ME558 Advanced Materials and Structural Design 2022/23 S1
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Assignment #1
ME558 Advanced Materials and Structural Design
2022/23 S1
Q1. The stress state at a point can be represented by a matrix as
「G | G = |Tyx |LTzx |
Txy Gyy Tzy |
T ] | Tyz | Gzz 」| |
The stress on any plane passing through this point can be given by
「G | LTzx |
Txy Gyy Tzy |
Txz ]「nx ] 「Gx ] Tyz | |nx | = |Gy | Gzz 」| |Lnx 」| |LGz 」| |
where n = nx ny nz T is the unit normal vector of that plane. For a 2D stress state
shown below, determine the normal stress and shear stress applied on the plane with normal as shown in Figure Q1 below.
y
x
Figure Q1
Q2. In real engineering components, stress (and strain) are 3-D tensors but in prismatic structures such as a long metal billet, the length of the structure is much greater than the other two dimensions (see figure below). The strains associated with the length direction, i.e., the normal strain czz and the shear strains cxz and cyz (if the length is the z-direction) are constrained by nearby material and are small compared to the cross- sectional strains. Plane strain is then an acceptable approximation. The strain tensor for
「c c 0]
plane strain is written as: ε =
Figure Q2
For isotropic material at plane strain state, demonstrate that the stress along the length
direction (z-direction) is given by
装zz =v(装xx + 装yy )= 入(cxx + cyy ),
where Lame’s constant 入= with E and v being the Young’s modulus
and Poisson’s ratio, respectively.
Q3. Four stress states are shown by 4 stress elements (a-d) as follows. (a)Determine which two stress elements describe the same state and justify your
answer. (10 marks)
Figure Q3
(b)If the stress state you identify in (a) is a plane strain state, determine the normal stress out of the plane in terms of T0 . (10 marks)
Q4. A cylinder pipe with internal and external radii a = 5 cm and b =6 cm is subjected to a pressure ofp = 50 MPa inside as shown in Figure Q4. The Young’s modulus and Possion’s ratio of the material are E = 200GPa , and v = 0.3 respectively. Based on the displacement solution provided in lecture notes, determine:
a) The strain compoments ETT (T), Eee (T) as functions of r. (6 marks)
b) The stress compoments GTT (T), Gee (T) as functions of r. (6 marks)
c) If the pipe is constrained at both ends, the problem can be deemed as a plane strain problem. Determine the normal stress out of the plane as a function of
r. (8 marks)
Figure Q4
Q5. For a 3D stress state given in matrix 「 | G = |Tyx |LTzx |
Txy yy Tzy |
T ] | Tyz | zz 」| |
demonstrate that the eigen values and eigen vectors of this matrix represent the principal stresses and principal directions (i.e., the normals of the principal planes).
2022-09-17