# How Bridges Work 6, Exercise 3: counterweight for an anchor arm
# A cantilever arm of 100 m carries 80 kN/m and a 2000 kN tip load. On the other
# side of the pier an anchor arm of length b carries only its own 80 kN/m. For
# b = 60, 80, 100 and 120 m find the overturning moment about the pier, the
# restoring moment of the anchor arm alone, the factor against tipping without a
# counterweight, and the counterweight W (kN) that must sit at the anchor end to
# give a factor of at least 1.5.
w, a, tip = 80, 100, 2000
over = w * a**2 / 2 + tip * a
print("overturning", over)

for b in (60, 80, 100, 120):
    restore = w * b**2 / 2
    factor = restore / over
    needed = max(0.0, (1.5 * over - restore) / b)
    print(b, restore, round(factor, 2), round(needed))

# Output when run:
# overturning 600000.0
# 60 144000.0 0.24 12600
# 80 256000.0 0.43 8050
# 100 400000.0 0.67 5000
# 120 576000.0 0.96 2700
#
# Columns: anchor arm length (m), its restoring moment (kN m), the factor
# without a counterweight, the counterweight needed at the anchor end (kN) for
# a factor of 1.5. A short anchor arm (60 m) gives only 0.24 on its own and
# needs 12,600 kN (about 1,280 tonnes) of counterweight; lengthening it to
# 120 m brings the factor to 0.96 and the counterweight down to 2,700 kN. The
# moments match when the anchor arm is about 122 m long (sqrt(2 x 600,000 /
# 80)), so a long enough anchor arm balances the main arm with no
# counterweight at all. That is the trade-off behind every balanced cantilever:
# either weight or length on the anchor side.