CELE Reinforced & Prestressed Concrete — Reinforced Concrete Beams: FlexureStudy Notes

A singly reinforced rectangular beam is the most common beam type encountered in practice. It has tension reinforcement (steel bars $A_s$) placed near the bottom (or the extreme tension fiber), while the compression zone in concrete resists the compressive stress. The beam width is $b$, the effective depth (distance from compression face to centroid of tension steel) is $d$, and the concrete strength is $f'_c$ with yield strength $f_y$ for the steel.

Stress Block and Force Equilibrium

When the beam reaches ultimate load, the concrete in compression develops a stress distribution that ACI 318 and NSCP 2015 simplify into an equivalent rectangular stress block:

• Stress intensity: $0.85 f'_c$ (uniform)
• Depth of block: $a$ (measured from the compression face)
• This block replaces the more complex parabolic stress distribution

The equivalent stress block simplification allows engineers to quickly calculate moment capacity without complex integration. The relationship between the depth of the stress block ($a$) and the neutral axis depth ($c$) is:

where $\beta_1 = 0.85$ for $f'_c \le 28\ \text{MPa}$, decreasing slightly for higher strength concrete (as per NSCP 2015).

Force Equilibrium (Compression = Tension)

The compressive force in the concrete stress block is:

The tensile force in the steel is:

At equilibrium, $C = T$:

Solving for $a$:

This is the fundamental equation for singly reinforced beams. Notice that $a$ depends only on the amount of steel ($A_s$), the steel strength ($f_y$), and the concrete properties and section geometry.

Nominal Moment Capacity

Once $a$ is known, the nominal moment $M_n$ is calculated as the tension force times the lever arm (distance from the line of action of the compressive force to the line of action of the tensile force):

Alternatively, using the compressive force:

Both forms are equivalent and commonly used. The term $(d - a/2)$ is the internal lever arm, often denoted as $j_d$ or simply the distance between the centroids of the compression and tension forces.

Design Moment Capacity (Factored)

The design moment capacity is:

The strength reduction factor $\phi = 0.90$ applies when the section is tension-controlled, meaning the net tensile strain in the extreme tension steel reaches or exceeds 0.005 (0.5%) at ultimate. This ductile behavior is required by NSCP 2015 and ACI 318 to ensure the beam warns of impending failure through visible cracking and deflection before sudden collapse.

Ductility Check: Net Tensile Strain

The net tensile strain at ultimate is calculated from strain compatibility:

where $0.003$ (0.3%) is the crushing strain of concrete at the extreme compression fiber. The section is:

• Tension-controlled if $\varepsilon_t \ge 0.005$ (φ = 0.90, ductile)
• Transition zone if $0.002 < \varepsilon_t < 0.005$ (φ varies, less common in design)
• Compression-controlled if $\varepsilon_t \le 0.002$ (φ = 0.65, brittle failure)

For a well-designed singly reinforced beam, the engineer should aim for the tension-controlled range to guarantee ductile failure.

The steel ratio is defined as:

It represents the percentage of longitudinal tension steel relative to the cross-sectional area of the beam ($b \times d$). Steel ratio limits are critical control parameters in beam design; they ensure the beam fails in a ductile manner and prevent over-reinforcement (compression-controlled failure) or under-reinforcement (sudden cracking without warning).

Balanced Steel Ratio (ρ_b)

The balanced ratio is the steel percentage at which the concrete reaches its crushing strain (0.003) and the steel simultaneously reaches its yield strain ($f_y/E_s$) at the same moment. At this point, the section transitions from tension-controlled to compression-controlled behavior.

For typical Philippine materials ($f'_c = 28\ \text{MPa}$, $f_y = 415\ \text{MPa}$, $\beta_1 = 0.85$):

If the actual steel ratio exceeds $\rho_b$, the beam becomes over-reinforced (compression-controlled), and the concrete crushes before the steel yields, leading to sudden brittle failure without warning.

Maximum Steel Ratio (ρ_max) — NSCP 2015 / ACI 318

NSCP 2015 and ACI 318 limit the maximum steel ratio to ensure the section remains tension-controlled with net tensile strain $\varepsilon_t \ge 0.005$:

For the standard case ($\beta_1 = 0.85$):

For $f'_c = 28\ \text{MPa}$ and $f_y = 415\ \text{MPa}$:

Note: The older code (ACI 318-05) used $\rho_{\max} = 0.75\rho_b$, but NSCP 2015 and modern ACI now use the strain-based formula above, which is more rigorous and ensures $\varepsilon_t \ge 0.005$.

