CELE Reinforced & Prestressed Concrete — Reinforced Concrete Fundamentals: WSD and USDStudy Notes

Reinforced concrete (RC) is a composite material where concrete resists compression and steel bars resist tension. Concrete alone is brittle and weak in tension; steel alone is expensive and prone to corrosion. Together, they create a material that is durable (when properly detailed), economical, and highly versatile—the backbone of Philippine infrastructure. Two competing design frameworks exist: **Working Stress Design (WSD, also called Alternate Design or Elastic Method):** - Assumes linear-elastic behavior under **service loads** (unfactored dead load D and live load L). - Keeps computed stresses below allowable fractions of the material strength (e.g., concrete stress ≤ 0.45f'_c, steel stress ≤ 0.5f_y or 0.4f_y depending on loading). - Uses the **modular ratio** n = E_s/E_c to convert steel area into an equivalent concrete section. - Conservative and conceptually simple, but does not directly address safety factors against ultimate failure. - Still encountered in older Philippine designs and legacy code references (e.g., earlier ACI editions). **Ultimate Strength Design (USD, also LRFD—Limit State Design or Load and Resistance Factor Design):** - Factors **up** the loads: $M_u = 1.2M_D + 1.6M_L$ (and other combinations per NSCP 2015 Section 203). - Reduces the nominal (ultimate) strength by a **strength-reduction factor** φ: $$\phi M_n \ge M_u, \quad \phi V_n \ge V_u$$ - Replaces the real curved concrete stress distribution with an **equivalent rectangular stress block** of intensity 0.85f'_c and depth a = β₁c. - Directly designed for failure; the φ factors account for variability in material and construction, and differences between assumed and actual stress distributions. - **NSCP 2015 mandates USD** for all new design; WSD is an alternative but secondary method, retained for educational and legacy-code comparisons. In practice, USD is more realistic because it acknowledges that materials do not behave perfectly elastically at high stress levels and that failure is a discrete, observable event. The PRC Civil Engineer Licensure Examination expects fluency in both approaches, with heavy emphasis on USD.

**Concrete Compressive Strength f'_c (MPa):** The 28-day cylinder compressive strength is the standard design parameter in NSCP 2015. Common grades in Philippines: f'_c = 20.7, 24.5, 27.6, 34.5 MPa (and metric equivalents). In design, always use the specified strength; do not assume "typical" or "average" unless explicitly stated in a problem. **Modulus of Elasticity E_c:** For normal-weight concrete (density ρ ≈ 2400 kg/m³), NSCP 2015 Section 421 (based on ACI 318) gives: $$E_c = 4700\sqrt{f'_c} \text{ (MPa, where } f'_c \text{ is in MPa)}$$ This formula is empirical and applies to normal-weight concrete. For lightweight concrete, a reduction factor is used. In the range f'_c = 20–55 MPa, E_c typically ranges from ~21,000 to ~35,000 MPa. **Steel Reinforcement Yield Strength f_y (MPa):** Two main grades are used in Philippines (per ASTM A615 / PH equivalent): - Grade 275 MPa (older, less common now): f_y = 275 MPa, E_s = 200,000 MPa. - Grade 415 MPa (standard): f_y = 415 MPa, E_s = 200,000 MPa. In USD design, the stress-strain curve is idealized as elastic-perfectly-plastic: linear up to f_y, then constant at f_y. The yield strain is: $$\varepsilon_{ty} = \frac{f_y}{E_s} \quad \text{(typical: } \varepsilon_{ty} = \frac{415}{200,000} = 0.002075 \approx 0.0021 \text{)}$$ A reinforcing bar is said to be "at yield" when the strain in the bar reaches ε_ty. In a beam under positive moment, the bottom steel yields first. **Modular Ratio n (WSD only):** WSD uses the modular ratio to transform a section into an equivalent concrete section: $$n = \frac{E_s}{E_c}$$ For example, one bar with area A_s transforms to an equivalent concrete area nA_s, because concrete and steel have different stiffnesses. This allows treating the composite section as if it were all concrete, with the "transformed" steel area. The modular ratio depends on f'_c and must be calculated for each strength grade (unlike f_y, which is fixed by the reinforcement manufacture).

