# How Bridges Work 10, Exercise 2: design the girders for a 35 m span
# A 35 m road span carries a total deck load of 144 kN/m (deck, surfacing and traffic)
# shared by 6 equal steel I-girders. Try each section below in order
# (flange width, flange thickness, web thickness, depth in mm), adding the girder's
# own weight to its share of the load. The stress limit is 165 MPa and the sag limit
# span/360. Print stress, sag and pass or fail for each, report the first section that
# passes and say which limit (stress or sag) governed the failures on the way.
import math

E, ALLOW = 200e3, 165
RHO, G = 7850, 9.81

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)

SECTIONS = [(350, 22, 14, 1000), (400, 25, 16, 1200), (450, 28, 18, 1400),
            (500, 32, 20, 1600), (550, 36, 22, 1800)]

def check(span_m, w_load, girders):
    for bf, tf, tw, d in SECTIONS:
        w = w_load / girders + RHO * G * area_i(bf, tf, tw, d) * 1e-6 / 1000
        i = i_section(bf, tf, tw, d)
        m = w * span_m**2 / 8
        stress = m * 1e6 * (d / 2) / i
        sag = 5 * w * (span_m * 1000) ** 4 / (384 * E * i)
        limit = span_m * 1000 / 360
        ok = stress <= ALLOW and sag <= limit
        why = "pass" if ok else ("stress" if stress > ALLOW else "sag")
        print(d, round(w, 1), round(stress, 1), round(sag, 1), round(limit, 1), why)
        if ok:
            return d
    return None

print("first passing depth:", check(35, 144, 6))

# Output when run:
# 1000 26.2 426.8 544.7 97.2 stress
# 1200 27.0 277.3 294.9 97.2 stress
# 1400 27.8 192.2 175.2 97.2 stress
# 1600 28.8 137.4 109.5 97.2 sag
# 1800 30.0 102.6 72.7 97.2 pass
# first passing depth: 1800
#
# Columns: depth (mm), load per girder including its own weight (kN/m), stress
# (MPa), sag (mm), sag limit (mm), result. The three shallowest sections fail on
# stress (426.8, 277.3 and 192.2 MPa against 165). The 1600 mm section passes
# on stress (137.4 MPa) but sags 109.5 mm against a limit of 97.2 mm, so sag
# governs; the 1800 mm girder passes both with 102.6 MPa and 72.7 mm. The
# first passing girder is 1,800 mm deep, a depth of about 1/19 of the span.
# Stiffness, not strength, decided the last step, as in How Bridges Work 2.