# How Bridges Work 2, Exercise 3: how far can a beam span?
# For I-sections with flange 150 x 12 mm and web 8 mm, depths 300, 450 and
# 600 mm, find the longest simply supported span when the stress limit is
# 165 MPa, (a) for the beam's own weight alone and (b) with a further
# 10 kN/m of load. Steel density 7850 kg/m3.
import math

RHO, G, ALLOW = 7850, 9.81, 165

def i_section(bf, tf, tw, depth):
    inner = depth - 2 * tf
    return (bf * depth**3 - (bf - tw) * inner**3) / 12

def area_i(bf, tf, tw, depth):
    return 2 * bf * tf + tw * (depth - 2 * tf)

def max_span(w_kn_m, m_cap_knm):
    """Span at which w L^2 / 8 reaches the moment capacity."""
    return math.sqrt(8 * m_cap_knm / w_kn_m)

for depth in (300, 450, 600):
    a = area_i(150, 12, 8, depth)
    i = i_section(150, 12, 8, depth)
    cap = ALLOW * (i / (depth / 2)) / 1e6        # kN m
    w = RHO * G * a * 1e-6 / 1000                # kN/m
    print(depth, round(cap, 1), round(w, 3),
          round(max_span(w, cap), 1), round(max_span(w + 10, cap), 1))

# Output when run:
# 300 97.6 0.447 41.8 8.6
# 450 164.4 0.54 49.4 11.2
# 600 241.2 0.632 55.3 13.5
#
# Columns: depth (mm), moment capacity (kN m), self-weight (kN/m),
# span with self-weight only (m), span with +10 kN/m (m).
# Doubling the depth from 300 to 600 mm multiplies the moment capacity by
# about 2.5 but the span carrying the extra 10 kN/m only by about 1.6
# (8.6 m to 13.5 m), and the span for self-weight alone by 1.3 (41.8 m to
# 55.3 m). The beam has to carry more steel the longer it gets, so each
# extra metre costs more than the last. That is the span limit of a plain
# beam, and the reason for the arches, trusses and cables that follow.