1.050: Energy Bounds in Elasticity (HW#10) Due: Friday, November 30, 2007 MIT – 1.050 (Engineering Mechanics I) Fall 2007 Instructor: Markus J. BUEHLER Team Building and Team Work: We strongly encourage you to form Homework teams of three students. Each team only submits one solution for correction. We expect true team work, i.e. one where everybody contributes equally to the result. This is testified by the team members signing at the end of the team copy a written declaration that "the undersigned have equally contributed to the homework". Ideally, each student will work first individually through the homework set. The team then meets and discusses questions, difficulties and solutions, and eventually, meets with the TA or the instructor. Important: Specify all resources you use for your solution. The following set of exercises is designed to familiarize you with the use of energy bounds in linear elasticity. This problem set is focused on 1D methods, and you will be able to apply the techniques you have learned in Lectures 27, 28 and 29. Please review the lecture notes and handouts carefully. For some of the solutions it is recommended to use Maple or Mathematica software to solve the resulting system of linear equations. 1. Truss structure: The sketch below shows a truss structure constructed with 6 vertical members and a horizontal rigid bar. The structure of subjected to two loads at points A and B (thus, this is a force-driven experiment). The top of the truss structure is connected to a fixed boundary. The goal of this problem is to solve for all displacements and axial forces using the structure’s energy content. L L L L L P Fixed boundary Rigid bar A B 2PDue: Friday – November 30, 2007 (in class) Page 2 of 7 r a. Proposed, approximate solution 1: Use the displacement field ξ '=ξ0'e r z , which is displayed in the following sketch: P L L L L L ze rr '' 0ξξ = 2P i. Confirm that this displacement field is kinematically admissible. ii. Compute the potential energy for the truss system, given this displacement field. iii. Use the theorem of minimum potential energy to determine ξ0 ' and the minimum potential energy, min Epot (ξ0'). ξ0' iv. Calculate the forces in each truss member. Check if the resulting force field is statically admissible (Hint: carry out a moment and force balance). Discuss the results. b. Proposed solution 2: Derive the displacement field which matches the kinematic conditions displayed in the following sketch: L L L L L P 0ξ1ξ 2PDue: Friday – November 30, 2007 (in class) Page 3 of 7 i. Compute the potential energy for the truss system, given this displacement field. ii. Use the theorem of minimum potential energy to determine ξ0,ξ1 and the minimum potential energy, min E (ξ0 ). ξ0 pot iii. How does the solution for the potential energy for ξ0 compare with the one obtained in Part a? Discuss your finding. iv. Calculate the forces in each truss member. Check if the resulting force field is statically admissible. Discuss the results. c. For the same force-driven experiment of Parts a and b, i. Compute the complementary energy for the truss system. ii. Use the theorem of minimum complementary energy to determine the axial forces and the minimum complementary energy, min Ecom (Ni ) . Ni iii. Compare the solution of Part c,ii to the solution of Part b,ii. What do we conclude about the quantities ξ0,ξ1, Ni ? Is the solution from Part a for ξ0' an upper or a lower bound to the actual solution ξ0?Due: Friday – November 30, 2007 (in class) Page 4 of 7 2. Minimum potential energy approach: The photograph below shows a section of a suspension bridge. We will use what we have learned about truss structures, beam deformation, and energy methods to analyze the deformation behavior of the top cord and the vertical cables. The engineering model of a simplified structure is presented in the next sketch. Photograph adapted from: O'Donnell, P.S. (2007) Akashi Kaikyo Suspension Bridge, in David Denenberg, www.bridgemeister.com, http://www.bridgemeister.com/pic.php?pid=996. Photograph removed due to copyright restrictions.Note: Please take a look at the handout of Lecture 29, which summarizes the approach for another example. Note that the total length of the elastic beam is 2L .Due: Friday – November 30, 2007 (in class) Page 5 of 7 Solve this problem using the potential energy approach. Assume that the displacement of the elastic beam is of the form: ⎛x − L ⎞2 ξz (x) =α+β⎜ ⎟ ⎝ L ⎠ a. Write the free energy of the system. To do this, i. First write the displacements of the three points: 1) at member 1, 2) at member 2, and 3) where the load is applied, as a function of the displacement function parameters α, β. ii. Express the free energy of the two trusses as a function of the parameters in the displacement function. iii. Express the free energy of the elastic beam as a function of the parameters in the displacement function. Start from the free energy expression of an elastic beam: 2 L h /2 b /2 10 02 ψB =∫∫ ∫ 2 E(εxx +ϑyz)dydzdx (see handout for Lecture 29) x=0 z=−h /2 y=−b /2 iv. Express the external work as a function of the parameters in the displacement function. v. Write the expression for the potential energy E pot(α, β). b. Using the minimum potential energy theorem, find the solution for α,βthat approximates the structure deformation solution. Calculate the potential energy for this set of parameters. Is the potential energy an upper or lower bound of the absolute minimum of the potential energy? c. Plot the displacement function, for E = 200 GPa, h = 0.05 m, b = 0.05 m, L = 1 m (h,b,L are dimensions of the beam element), S = b2 , H = 2 m (S,H are dimensions of the truss elements), P = 100 kN, using Matlab, Excel, Maple, or Mathematica. Do not hand in a hand-drawn sketch. Also provide the maximum displacement at the point where the force is applied. d. Is the displacement ξ0 an upper bound or lower bound to the actual solution? (Do not calculate the actual solution; simply discuss this point). e. Calculate the forces in each truss member, and check if the solution is statically admissible. Discuss the results. f. In a few sentences, discuss a possible strategy to arrive at the exact solution to this problem.Due: Friday – November 30, 2007 (in class) Page 6 of 7 3. Solving a statically