Solving Impossible Integral for MS Thesis - Civil Engineering

[jrautenb]

Hello all,

I'm working on a problem related to my MS thesis (in Civil Engineering, not mathematics). I have come across an integral on which I have spent entirely too long.

If I could get some pointers, I'd greatly appreciate it.


$$\int [E(x) \cdot (u''(x))^2 \cdot dx]$$


It is a definite integral over the length, L, of a beam, but that doesn't really matter too much.
E(x) = Eo*(1+x/L) with Eo and L being constants
u(x) = a2*x^2+a3*x^3+a4*x^4 with a1, a2, a3, and a4 being constants
u''(x) is the second derivative of u(x) with respect to x.

Any help would be greatly appreciated.

Thanks,
JRautenb

 

[Citan Uzuki]

I'm not sure why you're having trouble here -- according to the definitions of E and u you have given, the integrand is a polynomial. Just expand the polynomial and integrate term by term, and don't worry about doing anything fancy.

 

[吴黎明]

just look up the wolfram integrator online, and do it there.

 

[Tomtom]

I agree with citan. There's nothing difficult there at all.

Derivating u(x) gives $$u''(x)=12 \cdot [a4] \cdot x^2+6 \cdot [a3] \cdot x$$

$$(u''(x))^2=144 \cdot [a4]^2 \cdot x^4+144 \cdot [a4] \cdot [a3]x^2+36 \cdot [a3]^2 \cdot x^2$$

Multiply this with your E(x) (and E(x) is simplified to Eo + (Eo/L)*x).

Then you simply integrate it as you would with any polynomial. Remember adding the constant at the end of the integration.