Singly reinforced beam in bending

Design the bottom steel of a rectangular concrete beam to SANS 10100-1. You work one beam by hand, then enter the same beam in the STATIX RC beam calculator and compare the answers.

1. Learning outcomes

After this lesson you can:

find the ultimate design load and moment of a simply supported beam (SANS 10160-1);
work out the effective depth d from the cover, links and main bars;
use K, K′ and the lever arm z to decide if a beam needs only tension steel;
calculate the tension steel A_s, choose bars and check that they fit;
check minimum and maximum steel, and check your hand answer in STATIX.

2. Before you start

The free STATIX RC beam calculator, open in a second tab.
A pocket calculator, paper and a pencil.
About 45 minutes.
Your own copy of SANS 10100-1 is useful. This lesson gives clause numbers only, not the code's text.

3. Theory

Before the formulas, look at what happens inside the beam as the load grows.

The same section three times as the load grows. 1: at small loads the whole section works and the concrete at the bottom is in tension. 2: the bottom concrete cracks up to near the neutral axis, and the bars take all the tension. 3: at the ultimate load the concrete stress is close to a rectangle, which the code simplifies to 0.45 f_cu over a depth 0.9x. Lesson 1 designs for stage 3.

The stress block. At the ultimate limit state the concrete in compression is taken as a simple rectangle. Its stress is 0.45 f_cu and its depth is 0.9x, where x is the depth to the neutral axis. The steel in tension works at its design strength, 0.87 f_y. Concrete in tension is ignored. (SANS 10100-1, 4.3.3)

K, the bending ratio. K tells you how hard the concrete section works:

K = M / (f_cu b d²)

K′ is the upper limit for a section with tension steel only. With no moment redistribution, K′ = 0.156. If K ≤ K′, tension steel alone is enough. If K > K′, you need compression steel or a bigger section (Lesson 2).

Why K′ matters. A beam with a modest amount of steel fails slowly: the bars yield, the beam sags and cracks, and people have time to leave. A beam crammed with steel fails when the concrete crushes, suddenly. Keeping K ≤ K′ keeps the neutral axis at or above 0.5d, so the steel yields first.

The lever arm z is the distance between the compression force in the concrete and the tension force in the steel:

z = d [0.5 + √(0.25 − K/0.9)] but not more than 0.95d

The 0.95d limit stops z from getting unrealistically large when K is very small.

The steel area follows from moment = force × lever arm:

A_s = M / (0.87 f_y z)

Minimum and maximum steel. The detailing rules (SANS 10100-1, 4.11) set limits on the tension steel. For a rectangular beam with f_y = 450 MPa this lesson uses 0.13% of b × h as the minimum and 4% of b × h as the maximum, the same values as the RC beam worked example. Check the values for other cases in your copy of the code.

Units: work in N and mm. Then 1 kNm = 10⁶ Nmm and stresses come out in MPa (N/mm²).