Reinforced and Prestressed Concrete Design: Singly Reinforced Beam Analysis

Principles of Reinforcement Detailing

• General Detailing Rules (CENG325):

  • Continuous and Discontinuous Ends: Specific bar placements are required for top and bottom reinforcement at supports and midspans.

  • Clearance and Cover: Typical cover for reinforcement is specified as $40\,mm$.

  • Cantilevers: Requirements include top bars at the cantilever's discontinuous end and bottom bars at the discontinuous end.

  • Spacing and Development:

    • Span length is designated as $L$ (Clear Distance).

    • Top bars at support: $L1/3$.

    • Top bars at midspan: $L1/2$.

    • Bottom bars at support: $L1$ with splice lengths as per specific tables.

    • Cantilever bars: $Lo$, marked by $3/4 Lo$ or $L 1/4$ (where the greater value governs).

  • Hoop Reinforcement Spacing:

    • Maximum at support: $50\,mm$.

    • $S1 = 4 \times 50\,mm$.

    • $S2 = 4 \times 100\,mm$.

    • $S3 = \text{Rest } 150\,mm$.

Fundamental Assumptions in Flexural Analysis

• Assumption 1: The maximum concrete strain ($\epsilon_c$) at the extreme compression fiber shall be assumed equal to $0.003$.

• Assumption 2: The tensile strength of concrete shall be neglected in flexural calculations.

• Assumption 3: The relationship between concrete compressive stress and strain may be represented by a rectangular, trapezoidal, parabolic, or any other shape that predicts strength in agreement with compressive test results.

• Assumption 4: A uniform concrete stress of $0.85 f'_c$ shall be assumed. This stress is distributed over an equivalent compression zone bounded by the cross-section and a line parallel to the neutral axis, located at a distance $a$ from the fiber of maximum compressive strain.

• Assumption 5: The distance from the maximum compressive strain fiber to the neutral axis is denoted as $c$, measured perpendicular to that axis. The relationship between the stress block depth ($a$) and the neutral axis depth ($c$) is defined as:

  • $a = \beta c$

Mechanics and Notations of Singly-Reinforced Beams

• Notation Definitions:

  • $A_s$: Area of steel reinforcement.

  • $d$: Effective depth (distance from extreme compression fiber to the centroid of tension reinforcement).

  • $b$: Width of the beam.

  • $f'_c$: Specified compressive strength of concrete.

  • $c$: Depth from the extreme compression fiber to the neutral axis.

  • $a$: Depth of the equivalent rectangular stress block.

  • $\epsilon_s$: Strain in the steel reinforcement.

  • $\epsilon_c$: Strain in the concrete ($0.003$ at ultimate limit state).

  • $E_s$: Modulus of elasticity of steel, taken as $200,000\,MPa$ (also noted as $200,000\,GPa$ in some slides).

  • $f_y$: Specified yield strength of steel reinforcement.

  • $f_s$: Actual stress in the steel.

• Equilibrium and Force Vectors:

  • Compression Force ($C$): $C = 0.85 f'_c a b$

  • Tension Force ($T$): $T = A_s f_s$

  • Summation of Horizontal Forces: $C = T$

  • $0.85 f'_c a b = A_s f_s$

Moment Capacity Equations

• Nominal Moment Capacity ($M_n$):

  • The nominal moment can be calculated via summation of moments at the level of the steel or concrete resultant force.

  • Lever arm between $C$ and $T$ is $(d - \frac{a}{2})$.

  • $M_n = C(d - \frac{a}{2})$

  • $M_n = T(d - \frac{a}{2})$

  • Substituting force definitions:

    • $M_n = 0.85 f'_c a b (d - \frac{a}{2})$

    • $M_n = A_s f_s (d - \frac{a}{2})$

Strain Compatibility and Steel Stress

• Strain-Stress Relationship: $Strain = \frac{Stress}{\text{Modulus of Elasticity}}$.

• Compatibility Analysis (Ratio and Proportion):

  • Using the strain diagram: $\frac{0.003}{c} = \frac{\epsilon_s}{d - c}$

  • Substituting $\epsilon_s = \frac{f_s}{200,000}$:

  • $\frac{f_s}{200,000} = 0.003 \left( \frac{d - c}{c} \right)$

  • $f_s = 600 \left( \frac{d - c}{c} \right)$

• Yielding Criteria:

  • If $f_s \geq f_y$: The steel has yielded. Use $f_y$ in strength calculations.

  • If f_s < f_y: The steel has not yielded. Use the actual computed value of $f_s$.