One of the most important simplifications in USD is the **equivalent rectangular stress block** (also called the Whitney stress block, after one of its developers). Instead of using the real, curved concrete stress–strain distribution, USD assumes: - **Constant stress intensity:** 0.85f'_c throughout the compression zone (not the full f'_c, but 0.85 times it). - **Depth of the block:** a = β₁c, where c is the neutral-axis depth and β₁ is a factor that accounts for the shape and nonlinearity of the real stress distribution. - **Location of the resultant force:** at the centroid of the block, which is a/2 from the top fiber (or c from the neutral axis, since a = β₁c). This simplification is valid because: 1. Concrete does not behave as a linear-elastic material at near-failure stresses; the stress–strain curve is nonlinear and curved. 2. The factor 0.85 accounts for the difference between the peak stress in the real distribution and the rectangular approximation. 3. The factor β₁ adjusts for the varying shape as concrete strength changes. **The β₁ Factor:** The value of β₁ depends on f'_c and decreases as concrete strength increases. NSCP 2015 Section 421.1.3.3.1 specifies: $$\beta_1 = 0.85 \quad \text{for } f'_c \le 28 \text{ MPa}$$ $$\beta_1 = 0.85 - 0.05 \frac{f'_c - 28}{7} \quad \text{for } 28 < f'_c \le 55 \text{ MPa}$$ $$\beta_1 = 0.65 \quad \text{for } f'_c \ge 55 \text{ MPa}$$ **Why does β₁ decrease?** Higher-strength concrete has a stiffer, narrower stress block. The neutral axis moves closer to the compression face, so the stress block depth a becomes a smaller fraction of c. Using a constant β₁ = 0.85 for all strengths would overestimate the block depth and thus overestimate moment capacity at high f'_c. **Practical Insight:** For f'_c ≤ 28 MPa, β₁ = 0.85 is constant. Above 28 MPa, β₁ decreases linearly at a rate of 0.05 per 7 MPa of increase in f'_c, reaching 0.65 at f'_c = 55 MPa. Above 55 MPa, β₁ remains 0.65. This formula is critical in computing neutral-axis depth c and moment capacity M_n.

The strength-reduction factor φ ("phi") is a multiplier applied to the nominal (ultimate) strength to account for: 1. Uncertainty in material properties (concrete strength varies, steel yield varies). 2. Variability in construction (reinforcement placement, concrete compaction). 3. Differences between assumed and actual stress distributions. 4. Type of failure mode (sudden brittle vs. ductile behavior). Different failure modes and member types have different φ values because they have different levels of warning before failure. Ductile failures (e.g., beam flexure where steel yields first) are more forgiving; brittle failures (e.g., shear, column buckling) are sudden. Thus: $$\phi M_n \ge M_u \quad (\text{flexure}), \quad \phi V_n \ge V_u \quad (\text{shear}), \quad \phi P_n \ge P_u \quad (\text{columns})$$ **NSCP 2015 Strength-Reduction Factors:** For **flexure (bending):** - Tension-controlled sections: φ = 0.90. This includes most beam designs where the bottom steel is in tension and yields prior to concrete crushing. The beam bends visibly before failure, providing warning. - Compression-controlled sections: φ = 0.65 (tied columns) or 0.75 (spiral columns). These fail when concrete is crushed; less ductile, more sudden. - Transition zone (between tension- and compression-controlled): φ is interpolated linearly between 0.65/0.75 and 0.90 as the net tensile strain ε_t increases. For **shear and torsion:** - φ = 0.75 (all cases). Shear and torsional failures are typically sudden and brittle, offering little warning. For **compression (columns):** - Spiral reinforcement: φ = 0.75. Spiral confinement provides lateral restraint, increasing ductility. - Tied reinforcement: φ = 0.65. Ties are less effective at confinement; failure is more brittle. For **bearing on concrete:** - φ = 0.65. Bearing is a localized compression failure, often sudden. For **plain concrete:** - φ = 0.60. Plain (unreinforced) concrete has no ductility reserve. **Transition Between Compression- and Tension-Controlled (ACI/NSCP):** A section is: - **Compression-controlled** if the net tensile strain ε_t ≤ ε_ty (the yield strain of the reinforcement). In this case, φ = 0.65 (tied) or 0.75 (spiral). - **Tension-controlled** if ε_t ≥ 0.005 (5 times the yield strain). In this case, φ = 0.90. - **In transition** if ε_ty < ε_t < 0.005. In this case, φ is linearly interpolated. For Grade 415 steel: ε_ty = 415/200,000 = 0.002075. Thus a section is tension-controlled if the tensile strain in the extreme steel layer exceeds 0.005 = 5%. For **tied columns**, the φ interpolation formula is: $$\phi = 0.65 + 0.25 \frac{\varepsilon_t - \varepsilon_{ty}}{0.005 - \varepsilon_{ty}}$$ This is tested frequently on the licensure exam, especially in column design problems.

