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Deflection: 1- If two beams are the same length, and material, but one beam is twice as thick as...

Question:

Deflection:

1- If two beams are the same length, and material, but one beam is twice as thick as the other, by what factor will the deflection of the thicker beam be less than that of the thinner one?

2- Keeping everything else constant if the length of a beam is doubled, how much greater do I need to make the Young's modulus to keep the deflection for an applied force the same? Answer the question in details

Deflection of Beams:

Beams are generally a horizontal member that is subjected to various types of load in the vertical direction. The movement of the beam in the vertical from its regular position due to the loading is called the deflection of the beam and the angle made by the new axis with the initial position is known as the slope of the beam.

Answer and Explanation:

The differential equation of the elastic line is,

{eq}\rm{EI} \dfrac{\rm{d}^{2} \rm{y}}{\rm{d} \rm{x}^{2}}= M {/eq}

Where

  • {eq}EI {/eq} is the flexural rigidity.


Integrating both sides we get,

{eq}\int \rm{EI} \dfrac{\rm{d}^{2} \rm{y}}{\rm{d} \rm{x}^{2}} =\int M \\ \rm{E I} \dfrac{d y}{d x}= Mx + C_1 \ldots \ldots (1) {/eq}


By the way of integrating the above equation again, we get:

{eq}\rm{E I} y = M \dfrac{x^2}{2} + C_1 x + C_2 \ldots \ldots (2) {/eq}


(1)

From equation (2), we can see that the deflection of the beam is inversely proportional to the moment of inertia of the beam, i.e. {eq}y \ \ \alpha \ \ \dfrac{1}{I} {/eq}


Now, the moment of inertia of the beam is, {eq}I = \dfrac{bd^3}{12} {/eq}


From the above formula we can see that, by increasing the depth by 2 times, the moment of inertia will be increased by 8 times.


Hence, the deflection of the thicker beam will be {eq}\left( \dfrac{1}{8}\right)^{\rm{th}} {/eq} of the thinner beam.


(2)

Using equation (2), we can say that the deflection of any beam is directly proportional to the cube of the length of the beam and inversely proportional to Young's modulus of the beam.


For example:

Deflection of the cantilever beam subjected to point load at the free end is, {eq}y = \dfrac{PL^3}{3EI} {/eq}


To keep the deflection for an applied force the same, if we double the length of the beam, we need to increase Young's modulus by 8 times.


Learn more about this topic:

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Double Integration: Method, Formulas & Examples

from AP Calculus AB & BC: Help and Review

Chapter 12 / Lesson 15
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