Bonus Problems

# Bonus Problems - Bonus Problems Bonus problems are intended...

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Bonus Problems Bonus problems are intended to encourage the students to think about the physical principles used in the course and invent problems that probe the understanding of these physical principles. Bonus problems are to be problems that require some calculations or algebra but only a minimal amount if use is made of some physical principle or understanding. Ideally they can be also solved with much more work by using formulas without exercising understanding. Acceptable problems: Only one problem from each chapter (2-7) will be accepted. Problems that are very similar to problems already given on exams, homework, or in the class slides will not be accepted. Problems about principal stresses will not be accepted. This file will be updated periodically with the list of problems submitted and accepted. No similar problems will be accepted. Credit: 1. The 2% requires the existence of the more complex solution. Without it the credit will be reduced to 1%. 2. If very similar problems are provided by more than one student, the credit will be reduced by one half. . Format for proposed problems: 1. What physical principle or understanding makes it easy to solve the problem. 2. Problem 3. Solution 4. More complex solution needed without taking advantage of physical understanding or principle. Examples of physical principles or understanding including submitted problems. CAUTION: Many of the principles below can be implemented in problems that will not be accepted as too trivial. If on the other hand, you show that there is a likely- to-be-used alternative that is much more time consuming the problem is likely to be accepted. Also, please avoid going after minor concepts that we should not expect students to remember (such as elastic section modulus). Chapter 2: a. The physical meaning of the indices of the stress components b. Taking advantage of the symmetry of the stress or strain tensors. c. Understanding traction and its normal and shear components

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Given the stress matrix for a structure is, , calculate the traction components on the x - y plane. Solution 1: traction on the x - y plane is in the direction x , , since we have (no shear components in the z direction), the traction has then no shear components. Solution 2: the traction component on the x-y plane is of the form , the normal is in the z direction ( l = 0, m = 0, n = 1) The shear component is defined as , where is the normal stress d. Understanding the physical meaning of transformed stress components. e. Physical meaning of deviator stresses. Given the following cylinder subject to a uniaxial compression of 3MPa find the deviator stress tensor.
Solution: With uni-axial compression σ 2 = σ 3 =0. Then: 123 3 1 33 m MPa σ σσ + + == = 20 0 010 00 1 xx m xy xz dx y y y m y z xz yz zz m TM ⎡⎤ ⎢⎥ =−= −− ⎣⎦ P a ) 3 Rrelationship between the stress invariants and the principal stresses.

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Bonus Problems - Bonus Problems Bonus problems are intended...

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