Reinforced Concrete Beam

Why concrete needs steel

Concrete is a ceramic-like material: brilliant in compression, hopeless in tension. A typical structural mix reaches a compressive strength f'c of 20 to 40 MPa, but its direct tensile strength is only about a tenth of that — roughly 2 to 4 MPa. When a beam spans between two supports and carries gravity load, it sags. Sagging puts the top fibres into compression and stretches the bottom fibres in tension. The bottom is exactly where concrete is weakest.

Load a plain concrete beam and it cracks across the underside at a tiny fraction of the load the top could survive, then snaps — a brittle, sudden failure with no warning. The fix is structurally elegant: cast steel reinforcing bars (rebar) into the tension zone near the bottom. Concrete cracks there as before, but once it does, the steel bridging the crack picks up the tensile force. Concrete keeps resisting compression in the top, steel resists tension in the bottom, and each material does only what it is good at. That division of labour is the entire idea behind reinforced concrete.

Composite action: why it works

Three happy accidents make steel and concrete partners rather than rivals:

• Bond. Deformed rebar has rolled-in ribs that key into the surrounding concrete, transferring force between the two without slip. This bond is what lets a cracked beam carry load — the steel only helps if the concrete grips it.
• Matched thermal expansion. Steel expands at about 12 × 10⁻⁶ /°C and concrete at about 10 × 10⁻⁶ /°C. They are close enough that temperature swings don't tear the bond apart.
• Corrosion protection. The high alkalinity of concrete (pore-water pH ≈ 12.5–13) passivates the steel, stopping it rusting as long as the concrete stays sound and uncarbonated.

The two materials are tied together by the assumption of strain compatibility: at any depth the steel and the concrete stretch by the same amount, because the bars are locked into the concrete. That single assumption, plus the rule that plane sections stay plane in bending, is enough to derive everything that follows.

The neutral axis and the cracked section

In bending, strain varies linearly across the depth: maximum compression at the top, maximum tension at the bottom, zero somewhere in between. That zero-strain line is the neutral axis. In a plain elastic beam it sits at mid-height. In a reinforced concrete beam, the cracked tension concrete carries essentially no stress, so the neutral axis migrates upward into the intact compression zone.

To locate it we replace the steel with an equivalent area of concrete using the modular ratio n = E_s / E_c (about 7–8 for normal-weight concrete, since E_s ≈ 200 GPa and E_c ≈ 25–30 GPa). Setting the first moment of the compression area equal to the first moment of the transformed steel about the neutral axis gives its depth. Above the axis, concrete in compression; below it, cracked concrete contributing nothing and steel carrying all the tension.

Flexural strength: the design equation

At ultimate load, design codes simplify the curved concrete compression stress into an equivalent rectangular block of intensity 0.85 f'c and depth a. Force balance across the section drives the whole design:

Tension in steel (yielded):   T = A_s · f_y
Compression in concrete:      C = 0.85 · f'c · b · a

Horizontal equilibrium T = C gives the stress-block depth:
  a = A_s · f_y / (0.85 · f'c · b)

Nominal moment capacity (lever arm = d − a/2):
  M_n = A_s · f_y · (d − a/2)

Design strength:
  φ M_n   with φ = 0.90 (tension-controlled flexure, ACI 318)

where:
  A_s  = area of tension steel  (mm²)
  f_y  = rebar yield strength   (420 MPa for Grade 60)
  f'c  = concrete strength      (MPa)
  b    = section width          (mm)
  d    = effective depth, top fibre to steel centroid (mm)

The lever arm (d − a/2) is the distance between the centroid of the steel tension and the centroid of the concrete compression block. The whole couple is just two equal-and-opposite forces a short distance apart — exactly like a wrench turning a bolt.

Worked example: capacity of a singly reinforced beam

A rectangular beam is 300 mm wide and 550 mm deep, with three 20 mm diameter Grade 60 bars (A_s = 3 × 314 = 942 mm²) at an effective depth d = 500 mm. Concrete f'c = 30 MPa. What is the design moment capacity?

a = A_s f_y / (0.85 f'c b)
  = 942 × 420 / (0.85 × 30 × 300)
  = 395,640 / 7,650
  = 51.7 mm

M_n = A_s f_y (d − a/2)
    = 942 × 420 × (500 − 25.9)
    = 942 × 420 × 474.1
    = 1.876 × 10⁸ N·mm
    = 187.6 kN·m

φ M_n = 0.90 × 187.6 = 169 kN·m

The steel ratio ρ = A_s/(b·d) = 942/(300×500) = 0.63%, comfortably below the balanced ratio (~3.0% for this material pairing) and even below the ~1.9% tension-controlled limit that ACI 318 actually enforces, so the beam is under-reinforced: the steel yields first and the failure is ductile.

Plain vs. reinforced vs. prestressed concrete

	Plain concrete	Reinforced concrete (RC)	Prestressed concrete
Carries tension via	Concrete only (cracks early)	Passive rebar after cracking	Pre-tensioned tendons clamp concrete in compression
Behaviour at service load	Cracks at ~2–4 MPa tension	Cracks, steel holds it together	Often crack-free (concrete stays in compression)
Failure mode (designed)	Brittle, sudden	Ductile — yields & sags first	Ductile if under-reinforced; tendon-controlled
Typical use	Footings, mass walls, paving	Beams, slabs, columns, frames	Long-span floors, bridge girders, sleepers
Relative cost	Lowest	Moderate	Higher (jacks, anchorages, high-strength steel)
Span efficiency	Poor in bending	Good	Excellent — long, shallow spans

Shear, stirrups and detailing

Flexure is only half the story. Near the supports, vertical shear is highest, and a beam can fail by a diagonal tension crack running up from the support at roughly 45°. Concrete carries some shear (V_c), but the rest is taken by stirrups — closed loops of smaller-diameter bar wrapped around the longitudinal steel at regular spacing. They act like the diagonal members of a hidden truss (the truss analogy), stitching the diagonal crack closed.

• Concrete cover (25–50 mm) protects bars from corrosion and fire, and develops bond — but more cover means a smaller effective depth, so it trades durability against capacity.
• Development length ensures a bar can reach yield before it pulls out; hooks and laps extend it where straight embedment is short.
• Minimum and maximum steel bracket the design: minimum prevents sudden failure the instant the concrete cracks; maximum forces ductile, under-reinforced behaviour.
• Crack-width control at service load is handled by bar spacing and stress limits, keeping cracks below ~0.3 mm so water and chlorides stay out.

Failure modes and trade-offs

• Ductile flexural failure (designed-for). In an under-reinforced beam the steel yields, the beam sags and cracks widen visibly, and only later does the concrete crush. The warning is the whole point — occupants and inspectors see it coming.
• Brittle compression failure (avoided). An over-reinforced beam crushes the concrete top while the steel is still elastic — sudden, no warning. Codes cap the steel ratio (ACI 318 requires a tension strain of at least 0.005 at nominal strength) precisely to outlaw this.
• Diagonal shear failure. Insufficient or too-widely-spaced stirrups let a diagonal crack open near a support — often explosive. Shear governs many real failures more than flexure.
• Bond / anchorage failure. If a bar's development length is too short, it slips before yielding and the section never reaches its calculated capacity.
• Rebar corrosion and spalling. Chloride ingress or carbonation breaks the passive film; rust occupies up to ~6× the steel's volume and splits off the cover. This is the dominant durability failure for bridges and parking structures.
• Creep and shrinkage. Concrete keeps deforming under sustained load (creep) and shrinks as it dries, increasing long-term deflection — often by a factor of two or more over the instantaneous value.