USD design requires that the section be proportioned so that the nominal (ultimate) strength, reduced by φ, exceeds the factored (design) loads. The factored loads are obtained by multiplying service loads (dead D, live L, earthquake E, wind W, etc.) by prescribed load factors. NSCP 2015 Section 203 specifies the combinations. **Primary Load Factors (Limit State of Strength):** For **gravity loads only** (most common in building design): $$U = 1.2D + 1.6L \quad (\text{Combination 1, all positive})$$ Alternative gravity combinations: $$U = 1.2D + 1.0L + 1.0E \quad (\text{with seismic})$$ $$U = 1.2D + 1.0W + 1.0L \quad (\text{with wind, reduced live})$$ $$U = 0.9D + 1.6W \quad (\text{wind-only, minimal dead load})$$ $$U = 0.9D + 1.0E \quad (\text{seismic-only, minimal dead load})$$ Where: - **D** = dead load (self-weight, permanent fixtures). - **L** = live load (occupancy, temporary loads). - **E** = earthquake load (seismic forces). - **W** = wind load. For **design**, the largest U (or M_u, V_u, P_u) from all applicable combinations governs. In most residential and office buildings, the 1.2D + 1.6L combination controls for positive moment, while 0.9D + 1.6W or 0.9D + 1.0E might control for negative moment or stability. **Physical Meaning:** The factors 1.2 and 1.6 are chosen such that: - Dead load is loaded at 1.2 (fairly high confidence in self-weight estimate, hence lower factor). - Live load is loaded at 1.6 (greater uncertainty in live load, hence higher factor). - The combination represents a mean-plus-variability approach: a rare event where D and L occur simultaneously at their mean plus some multiple of standard deviation. **Application in Reinforced Concrete:** For a beam or slab: $$M_u = 1.2M_D + 1.6M_L \quad (\text{moment due to factored loads})$$ $$V_u = 1.2V_D + 1.6V_L \quad (\text{shear due to factored loads})$$ The section is then designed so that: $$\phi M_n \ge M_u, \quad \phi V_n \ge V_u$$ where M_n and V_n are computed based on the geometric and material properties of the section.

