Attempt to solve the problem of torsion of a rectangular cross section using the boundaryequation method, for which you should postulate a Prandtl stress function of the form =Kf(x1, x2), where f(x₁, x2) is the product of four functions, each of which vanishes on oneof the sides £₁ = ±a and x2 = ±b (assuming the width of the bar is 2a in î₁ and 26 in x2.Show that you cannot satisfy the compatibility relationship with this stress function. (Note:This illustrates that the boundary equation method is not a general approach for all crosssections and motivates the more general approach in which a solution is sought to the Poissonequation, as we did most generally using Fourier expansions.)

By assuming a Prandtl stress function with the following form, we can use the boundary equation…

x2 Attempt to solve the problem of torsion of a rectangular cross section using the boundary equation method, for which you should postulate a Prandtl stress function of the form = Kf(x1, x2), where f(x1, x2) is the product of four functions, each of which vanishes on one of the sides x₁ = ±a and x₂ = ±b (assuming the width of the bar is 2a in ₁ and 26 in x2. Show that you cannot satisfy the compatibility relationship with this stress function. (Note: This illustrates that the boundary equation method is not a general approach for all cross sections and motivates the more general approach in which a solution is sought to the Poisson equation, as we did most generally using Fourier expansions.)

x2 Attempt to solve the problem of torsion of a rectangular cross section using the boundary equation method, for which you should postulate a Prandtl stress function of the form = Kf(x1, x2), where f(x1, x2) is the product of four functions, each of which vanishes on one of the sides x₁ = ±a and x₂ = ±b (assuming the width of the bar is 2a in ₁ and 26 in x2. Show that you cannot satisfy the compatibility relationship with this stress function. (Note: This illustrates that the boundary equation method is not a general approach for all cross sections and motivates the more general approach in which a solution is sought to the Poisson equation, as we did most generally using Fourier expansions.)

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Concept explainers

Axial Load

When a bar of the uniform cross-section is acted by a load, such that the load acts along the longitudinal axis of the bar, such kinds of loadings are known as axial loadings. In general terms, a load is an external force acting on a component. An axial load acts perpendicular to the geometric cross-section of the component. Based on the direction of the application of the load, an axial load can be tensile or compressive. The analysis of bodies under the action of axial loads is generally studied under the branch of mechanics, known as statics.

Question

Attempt to solve the problem of torsion of a rectangular cross section using the boundary
equation method, for which you should postulate a Prandtl stress function of the form =
Kf(x1, x2), where f(x₁, x2) is the product of four functions, each of which vanishes on one
of the sides £₁ = ±a and x2 = ±b (assuming the width of the bar is 2a in î₁ and 26 in x2.
Show that you cannot satisfy the compatibility relationship with this stress function. (Note:
This illustrates that the boundary equation method is not a general approach for all cross
sections and motivates the more general approach in which a solution is sought to the Poisson
equation, as we did most generally using Fourier expansions.)

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