beam - 5

[logistic_guy]

Determine the slope and deflection of end \(\displaystyle A\) of the cantilevered beam. \(\displaystyle E = 200 \ \text{GPa} \ \) and \(\displaystyle \ I = 65.0(10^6) \ \text{mm}^4\).

 

[khansaheb]

Determine the slope and deflection of end \(\displaystyle A\) of the cantilevered beam. \(\displaystyle E = 200 \ \text{GPa} \ \) and \(\displaystyle \ I = 65.0(10^6) \ \text{mm}^4\).
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[logistic_guy]

Beam deflection equation:

\(\displaystyle EI\frac{d^2y}{dx^2} = M_x\)

Looking at the beam, we know that \(\displaystyle M_x\) must be \(\displaystyle 30000 \ \text{Nm}\).

Then,

\(\displaystyle EI\frac{d^2y}{dx^2} = 30000\)

Or

\(\displaystyle EIy = 15000x^2 +C_1x + C_2\)

 

[logistic_guy]

The goal is now how to find these two constants. We know that at the supports there is no deflection. In other word, at \(\displaystyle x = 3\), we have:

\(\displaystyle y = 0\)
And
\(\displaystyle \frac{dy}{dx} = 0\)

If we substitute the first condition, we get:

\(\displaystyle 0 = 15000(9) + 3C_1 + C_2\)

\(\displaystyle C_2 = -135000 - 3C_1\)

Then we have:

\(\displaystyle EIy = 15000x^2 + C_1x - 135000 - 3C_1\)

Take the derivative.

\(\displaystyle EI\frac{dy}{dx} = 30000x + C_1\)

Use the second condition.

\(\displaystyle 0 = 30000(3) + C_1\)

\(\displaystyle C_1 = -90000\)

Then, we have:

\(\displaystyle EIy = 15000x^2 - 90000x - 135000 + 3(90000)\)

Or

\(\displaystyle EIy = 15000x^2 - 90000x + 135000\)

Determine the slope

The question is asking for the slope at \(\displaystyle x = 0\), so we have to take the derivative of the equation above and substitute zero.

\(\displaystyle EI\frac{dy}{dx} = 30000x - 90000\)

Substitute \(\displaystyle x = 0\).

\(\displaystyle EI\frac{dy}{dx} = 30000(0) - 90000\)

\(\displaystyle EI\frac{dy}{dx} = -90000\)

\(\displaystyle \frac{dy}{dx} = -\frac{90000}{EI} = -\frac{90000}{200 \times 10^9 \times 65 \times 10^6 \times (1/1000)^4} = \textcolor{blue}{-0.006923 \ \text{rad}}\)

 

[logistic_guy]

Determine the deflection of end \(\displaystyle A\)

\(\displaystyle EIy = 15000x^2 - 90000x + 135000\)

Or

\(\displaystyle y = \frac{15000x^2 - 90000x + 135000}{EI}\)

Plug in numbers.

\(\displaystyle y = \frac{15000(0)^2 - 90000(0) + 135000}{200 \times 10^9 \times 65 \times 10^6 \times (1/1000)^4} = 0.010385 \ \text{m} = \textcolor{blue}{10.385 \ \text{mm}}\)