Though **NSCP 2015 mandates USD for new designs**, WSD remains important because: 1. Older Philippine buildings (pre-2000s) were designed to WSD codes. 2. The PRC Civil Engineer Licensure Examination tests both methods for comparison and historical knowledge. 3. WSD provides intuitive understanding of how concrete and steel work together under service loads. 4. Some designers still use WSD for preliminary sizing due to its simplicity. **WSD Philosophy:** WSD assumes the structure remains in the **linear-elastic range** under service loads (D + L, unfactored). Concrete and steel are treated as elastic materials with their respective moduli E_c and E_s. The key steps are: 1. Assume a cracked section (most practical for beams in tension). 2. Use the modular ratio n = E_s/E_c to convert steel area into equivalent concrete area. 3. Compute the neutral axis depth c from the condition that the first moment of area = 0. 4. Calculate concrete stress f_c and steel stress f_s from the moment using the elastic formula. 5. Check that f_c ≤ allowable f_c and f_s ≤ allowable f_s. **Typical Allowable Stresses (WSD):** Allowable concrete compressive stress: $f_c \le 0.45 f'_c$ (maximum at extreme fiber). Allowable steel stress: $f_s \le 0.4 f_y$ or $0.5 f_y$ (varies by loading and rebar type). **Transformed Section (WSD):** The composite section is transformed into an "equivalent concrete section" by replacing steel area A_s with nA_s. The neutral axis is then found from the centroid of the transformed area. Moment capacity is computed using $f = M \cdot y / I$, where y is the distance from the neutral axis and I is the moment of inertia of the transformed section. **Limitations of WSD:** - Does not directly address failure; relies on indirect "factors of safety" embedded in the allowable stresses. - Cannot easily account for variations in material strength or construction variability. - Linear-elastic assumption breaks down near failure; high-strength concrete and ductility are poorly represented. - Allowable stresses vary by code and year, creating confusion. Despite these limitations, understanding WSD is essential for: - Analyzing heritage structures. - Comparing with USD to see why the shift was made. - Solving exam problems that explicitly ask for WSD design or analysis. **Key Difference from USD:** WSD: service loads → check stresses ≤ allowables. USD: factored loads → check φ × nominal strength ≥ factored demand.

This section consolidates the essential formulas and concepts from the chapter. Mastery of these relationships is critical for both theory questions and numerical problems on the PRC Civil Engineer Licensure Examination. **Material Properties:** $$E_c = 4700\sqrt{f'_c} \quad (\text{MPa; normal-weight concrete})$$ $$n = \frac{E_s}{E_c} = \frac{200,000}{E_c} \quad (\text{modular ratio, WSD})$$ $$\varepsilon_{ty} = \frac{f_y}{E_s} \quad (\text{yield strain, typically } 0.002\text{–}0.003)$$ **Stress-Block Factor β₁:** $$\beta_1 = \begin{cases} 0.85 & \text{if } f'_c \le 28 \text{ MPa} \\ 0.85 - 0.05 \frac{f'_c - 28}{7} & \text{if } 28 < f'_c \le 55 \text{ MPa} \\ 0.65 & \text{if } f'_c \ge 55 \text{ MPa} \end{cases}$$ Equivalent rectangular stress block depth: $$a = \beta_1 c \quad (\text{depth of 0.85} f'_c \text{ intensity block})$$ **Strength-Reduction Factors φ (NSCP 2015):** | Component | Condition | φ | |---|---|---| | Flexure | Tension-controlled (ε_t ≥ 0.005) | 0.90 | | Flexure | Compression-controlled (ε_t ≤ ε_ty) | 0.65 (tied) / 0.75 (spiral) | | Flexure | Transition (ε_ty < ε_t < 0.005) | Linear interpolation | | Shear | All | 0.75 | | Columns | Tied | 0.65 | | Columns | Spiral | 0.75 | | Bearing | All | 0.65 | **Load Factoring (NSCP 2015, Gravity Loads):** $$M_u = 1.2 M_D + 1.6 M_L \quad (\text{primary combination})$$ $$V_u = 1.2 V_D + 1.6 V_L$$ Other combinations with wind, seismic, or reduced live load apply; use governing (largest) factored demand. **USD Design Inequality (All Member Types):** $$\phi M_n \ge M_u, \quad \phi V_n \ge V_u, \quad \phi P_n \ge P_u$$ where M_n, V_n, P_n are the nominal (ultimate) capacities based on the section geometry and reinforcement. **WSD Allowable Stresses:** $$f_c \le 0.45 f'_c \quad (\text{concrete compression})$$ $$f_s \le 0.4 f_y \text{ or } 0.5 f_y \quad (\text{steel tension, varies by code})$$ Under service loads M = M_D + M_L (unfactored).

