MECH321 Midterm 1 Winter 2017

MECH 321 Winter 2017 Midterm 1 (3 problems, 20 points, 1h 20 minutes, two formula sheets are provided) Problem 1 (4 points): For each of these four questions, use a maximum of one line to answer. a) What is the difference between a matrix and a tensor? b) Why is the stress tensor symmetric? c) What does “invariant” mean in mechanics? d) What are the condition(s) to use small strains for deformations?

Problem 2: Shaft (8 points) Consider this shaft of diameter d=100 mm and length L=2 m. A force P=200 kN and a torque T1 =10 kN.m are applied at the end of the shaft. Another torque T2 =2 kN.m is applied half way along the shaft, as shown. Compute the maximum tensile stress in the system, determine where it occurs and along which direction(s) (neglect all stress concentrations).

d T1 T2 L

P

1

Problem 3: Three point bending test (8 points) In the so called three point bending test, a transverse force is applied in the middle of a simply supported beam-like sample, as shown.

A sample of unknown material was tested yesterday. When the force F= 60 N was applied, the strains were recorded with a rosette underneath the sample and half way along the sample. Someone already computed the strains from the rosette:

x = 300 y = -20 xy = 150 …but they forgot to report the orientation of the coordinate system xy on the surface of the sample. However this data is still usable. Assume that the material is isotropic and linear elastic, and compute its elastic modulus and Poisson’s ratio.

2

Formula sheet (1/2) Stress transformation  ij '  a im a jn mn Properties of the transformation matrix: a ij a kj   ik and a ji a jk   ik

 xx   yy  xx   yy  cos 2   xy sin 2   x ' x '  2 2   xx   yy  xx   yy  cos 2   xy sin 2  Mohr’s Formulae  y ' y '  2 2   yy   xx  sin 2   xy cos 2  x ' y '  2  ________________________________________________________________________

Equations of elasticity Stress Equilibrium:  ij , j  f i  0 Strains:  ij 

1 ui, j  u j ,i  2

or





1 (1   ) ij   kk  ij E  E    ij   kk  ij    ij  1   1  2 

Hooke’s law:  ij 

Strain compatibility equations: 2  2 xy  2 xx   yy  2 y 2 x 2 xy

2 2  2 zz   xy   yz  2 xz    xy z 2 xz yz

 2 xx  2 zz  2 xz   2 z 2 x 2 xz 2   yy  2 zz  2 yz  2 z 2 y 2 yz

 2 yy

2 2  2 xz   xy   yz    xz y 2 yz xy

2 2  2 xx   yz  2 xz   xy    yz x 2 xy xz

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