Minimum Steel Ratio (ρ_min)

The minimum ratio ensures that the beam can develop a nominal moment at least equal to the cracking moment. Without minimum reinforcement, the concrete cracks suddenly at low loads, and the beam can collapse suddenly without plastic deformation.

For $f'_c = 28\ \text{MPa}$ and $f_y = 415\ \text{MPa}$:

Design Range

For a ductile, tension-controlled, economical design, keep:

If the required steel ratio $\rho$ exceeds $\rho_{\max}$ for a given beam size, the designer must either (1) increase the beam depth $d$, (2) increase the width $b$, (3) add compression reinforcement (doubly reinforced), or (4) use higher strength concrete or steel.

In beam design, the factored applied moment $M_u$ is known (from load analysis), and the engineer must determine the required tension steel area $A_s$. The typical process is:

1. Assume a reasonable beam size ($b$, $d$) based on span and support conditions
2. Calculate the resistance coefficient $R_n$
3. Use the quadratic formula to solve for $\rho$
4. Convert $\rho$ to $A_s$
5. Select bar sizes and check $\rho_{\min} \le \rho \le \rho_{\max}$

Resistance Coefficient Method

Starting from the design moment equation:

Substituting $a = \rho f_y d / (0.85 f'_c)$:

Dividing both sides by $\phi b d^2$:

Define the resistance coefficient:

This simplifies the design equation to:

Rearranging into standard quadratic form:

Solving for Steel Ratio

Using the quadratic formula and keeping the smaller root (to avoid over-reinforcement):

This is the design formula most commonly used on Philippine board exams. It avoids iterative calculations and directly yields $\rho$ from $R_n$.

Practical Design Steps

1. Assume beam dimensions ($b$, $d$) — typically $d \approx L/15$ to $L/20$ for simply supported spans
2. Calculate $R_n$: $R_n = M_u / (\phi b d^2)$ where $\phi = 0.90$
3. Calculate $\rho$: Use the quadratic formula above

4. Check limits: Verify $\rho_{\min} \le \rho \le \rho_{\max}$

   • If $\rho < \rho_{\min}$: use $A_s = \rho_{\min} b d$ (minimum reinforcement governs)
   • If $\rho > \rho_{\max}$: increase $b$ or $d$ (beam is too small)
5. Calculate $A_s$: $A_s = \rho b d$
6. Select bars: Choose combinations of standard diameters (16, 20, 25, 32 mm in the Philippines) that provide at least $A_s$ and verify spacing requirements per NSCP 2015

Important Note on φ = 0.90

The design method assumes $\phi = 0.90$. This is valid only if the section is tension-controlled ($\varepsilon_t \ge 0.005$). After selecting the bar size, the engineer should verify:

If the check fails, the section is over-reinforced, and the design must be revised.

Beam problems in the PRC Civil Engineer Licensure Exam fall into two broad categories: analysis (given section, find capacity) and design (given moment, find steel). Understanding the distinction and the solution pathway for each is critical for exam success.

Analysis Problems

Given: Beam dimensions ($b$, $d$), reinforcement ($A_s$), material strengths ($f'_c$, $f_y$)

Find: Design moment capacity $\phi M_n$

Solution pathway:

1. Calculate stress block depth: $a = A_s f_y / (0.85 f'_c b)$
2. Calculate nominal moment: $M_n = A_s f_y (d - a/2)$
3. Check ductility: Calculate $c = a/\beta_1$ and $\varepsilon_t = 0.003(d-c)/c$
4. Determine $\phi$: Use 0.90 if tension-controlled ($\varepsilon_t \ge 0.005$), otherwise use lower $\phi$
5. Calculate design moment: $\phi M_n = \phi \times M_n$

Analysis is straightforward: you proceed through the calculations in sequence, and there is typically a unique answer.

