# How Bridges Work 4, Exercise 3: how deep must the truss be?
# A 40 m truss carries a uniform load of 20 kN/m. The top chord is a strut of
# I = 8.0e6 mm4 and a panel length of 4 m (pin-ended). E = 200 000 MPa.
# For truss depths of 4, 5, 8 and 10 m find the chord force (beam moment divided
# by the depth), the strut's Euler buckling load, and whether the strut passes with a
# factor of safety of 2. Then find the smallest depth in 0.5 m steps that passes.
import math

E = 200e3

def euler_kn(i_mm4, length_m):
    return math.pi**2 * E * i_mm4 / (length_m * 1000) ** 2 / 1000

m = 20 * 40**2 / 8
cap = euler_kn(8.0e6, 4)
print("moment", m, "buckling load", round(cap), "allowed", round(cap / 2))
for depth in (4, 5, 8, 10):
    force = m / depth
    print(depth, force, force <= cap / 2)

depth = 4.0
while m / depth > cap / 2:
    depth += 0.5
print("smallest depth:", depth, round(m / depth, 1))

# Output when run:
# moment 4000.0 buckling load 987 allowed 493
# 4 1000.0 False
# 5 800.0 False
# 8 500.0 False
# 10 400.0 True
# smallest depth: 8.5 470.6
#
# The strut would buckle at 987 kN, or 493 kN with a factor of safety of 2. At
# 4, 5 and 8 m deep the chord force (1000, 800, 500 kN) is more than that; at
# 10 m (400 kN) it passes. In 0.5 m steps the smallest passing depth is 8.5 m
# (470.6 kN). Making the truss deeper reduces the chord force in inverse
# proportion, exactly as a deeper I-beam reduces the flange stress in How Bridges Work 2.