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CONFIDENTIAL EXAM PAPER

MECH3361
MECHANICS OF SOLIDS 2

Final Examination
Semester 2 – 2021

Total Duration: 2 hours and 30 minutes

Writing Time: 2 hours
Reading Time: 10 minutes
Organising answer document for submission: 20 minutes

INSTRUCTIONS TO CANDIDATES

There are two sections:
10 × Multiple Choice / Short Answer Questions (50 marks)
2 × Long Answer Questions with Detailed Derivation/Calculation (50 marks)

Instructions:
This is an open book exam and you are allowed to access all teaching materials as well as computer system (including Matlab/Ansys, etc) and/or calculator.
Note that the questions are not equally weighted between the different sections and please carefully read through and attempt all the questions.
Please show all working/reasoning if required.
The exam question sheet can be downloaded to your local devices. Your answer should be arranged sequentially and organized in a single document for submission.
Your answer document can be organized and presented in the following ways:
Option #1: Download the MS Word exam paper (this document) to your own computer and work on the questions in computer including insertion of equations and figures electronically. Finally, submit your answer in the Word file with a filename of “Your surname + First name + SID” (e.g. Li Qing 493518607)
Option #2: Download the MS Word Quiz paper (this document) to your own computer and print the quiz sheets to blank paper. Then work on the questions by hand-writing on the printed exam answer sheet. Finally, scan/photograph the hand-writing exam answer sheet to upload in a single file in right order (similar filename to Option #1).
Option #3: Download the MS Word Quiz paper (this document) to your own computer. Then work on the questions in separate sheets (exam answer sheet) with clear indication of question number and page number on each page. Finally, scan/photograph the hand-writing quiz answer sheets and merge with your MS Word Quiz paper in a single file for submission (similar filename similar to Option #1).
Option #4: You can combine these above three options. For example, you can type the answers in Word for some questions (e.g. multiple choice questions), and insert the photos of your hand-writing answers for the other questions (e.g. long answer calculation questions Q11 and Q12) in the MS Word Quiz paper, finally submit as a single document (similar filename similar to Option #1).

Student Signature: _______________________________________________ Date: _____________?
SECTION 1 (50 Marks) – MULTIPLE CHOICE / SHORT ANSWER QUESTIONS
All questions in this section are worth five (5) marks equally.

Question 1 (5 marks)
For the structure made of isotropic material with a certain Young’s modulus E and Poisson’s ratio ?. A very long circular tunnel and block is modelled as a 2D problem in Ansys. The stress results are listed through a screenshot as shown below.

If there is concern on the stress status at node #4 [σ_4], please refer to the stress result listed above to choose a proper infinitesimal element from the following answers (in MPa).

(f) None of these above choices, please draw your own in the blank cube above.
Question 2 (5 marks).
A FE model was created to solve the mechanical problem for a very “long” circular tunnel on a large block made of an isotropic material (Young’s modulus E=120 GPa and Poisson’s ratio ?=0.3). The strain status of square element A was shown (in micro-strain ?? i.e. 10^(-6)).

(1) Which of the following strain tensors is correct?

(2) Please use the conditions given to calculate the hydrostatic stress of this element.

?

Question 3 (5 marks).
A thin aluminum disk A (inner disk) has been thermally shrink-fitted into another thin steel disk B (outer tube). The final dimensions of the hybrid composite disk are R_Ai=25 mm, R_Ao=50 mm, R_Bi=50 mm, and R_Bo=75 mm. After fitting, a radial pressure p=100 MPa is expected to generate in the interface between the inner (A) and outer (B) disks. If the material properties are aluminum E_Al=70 GPa, ?_Al=0.3, Yielding strength 〖[σ〗_Al]=250 MPa; and mild steel E_St=210 GPa, ?_St=0.3, Yielding strength 〖[σ〗_St]=350 MPa.
Please use von Mises criterion to determine whether or not the inner disk is free of yielding.

?

Question 4 (5 marks).
To simulate a bi-layered cylindrical material system (aluminum in the upper part and mild steel in the bottom part) with a circumferential traction p, a 2D finite element model can be created. For the adopted mesh of Q4 elements as shown, the elemental stiffness matrix can be expressed as: [k]=1/2 ∫_V?〖[B]^T [E][B] 〗 dV.

Which of the following matrices is correct for the problem modelled and why?

[E]=E/(1-?^2 ) [■(1&?&0@?&1&0@0&0&((1-?))?2)]
[E]=E/(1+?)(1-2?) [■(1-?&?&0@?&1-?&0@0&0&((1-2?))?2)]
[E]=E/(1+?)(1-2?) [■(1-?&?&?&0@?&1-?&?&0@?&?&1-?&0@0&0&0&((1-2?))?2)]
[E]=E/(1+?)(1-2?) [■(1-?&?&?&0&0&0@?&1-?&?&0&0&0@?&?&1-?&0&0&0@0&0&0&(1-2?)/2&0&0@0&0&0&0&(1-2?)/2&0@0&0&0&0&0&(1-2?)/2)]
None of these, and write your own material matrix [E] below

?