Design Problems

Given: Factored moment $M_u$, preferred beam size ($b$, $d$), material strengths ($f'_c$, $f_y$)

Find: Required tension steel area $A_s$ and bar selection

Solution pathway:

1. Calculate resistance coefficient: $R_n = M_u / (\phi b d^2)$
2. Solve for steel ratio: $\rho = \frac{0.85 f'_c}{f_y}\left(1 - \sqrt{1 - \frac{2R_n}{0.85f'_c}}\right)$

3. Check limits:

   • If $\rho < \rho_{\min}$: use $\rho_{\min}$ (beam is lightly loaded)
   • If $\rho > \rho_{\max}$: STOP — increase beam size; do not add more steel
4. Calculate required area: $A_s = \rho b d$
5. Select standard bar sizes to provide at least $A_s$
6. Verify the selected reinforcement is tension-controlled by checking $\varepsilon_t$

Design requires engineering judgment: the engineer must choose reasonable beam dimensions, select bar combinations, and verify assumptions.

Common Pitfalls in Board Exams

1. Assuming $\phi = 0.90$ without checking $\varepsilon_t$ — over-reinforced sections fail in compression, and $\phi$ is lower (0.65–0.80). Always verify tension-controlled behavior.
2. Using the wrong formula for $\rho_{\max}$ — the modern NSCP 2015 and ACI 318 use the strain-based formula $\rho_{\max} = 0.85\beta_1(f'_c/f_y)(0.375)$, not the older $0.75\rho_b$.
3. Confusing $0.85f'_c$ with $f'_c$ in the stress block — the equivalent rectangular stress block always uses $0.85f'_c$, not the full concrete strength.
4. Forgetting to check $\rho_{\min}$ — even a lightly reinforced beam must have at least the minimum ratio to control shrinkage and temperature cracking.
5. Selecting final bar size without re-checking ductility — the formula uses $\phi = 0.90$, but the final design must satisfy $\varepsilon_t \ge 0.005$.
6. Misunderstanding the effective depth $d$ — it is the distance from the extreme compression fiber to the centroid of the tension steel, not the overall beam depth. Typical clearances account for 40–50 mm of cover plus half a bar diameter.

A doubly reinforced beam has both tension reinforcement ($A_s$) near the bottom and compression reinforcement ($A'_s$) near the top. Compression steel is added when:

1. The required moment $M_u$ exceeds the capacity of a tension-controlled singly reinforced section of the given dimensions
2. The designer wants to reduce long-term deflection (compression steel reduces deflection because it reduces the effective stress on tension steel)
3. The beam is subjected to live load reversals or earthquake (negative moment regions in continuous beams)

Force Equilibrium in Doubly Reinforced Beams

The beam carries moment in two ways:

1. Singly reinforced couple: Concrete compression stress block ($C_c$) balances tension force from the lower steel ($T_s$)
2. Steel-steel couple: Compression reinforcement ($C_s = A'_s f'_s$) in the compression zone balances additional tension steel ($T'_s = T - T_s$)

Force balance:

Moment equation:

where $d'$ is the distance from the compression face to the centroid of compression steel (typically 40–50 mm for clear cover).

Key Consideration: Does Compression Steel Yield?

The strain in compression steel at ultimate is:

The stress in compression steel is $f'_s = E_s \varepsilon'_s$ (if $\varepsilon'_s < f_y / E_s$) or $f'_s = f_y$ (if $\varepsilon'_s \ge f_y / E_s$).

For typical beam design:

• If $c$ is large (over-reinforced), the compression strain is large, and compression steel yields: $f'_s = f_y$
• If $c$ is small (under-reinforced), the compression strain may be low, and $f'_s < f_y$ (inelastic behavior)

Practical Design for Doubly Reinforced Beams

1. Calculate the capacity of a singly reinforced section with $\rho = \rho_{\max}$:

   $$M_{\text{single,max}} = \phi \times A_{s,\max} \times f_y \left(d - \frac{a_{\max}}{2}\right)$$

2. If $M_u > M_{\text{single,max}}$, the additional moment is:

   $$M_{\text{additional}} = M_u - M_{\text{single,max}}$$

3. Design the steel-steel couple to resist the additional moment:

   $$M_{\text{additional}} = A'_s f'_s (d - d')$$

   Solving for $A'_s$ (assuming $f'_s = f_y$ for simplicity, which is conservative):

   $$A'_s = \frac{M_{\text{additional}}}{\phi f_y (d - d')} + \text{(tension steel to balance)}$$

4. The total tension steel includes the singly reinforced portion plus additional steel to balance $A'_s$:

   $$A_s = A_{s,\max} + A'_s$$

Common Board Exam Approach

For simplicity, many exam problems assume compression steel yields ($f'_s = f_y$) and use a simplified formula:

where $M_{n,\text{single}}$ is the nominal moment from the singly reinforced portion.

