# How Bridges Work 3, Exercise 1: how long can a stone slab span?
# A clapper bridge is a stone slab laid across a stream. Stone is weak in
# tension, so the slab fails when the bending tension on its underside
# reaches the working tension of the stone. For a slab 1 m wide, find the
# longest span when it carries (a) only its own weight and (b) its own
# weight plus 3 kN/m of people and animals. Take thicknesses of 0.2, 0.3 and
# 0.4 m and working tensions of 1, 2 and 3 MPa (illustrative values), with
# stone density 2600 kg/m3.
import math

RHO, G = 2600, 9.81

def slab(thick_m, allow_mpa, extra=0.0, width_m=1.0):
    w = RHO * G * thick_m * width_m / 1000           # own weight, kN/m
    z = width_m * 1000 * (thick_m * 1000) ** 2 / 6   # section modulus, mm3
    m_cap = allow_mpa * z / 1e6                      # kN m
    return (round(math.sqrt(8 * m_cap / w), 1),
            round(math.sqrt(8 * m_cap / (w + extra)), 1))

for allow in (1, 2, 3):
    for t in (0.2, 0.3, 0.4):
        print(allow, t, slab(t, allow, 3.0))

# Output when run (columns: working tension MPa, thickness m,
# (span with own weight only, span with +3 kN/m), both in metres):
# 1 0.2 (3.2, 2.6)
# 1 0.3 (4.0, 3.4)
# 1 0.4 (4.6, 4.0)
# 2 0.2 (4.6, 3.6)
# 2 0.3 (5.6, 4.7)
# 2 0.4 (6.5, 5.7)
# 3 0.2 (5.6, 4.4)
# 3 0.3 (6.9, 5.8)
# 3 0.4 (7.9, 7.0)
#
# Stone is weak in tension, so a plain slab manages only a few metres, and
# the span depends on the square root of the thickness: doubling the
# thickness from 0.2 to 0.4 m gains only about 40% (5.6 m to 7.9 m at 3 MPa),
# because the slab is also heavier. The working tensions are illustrative;
# real stone varies, and a flaw in a slab can break it at much less. That
# is why clapper bridges use short spans (several slabs on stone piers).