Two points Catenary equation [1/2]

There are two points in xz plane. A(0,0) , B(D,h) q is distributed weight per unit.

 

Horizontal Force H is equal everywhere on AB curve. (q have only vertical direction) Following is very small segment of AB curve.

\[\frac{dz}{dx}=\frac{qds}{H}\]

 

Differentiating left & right side with respect to x

 

\[\frac{d^2z}{dx^2}=\frac{q}{H}\frac{ds}{dx}\]

 

Geometric condition \[(\frac{dx}{ds})^2+(\frac{dz}{ds})^2=1\] Then

Therefore \[\frac{d^2z}{dx^2}=\frac{q}{H}\sqrt{1+(\frac{dz}{dx})^2}\] Using WolframAlpha

 

c1 & c2 are determined by boundary condition A(0,0) , B(D,h)

 

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