Deflection Benefits

Compression steel reduces the neutral-axis depth $c$ for the same applied moment, which increases the effective moment of inertia and reduces deflection. Long-span beams (where deflection governs) often use compression steel even if the moment capacity is adequate.

A T-beam is a beam where a concrete slab acts as a compression flange, with a web (narrow section) below. T-beams are typical in floor systems where the slab is poured integrally with the supporting beams. The flange increases the compression capacity without significantly increasing the weight, allowing engineers to design economical long-span systems.

T-Beam Geometry

• Flange width: $b_f$ (effective width of slab acting with the beam)
• Flange thickness: $t_f$
• Web width: $b_w$ (width of the beam stem)
• Effective depth: $d$ (to centroid of tension steel, as usual)

The effective flange width $b_f$ is limited by NSCP 2015 and ACI 318:

where $L$ is the span length. The limitation prevents assuming an unrealistically large slab width.

Analysis: When Does the Stress Block Stay Within the Flange?

The critical check is whether the stress block depth $a$ exceeds the flange thickness $t_f$.

Case 1: $a \le t_f$ (stress block in flange only)

If the stress block stays entirely within the flange, the T-beam behaves like a rectangular beam with width $b_f$:

Design proceeds as for a singly reinforced rectangular beam, but using $b_f$ instead of $b$.

Case 2: $a > t_f$ (stress block extends into web)

If the stress block exceeds the flange thickness, the compression force must be split:

1. Flange compression: $C_f = 0.85 f'_c \times t_f \times b_f$ (force in the full flange)
2. Web compression: $C_w = 0.85 f'_c \times (a - t_f) \times b_w$ (additional force in the web)
3. Total compression: $C = C_f + C_w = 0.85 f'_c (t_f b_f + (a - t_f) b_w)$

Force equilibrium gives:

Solving for $a$:

The nominal moment is:

This can be simplified by using the centroid of the compression force, but the formula is more complex.

Practical Design Strategy for T-Beams

1. Estimate effective flange width $b_f$ using NSCP 2015 limits

2. Assume $a \le t_f$ and design as a rectangular beam with width $b_f$:

   $$\rho = \frac{0.85 f'_c}{f_y}\left(1 - \sqrt{1 - \frac{2R_n}{0.85 f'_c}}\right), \quad R_n = \frac{M_u}{\phi b_f d^2}$$

3. Calculate $a$ from the design and check: is $a \le t_f$?

   • If YES, the assumption is correct; proceed with the rectangular design
   • If NO, go back and recalculate using Case 2 (more complex)

Common Board Exam Problem

T-beam problems typically ask: "Is the stress block within the flange? Design the beam." The student must:

1. Assume rectangular behavior first
2. Calculate $a$
3. Check against $t_f$
4. If $a > t_f$, provide the more detailed formula

Most exam problems are designed so that $a \le t_f$, making the analysis straightforward.

Based on years of PRC Civil Engineer Licensure Examination results, certain errors recur frequently. Recognizing and avoiding these pitfalls is essential for earning high scores.

Error 1: Assuming φ = 0.90 Without Verifying Ductility

The mistake: Calculating $\phi M_n = 0.90 M_n$ automatically, without checking $\varepsilon_t \ge 0.005$.

Why it's wrong: An over-reinforced section fails in compression, and the strength reduction factor is φ ≈ 0.65–0.75, not 0.90. The calculated capacity is overstated.

How to avoid it: Always compute:

If $\varepsilon_t < 0.005$, the section is NOT tension-controlled. Depending on $\varepsilon_t$, the correct $\phi$ lies between 0.65 and 0.90. For exam purposes, if you find $\varepsilon_t < 0.005$, flag the design as problematic and note that the actual capacity is lower than calculated with φ = 0.90.

