Massachusetts Institute of Technology Department of Aeronautics and Astronautics Cambridge, MA 02139 16.01/16.02 Unified Engineering I, II Fall 2003 Problem Set 12 Name: Due Date: 11/25/03 Time Spent (min) F12 F13 F14 M13 M14 M15 M16 Study Time Announcements:Unified Engineering Fall 2003 Fluids Problems F12–F14 F12. For the two flows given by . . . 2�(x, y) = arctan (y/x) �(x, y) = x 2 + y a) Determine the velocity fields, and sketch the streamlines. b) Determine the volume flow rate through a circle of radius r. c) Which of these flows is not feasible to set up in a lab? Explain. F13. A venturi has a minimum throat area of 0.7 times the inlet/outlet areas. The water tank is open to ambient atmospheric pressure. Determine the sea-level wind speed V which is needed at the inlet to raise the water column 10 cm. V 10 cm F14. �1(x, y) and �2(x, y) are known to be physically-possible flows (i.e. satisfy mass con-servation), and their corresponding pressure fields p1(x, y) and p2(x, y) are known via the Bernoulli equation. a) A third flow is now defined by �3(x, y) = �1 + �2. Explain how you would obtain its corresponding pressure field p3. b) Yet another flow �4 = ��1/�x is defined. Is this a physically-possible flow?† † † † † † † † Unified Engineering Fall 2003 Problem M13 (Materials and Structures) a) The state of axial stress through the thickness of a beam in pure bending (i.e. loaded only by a moment) is given by: Ê M ˆ ˜x3 for –h < x3 < +hs11 =ËÁ I ¯ M is the bending moment (which is constant in x1), and I is the second moment of area of the cross-section of the beam (which is also constant). If s12 =s32 =s22 =s33 = 0 what can you say about the variation of the shear stress s13 with x1, x2 and x3? b) The bending moment M now varies as a function of x1 according to M = cx1 The axial Ê M ˆ stress is still give by s11 =Á ˜x3. Again, s12 =s32 =s22 =s33 = 0. How does s13Ë I ¯ vary with x1, x2 and x3? Note s13 = 0 for –h = x3 and x3 = +h (the top and bottom surfaces of the bean are free surfaces and do not have any stress acting on them).Unified Engineering Fall 2003 Problem M14 For each of the states of plane stress shown acting on the differential elements drawn below do the following: 1) Draw a Mohr's circle describing the stress state. 2) Determine the principal stresses and the maximum shear stress 3) Calculate the angle from the x1 direction as shown to the more tensile principal stress direction. Note whether the angle is clockwise or counterclockwise. 2† † † Problem M15 (Materials and Structures) i) By considering the change in volume of an infinitessimal element undergoing small elongational strains show that the volumetric strain Ë ÊDVV ¯ ˆ = e1 +e2 +e3 ii) A continuous body experiences a displacement field, un that is described by: ( 2 2u1 =[0.5 x1 - x2 ) + 0.5x1x2 ]10-3 mm ( 2 2u1 =[0.25 x1 - x2 ) - x1x2 ]10-3 mm u3 = 0. Determine: a) The 6 components of the strain tensor as a function of position (i.e. in terms of x1, x2, x3) b) The rigid body rotation about x3 as a function of position (i.e. in terms of x1, x2, x3). c) The principal strains and the principal strain directions at x1= 5mm and x2=7 mm. d) The volumetric strain at x1= 5mm and x2=7 mm. 3Unified Engineering I Fall 2003 Problem M16 The purpose of this question is to demonstrate the equivalence of the two methods at our disposal for transforming strain (and stress) and for calculating the principal values and directions. Given a state of plane strain: e11 = -0.000200, e22 = +0.000400, e12 = -0.000200, do the following: a) Draw a Mohr's circle for the strain state. Note you may find it convenient to work in terms of "micro-strain" (strain/10-6) b) From the Mohr's circle determine the principal strains eI and eII and principal directions, x˜1 and x˜ 2 . You should specify the directions as counterclockwise angles with respect to the original x1 and x2 coordinate system. c) Determine the direction cosines for the transformation between x1,x2 and x˜1 ,x˜ 2 d) Using the appropriate tensor operation show that the original strain tensor (e)nm transforms to the principal strain tensor (e˜ pq).
View Full Document
Unlocking...