3 032 Quiz 3 Fall 2006 DO NOT TURN THIS PAGE OVER AND START THE QUIZ UNTIL YOU ARE ASKED TO DO SO Guidelines 1 Show all your work on the sheets included in this stapled document 2 Use partial credit to your advantage If you re running short on time solve algebraically and then solve numerically plugging in numbers later 3 If there is not much space given for you to provide an answer we want you to be brief 4 You may not need to use all the information given e g dimensions to reach your conclusions 5 Enjoy your dark chocolate tru e before getting started It lowers blood pressure JAMA 2003 contains antioxidants Nature 2003 and supposedly helps neural synapses responsible for memory to re faster Nature Health 2003 NAME PRINTED I agree that this document represents my own independent work on this quiz using only my own brain my allowed single crib sheet of equations and notes and my pen pencil calculator protractor compass ruler sliderule SIGNATURE GOOD LUCK 1 1 I am often reminded and with good reason that MIT students should not be treated like brains on sticks Fair enough but let s consider for a moment the mechanical consequences of this mental picture My rst impulse is to wonder what such a stick should be made of so that it should not buckle under the weight of a human brain mass 1 4 kg Let s assume the stick is stuck rmly in the ground clamped is 180 cm in height approximately 6 and is cylindrical in cross section with the radius of a rat femur 1 9 mm Horcajada et al J Endocrin 165 2000 Strangely much easier to nd documented rat femur radii than human femur radii a What are the magnitudes of all the reaction forces and moments at the bottom of the brain stick Be sure to include even those reaction forces and moments that are equal to zero and state as such Figure by MIT OpenCourseWare Adapted from http upload wikimedia org wikipedia commons 6 6d Brain stem normal human svg Courtesy of Patrick J Lynch medical illustrator and C Carl Jaffe MD cardiologist b What is the minimum Young s elastic modulus E that this brain stick material must exhibit so that it remains stable under the prodigious weight of the typical brain c Given this requirement what material do you suggest the brain stick comprise There are several possible answers and yours should be well justi ed by your data and the intended application Material cost for the brain stick is not a constraint as brains are invaluable 2 d Motivated by the aesthetics of the human body I decide to make these sticks of 7075 T6 alu minum alloy rod with a radius of 6 mm about that of the human spinal cord I then realize that because I can only t so many brain sticks in a lecture hall I ll probably be making cyclic use of these sturdy brain sticks taking o a tired brain and putting on a fresh brain once per week Luckily I know the steady state crack growth behavior of this alloy under these conditions as indicated in the graph below da dN m cycle 100 10 1 0 1 0 01 7075 T6 Aluminum R 0 5 10 20 K MPa m 50 100 Figure by MIT OpenCourseWare Figure 1 Crack growth per fatigue cycle as a function of K for 7075 T6 aluminum What is the value of R for this cyclic loading e What are the values of C and m that characterize the steady state crack growth for this 7075 Al under this R Be sure to include units 3 f I decide I can visually inspect each brain stick to detect cracks on the surface that are about 100 m in length For the cyclic stress involving one brain per brain stick am I in the steady state crack growth regime g Since I did not acquire these da dN vs K data myself I decide to gure out if I need a safety factor by measuring the fatigue life of one such aluminum brain stick for a stress range of 1000 MPa For how many lecture weeks can I use each brain stick under this loading scheme before I should retire it for fear of fatigue failure h As the brain stick is made of 7075 Al and is under cyclic axial loading how do you expect the fatigue failure surface to appear i This material is already an alloy but I want to further extend the fatigue lifetime of the brainstick Brie y but completely describe two options to improve the fatigue life of this speci c alloy i e further alloying is not an option 4 j Given that I m less than 180 cm tall and will have a tough time securing brains to the tops of these brain sticks let s take a di erent tack It may be easier for the brains if they are stuck onto a wooden beam suspended from the ceiling by wires on each end a sort of bench that can be set up easily in all lecture halls The brains will be equally spaced three to a bench as shown Wire 0 5 m 0 5 m 0 5 m 0 5 m Figure by MIT OpenCourseWare Adapted from http upload wikimedia org wikipedia commons 6 6d Brain stem normal human svg Courtesy of Patrick J Lynch medical illustrator and C Carl Jaffe MD cardiologist Figure 2 Brain bench A wooden beam 1 5 m x 1 m x 0 25 m suspended at each end by wires along which 3 brains can be spaced equally Brain graphic source science howstu works com brainport htm Determine the reaction forces at the ends of the bench If the suspension wires are made of hotrolled AISI 1020 steel with a true tensile fracture strength of 800 MPa what is the minimum wire diameter I can use for this purpose The density of wood is approximately 900 kg m3 and beam dimensions are given in the gure caption k Determine the shear force and bending moment diagram for this wooden beam using your method of choice to nd the magnitude of V and M at end points transition points and min ima maxima along the beam length The weight of the wooden beam should not be neglected but partial credit will be given if you do neglect it 5 2 In addition to predicting the mechanical responses of linear viscoelastic materials like amorphous polymers springs and dashpots can be used to created models of creep and recovery in crystalline materials a Although springs and dashpots are used to predict the behavior of amorphous polymers and crystalline materials in response to mechanical loading they are predicting two very di erent ways of dissipating mechanical energy Explain the di erence precisely but concisely in terms of elasticity plasticity and atomic molecular mechanisms b Below is a model that accurately captures the creep and recovery response of several micro crystalline materials For a step stress applied at t …