [18 marks]

c) Use your solution to part b) to solve for the displacements using the following global matrix system. If you have not answered b), use the vector provided in equation 2.

The force vector - ONLY TO BE USED IF YOU HAVE NO ANSWER FOR PART b). Note this has no relation to the answers to part b).

[10 marks]

d) An additional body force of B kN/m3 in the y direction is placed evenly across the element. Determine the maximum value of B such that the displacement does not exceed 0.0001m in the negative y direction. [10 marks]