Reinforced Concrete Cantilever Canopy

What is modelled

A cantilever carries its load by bending that is largest at the wall. Concrete is weak in tension, so where the moment exceeds the cracking moment the section cracks and is much less stiff, and the beam sags more than a simple elastic calculation says. The page works out the moment from the self weight and the extra load, the cracking moment and the capacity of each section along the tapered span, the effective stiffness by the Branson interpolation between the uncracked and the cracked section, and then integrates the curvature twice to get the deflection. The load factor tells how far the load could be scaled before some section reaches its flexural capacity.

Equations

• Moment M(x) = ∫xL w(s)(s − x) ds with w = γ h(x) + q.
• Cracking Mcr = 0.62 √fc · b h2 / 6, capacity Mn = As fy (d − a/2), a = As fy / (0.85 fc b).
• Effective stiffness Ie = (Mcr/M)3 Ig + [1 − (Mcr/M)3] Icr.
• Deflection y'' = M / (E Ie) times the long-term factor.

Checks run in the page

• Uncracked prismatic cantilever against w L4 / (8 E I).
• Shear and moment at the wall against the load and its first moment.
• Neutral axis of the cracked section by bisection against the quadratic formula.
• Load factor by bisection against the smallest capacity-to-moment ratio.

Honest simplifications

The wing is treated as a one-metre strip with a constant steel ratio, the concrete and steel properties are typical round values, shear, anchorage, the counterweight of the real building, wind and dynamic effects are ignored and the load factor has no code partial factors. It is a teaching model, not a structural design.

• Press Thin and long and watch the cracked zone and the deflection grow.
• Press Under-reinforced and read the load factor.
• Press Heavy snow or rain and see how the margin shrinks.