buckling.rkt

#lang racket

(require tabular)

;; A floor of original length x
;; that expands by n
;; but is constrained to fit in a space of length x
;; will buckle.
;;
;; Let's approximate by treating the resulting shape as an isosceles
;; triangle instead of whatever curve will actually result. Call the
;; height of the triangle h.
;;
;; Split the triangle in half to get two right triangles; consider
;; just one. It will have sides ½x, h, and ½x + ½n.
;;
;;          (½x + ½n)² = (½x)² + h²
;; (½x)² + ½xn + (½n)² = (½x)² + h²
;;         ½xn + (½n)² = h²
;;          ½n(x + ½n) = h²

(define (height x n)
  (define half-n (/ n 2))
  (sqrt (* half-n (+ x half-n))))

;; Units: metres.
;;
(->table #:columns (list 'x 'n 'h)
         (for*/list [(n '(0.001 0.002 0.005 0.01 0.02))
                     (x '(1 5 10))]
           (list x n (height x n))))

;; Huh, wow!
;;
;; x |  n  |         h
;; -----------------------------
;;
;; 1 |0.001|0.02236626924634504
;; 5 |0.001|0.05000249993750312
;; 10|0.001|0.07071244586351119
;;
;; 1 |0.002|0.031638584039112745
;; 5 |0.002|0.07071774883294858
;; 10|0.002|0.10000499987500625
;;
;; 1 |0.005|0.050062460986251965
;; 5 |0.005|0.11183134623172522
;; 10|0.005|0.15813364600868468
;;
;; 1 |0.01 |0.07088723439378912
;; 5 |0.01 |0.15819292019556375
;; 10|0.01 |0.22366269246345044
;;
;; 1 |0.02 |0.1004987562112089
;; 5 |0.02 |0.2238302928559939
;; 10|0.02 |0.3163858403911275