Error 2: Using 0.85 f'_c Instead of the Full f'_c Inconsistently

The mistake: Calculating the stress block as $0.85 f'_c \times a \times b$, but then using $f'_c$ (not $0.85 f'_c$) in subsequent formulas, or vice versa.

Why it's wrong: Force and moment calculations become inconsistent, leading to algebraic errors.

How to avoid it: Remember: the equivalent rectangular stress block always uses 0.85 f'_c. Write it out explicitly in every calculation:

Error 3: Confusing Effective Depth d with Overall Depth h

The mistake: Using the overall beam depth as $d$, instead of the distance to the centroid of the tension steel.

Why it's wrong: The effective depth $d$ is smaller due to concrete cover (typically 40–50 mm) and half the bar diameter (10–15 mm for 20–25 mm bars). Using the wrong $d$ overstates the lever arm $(d - a/2)$ and leads to incorrect moment calculations.

How to avoid it: In problems, $d$ is usually given explicitly. If you must estimate:

Typically $d \approx 0.85 h$ to $0.90 h$ for typical beam sections.

Error 4: Using the Wrong Formula for ρ_max

The mistake: Using the older formula $\rho_{\max} = 0.75 \rho_b$ instead of the modern NSCP 2015 formula.

Why it's wrong: NSCP 2015 and ACI 318-14+ use a strain-based formula $\rho_{\max} = 0.85\beta_1(f'_c/f_y)(0.375)$ that directly enforces $\varepsilon_t \ge 0.005$. The old $0.75\rho_b$ is conservative and differs numerically.

How to avoid it: Use the NSCP 2015 formula:

Make a note card with this formula to bring into the exam (if reference materials are allowed).

Error 5: Forgetting to Check Minimum Steel Ratio

The mistake: Designing a beam with $\rho < \rho_{\min}$ (lightly loaded beam) and not correcting it.

Why it's wrong: A beam below minimum reinforcement cannot develop cracking-moment capacity and will fail suddenly when the concrete cracks, without the steel being fully stressed.

How to avoid it: Always verify:

If the design gives $\rho < \rho_{\min}$, set $A_s = \rho_{\min} b d$ instead.

Error 6: Misinterpreting Effective Flange Width in T-Beams

The mistake: Using the full slab width $b_f$ without applying the NSCP 2015 limitations.

Why it's wrong: The effective flange width is limited to prevent unrealistic assumptions about the slab acting together with the beam. A slab far from the beam contributes little to the compression capacity.

How to avoid it: Always apply the limits:

Use the minimum of the three, not the actual slab width.

Error 7: Assuming Compression Steel Yields Without Checking

The mistake: In doubly reinforced design, assuming $f'_s = f_y$ without calculating the strain in the compression steel.

Why it's wrong: If the neutral axis is shallow, the compression steel may not reach yield strain, and $f'_s < f_y$. The moment contribution is then smaller.

How to avoid it: For exam purposes, most problems implicitly assume compression steel yields (or state it). If in doubt, calculate:

If $\varepsilon'_s \ge f_y / E_s \approx 0.00207$ (for $f_y = 415\ \text{MPa}$), then $f'_s = f_y$. Otherwise, $f'_s = E_s \varepsilon'_s = 200\,000 \times \varepsilon'_s\ \text{(MPa)}$.

Error 8: Misreading Bar Diameters or Areas

The mistake: Confusing bar size (10, 12, 16, 20, 25, 28, 32 mm) with area values, or reading areas from a reference table incorrectly.

Why it's wrong: A single 25 mm bar provides ~490.9 mm² (not 625 mm² = 25²), and selecting the wrong size leads to under- or over-reinforcement.

How to avoid it: Memorize the common Philippines bar sizes and areas (or bring a reference card):

• 10 mm: 78.5 mm²
• 12 mm: 113.1 mm²
• 16 mm: 201.1 mm²
• 20 mm: 314.2 mm²
• 25 mm: 490.9 mm²
• 28 mm: 615.8 mm²
• 32 mm: 804.2 mm²

When selecting bars for the design, aim to provide slightly more area than required (e.g., if $A_s = 1299\ \text{mm}^2$ is required, use three 25 mm bars = 1473 mm² ≈ 1300 mm²).