tions to the biharmonic equation in polar coordinates. In certain structural and mechanical engineering applications, curved beams are very common. The purpose of this exercise is to demonstrate that the solution methods we developed in class can be applied to such problems. (a) Determine the differential equation that results from using the Airy stress function, o(r, 0) = f(r) sin 0, in the biharmonic equation. (b) Using the differential equation from (a), make a substitution f(r) = rm to arrive at an equation for r and m. Your equation should have roots m = 1, m = 1, m = -1, m = 3. You may need to use a computer to simplify the equation. (c) Substitute these roots back into the Airy stress function (Note that for terms in f(r) containing multiple roots, the second such term should be multiplied by log r).
tions to the biharmonic equation in polar coordinates. In certain structural and mechanical engineering applications, curved beams are very common. The purpose of this exercise is to demonstrate that the solution methods we developed in class can be applied to such problems. (a) Determine the differential equation that results from using the Airy stress function, o(r, 0) = f(r) sin 0, in the biharmonic equation. (b) Using the differential equation from (a), make a substitution f(r) = rm to arrive at an equation for r and m. Your equation should have roots m = 1, m = 1, m = -1, m = 3. You may need to use a computer to simplify the equation. (c) Substitute these roots back into the Airy stress function (Note that for terms in f(r) containing multiple roots, the second such term should be multiplied by log r).
Question
Transcribed Image Text:0
a
r
F
Wall
Figure 1: Curved beam.
![tions to the biharmonic equation in polar coordinates. In certain structural and mechanical
engineering applications, curved beams are very common. The purpose of this exercise is to
demonstrate that the solution methods we developed in class can be applied to such problems.
(a) Determine the differential equation that results from using the Airy stress function,
(r, 0) = f(r) sin, in the biharmonic equation.
(b) Using the differential equation from (a), make a substitution f(r) = rm to arrive at an
equation for r and m. Your equation should have roots m = 1, m = 1, m = -1, m = 3.
You may need to use a computer to simplify the equation.
(c) Substitute these roots back into the Airy stress function (Note that for terms in f(r)
containing multiple roots, the second such term should be multiplied by log r).
(d) Find the stress components orr, 000, and ore from this Airy stress function.
(e) Write down boundary conditions for the problem shown in Fig. 1.
(f) Apply the boundary conditions to determine final equations for Orr, 000, and ore.
Show that the displacement field takes the form
Up
Uj
=
2D
2D
E
-0 cos 0 +
-0 sin 0
E
+ D
sin 0
E
[Ar²(1
Ar² (1-3v) + B
1+v
p2
+ D(1 - v) log r + K sin 0 + L cos 0
- 1) logr] +
cosº [Ar² (5 + v) + B¹ +¹ − D(1 − 1) log
E
p2
cos 0 + K cos 0 L sin 0 + Hr
1+v
E
where A, B, D, K, L, H are constants and E, v are elastic constants.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff0e69105-edad-4e40-a986-7d2af8f1b9bd%2F2392354d-6c3f-4725-83f1-a4f94a106317%2Fcsb8ngi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:tions to the biharmonic equation in polar coordinates. In certain structural and mechanical
engineering applications, curved beams are very common. The purpose of this exercise is to
demonstrate that the solution methods we developed in class can be applied to such problems.
(a) Determine the differential equation that results from using the Airy stress function,
(r, 0) = f(r) sin, in the biharmonic equation.
(b) Using the differential equation from (a), make a substitution f(r) = rm to arrive at an
equation for r and m. Your equation should have roots m = 1, m = 1, m = -1, m = 3.
You may need to use a computer to simplify the equation.
(c) Substitute these roots back into the Airy stress function (Note that for terms in f(r)
containing multiple roots, the second such term should be multiplied by log r).
(d) Find the stress components orr, 000, and ore from this Airy stress function.
(e) Write down boundary conditions for the problem shown in Fig. 1.
(f) Apply the boundary conditions to determine final equations for Orr, 000, and ore.
Show that the displacement field takes the form
Up
Uj
=
2D
2D
E
-0 cos 0 +
-0 sin 0
E
+ D
sin 0
E
[Ar²(1
Ar² (1-3v) + B
1+v
p2
+ D(1 - v) log r + K sin 0 + L cos 0
- 1) logr] +
cosº [Ar² (5 + v) + B¹ +¹ − D(1 − 1) log
E
p2
cos 0 + K cos 0 L sin 0 + Hr
1+v
E
where A, B, D, K, L, H are constants and E, v are elastic constants.
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