Types of Failure and Steel Ratios

Failure types are determined by the percentage of steel in the section relative to the concrete.

• Tension-Controlled Section:

  • Steel reaches a strain $\epsilon_s \geq 0.005$ before concrete reaches its maximum strength.

  • Occurs when there is a small percentage of steel.

  • Failure is characterized as Ductile Failure.

  • Section is considered Under-Reinforced.

• Balanced Section:

  • Concrete reaches $\epsilon_c = 0.003$ at the exact same time the steel reaches its yielding strain.

  • Steel and concrete fail simultaneously.

• Compression-Controlled Section:

  • Concrete reaches $\epsilon_c = 0.003$ before the steel reaches its yielding point.

  • Occurs when there is a large percentage of steel.

  • Failure is characterized as Brittle Failure.

  • Section is considered Over-Reinforced.

Strength Reduction Factors and Reinforcement Limits

• Strength Reduction Factor ($\phi$):

  • Tension-Controlled ($f_s \geq 1000$ or $\epsilon_s \geq 0.005$): $\phi = 0.9$

  • Compression-Controlled ($f_s \leq 400$ or $\epsilon_s \leq \epsilon_y$): $\phi = 0.65$

  • Transition Zone:

    • $\phi = 0.65 + 0.25 \left( \frac{\epsilon_s - \epsilon_y}{0.005 - \epsilon_y} \right)$

    • $\phi = 0.65 + 0.25 \left( \frac{f_s - f_y}{1000 - f_y} \right)$

• Balanced Steel Ratio Formula ($\rho_{bal}$):

  • $c_b = \left( \frac{600}{600 + f_y} \right) d$

  • $\rho_{bal} = \frac{0.85 f'_c \beta}{f_y} \left( \frac{600}{600 + f_y} \right)$

  • If \rho_{actual} < \rho_{bal}, the steel yields.

  • If \rho_{actual} > \rho_{bal}, the steel does not yield (DNY).

• Maximum Conditions:

  • Maximum Condition (f_s > f_y): $c_{max} = \frac{3}{7}d$.

  • Tension-Controlled Limit: $c_t = \frac{3}{8}d$.

  • $\rho_{max} = \frac{0.85 f'_c \beta}{f_y} \left( \frac{3}{7} \right)$

  • $\rho_t = \frac{0.85 f'_c \beta}{f_y} \left( \frac{3}{8} \right)$

• Minimum Flexural Reinforcement ($A_{s,min}$):

  • Ref: NSCP/ACI Section 409.6.1.

  • $A_{s,min}$ is the greater of:

    • (a) $\frac{0.25 \sqrt{f'_c}}{f_y} b_w d$

    • (b) $\frac{1.4}{f_y} b_w d$

Worked Examples

Sample Problem 1

• Given: $b = 300\,mm$, $d = 500\,mm$, $3 - 25\,mm$ RSB, $f'_c = 28\,MPa$, $f_y = 414\,MPa$.

• Results:

  1. Depth of Stress Block ($a$): $69.908\,mm$

  2. Nominal Moment Capacity ($M_n$): $283.5\,kNm$

  3. Design Moment Capacity ($\phi M_n$): $255.17\,kNm$

  4. Balanced Steel Ratio ($\rho_{bal}$): $0.0335$

  5. Mode of Design: Tension-Controlled

Sample Problem 2

• Given: $b = 300\,mm$, $d = 500\,mm$, $9 - 28\,mm$ RSB, $f'_c = 34\,MPa$, $f_y = 414\,MPa$.

• Results:

  1. Balanced Steel Ratio: $0.0333$

  2. Actual Steel Ratio: $0.037$

  3. Depth of Stress Block ($a$): $245.87\,mm$

  4. Minimum Steel Ratio: $0.003383$

  5. Mode of Design: Compression-Controlled

  6. Nominal Moment Capacity ($M_n$): $803.79\,kNm$

  7. Design Moment Capacity ($\phi M_n$): $522.46\,kNm$

Sample Problem 3

• Given: $b = 300\,mm$, $d = 440\,mm$, $f'_c = 27.58\,MPa$, $f_y = 413.80\,MPa$, $M_u = 262.56\,kNm$.

• Results:

  1. Maximum Depth of Compression ($c$): $165\,mm$

  2. Maximum Steel Ratio ($\rho_{max}$): $0.01806$

  3. Area of Steel Required for $M_u$: $1906.078\,mm^2$

  4. Number of $20\,mm$ RSB: $7\,PCS$