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Question 4 1 pts A small aspect ratio beam is illustrated below, with cantilevered boundary conditions and tip load. b=0.1 m 18-1398 L P If the Euler-Bernoulli beam theory is followed, the governing differential equations result in: $$w(x) = \frac{P L^2}{3EI} - \frac{P x}{2EI} (L^2 - \frac{x^2}{3})$$ This results in the following displacement field: $$ \{u, v, w\} = \{-z \phi(x), 0, w(x)\} = \{ \frac{P x}{2EI} (L^2 - x^2), 0, \frac{P L^2}{3EI} - \frac{P x}{2EI} (L^2 - \frac{x^2}{3})\}$$ However, Euler-Bernoulli beam theory is generally considered invalid for insufficiently slender beams, i.e. their length to width ratio should be at least 10. In this problem this ratio is 2, so it may be more appropriate to use Timoshenko beam theory, whose assumed displacement field allows for the formation of shear strains. For Timoshenko beam theory, a new degree of freedom $\phi$ is introduced such that plane sections are not necessarily perpendicular to the neutral axis. For cantilevered boundary conditions and tip load, the governing differential equations result in: $$w(x) = \frac{P L (L-x)}{KAG} - \frac{P x}{2EI} (L^2 - \frac{x^2}{3}) + \frac{P L^2}{3EI}$$ $$\phi(x) = \frac{P}{2EI} (x^2 - L^2)$$ This results in the following displacement field: $$ \{u, v, w\} = \{-z \phi(x), 0, w(x)\} = \{ \frac{P x}{2EI} (L^2 - x^2), 0, \frac{P L (L-x)}{KAG} - \frac{P x}{2EI} (L^2 - \frac{x^2}{3}) + \frac{P L^2}{3EI} \}$$ where, P= 17 MN is the applied tip load, E = 70 GPa is Young's modulus, I = $\frac{1}{12}bh^3$ is the appropriate 2nd area moment, G = 26.25 GPa is the shear modulus, A= bh is the cross sectional area, L = 2 m is the beam length, and K = 0.897 is the shear correction factor. What is the value of $e_{zz}$ at {x, y, z} = {L, 0,0} if the Timoshenko beam theory is followed?

Name: question 4 1 pts a small aspect ratio beam is illustrated below with cantilevered boundary conditions and tip load b01 m 18 1398 l p if the euler bernoulli beam theory is followed the govern 68547
Uploaded: 2022-06-04T10:14:42-08:00
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Description: question 4 1 pts a small aspect ratio beam is illustrated below with cantilevered boundary conditions and tip load b01 m 18 1398 l p if the euler bernoulli beam theory is followed the govern 68547

          Question 4
1 pts
A small aspect ratio beam is illustrated below, with cantilevered boundary conditions
and tip load.
b=0.1 m
18-1398
L
P
If the Euler-Bernoulli beam theory is followed, the governing differential equations
result in:
$$w(x) = \frac{P L^2}{3EI} - \frac{P x}{2EI} (L^2 - \frac{x^2}{3})$$
This results in the following displacement field:
$$ \{u, v, w\} = \{-z \phi(x), 0, w(x)\} = \{ \frac{P x}{2EI} (L^2 - x^2), 0, \frac{P L^2}{3EI} - \frac{P x}{2EI} (L^2 - \frac{x^2}{3})\}$$
However, Euler-Bernoulli beam theory is generally considered invalid for insufficiently
slender beams, i.e. their length to width ratio should be at least 10. In this problem this
ratio is 2, so it may be more appropriate to use Timoshenko beam theory, whose
assumed displacement field allows for the formation of shear strains.
For Timoshenko beam theory, a new degree of freedom $\phi$ is introduced such that plane
sections are not necessarily perpendicular to the neutral axis. For cantilevered boundary
conditions and tip load, the governing differential equations result in:
$$w(x) = \frac{P L (L-x)}{KAG} - \frac{P x}{2EI} (L^2 - \frac{x^2}{3}) + \frac{P L^2}{3EI}$$
$$\phi(x) = \frac{P}{2EI} (x^2 - L^2)$$
This results in the following displacement field:
$$ \{u, v, w\} = \{-z \phi(x), 0, w(x)\} = \{ \frac{P x}{2EI} (L^2 - x^2), 0, \frac{P L (L-x)}{KAG} - \frac{P x}{2EI} (L^2 - \frac{x^2}{3}) + \frac{P L^2}{3EI} \}$$
where, P= 17 MN is the applied tip load, E = 70 GPa is Young's modulus, I = $\frac{1}{12}bh^3$
is the appropriate 2nd area moment, G = 26.25 GPa is the shear modulus, A= bh is the
cross sectional area, L = 2 m is the beam length, and K = 0.897 is the shear correction
factor.
What is the value of $e_{zz}$ at {x, y, z} = {L, 0,0} if the Timoshenko beam theory is
followed?

question 4 1 pts a small aspect ratio beam is illustrated below with cantilevered boundary conditions and tip load b01 m 18 1398 l p if the euler bernoulli beam theory is followed the govern 68547

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