Concrete Beam Flexure Basics

Concept

Unlike steel, which is strong in both tension and compression, concrete is strong in compression but exceedingly weak in tension. When a concrete beam is subjected to a bending moment, it develops compression on one side of its neutral axis and tension on the other. Because the concrete in the tension zone will quickly crack, steel reinforcement bars (rebar) must be placed in the tension zone to carry the pulling forces. The foundational design of a singly reinforced concrete beam ensures that the compressive strength of the concrete block is perfectly balanced against the tensile yielding strength of the steel, providing adequate moment resistance safely.

Formula & Method

Modern concrete design (such as ACI 318) uses the equivalent rectangular compressive stress block (often called the Whitney stress block) to simplify the non-linear stress distribution in the concrete.

For a rectangular beam of width $b$ and effective depth $d$ (distance from extreme compression fiber to the centroid of the tension steel), the internal force equilibrium requires that the total tension in the steel ($T$) equals the total compression in the concrete ($C$):

$T = A_s f_y$ $C = 0.85 f_c' a b$

Setting $T = C$, the depth of the equivalent rectangular stress block ($a$) is:

$a = \frac{A_s f_y}{0.85 f_c' b}$

The nominal moment capacity ($M_n$) is the internal force multiplied by the lever arm distance between the tension and compression forces:

$M_n = A_s f_y \left( d - \frac{a}{2} \right)$

The design moment capacity ($\phi M_n$) must be greater than or equal to the ultimate factored moment ($M_u$), where $\phi$ is the strength reduction factor (typically 0.90 for tension-controlled flexural members).

Variables & Units

• $M_n$ = Nominal flexural strength, in N·mm or kN·m.
• $A_s$ = Area of tension steel reinforcement, in mm$^2$.
• $f_y$ = Yield strength of the steel reinforcement, in MPa.
• $f_c'$ = Specified compressive strength of concrete, in MPa.
• $b$ = Width of the compression face of the beam, in mm.
• $d$ = Effective depth of the beam, in mm.
• $a$ = Depth of the equivalent rectangular stress block, in mm.
• $\phi$ = Strength reduction factor.

Worked Example

A detailed interactive calculation example for concrete beam flexure can be found at Worked Example: Concrete Beam Flexure Basics.

Engineering Meaning

The equation ensures a ductile failure mode. By keeping the steel area ($A_s$) low enough, engineers guarantee that the steel will yield (stretch significantly) before the concrete crushes abruptly in compression. This yielding provides ample visible warning (large deflections and wide cracks) to occupants that the beam is overloaded. If too much steel is used (an over-reinforced beam), the concrete would crush suddenly without warning, which is a catastrophic failure mode strictly prohibited by design codes.

Engineering Check

Always verify that the calculated reinforcement ratio ($\rho = A_s / (bd)$) satisfies both the code-mandated minimum (to prevent sudden failure immediately upon concrete cracking) and maximum limits (to ensure tension-controlled ductile behavior). Furthermore, ensure there is adequate concrete cover to protect the steel from corrosion and fire.

Explicit Exclusions

This foundational article excludes the analysis of doubly reinforced beams (beams with compression steel), T-beam sections, and shear reinforcement (stirrup) design. It focuses purely on foundation-level, singly reinforced rectangular flexure.