Design of a Rectangular Singly-Reinforced Beam

This flashcard set covers the fundamental vocabulary, strain-based section classifications, and design formulas for singly-reinforced rectangular concrete beams based on the provided lecture notes.

Strength Design Method

A design methodology, formerly known as the Ultimate Strength Design (USD) Method, which follows the principle that the design strength must be greater than or equal to the strength required to carry factored loads (e.g., $\phi M_n \geq M_u$).

Nominal flexural strength ($M_n$)

The theoretical flexural strength of a member, calculated as $M_n = C(d - \frac{a}{2})$ or $M_n = T(d - \frac{a}{2})$.

Design flexural strength ($\phi M_n$)

The nominal flexural strength multiplied by a strength reduction factor ($\phi$).

Strength reduction factor ($\phi$) for Flexure (tension-controlled)

The factor applied to nominal flexural strength for tension-controlled sections, which is equal to $0.90$.

Tension-Controlled Section

A section in which the net tensile strain in the extreme tension steel ($\varepsilon_t$) is greater than or equal to $0.005$ when the concrete in compression reaches its assumed limit of $0.003$.

Compression-Controlled Section

A section in which the net tensile strain in the extreme tension steel ($\varepsilon_t$) is less than or equal to the yield strain ($\varepsilon_{ty}$) at the time the concrete in compression reaches its strain limit of $0.003$.

Transition Section

A section where the net tensile strain in the extreme tension steel ($\varepsilon_t$) lies between the compression-controlled limit ($\varepsilon_{ty}$) and the tension-controlled limit ($0.005$).

Whitney's rectangular stress diagram

A model that replaces the actual concrete stress distribution with an equivalent rectangular block having a stress of $0.85 f'_c$ and a depth $a = \beta_1 c$.

$$\beta_1$$

A factor for the depth of the stress block; it is $0.85$ for $f'_c \leq 28\,MPa$. For $f'_c > 28\,MPa$, it is calculated as $0.85 - 0.05(\frac{f'_c - 28}{7})$, but not less than $0.65$.

Balanced Strain

The condition where the net tensile strain in the extreme tension steel ($\varepsilon_t$) reaches the yield strain ($\varepsilon_{ty}$, which is $\frac{f_y}{E_s}$) exactly as the concrete reaches its strain limit of $0.003$.

Minimum Strain Limit

For nonprestressed flexural members with factored axial compressive load less than $0.10 f'_c A_g$, the net tensile strain ($\varepsilon_t$) at nominal strength shall not be less than $0.004$.

Steel Ratio ($\rho$)

The ratio of the area of tension reinforcement ($A_s$) to the effective area of the concrete cross-section ($bd$), given by $\rho = \frac{A_s}{bd}$.

Coefficient of resisting moment ($R_n$)

A coefficient used in beam design, calculated as $R_n = \frac{M_u}{\phi b d^2}$ or expressed in terms of the steel ratio as $R_n = \rho f_y - \frac{(\rho f_y)^2}{1.7 f'_c}$.

Balanced Steel Ratio ($\rho_b$)

The steel ratio that produces balanced strain conditions, calculated as $\rho_b = \frac{0.85 f'_c \beta_1}{f_y} (\frac{600}{600 + f_y})$.

Maximum Steel Ratio for Tension-Controlled Sections ($\rho_{max}$ at $\varepsilon_t = 0.005$)

The steel ratio required to ensure a tension-controlled section where $\varepsilon_t = 0.005$, calculated as $\rho_{max} = \frac{0.85 f'_c \beta_1}{f_y} (\frac{3}{8}) (\frac{d_t}{d})$.

Effective depth ($d$)

The distance from the extreme compression fiber to the centroid of the tension reinforcement.

Net tensile strain ($\varepsilon_t$) calculation

The strain in the extreme tension steel derived from the strain profile as $\varepsilon_t = 0.003 (\frac{d_t - c}{c})$.