MECH 321 Winter 2013 Midterm 1

MECH 321 Winter 2013 Midterm 1 Closed book, Calculators allowed, 1:20 min Detach the first page from the booklet and put your name on page 3. Only what is inside the frames will be graded 1) Short questions: answer in the boxes provided on page 3 (5 points) a) What is a “uniform” stress field? b) What is an isotropic material? c) What are the condition(s) for small strain approximation? d) What is an “invariant”? e) What is the use of the compatibility equations? 2) Windy day (8 points) This round post (diameter d) holds a sign. On a windy day, the wind applies a steady pressure p =20 kP on the sign, as shown. Neglect gravity, neglect stress concentrations. a) Show the stress element (with all the values of stresses) at the location where the stresses are the highest in the post. b) What are the maximum shear and maximum tensile stress at that location? c) The wind has picked up and the pressure on the sign has doubled to p=40 kP. What are the maximum shear and maximum tensile stress now? d=400 mm, h1=3 m, h2=0.5 m, w=5 m 3) A heavy beam (7 points) This homogenous, simply supported beam is subjected to its own weight. The width of the beam (along the z axis) is 2w. A proposed solution is shown below:

a) b) c) d)

  3 3  x 2  L2  y 2    2   xx  gy    2  h   5 2  h  2   1 gy  y  1 yy  2  2  h   2   3 gx 1  y   2   xy 2 Does this solution satisfy equilibrium everywhere?  h   Does this solution satisfy boundary conditions on the       0 xz yz  zz upper and lower edge (y=±h)? Is this solution consistent with the forces exerted at the two ends of the beam? Is it possible to compute the deflection of the beam in its center? If yes explain briefly how.

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Formula sheet MECH 321  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  Stress Equilibrium:  ij , j  f i  0 1 ui, j  u j ,i  2 1 Hooke’s law:  ij  (1   ) ij   kk  ij E Strain compatibility equations: 2  2 xy  2 xx   yy   2 y 2 x 2 xy

Strains:  ij 



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



 ij 

or

E 1 

    kk  ij    ij  1  2  

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

 2 yy xz



2 2  2 xz   xy   yz   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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MECH 321 Winter 2013 Midterm 1 Name: ___________________________________ 1 a) 1 b) 1 c) 1 d) 1 e) Problems 2 & 3

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