# How Bridges Work 2, Exercise 1: a timber joist, flat or on edge
# A simply supported timber beam spans 4 m and carries a uniform load
# of 6 kN/m (its own weight included). Its section is 75 mm x 225 mm.
# Find the maximum moment and shear, then the maximum bending stress
# with the beam on edge (225 mm deep) and laid flat (75 mm deep).

def udl_beam(w, span):
    """Max moment (kN m) and max shear (kN) for a uniform load w (kN/m)."""
    return w * span**2 / 8, w * span / 2

def rect_i(b, h):
    """Second moment of area of a rectangle, in mm^4."""
    return b * h**3 / 12

def bend_stress(m_knm, i_mm4, y_mm):
    """Bending stress in MPa from a moment in kN m."""
    return m_knm * 1e6 * y_mm / i_mm4

m, v = udl_beam(6, 4)
print(m, v)

for name, b, h in [("on edge", 75, 225), ("flat", 225, 75)]:
    i = rect_i(b, h)
    print(name, round(i), round(bend_stress(m, i, h / 2), 1))

# Output when run:
# 12.0 12.0
# on edge 71191406 19.0
# flat 7910156 56.9
#
# Same timber, same load. Laid flat, I is 9 times smaller (the depth is 3
# times smaller and it is cubed, with the width 3 times bigger, so 27/3 = 9),
# while the distance y to the outer face is only 3 times smaller. Stress is
# M * y / I, so it comes out 9/3 = 3 times higher (56.9 against 19.0 MPa).
# Always lay a joist on edge.