CELE Reinforced & Prestressed Concrete — Reinforced Concrete Beams: Shear and TorsionSummary

WSD assumes elastic behavior under service loads and limits stress to safe fractions of material strength (e.g., concrete stress ≤ 0.45f'c, steel stress ≤ 0.50fy). It employs the modular ratio n = Es/Ec to transform steel area into equivalent concrete area, allowing the use of standard elastic bending formulas. The design criterion is simple: actual service stresses must remain below allowables. Although NSCP 2015 does not recommend WSD for new design, it remains testable for comparative knowledge and is still used in some jurisdictions. Example: for a beam carrying 80 kN⋅m service load, WSD calculates the actual stress at midspan and verifies it does not exceed 0.45f'c.

USD is the NSCP 2015 default. It factors up the service loads by design factors (1.2D + 1.6L for gravity, with other combinations for wind, seismic, etc.), finds the nominal (unfactored) strength Mn of the section, applies a strength-reduction factor φ based on failure mode, and verifies φMn ≥ Mu. The approach is safety-based: it assumes reserves of strength (φ < 1) to account for variability in materials and workmanship. The nominal strength is determined from the equivalent rectangular stress block (Whitney block) assuming concrete stress 0.85f'c and neutral-axis depth a = β₁c. Design equation: φMn ≥ Mu, where φ ≈ 0.90 for tension-controlled flexure, 0.75 for shear, and 0.65–0.75 for compression depending on column type.

Concrete strength f'c (in MPa) is determined by 28-day compressive cylinder testing per ASTM C39. Typical values in the Philippines range from 21 MPa (standard) to 35, 42, and 55 MPa (high-strength). The elastic modulus for normal-weight concrete is empirically given by Ec = 4700√f'c (in MPa), which accounts for the increasing modulus with higher strength. For f'c = 28 MPa, Ec ≈ 24,872 MPa ≈ 24,900 MPa; for f'c = 35 MPa, Ec ≈ 27,749 MPa. Concrete is not perfectly elastic—the Ec formula is a lower-bound estimate. The relationship is essential for calculating the modular ratio n in WSD and predicting member stiffness in both design methods.

Reinforcing steel is typically Grade 275 (fy = 275 MPa) or Grade 415 (fy = 415 MPa) per NSCP 2015, with nominal yield stress defined as the stress at 0.2% permanent strain (off-set method). The elastic modulus Es = 200,000 MPa is constant regardless of grade. The yield strain is εty = fy/Es: for Grade 415, εty = 415/200,000 = 0.00208 (or 0.208%). Steel behavior is elasto-plastic: below fy, stress = E·strain; at and beyond fy, strain increases without significant stress increase (plastic plateau). For design, we assume steel yields when required (in tension-controlled sections) or may not yield (in compression-controlled sections).

The modular ratio n is the ratio of steel elastic modulus to concrete elastic modulus: n = Es/Ec = 200,000/Ec. It is used exclusively in WSD (elastic) design to transform steel area As into equivalent concrete area nAs. This allows the designer to treat the beam as if it were entirely concrete, simplifying moment-of-inertia and stress calculations. For f'c = 28 MPa with Ec ≈ 24,872 MPa, n ≈ 8.04, often rounded to 8 for calculation. The modular ratio is a key parameter in the transformed section: the neutral axis is found by balancing first moments, and section properties are computed about the neutral axis. Example: A 300 mm wide beam with 2 bars of 20 mm diameter (As = 628 mm²) and n = 8 has equivalent concrete area nAs = 5,024 mm².

Under ultimate load, the actual concrete compression stress distribution is curved (parabolic or similar). USD simplifies this with the equivalent rectangular (Whitney) block: uniform stress intensity 0.85f'c extending from the extreme compression fiber to depth a = β₁c, where c is the neutral-axis depth measured from the extreme compression fiber. This block has the same total compressive force and similar moment effect as the actual curve, simplifying hand calculations and nominal-strength derivations. The factor β₁ adjusts the block depth to account for concrete strength: at low strength (f'c ≤ 28 MPa), β₁ = 0.85 (block depth is 85% of c); at higher strength, β₁ decreases because high-strength concrete's stress distribution is more concentrated near the fiber. For 28 < f'c ≤ 55 MPa: β₁ = 0.85 − 0.05(f'c − 28)/7; for f'c ≥ 55 MPa: β₁ = 0.65 (minimum).

