Differentiating the equation of curvature equation

Jan 2020
3
0
london
Hi everyone,

The equation for the curvature(k) is the following:

(1) k = (y'')/((1+(y')^2)^3/2)
For small deflections, this is just simplified to y'' (the numerator)

In beam theory the following equation is used:
(2) EIy'' + Py = 0, where the y'' is referring to the simplified curvature equation. If we use the proper equation, it will be

(3) EIk + Py = 0, where k is the equation above.

There is also another equation used that is the 2nd differential of equation (2):

(4) EIy'''' + Py'' = 0

I would like to know the second differential of equation (3), using k. Basically how to differentiate the equation (1) twice and plug it into equation (3)

Many Thanks
 
Dec 2015
1,078
166
Earth
Have you found the derivative of (1) yet ?
 
Dec 2015
1,078
166
Earth
Is this what you are searching for ?
\(\displaystyle k'=0=\dfrac{y'''(1+y'^2 )^{3/2}-3\sqrt{1+y'^2 } y'y''^2 }{1+y'^3 } \).

\(\displaystyle y'''(1+y'^2 )^{3/2}-3\sqrt{1+y'^2 } y'y''^2=0\).
 
Jan 2020
3
0
london
i do not know what the correct answer should be, but i believe this is correct. Are you able to differentiate this equation again, i.e find k'' ?. Because i require an equation with y'''' in it