The following errors are frequently committed by exam takers. Awareness and avoidance are essential for success. **Pitfall 1: Confusing WSD and USD Logic** Error: Computing service-load stresses (like in WSD) but then comparing to φ-reduced strengths (like in USD). Correct Approach: **WSD uses unfactored service loads and allowable stresses**. **USD uses factored loads and φ-reduced nominal strengths**. Choose one method and apply it consistently. **Pitfall 2: Wrong β₁ for a Given f'_c** Error: Using β₁ = 0.85 for all concrete strengths, or incorrectly computing β₁ for 28 < f'_c ≤ 55 MPa. Correct Approach: Always check f'_c first. If f'_c ≤ 28, use β₁ = 0.85. If f'_c > 28, use the linear formula β₁ = 0.85 − 0.05(f'_c − 28)/7. Many students forget to check the threshold and apply the formula when f'_c = 21 or 24 MPa, yielding an incorrect (too-low) β₁. **Pitfall 3: Using 0.85 Instead of 0.85f'_c in the Stress Block** Error: Computing the compression force as C = 0.85 × b × a, forgetting to multiply by f'_c. Correct Approach: The block intensity is **0.85f'_c** (in MPa), so the compression force is $C = 0.85 f'_c \cdot b \cdot a$ (in newtons, when b and a are in mm and f'_c is in MPa, divide by 1000). **Pitfall 4: Incorrect Strength-Reduction Factor φ for Columns** Error: Using φ = 0.90 for a column (beams use 0.90, so students assume columns do too). Correct Approach: Columns are NOT tension-controlled in flexure. Use φ = 0.65 for tied columns and φ = 0.75 for spiral columns. Only flexure (beams, slabs, deep beams) use φ = 0.90 when tension-controlled. **Pitfall 5: Not Checking All Load Combinations** Error: Assuming the 1.2D + 1.6L combination always governs. Correct Approach: NSCP 2015 requires checking multiple combinations. Depending on the relative magnitudes of D, L, W, and E, the governing factored load might come from 0.9D + 1.6W or other combinations. Always compute all applicable combinations and use the largest demand. **Pitfall 6: Misidentifying Tension-Controlled vs. Compression-Controlled** Error: Confusing the strain threshold (0.005) or misinterpreting what "tension-controlled" means. Correct Approach: A section is **tension-controlled** if the net tensile strain ε_t ≥ 0.005 (5 times the yield strain, for any bar size). At this high strain, the concrete is not the limiting factor; steel yields and strains extensively. **Compression-controlled** means ε_t ≤ ε_ty (the yield strain), so the concrete crushes before steel yields; failure is abrupt. **Pitfall 7: Wrong Modular Ratio or Confused Transformed Section** Error: Using n = E_s / E_c incorrectly in the transformed section, or forgetting that only the steel is transformed (not the concrete). Correct Approach: The modular ratio n = E_s / E_c (always > 1 because steel is stiffer than concrete). In the transformed section, concrete area remains b × h, and steel area A_s is replaced by n × A_s. The neutral axis depth is found by setting the first moment of area = 0. **Pitfall 8: Load Factor Order or Sign** Error: Applying 1.2 to live load and 1.6 to dead load (backwards), or using negative factors. Correct Approach: Load factors are always positive. The typical gravity combination is **1.2D + 1.6L** (not the reverse). Mnemonic: "Live load gets the bigger factor" because it is more uncertain than dead load. **Pitfall 9: Forgetting to Reduce Capacity by φ in the Design Check** Error: Comparing nominal capacity M_n directly to factored demand M_u, without multiplying M_n by φ. Correct Approach: The USD inequality is $\phi M_n \ge M_u$, not $M_n \ge M_u$. Always reduce the nominal capacity by φ before comparing to the factored demand. φ is always less than 1, so φM_n < M_n; this is the "penalty" for uncertainty.