Design methods for reinforced concrete
Reinforced concrete (RC) is concrete with steel bars inside. Concrete has a compressive strength fc (for example 25 MPa) but its tension strength is only about a tenth of that. Steel (rebar) has a yield strength fy (for example 500 MPa) and takes all the pull.
The modern way is limit state design. We increase the loads with load factors, then check that the member strength (reduced by safety factors) is bigger. Two limits are checked: strength (it must not break) and serviceability (cracks and sag must be small). The old working stress method instead kept stresses under allowed values.
Steel needs cover, a layer of concrete outside it (about 25–40 mm). Cover protects steel from rust and fire.
Design of beam structures
In a beam section (width b, effective depth d from the top to the centre of the bars) there are two forces. The concrete near the top gives a squeezing force C = 0.85 fc × b × a, where a is the depth of the squeezed block. The steel gives a pull T = As × fy.
Balance: C = T, so a = As fy / (0.85 fc b). The pair C and T make a couple with arm (d − a/2). So the moment capacity M = As fy (d − a/2).
To design: find the moment M from the loads, then find As ≈ M / (fy × 0.9 d), choose bars, and check. Too much steel is dangerous: the concrete crushes suddenly before the steel yields. Too little steel makes the beam crack and fail suddenly too. Add stirrups (loops of bar) to resist shear near the supports.
Design of column structures
A column mainly takes squeezing from the floors above. Concrete takes most of it and the vertical steel bars help. A simple short-column formula is P = 0.4 fc Ac + 0.67 fy Asc, where Ac is the concrete area and Asc the steel area.
Rules of thumb: steel is 0.8% to 4% of the section. The vertical bars are held by ties (rings) so they do not bulge outward. A tall slim column can buckle (bend sideways), so its strength is reduced for slenderness. Columns also feel bending when beams push on one side.
Design of prestressed concrete structures
In prestressed concrete, high-strength steel cables (tendons) are pulled tight and then anchored, so they squeeze the concrete before any load comes. The load then first has to cancel this squeeze before the concrete can be stretched. So the beam stays uncracked, can be thinner, and can span farther.
Net stress at the bottom: σ = P/A + P e y / I − M y / I in simple terms, prestress squeeze minus load stretch. If the answer is still squeeze (positive), there is no crack. Prestress is lost slowly over time (concrete shrinks and creeps, steel relaxes), so designers allow about 15–25% losses.
Pre-tensioning pulls the cables first and pours concrete around them (in factories). Post-tensioning pulls cables in ducts after the concrete hardens (on site).
Try it
At home: take a dry sponge or a block of chalk and bend it: it cracks at the stretched side. Now squeeze a row of books and lift them: that is prestress.
In the 3D: find the smallest number of bars that makes the label say SAFE (demand is 150 kNm).
Key formulas and definitions
- C = 0.85 f_c b a; T = A_s f_y; C = T
- a = A_s f_y / (0.85 f_c b)
- M = A_s f_y (d − a/2)
- A_s ≈ M / (f_y × 0.9 d) (first estimate)
- Bar area = π d² / 4 (20 mm bar = 314 mm²)
- Short column: P = 0.4 f_c A_c + 0.67 f_y A_sc
- Prestressed bottom stress = prestress squeeze − load stretch
Worked examples
1. Find the steel area of three 20 mm bars.
One bar: π × 20² / 4 = 314 mm². Three bars: 3 × 314 = 942 mm².
2. Beam b = 300 mm, f_c = 25 MPa, f_y = 500 MPa, A_s = 942 mm². Find the compression block depth a.
a = 942 × 500 / (0.85 × 25 × 300) = 471 000 / 6375 = 73.9 mm.
3. Same beam, d = 450 mm. Find the moment capacity.
M = 942 × 500 × (450 − 73.9/2) = 471 000 × 413.05 = 194.5 × 10⁶ N·mm = 194.5 kNm.
4. Design moment is 150 kNm, d = 450 mm, f_y = 500 MPa. Estimate A_s and the number of 20 mm bars.
A_s ≈ 150 × 10⁶ / (500 × 0.9 × 450) = 741 mm². Bars = 741 / 314 = 2.36, so use 3 bars (942 mm²).
5. A short column 300 × 300 mm has 1440 mm² of steel. f_c = 25, f_y = 500. Find its load capacity.
A_c = 90 000 − 1440 = 88 560 mm². P = 0.4 × 25 × 88 560 + 0.67 × 500 × 1440 = 885.6 kN + 482.4 kN = 1368 kN.
6. Bottom fibre: the load stretches by 10 MPa, the prestress squeezes by 12 MPa. Will it crack?
Net = 12 − 10 = 2 MPa squeeze. It is still squeezed, so no crack.
Common mistakes
- Putting the steel at the top of a simply supported beam. The bottom is stretched.
- Using the whole depth h instead of the effective depth d. d is measured to the centre of the bars.
- Adding too much steel and thinking it is always safer. Over-reinforced beams fail suddenly by crushing.
- Forgetting cover. Without it the steel rusts and the concrete cracks.