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Image of bowed beam and string

Suppose you have a uniform beam, such as a piece of plywood that is 4 feet long, six inches wide, and half an inch thick. Suppose you cause it to bend by attaching, for example, a string under tension between the two ends, similar to how one would string a bow for shooting arrows. What shape will the beam take?

My goal is to create a parabolic shape, for use in a trough-type solar concentrator. Ideally, I would like to be able to vary the focal length simply by changing the tension on the string. I want quite a long focal length - several times the length of the beam - so the beam will not be bent very much.

If the natural bending shape of a uniform beam is not a parabola, what profile could I use when cutting out the plywood so that it would take on a parabolic shape (see the bottom three profile shapes as examples)?

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    $\begingroup$ For very small deformation it could be $\sin$ function like in Euler beam buckling. $\endgroup$
    – Tomáš Létal
    Commented Feb 26 at 21:34
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    $\begingroup$ Based on your response (thank you), I looked up Euler beam buckling. It looks like I should have described my problem using more industry-standard terminology: that is, the beam is "pinned" at both ends, and subject to only axial compressive loads, causing "buckling". Also, from what I read, it looks like the shape of the buckling is deflection = A*cos(x), where x is the distance along the beam scaled from -pi/2 to +pi/2, and the amplitude of the buckling (A) increases with the amount of compressive force. Is that an accurate interpretation? $\endgroup$
    – David Rose
    Commented Feb 28 at 18:29
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    $\begingroup$ This is done with stiffness variations in Asali et al.'s "Bending parabolas: formwork for compression-only structures." The stiffness will scale essentially linearly with the width. $\endgroup$
    – Chemomechanics
    Commented Feb 28 at 19:17
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    $\begingroup$ The Asali et al paper appears to answer the question exactly. According to them, a uniform beam will form an almost exact parabola if the amount of buckling is less than half the length of the beam (which it will be in my case). For more precision or more pronounced bends, they provide equations that specify the required variations in width of the beam that would produce a parabola. The general shape seems to be a double fusiform (thin at the ends and middle, and thicker at the 1/4 and 3/4 points along the beam). I can't mark this as the right answer, but I think I now have what I need. $\endgroup$
    – David Rose
    Commented Feb 28 at 22:03
  • $\begingroup$ @DavidRose Yes, your interpretation is correct. It should also work for greater deformation (I was a little bit wrong there), only inaccuracy comes from different bending stiffness of curved beam, which is not included in the approach. Based on Chemomechanics comment, you could also create your solution an post it here as an answer and then mark it as answered. $\endgroup$
    – Tomáš Létal
    Commented Feb 29 at 18:56

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