The strength-reduction factor φ accounts for uncertainty in materials, workmanship, and model approximations. It multiplies the nominal strength Mn (or Vn, Pn) to give the design strength φMn. NSCP 2015 prescribes: (1) Tension-controlled flexure: φ = 0.90; (2) Shear and torsion: φ = 0.75; (3) Compression-controlled columns with spiral ties: φ = 0.75; (4) Compression-controlled columns with tied bars: φ = 0.65; (5) Bearing on concrete: φ = 0.65; (6) Plain (unreinforced) concrete: φ = 0.60. Between compression-controlled and tension-controlled limits (transition zone), φ is interpolated linearly as a function of net tensile strain εt. The higher φ for tension-controlled (0.90) reflects greater ductility and predictability; the lower φ for tied columns (0.65) reflects brittleness and sensitivity to geometric imperfections.

NSCP 2015 specifies design load combinations for gravity, wind, seismic, and other effects. The primary gravity combination is Pu or Mu = 1.2D + 1.6L, where D is dead load (self-weight) and L is live load (occupancy, storage). Additional combinations include: (1) 1.2D + 1.0L + 1.6W (wind); (2) 0.9D + 1.6W; (3) 1.2D + 1.0L + 1.0E (seismic, E is seismic effect); (4) 0.9D + 1.0E. The factors 1.2 and 1.6 represent the probability of simultaneous occurrence and inherent variability; 1.2 for dead (more predictable) and 1.6 for live (less predictable). The designer computes internal moments, shears, and axial forces (Mu, Vu, Pu) for each combination and selects the critical (maximum) values for design. Example: if gravity gives Mu = 192 kN⋅m and wind gives Mu = 110 kN⋅m, use 192 kN⋅m for flexural design.

A section's failure mode depends on the net tensile strain εt at the extreme tension steel when the extreme compression fiber reaches 0.003 strain. If εt ≥ 0.005, the steel is well into the plastic plateau and failure is ductile (tension-controlled, φ = 0.90). If εt ≤ εty = fy/Es, failure is brittle (compression-controlled, φ = 0.65 for tied, 0.75 for spiral). The transition zone occurs for εty < εt < 0.005, where φ interpolates linearly. For Grade 415 steel, εty = 415/200,000 = 0.00208; thus a section with net tensile strain 0.003 is in transition and requires interpolation. High reinforcement ratio leads to compression-controlled failure (limited rotation); low ratio leads to tension-controlled (ductile). NSCP 2015 and ACI 318 both use this strain-based classification.

The fundamental inequality of USD is φMn ≥ Mu (and similarly φVn ≥ Vu, φPn ≥ Pu for other limit states). Here, Mu is the factored design moment (from 1.2D + 1.6L, etc.); Mn is the nominal moment strength calculated using the equivalent stress block and strain compatibility; φ is the strength-reduction factor based on failure mode. The inequality is a margin of safety: the nominal strength is reduced by φ to account for uncertainties, and the reduced strength must exceed the factored demand. Rearranging: Mn ≥ Mu/φ, the required nominal strength. If this inequality is violated, the section must be enlarged or reinforcement increased. This is the design check; in preliminary design, the designer iterates to find a section that just satisfies (or slightly exceeds) the requirement.

WSD works with service loads (the actual loads expected in use: dead + live at their nominal values). USD factors up the service loads by design factors (1.2, 1.6, etc.) to create factored loads, which are then used to compute factored internal effects (Mu, Vu, Pu). This is a fundamental shift in thinking: WSD asks 'Will stresses at service load stay safe?' USD asks 'Can the nominal strength, reduced by φ, resist the factored load?' WSD is deterministic and uses a single load level; USD is probabilistic and uses amplified loads to create a margin. Example: a beam under 80 kN⋅m service moment is checked at 80 kN⋅m in WSD (allowable stress check) but at Mu = 1.2(0) + 1.6(80) = 128 kN⋅m in USD (assuming all service moment is live; if 40 kN⋅m is dead, then Mu = 1.2(40) + 1.6(40) = 112 kN⋅m).

Concrete modulus Ec and steel modulus Es are experimentally determined. Their ratio, the modular ratio n, allows steel to be converted to equivalent concrete in elastic analysis. This transformation is the entire basis of WSD hand calculations and ensures the transformed section obeys linear-elastic beam theory.

Concrete strength f'c uniquely determines β₁. The β₁ factor, in turn, positions the equivalent rectangular stress block (depth a = β₁c) relative to the neutral axis. The block position affects both the magnitude and lever arm of the internal couple, directly calculating the nominal moment strength Mn. Higher f'c → lower β₁ → shallower block → higher nominal strength (all else equal).

The limit state (flexure, shear, compression, bearing) and the section's control (tension-controlled or compression-controlled, determined by net tensile strain εt) together determine φ. Tension-controlled flexure gets φ = 0.90; compression-controlled tied column gets φ = 0.65. This mapping ensures φ reflects both the failure mode and the ductility available.

