A simply supported beam with unit thickness (in the direction into the paper), which is subjected...

 

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A simply supported beam with unit thickness (in the direction into the paper), which is subjected to uniform pressure, p, on the top surface, is shown below. Consider this a 2D problem. (a) (b) (c) (d) (e) Find all the exact boundary conditions as a function of stresses or displacements in order to formulate the exact (in the sense of 2D elasticity) boundary value problem (BVP) shown the figure below: р y -X▶ 2w 20 To make this problem tractable in 2D elasticity, we prefer to formulate this BVP as tractions prescribed BVP by assuming the reaction shear force, R(y), on faces x = ± l. Write the boundary conditions on all 4 faces of the beam as a function of stress only. Draw a distribution for R(y) on faces x = l, which is statically equivalent to the boundary conditions in the Figure above. What is a general Airy stress function for this class of problems? Explain which terms in this general Airy stress function are zero for this particular problem, and provide a simplified form for the Airy stress function. What mathematical condition should this Airy stress function satisfy and what is the origin of this condition? Use a simple graphic and invoke mechanics principles to show the segment of the beam for which the elasticity solution derived from your Airy stress function for the problem is valid with a few percent accuracy. Suggest a reasonable /w ratio so that your solution is applicable in 90% of the beam. Which components of the stress tensor in the elasticity solution for the relaxed problem (part ii) are expected to be non-zero? Which components of the stress tensor practically vanish for very large t/w (i.e., slender beam)? A simply supported beam with unit thickness (in the direction into the paper), which is subjected to uniform pressure, p, on the top surface, is shown below. Consider this a 2D problem. (a) (b) (c) (d) (e) Find all the exact boundary conditions as a function of stresses or displacements in order to formulate the exact (in the sense of 2D elasticity) boundary value problem (BVP) shown the figure below: р y -X▶ 2w 20 To make this problem tractable in 2D elasticity, we prefer to formulate this BVP as tractions prescribed BVP by assuming the reaction shear force, R(y), on faces x = ± l. Write the boundary conditions on all 4 faces of the beam as a function of stress only. Draw a distribution for R(y) on faces x = l, which is statically equivalent to the boundary conditions in the Figure above. What is a general Airy stress function for this class of problems? Explain which terms in this general Airy stress function are zero for this particular problem, and provide a simplified form for the Airy stress function. What mathematical condition should this Airy stress function satisfy and what is the origin of this condition? Use a simple graphic and invoke mechanics principles to show the segment of the beam for which the elasticity solution derived from your Airy stress function for the problem is valid with a few percent accuracy. Suggest a reasonable /w ratio so that your solution is applicable in 90% of the beam. Which components of the stress tensor in the elasticity solution for the relaxed problem (part ii) are expected to be non-zero? Which components of the stress tensor practically vanish for very large t/w (i.e., slender beam)?

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In the image provided you are asked several questions about a simply supported beam under a uniform pressure load Lets address each question one by one a Exact boundary conditions for the 2D elasticit... View the full answer