Question 5 (5 marks).
The cylindrical bearing is loaded by a force P as shown. It is assumed that the materials of the cylindrical rollers and bearings are the same.

(1) At which point is the cylindrical bearing likely to experience the highest stress σ_P (peak contact stress) and why?
Point A
Point B
Point C

(2) If the magnitude of force P was increased to 4P, which of the following is correct for the peak stress σ_4P?
σ_4P=4σ_P
σ_4P=2σ_P
σ_4P=σ_P
σ_4P=0.25σ_P
None of these, please write your own answer.

Question 6 (5 marks).
A plane stress problem consists of two constant strain triangular (CST) elements as shown below. The corresponding global nodal displacement vector is given as:
elements (1) and (2) are given in the symmetric forms below:
Which of the following global stiffness matrices can be correct?
None of these, please write your own.
Question 7 (5 marks).
(1) Which of the following plots is a possible distribution of the von Mises stress σ_vm for the two CST (T3) elements as shown, and why?
(2) Which of the followings is a possible distribution of the u-displacement (u) for the two LST (T6) elements as shown, and why?
Question 8 (5 marks).
For the following four massless spring (spring stiffness k=k_1=k_2=k_3=k_4) and two-mass (m) system as shown.
(1) Which of the following describes the global stiffness matrix?
(2) For the above spring-mass system (neglect the masses of springs), which of the following plots could be a proper mode shape of the system

Question 9 (5 marks).
For a quadrilateral thin plate structure as shown in the Cartesian coordinate system, a single isoparamatric Q4 element (ABCD) is used, in which side AB is constrained the y-displacement and size AD is constrained the x-displacement, as shown. The nodal displacements were obtained from finite element analysis as:
Node A: 〖{u_A,”??” v_A}〗^T=〖{0,0}〗^T. Node B: 〖{u_B,”??” v_B}〗^T=〖{4, 0}〗^T
Node C: 〖{u_c,”??” v_c}〗^T=〖{4,-4}〗^T, Node D: 〖{u_D,”??” v_D}〗^T=〖{0,-4}〗^T
It is assumed that a reduced numerical integration was used at a single Gaussian point.
Please calculate the displacements (u_G,”??” v_G) at this Gaussian point.

?

Question 10 (5 marks).
The strains ε_xx and ε_yy are plotted for the following two-CST finite element model as shown (in a unit of micro-strain με). It is assumed that there is no shear strain in the two elements, i.e. ε_xy =0.

Please calculate the elemental Tresca stresses in both the elements given that E=91GPa,?=0.3, and thickness t=1 for a plane stress problem.
SECTION 2 (50 Marks) – LONG ANSWER QUESTIONS WITH CALCULATION
Both the questions in this section are worth twenty five (25) marks. Show all relevant working.
Question 11 (25 marks).
To analyse a cantilever “thick” block as shown below, the experimental, theoretical and numerical methods are used. In the experimental study, a 45° strain rosette is mounted at point P as shown. The strain readings are ε_A=ε_(0°)=1×10^(-4), ε_B=ε_(45°)=2×10^(-4), and ε_C=ε_(90°)=3×10^(-4). The plate is made of mild steel with Young’s modulus E=200GPa, Poisson’s ratio ?=0.25 and Yielding strength 250MPa.
Determine the strain and stress tensors at this point (P);
Determine the “factors of safety” for von Mises and Tresca criteria respectively, and further justify which should be used for design of the structure;
If an Airy stress function of ?=ay^3+bx^3 y+cx^3 is attempted for the stress status in the x-y plane, use the experimental analysis results to evaluate constants a, b and c;
From the Ansys solution, a stress contour is shown below. Please estimate what specific stress component it is likely presenting (Please feel free to create an Ansys model if needed)?
(a) σ_xx (b) σ_yy (c) σ_zz (d) σ_xy

Question 12 (25 marks).
For the following questions, all structural members have the same material and geometric properties, with EI=1 and EA=12. Answers should be in terms of the given point load P.
In the fully-clamped horizontal structure (ABC) in Fig. Q12(a), a vertical force P is applied to the mid-span point B as shown. Use the finite element method to determine the deflection at point B, as well as the reaction forces and reaction moments at nodes A and C.
In order to enhance the stiffness of the horizontal beam structure, a vertical structural component (BD) is placed below the midpoint B through a pin joint and node D is fully fixed to the ground, as shown in Fig. Q12(b). Determine the deflection at midpoint point B and the reactions (forces and moments) at nodes A, C, and D.
To further increase the stiffness of the structure, a stopper is placed underneath midpoint B at a vertical distance of ? as shown in Fig. Q12 (c). It is assumed that midpoint B will touch the stopper under loading P. Determine the reaction force from the stopper and again, the reaction forces and moments at nodes A, C, and D.