ACI Design Criteria | mathify.us

// SECTION 01

Bar Placement Based on Bending Moment Diagrams

Concrete is strong in compression but very weak in tension — roughly 10× weaker. Steel bars (rebar) are placed in the beam wherever tension occurs. To know where tension is, you read the Bending Moment Diagram (BMD).

// GOLDEN RULE Where the BMD is positive (sagging) → bottom fibers are in tension → bars go at the BOTTOM.
Where the BMD is negative (hogging) → top fibers are in tension → bars go at the TOP.

INTERACTIVE — Select Beam Type to See Bar Placement

// EXPLANATION Simply supported beam with UDL. The beam sags downward at midspan → bottom fibers stretch (tension) → steel bars placed at the BOTTOM. The BMD is entirely positive (parabolic hump).

// SECTION 02

Effect of Bar Count on Force Per Bar

Here's a surprising insight: adding more bars increases the total moment capacity Mn, but the force each individual bar carries actually decreases. This is because the total tension force T is split among more bars.

From the Whitney stress block: $T = A_s f_y$ and $M_n = A_s f_y \left(d - \frac{a}{2}\right)$

Force per bar = $\frac{T}{\text{number of bars}}$. As more bars are added, T increases, but divides by a bigger number.

INTERACTIVE — Slide to Change Number of No.8 Bars (b=12in, d=15.5in, f'c=4ksi, fy=60ksi)

Number of Bars: 2

As (steel area)

—

in²

a (stress block)

—

Mn (nominal moment)

—

k·ft

T (total tension)

—

kips

Force / bar

—

kips/bar

ρ (rho)

—

ratio

// INSIGHT Notice: as you add bars, Mn ↑ but force/bar ↓. The stress per bar is lower with more bars.

// SECTION 03

Lightly vs Heavily Reinforced Beams

The reinforcement ratio $\rho = A_s / (bd)$ tells us how much steel is present relative to the concrete cross-section. This ratio controls the failure mode of the beam.

INTERACTIVE — Strain Diagram at Failure (slide ρ to see behavior)

ρ (Reinforcement Ratio): 0.010 | UNDER-REINFORCED ✓

c / d ratio

—

εt (steel strain)

—

εy (yield strain)

0.00207

Failure Mode

—

Why does ACI prefer under-reinforced beams? When steel yields first, it stretches significantly. This causes cracks to widen visibly — giving you a warning before collapse. The structure gives you time to evacuate.

// DANGER — OVER-REINFORCED If ρ > ρmax, concrete crushes suddenly with no warning. The steel never yields. The beam fails in a brittle, explosive manner. ACI 318 prohibits this by limiting ρ ≤ ρmax.

// ACI REQUIREMENT Under-reinforced: steel yields → large strains → visible crack widening → warning before failure. ACI requires $\varepsilon_t \geq 0.004$ at nominal strength (tension-controlled: $\varepsilon_t \geq 0.005$, φ = 0.90).

// SECTION 04

Failure by ACI USD Method

In the Ultimate Strength Design (USD) method, failure is defined precisely:

// ACI DEFINITION OF FAILURE Failure occurs when the compressive strain in concrete at the extreme top fiber reaches $\varepsilon_u = 0.003$. At this exact moment, the applied moment equals the Nominal Moment $M_n$.

The key formula chain: $\varepsilon_u = 0.003$ → Whitney stress block activates → $C = 0.85 f'_c \cdot a \cdot b$ → equilibrium $T = C$ → solve for $a$ → calculate $M_n$.

INTERACTIVE — Stress & Strain at Failure (adjust f'c and fy)

f'c: 4 ksi

fy: 60 ksi

β₁

—

ρb (balanced)

—

ρmax (ACI)

—

εy (yield strain)

—

// SECTION 05

ACI Reinforcement Ratio Range

The reinforcement ratio $\rho = A_s/(bd)$ must stay within ACI-specified limits. Too little steel → sudden brittle failure when concrete first cracks. Too much → sudden brittle crushing failure. The safe zone is between $\rho_{min}$ and $\rho_{max}$.

INTERACTIVE — Ratio Range Explorer (f'c=4ksi, fy=60ksi)

Your design ρ: 0.012 → ✓ VALID

// KEY FORMULAS $\rho_{min} = \max\left(\dfrac{3\sqrt{f'_c}}{f_y},\ \dfrac{200}{f_y}\right)$ (psi units)

$\rho_b = 0.85\beta_1 \dfrac{f'_c}{f_y} \cdot \dfrac{87000}{87000 + f_y}$ (psi units)

$\rho_{max}$: based on $\varepsilon_t \geq 0.004$ → typically $\approx 0.75\rho_b$

// SECTION 06

Derivation: Reinforcement Ratio at Failure (from scratch)

We derive $\rho_b$ step by step using two principles: equilibrium (T = C) and strain compatibility (similar triangles on the strain diagram).

STEP-BY-STEP DERIVATION — Click Next to Reveal Each Step

Step 0 / 8

// SECTION 07

Formula Reference

All key formulas from this topic. Hover each card for detail.

REINFORCEMENT RATIO

$$\rho = \frac{A_s}{bd}$$

As = steel area, b = beam width, d = effective depth to steel centroid

TENSION FORCE

$$T = A_s f_y$$

Steel tension force at yield. Valid when under-reinforced (εt ≥ εy)

COMPRESSION FORCE (Whitney)

$$C = 0.85 f'_c \cdot a \cdot b$$

Rectangular stress block. a = β₁c is the equivalent block depth

NOMINAL MOMENT

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

Moment arm is (d - a/2). This is the lever arm between C and T.

Mn (in terms of ρ)

$$M_n = \rho f_y b d^2\!\left(1 - 0.59\frac{\rho f_y}{f'_c}\right)$$

Useful for design when As is unknown. Derived by substituting a = ρfyd/(0.85f'c)

STRESS BLOCK DEPTH

$$a = \frac{A_s f_y}{0.85 f'_c b} = \beta_1 c$$

From T = C. β₁ relates the actual NA depth c to the equivalent block depth a

β₁ FACTOR

$$\beta_1 = \begin{cases}0.85 & f'_c \leq 4000\text{ psi}\\ 0.85-0.05\frac{f'_c-4000}{1000} & f'_c > 4000\end{cases}$$

Minimum value β₁ = 0.65. Accounts for the shape of actual concrete stress block

BALANCED RATIO

$$\rho_b = 0.85\beta_1\frac{f'_c}{f_y}\cdot\frac{87000}{87000+f_y}$$

psi units. Steel yields exactly when concrete crushes (simultaneous). NOT allowed as design point.

MINIMUM RATIO

$$\rho_{min} = \max\!\left(\frac{3\sqrt{f'_c}}{f_y},\frac{200}{f_y}\right)$$

psi units. Prevents sudden brittle failure when concrete first cracks (Mcr > Mn)

STRAIN COMPATIBILITY

$$\frac{c}{d} = \frac{\varepsilon_u}{\varepsilon_u + \varepsilon_t}$$

From similar triangles on strain diagram. εu = 0.003 at top fiber at failure.

DESIGN MOMENT

$$\phi M_n \geq M_u$$

φ = 0.90 (tension-controlled, εt ≥ 0.005). Mu = factored design moment from loads.

ρ AT ANY FAILURE STATE

$$\rho = 0.85\beta_1\frac{f'_c}{f_y}\cdot\frac{c}{d}$$

General form derived from equilibrium T=C. At balanced: c/d = cb/d from strain compat.