CELE Reinforced & Prestressed Concrete — Reinforced Concrete Beams: FlexureSummary

ACI 318 and NSCP 2015 replace the parabolic concrete stress distribution with a rectangular block of intensity $0.85f'_c$ extending from the extreme compression fiber to depth $a = \beta_1 c$, where $c$ is the neutral-axis depth and $\beta_1$ depends on concrete strength (typically 0.85 for $f'_c \leq 28$ MPa, decreasing for higher strengths). This simplification avoids complex calculus while maintaining accuracy and is mandated in all Philippine design codes.

From force equilibrium in the cross-section, compression force in concrete equals tension force in steel: $0.85f'_c \cdot a \cdot b = A_s f_y$. Solving yields $a = \frac{A_s f_y}{0.85 f'_c b}$. This is the horizontal distance from the extreme compression fiber to the neutral plane of the stress block.

Nominal moment is the internal resisting moment from the tension-steel couple: $M_n = A_s f_y(d - \frac{a}{2})$, where $d$ is the effective depth to the centroid of tension steel and the lever arm is $(d - a/2)$. For tension-controlled sections ($\varepsilon_t \geq 0.005$), the capacity reduction factor is $\phi = 0.90$, so design capacity is $\phi M_n = 0.90 M_n$. This accounts for material variability and analysis uncertainties per NSCP 2015.

At ultimate, if the net tensile strain in the steel is $\varepsilon_t \geq 0.005$ (0.5%), the section is classified as tension-controlled. The strain is calculated as $\varepsilon_t = 0.003 \frac{d-c}{c}$, where $c = a/\beta_1$ is the neutral-axis depth and 0.003 is the concrete crushing strain. NSCP 2015 enforces this limit to ensure ductile failure (steel yields first) rather than brittle concrete-controlled failure.

The balanced ratio is the amount of steel at which concrete and steel reach their ultimate limits simultaneously: concrete at crushing strain 0.003 and steel at yield strain $\varepsilon_y = f_y/E_s$. Derived from strain compatibility, $\rho_b = 0.85\beta_1\frac{f'_c}{f_y}\cdot\frac{600}{600+f_y}$. For $f'_c = 28$ MPa and $f_y = 415$ MPa (common in Philippines), $\rho_b \approx 0.0288$.

To guarantee tension control, NSCP 2015 limits the steel ratio such that $\varepsilon_t \geq 0.005$ at ultimate. This yields $\rho_{\max} = 0.85\beta_1\frac{f'_c}{f_y}(0.375)$, which is approximately 0.75$\rho_b$ but derived directly from the net-strain requirement. For $f'_c = 28$, $f_y = 415$: $\rho_{\max} \approx 0.0183$. Older codes used 0.75$\rho_b$; the 2015 version is more explicit and ties to the 0.005 strain limit.

Minimum reinforcement prevents sudden brittle failure immediately upon concrete cracking. NSCP 2015 requires $\rho_{\min} = \max\left(\frac{1.4}{f_y}, \frac{\sqrt{f'_c}}{4f_y}\right)$. For $f'_c = 28$ MPa, $f_y = 415$ MPa: $\rho_{\min} = \max(0.00337, 0.00330) = 0.00337$. This ensures that even in understrength beams, post-crack resistance exists.

In design, the moment demand is normalized by the cross-sectional strength: $R_n = \frac{M_u}{\phi b d^2}$, with units of stress (MPa). This dimensionless coefficient allows direct lookup or formula application. For given $R_n$, the required $\rho$ follows from the quadratic: $\rho = \frac{0.85f'_c}{f_y}\left(1 - \sqrt{1 - \frac{2R_n}{0.85f'_c}}\right)$. This avoids trial-and-error iteration.

When $M_u$ exceeds the capacity of a singly reinforced section at $\rho = \rho_{\max}$, or when deflection control requires additional stiffness, compression steel $A'_s$ is added. Moment capacity is superposed: the singly reinforced couple $(A_s - A'_s)$ at the singly reinforced limit, plus a steel-steel couple $A'_s(f'_s - f_y)(d - d')$, where $f'_s$ is the compression steel stress (check if it reaches $f_y$) and $d'$ is the centroid depth of compression steel. This extends design flexibility beyond the single-steel limit.

