# How Bridges Work 8, Exercise 2: the best stay angle
# A stay holds a vertical load of 5000 kN at a deck point 150 m from the tower
# base. For stay angles above the deck from 20 to 70 degrees in 5 degree steps
# find the tension, the horizontal push on the deck, and the steel volume of the
# stay at 600 MPa (volume = tension x length / stress; the horizontal distance
# is fixed, so the length is 150 / cos(angle)). Find the angle with the
# smallest volume, and say what is the cost of a shallower stay.
import math

p, x, stress = 5000, 150, 600e3
best = None
for deg in range(20, 75, 5):
    th = math.radians(deg)
    t = p / math.sin(th)
    horiz = p / math.tan(th)
    vol = t * (x / math.cos(th)) / stress
    print(deg, round(t), round(horiz), round(vol, 2))
    if best is None or vol < best[1]:
        best = (deg, vol)
print("smallest volume:", best[0], round(best[1], 2))

# Output when run:
# 20 14619 13737 3.89
# 25 11831 10723 3.26
# 30 10000 8660 2.89
# 35 8717 7141 2.66
# 40 7779 5959 2.54
# 45 7071 5000 2.5
# 50 6527 4195 2.54
# 55 6104 3501 2.66
# 60 5774 2887 2.89
# 65 5517 2332 3.26
# 70 5321 1820 3.89
# smallest volume: 45 2.5
#
# Columns: angle (degrees), tension (kN), horizontal push on the deck (kN),
# steel volume (m3). The volume is smallest at 45 degrees (2.5 m3) and rises
# either side: a steep stay (70 degrees) has low tension but is long and
# needs a tall tower; a shallow stay (20 degrees) pushes the deck with 13,737
# kN against 5,000 at 45 degrees and uses 3.89 m3, about 56% more steel.
# Real designs set the angles by the geometry of the tower and the deck, so the
# 45 degree rule is a guide, not a requirement.