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I have a strip of stainless steel encastree'd at one end to which is applied a constant pressure on one side, and I need to know what the deflection equation y = f(x) is at equilibrium. If the deflection was small I would be able to use one of the very well known deflection formulas which assume vertical loads, but it isn't.

I am trying to work out the bending moment M=f(x) using integrals to use: $$y(x)=\int\int_{beam} \frac{M}{EI}dx^2$$ But I'm getting more and more confused. What is the right approach to solving this problem?

Here is a diagram:

enter image description here

Air
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Mister Mystère
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1 Answers1

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For large deformation, there are changes to the strain-displacement relation and vertical equilibrium, but horizontal equilibrium remains unchanged.

Let N be the axial force and M the bending moment, for finite rotations, these are defined as: $$ N=EA\epsilon^o \\ M=EI\kappa $$ where $$\epsilon^o = \frac{du}{dx} + \frac{1}{2}\left(\frac{dw}{dx}\right)^{2} $$ and $$ \kappa = -\frac{\frac{d^{2}w}{dx^{2}}}{\left[1 + \left(\frac{dw}{dx}\right)^2\right]^{3/2}} $$ so vertical equilibrium is given by: $$ \frac{d^2M}{dx^2} + N\frac{d^{2}w}{dx^2} + P = 0 $$ where $w$ is the vertical deflection The solution is then the solution to this differential equation dependent upon your set boundary conditions

bern
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