Applied Calculus problem-1st year engineering

[greentlc]

Recall that the bending moment, M, at a distance, x meters, from the fixed end of a cantilever beam is given in terms of the second derivative of the deflection, y, of the beam. Determine the deflection,y, of the beam at the free end of the beam. Assume that the beam is perpendicular to the wall at the fixed end of the beam.


Given equations:

M=EId²y/dx²

M= -12x²+96x-192

E = 222,000 N/m²
I = 0.275m⁴

My Attempt:

Integrate this function:

d²y = -12x²+96x-192/EI to get

dy=1/EI*(-4x³+48x²-192x+C)

I know I need to solve for the constant of integration, and then integrate the function once more to solve for y. I don't know what values to use for x and M. Is M(0)=0??

Any help is greatly appreciated

greentlc









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[greentlc]

Sorry, the first integral would be the shear, so M(0)=0 means nothing here. I need to know the shear value for a certain x value.

thanks again

 

[SteamKing]

greentic: If y(x) represents the deflection of the beam at any point x along the length of the beam, then dy/dy is called the slope of the beam. If the slope is differentiated, so that d2y/dx2 is obtained, this quantity is proportional to the bending moment M(x). For this beam, M(x) is given in the second eqn. above. If the beam is built into and perpendicular to a wall at the fixed end, what must its slope be?

 

[greentlc]

SteamKing:

thank you for pointing me in the right direction. I was definitely a little confused. The slope at x=0 is 0. From there I can find the constant. Then differentiate once more. This will give me y(x), where y(0)=0 to find C. Then y(4)=deflection at the end of the beam.

Is this correct?

Thanks again
greentlc

 

[SteamKing]

greentic: Your selection of 0 slope at x = 0 is correct. However, you are given M(x) and asked to find y(x). Since M(x) is proportional to d2y/dx2, then you must integrate M(x) to find slope (x) and then integrate slope (x) to find y(x), using your boundary conditions at each step to solve for the constant of integration.