Power system analysis homework solutionsFull description
Full description
Economics Midterm
Full description
Full description
solved chemistry homework Gar VII
solucionarioFull description
AE 321 Homework 9 Due in class on November 15, 2013 Problem 1.
Consider a prismatic (rectangular cross-section) beam stretching due to its own weight, as shown in figure below. The beam is not attached to a firm support. a) Formulate the boundary value problem for this beam. b) Derive the displacements field (u,v,w), including Rigid Body Translations and Rotations. c) Simplify (u,v,w) by eliminating the contribution of RBT an d RBR. d) Sketch the deformed beam under the effect of its own weight. Make sure to include in your schematic the shape of the beam corners and its free edges after it deforms. Explain . e) If the undeformed beam was colored co lored with vertical stripes how would the stripes look like after deformation? Why? f) If you were to fully bond a fragile flat panel on a face of this bar which face would you choose? g) If the strength of the material comprising this bar is
what is the maximum length ℓ critical critical
σ f
before the bar will break due to its own weight? If the bar length is 1.2ℓ critical critical where exactly will it break? h) If you were to design a cable for a space elevator what should the cross-section of the cable look like as a function of distance from the ground in order to minimize the material you need to use? i) If this space elevator cable is made of steel with σ f = 900 MPa, and minimum acceptable diameter of 1mm, what should the diameter of the cable be at LEO (2,000 km) altitude? z
g ℓ
y
1
Problem 2.
A prismatic beam with dimensions (x,y,z) = (2ℓ, 2h, w), made of a homogeneous and isotropic material, is subjected to bending as shown below. a) Using simple geometric arguments derive an expression for the displacement in x-direction as a function of the local angular deflection, θ, of the beam. b) Assuming that the beam is made of a homogenous and isotropic material, start from the displacement function derived in part (a) to derive the stresses in the beam. c) Use this stress distribution to relate the applied moment at the beam ends to the stress on the same surfaces. What is the name for the equation you derived? d) What assumptions did you make to derive the equation in part (c)? e) Determine the BCs and stresses on each surface of the beam. f) Determine the displacement field in the beam including Rigid Body Translations and Rotations. g) Simplify the displacements you derived in part (f) by eliminating the contribution of Rigid Body Translations and Rotations. h) Sketch the deformed beam under the application of a moment. Make sure to include in your schematic the shape of the beam corners and its free edges after it deforms. Explain your answers to receive full marks.
i) If the undeformed beam was colored with vertical stripes how would the long (x-y) an d short (y-z) faces look like after bending? Explain your answers to receive full marks.