0910SEM1-ME2113

ME2113

NATIONAL UNIVERSITY OF SINGAPORE

ME2113 – MECHANICS OF MATERIALS I (Semester I : AY2009/2010) Time Allowed : 2 Hours

INSTRUCTIONS TO CANDIDATES:

1.

This examination paper contains FOUR (4) questions and comprises SIX (6) printed pages.

2.

Answer ALL FOUR (4) questions.

3.

All questions carry equal marks.

4.

This is a CLOSED-BOOK EXAMINATION.

5.

Programmable calculators are NOT allowed for this examination.

6.

Answer questions 1 and 2 in one booklet and questions 3 and 4 in another booklet.

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ME2113

Formulae The following formulae written using standard notations may be used.

Yield Criteria

τ max = 1 (σ max − σ min ) =

Tresca:

σY

2

2

[

Mises:

]

1 (σ 1 − σ 2 )2 + (σ 2 − σ 3 )2 + (σ 3 − σ 1 )2 ≤ σ Y 2

Hooke’s Law for isotropic elastic materials

1 (σ x −νσ y −νσ z ) E 1 ε y = (σ y −νσ x −νσ z ) E 1 ε z = (σ z −νσ y −νσ x ) E 1 1 ; γ yz = τ yz ; γ zx = τ zx G G

εx =

γ xy =

1 τ xy G

Strain – Displacement Relationships

εx =

∂u ∂x

εy =

∂v ∂y

εz =

∂w ∂z

γ xy =

∂v ∂u + ∂x ∂y

γ yz =

∂v ∂w + ∂z ∂y

γ xz =

∂u ∂w + ∂z ∂x

;

where G =

E 2(1 + ν )

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ME2113

QUESTION 1 A beam of flexural rigidity EI is fixed at both ends and is loaded by a point load P at point C and a uniformly distributed load w as shown in Figure 1a. Draw the composite M/EI diagram along the length of the beam and determine the reactions and bending moments (RA, RB, MA and MB) at points A and B using the moment-area method. Hence evaluate the deflection at point C. (25 marks) Y

P Load intensity w

X MA

A a

MB

B

C a

a

a

RA

RB

Figure 1a

The following information (usual notations apply) may be used: (i)

The area A0 under the parabolic curve KQ in Figure 1b is given by:

4 wa 3 A0 = − 3EI M/(EI) a/2 K

X

Centroid -2wa2/EI

Q 2a

Figure 1b

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ME2113

QUESTION 2 An unpressurized cylindrical storage tank, which has an inner diameter of 8.5 m and a wall thickness of 3 mm as shown in Figure 2, is made of steel having an allowable in-plane shearing stress of 200 MPa. (a)

Determine the depth h to which the tank can be filled with water if a factor of safety of 3.5 is desired. (8 marks)

(b)

If the tank is filled with water to a depth of h = 22 m, and the tank is subjected to a torque of magnitude T = 25 MNm, using the Mohr’s circle determine the maximum principal stress and the maximum in-plane shearing stress in the cylindrical wall. The weight density of water is 9.81 kN/m3. (17 marks)

8.5 m

T

25 m

h (Water depth)

Figure 2

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ME2113

QUESTION 3 A stepped bar consists of two sections with cross-sectional areas A1 and A2 respectively. Its ends A and B are fixed to rigid walls as shown in Figure 3. An axial load of magnitude P is applied at C. (a)

Determine the reaction RA at the end A in terms of the parameters P, A1, A2, L1, L2, E1, E2.

(b)

Calculate the stresses in section 1 and section 2, and the deflection at C for these values: P = 60 kN, A1 = 300 mm2, A2 = 150 mm2 , L1 = 400 mm, L2 = 400 mm, E1 = 100 GPa, E2 = 200 GPa.

(c)

Derive an expression for the horizontal displacement u(x) at an arbitrary distance x from A for the stepped bar. Use the values of P, A1, A2, L1, L2, E1, E2 given in Part (b). (25 marks)

A

E1

C

E2

P L2

L1 x Figure 3

B

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ME2113

QUESTION 4 (a)

The stress state in a thin film of a layered structure of an electronic device can be considered as plane stress, see Figure 4. Write down the non-zero stress components and the non-zero strain components.

(b)

Starting with the elastic stress-strain relations provided in the formulae sheet, invert these relations for the case of plane stress to obtain the stresses in terms of the strains.

(c)

At a particular point in the film, εx = 1200×10−6, εy = −150×10−6, γxy = 800×10−6. For E = 80 GPa and ν = 1/3, calculate the principal stresses at this point of the film.

(d)

Using Tresca yield criterion, check if yielding has occurred at this point. Take the film’s yield stress σY = 100 MPa. (25 marks)

Figure 4

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