The service load (D, L, W, E at their nominal values) is multiplied by code-prescribed factors (1.2, 1.6, etc.) to obtain the factored load. The factored load is used to compute the factored internal effect (Mu, Vu, Pu). The required nominal strength is then Mn ≥ Mu/φ. This cycle repeats for all governing load combinations, and the controlling (maximum) nominal strength demand is used to size the section.

A single reinforced concrete section can be analyzed (or designed) by either WSD or USD. WSD calculates stresses at service load and compares to allowables; USD calculates nominal strength at factored load and compares via φMn ≥ Mu. Both methods must yield safe designs for the same section. WSD is less efficient (more conservative) because it does not exploit the true strength available; USD is more refined and economical, hence preferred by modern codes.

For each limit state (flexure, shear, axial), the designer evaluates all governing load combinations (gravity, wind, seismic) and identifies which combination produces the maximum demand. That demand controls the design. For example, flexure might be controlled by 1.2D + 1.6L, while shear might be critical under 0.9D + 1.6W. The designer must check all and design for the envelope.

Given a beam span, dead load (self-weight + permanent fixtures), and live load, a design engineer first computes the critical factored moment Mu = 1.2D + 1.6L using bending theory. Then, assuming a trial width b and selecting steel grade (typically Grade 415), the engineer estimates depth d using a target reinforcement ratio (e.g., 1–2% for typical beams). The equivalent stress block is set up: a = β₁c. Strain compatibility and force equilibrium yield neutral-axis depth c, block depth a, and nominal moment Mn. The check φMn ≥ Mu is then verified. If it fails, depth is increased and the iteration repeats. This is the bread-and-butter design loop taught in every RC course and tested on the PE exam.

An engineer is asked to evaluate an existing beam in a building or to verify its capacity after a use change increases live load. The section dimensions (b, h, d, As) are known from drawings. Using USD, the engineer computes the nominal moment strength Mn from the equivalent stress block, applies φ = 0.90 (assuming tension-controlled), and determines φMn. The new load combination is factored (1.2D_new + 1.6L_new), and Mu is computed. If φMn ≥ Mu, the beam is adequate; if not, retrofitting (adding steel, widening, propping) may be needed. This is routine work in professional practice and a frequent exam scenario.

Older buildings in the Philippines (pre-2010) may have been designed under the 1992 NSCP or earlier codes using WSD. To evaluate or modify such structures, engineers must sometimes revert to WSD logic. The transformed section is drawn: actual concrete area plus nAs for steel. The neutral axis is located by setting the first moment of area about the axis to zero. Stresses are computed at service load using fc = Mc/I and fs = nMc/I. The stresses are checked against allowables: fc ≤ 0.45f'c, fs ≤ 0.50fy (typical). Understanding WSD is thus a practical necessity for field engineers working on older structures and is often tested on the PE exam as a comparative method.

A building column supports floor loads (D, L), plus equipment loads and lateral loads from wind or seismic. The NSCP 2015 specifies multiple load combinations: 1.2D + 1.6L (gravity), 1.2D + 1.0L + 1.6W (gravity + wind), 0.9D + 1.6W, 1.2D + 1.0L + 1.0E (seismic), etc. The design engineer computes the internal effects (axial force, moment, shear) for each combination, often using structural analysis software. The critical combination for each effect is identified: for example, axial compression might peak under 1.2D + 1.6L, while axial tension might occur under 0.9D + 1.6W. The column is then designed for the envelope of demands. This multi-combination thinking is essential in modern practice and frequently examined.

A large building in Manila uses 42 MPa concrete to minimize column size. The design engineer must use the correct β₁: β₁ = 0.85 − 0.05(42 − 28)/7 = 0.85 − 0.05(2) = 0.75. This affects every nominal-strength calculation. A careless engineer using β₁ = 0.85 (appropriate for 28 MPa) would overestimate capacity and risk under-design. Conversely, using β₁ = 0.65 (applicable only for f'c ≥ 55 MPa) would over-design and waste concrete and steel. Correct β₁ selection is a detail-oriented but critical task in design, and it is a common source of error in student solutions.

A 400 mm square tied reinforced concrete column in a multi-story building carries axial load and moment. The engineer must determine whether the section is tension-controlled or compression-controlled by computing the net tensile strain εt. If εt < εty = 415/200,000 ≈ 0.00208, the section is compression-controlled and φ = 0.65 (tied column). If εt > 0.005, it is tension-controlled and φ = 0.90. If in between, φ is interpolated. This classification dramatically affects the nominal strength: a section with φ = 0.65 can resist less than one with φ = 0.90 for the same nominal strength Mn. The engineer iterates: trial section → compute εt → select φ → check φPn ≥ Pu. Wrong φ selection invalidates the design, so this is a frequent source of exam failure.