In monolithic construction, the slab above the beam acts as a compression flange. If the stress block depth $a$ stays within the flange thickness $t_f$, the section behaves as a rectangular beam of width $b_f$ (flange width). If $a > t_f$, the effective compression zone includes both flange and web, requiring separate calculation: moment from flange couple plus moment from web couple. For design, first assume $a \leq t_f$; if violated, reformulate with $a > t_f$.

$d$ is the perpendicular distance from the extreme compression fiber to the centroid of the tension steel, measured along the beam axis. It is NOT the full beam depth $h$ or the bottom cover; it accounts for concrete cover (typically 40–50 mm), stirrup diameter, and bar diameter. Correct identification of $d$ is vital: too large an assumed $d$ overstates capacity; too small understates it. Always sketch the cross-section and measure or compute $d$ carefully.

The equivalent stress block extends from the top compression fiber to depth $a$. The neutral axis is at depth $c = a/\beta_1$ from the compression fiber. At this depth, concrete strain is 0.003 (crushing). Below the neutral axis, concrete is in tension but typically cracked and ignored in stress calculations. Steel strain is found from strain compatibility: $\varepsilon_s = 0.003(d-c)/c$. If $\varepsilon_s \geq f_y/E_s$, steel is yielding.

As $\rho$ increases from $\rho_{\min}$ to $\rho_b$, the section transitions from tension-controlled (steel yields first) to balanced (both yield simultaneously) to over-reinforced (concrete crushes first). NSCP 2015 limits design to $\rho \leq \rho_{\max} \approx 0.75\rho_b$ to guarantee $\varepsilon_t \geq 0.005$ and ensure $\phi = 0.90$. Beyond $\rho_{\max}$, compression steel is needed.

The design workflow is a chain: given $M_u$ (ultimate moment from factored loads), compute $R_n = M_u/(\phi b d^2)$ to normalize the demand. The $\rho$ formula then yields the required steel ratio, and $A_s = \rho b d$ gives the steel area. This sequential approach eliminates guessing and ensures consistency.

Tension-controlled sections ($\varepsilon_t \geq 0.005$) use $\phi = 0.90$: ductile, steel yields with warning. Transition sections ($\varepsilon_t = 0.002$ to $0.005$) use $\phi = 0.65 + 0.25(\varepsilon_t - 0.002)/0.003$ (linear interpolation per NSCP 2015). Compression-controlled sections ($\varepsilon_t \leq 0.002$) use $\phi = 0.65$: brittle, concrete crushes suddenly. Design code restricts ordinary beams to tension-controlled ($\phi = 0.90$) to avoid brittle failure.

Without adequate reinforcement, a concrete beam cracks and loses strength suddenly. Minimum steel ensures post-crack tensile capacity: the cracked section can still carry load via the steel. The two formulas for $\rho_{\min}$ represent different safety philosophies: $1.4/f_y$ is a fraction of steel-yield strength; $\sqrt{f'_c}/(4f_y)$ is a fraction of concrete tensile capacity. Taking the larger protects against both brittle failure and excessive deflection.

Singly reinforced (tension steel only) is the simplest, used for most beams if capacity is adequate. When $M_u$ exceeds singly reinforced capacity at $\rho_{\max}$, add compression steel to create a doubly reinforced section: capacity is superposed from a singly reinforced couple and a steel-steel couple. T-beams leverage the slab as a flange to increase compression area; if the flange is insufficient, T-beam capacity approaches doubly reinforced behavior. Each is a progression in design complexity and load-carrying potential.

Effective depth $d$ determines how far tension steel is from the compression zone. Larger $d$ increases the lever arm $(d - a/2)$ and thus moment capacity $M_n = A_s f_y(d - a/2)$. For a given $M_u$, larger $d$ allows smaller $A_s$, reducing cost and congestion. Deeper beams are more economical, but must satisfy deflection and other serviceability limits (topics beyond this chapter).