Differentiating the equation of curvature equation (NOT SOLVED YET)

[suren]

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

 

[Deleted member 4993]

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

You say:

Basically how to differentiate the equation (1) twice and plug it into equation (3)

For what purpose? Are you trying derive equation (2)?

In that case the process will be totally different.

Are you taking class where beam bending/deflection is used? In that case, what textbook are you using?

 

[suren]

Hi there,

i am using the following textbook : Stability of Structures: Principles and and applications, from Chai H. Yoo. The pdf of this textbook can be found easily online. In this book (page 9), they differentiate equation (2) to get (4). However, this equation is the equation for small deflections (where they use k = y'', instead of equation (1). Therefore, using equation (3), (where the full equation of k is used), i would like to differentiate it twice to get a new 4th order equation.