Sources and References

Primary texts — MacGregor, J. G. and Bartlett, F. M. Reinforced Concrete: Mechanics and Design, Canadian ed. Pearson, 2000. Kulak, G. L. and Grondin, G. Y. Limit States Design in Structural Steel, 10th ed. Canadian Institute of Steel Construction, 2018.

Supplementary texts — Salmon, C. G., Johnson, J. E., and Malhas, F. A. Steel Structures: Design and Behavior, 5th ed. Pearson, 2008. Nilson, A. H., Darwin, D., and Dolan, C. W. Design of Concrete Structures, 15th ed. McGraw-Hill, 2015. Nawy, E. G. Reinforced Concrete: A Fundamental Approach, 6th ed. Pearson, 2008.

Online resources — CISC Handbook of Steel Construction technical bulletins (public portions); American Concrete Institute open ACI 318 companion educational material; AISC Steel Construction Manual design guide excerpts; MIT OpenCourseWare 1.054 Mechanics and Design of Concrete Structures; Portland Cement Association design aids.

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Chapter 1: Structural Systems and Loads

1.1 Systems for Gravity Loads

Gravity (dead, live, snow, equipment) loads flow downward through slabs into beams, from beams into girders, from girders into columns, and from columns into foundations. In a typical steel-framed building, floor decking spans to secondary beams, which span to girders at column lines. Reinforced concrete variants include flat plates (slabs supported directly by columns), waffle slabs (two-way ribbed), one-way slabs on beams, and post-tensioned slabs.

1.2 Systems for Lateral Loads

Lateral (wind, seismic) loads are resisted by dedicated systems:

• Moment frames: beams and columns rigidly connected; resist lateral load by flexure and shear in the frame.
• Braced frames: diagonal members carry lateral load as axial forces in a truss.
• Shear walls: solid walls of concrete or masonry resist in-plane lateral loads.
• Dual systems: combinations, with moment frames engaged by floor diaphragms to share lateral load with shear walls or braced frames.

Floor diaphragms (slabs) distribute lateral load to the vertical lateral-resisting elements in proportion to their relative stiffness.

1.3 Limit States Design

Modern codes (CSA, ACI, AISC, Eurocode) use limit states design, factoring loads upward and factoring resistances downward. The safety inequality is

\[ \phi R \ge \sum \gamma_i Q_i, \]

where \(R\) is the nominal resistance, \(\phi\) is the resistance factor, \(Q_i\) are the nominal load effects, and \(\gamma_i\) are load factors. Typical Canadian factors are \(\gamma_D = 1.25\), \(\gamma_L = 1.5\), and \(\phi = 0.65\) to 0.9 depending on failure mode.

Chapter 2: Reinforced Concrete Beams — Flexure

2.1 Behaviour Under Bending

A reinforced concrete beam passes through three stages as load increases: uncracked elastic (concrete tension region intact), cracked elastic (tension concrete cracked but steel elastic), and nominal flexural strength (steel yields and concrete crushes). Design at ultimate limit state assumes the nominal flexural strength stage.

2.2 Rectangular Stress Block and Flexural Strength

The Whitney rectangular stress block replaces the actual parabolic concrete compression distribution with a uniform stress of \(\alpha_1 f'_c\) over a depth \(a = \beta_1 c\), where \(c\) is the neutral axis depth. For \(f'_c \le 30\) MPa, CSA takes \(\alpha_1 = 0.85\) and \(\beta_1 = 0.85\).

Equilibrium for a singly reinforced rectangular beam of width \(b\) with tension steel area \(A_s\) at yield:

\[ A_s f_y = \alpha_1 f'_c b a. \]

Solving for \(a\) and taking moment about the steel centroid,

\[ M_r = \phi_s A_s f_y\left(d - \frac{a}{2}\right), \]

where \(d\) is the effective depth. Design requires \(M_r \ge M_f\), with \(M_f\) the factored moment demand.

2.3 Tension, Balanced, and Compression Failure

Ductile flexural failure requires steel to yield before concrete crushes. The balanced reinforcement ratio \(\rho_b\) corresponds to simultaneous yield and crush; design must use \(\rho < \rho_b\) to guarantee ductile failure. Codes prescribe an upper limit \(\rho_{max} \approx 0.75\rho_b\) and a lower limit \(\rho_{min} = 1.4/f_y\) (or \(\sqrt{f'_c}/(4f_y)\) if larger).

2.4 One-Way Slabs

A one-way slab spans between parallel supports and is analyzed per metre of width as a shallow beam. Minimum steel for shrinkage and temperature (\(\rho = 0.002\)) is required transverse to the span. Cover requirements (20 mm interior, 40 mm weather-exposed) protect reinforcement from corrosion and fire.

Chapter 3: Reinforced Concrete Beams — Shear

3.1 Diagonal Tension

Under combined shear and flexure, principal tension stresses form at roughly 45 degrees to the beam axis, producing diagonal cracks near supports. Unreinforced concrete can carry a nominal shear

\[ V_c = 0.2\lambda\phi_c\sqrt{f'_c}b_w d \]

(simplified Canadian equation), where \(\lambda\) accounts for concrete density. When the factored shear demand exceeds \(V_c\), transverse reinforcement (stirrups) is required.

3.2 Stirrup Design

Vertical stirrups of area \(A_v\) at spacing \(s\), with yield \(f_{yv}\), contribute

\[ V_s = \frac{\phi_s A_v f_{yv}d}{s}. \]

Design requires \(V_c + V_s \ge V_f\). Minimum stirrup spacing is \(s_{max} = \min(d/2, 600\text{ mm})\) when \(V_s \le 0.25\phi_c\sqrt{f'_c}b_w d\); closer spacing is required for higher shear. Stirrup yield stress is usually capped at 500 MPa to prevent excessive crack widths at service loads.

Chapter 4: Development, Bond, and Anchorage

Reinforcement must develop its yield strength by bond to the concrete over an adequate length. The development length for a bar of diameter \(d_b\) in tension is

\[ \ell_d = \frac{1.15\,k_1k_2k_3k_4 f_y}{\sqrt{f'_c}}\,d_b, \]

with modifiers for bar location, coating, size, and concrete density. Hooks (90° or 180°) reduce the required length. Splices and cutoffs must be located and detailed so that at every section the bars can develop the required stress.

Chapter 5: Serviceability — Deflection and Cracking

5.1 Deflection

At service load, concrete beams are usually cracked but reinforcement is elastic. Effective moment of inertia for deflection is given by Branson’s formula

\[ I_e = \left(\frac{M_{cr}}{M_a}\right)^3 I_g + \left[1 - \left(\frac{M_{cr}}{M_a}\right)^3\right]I_{cr}, \]

interpolating between gross and fully cracked section stiffness. Long-term deflection multiplies the immediate value by a creep-and-shrinkage factor of 2 to 3 depending on duration and compressive steel content.

5.2 Crack Control

Cracks wider than about 0.3 mm allow corrosion-driving moisture to reach reinforcement. Codes control cracks by limiting bar spacing and specifying reinforcement arrangement; the Gergely-Lutz equation estimates probable crack width and is satisfied indirectly by code detailing rules.

Chapter 6: Steel Beams

6.1 Plastic Bending

Steel yields in tension and compression without tearing, allowing fully plastic hinges. The plastic moment of a rectangular section is \(M_p = F_y Z\), where \(Z\) is the plastic section modulus. For a wide-flange shape, \(Z/S\) (shape factor) is typically 1.12 to 1.15.

6.2 Classification and Local Buckling

Cross sections are classified by the width-to-thickness ratios of their plate elements. Class 1 (plastic) develops \(M_p\) and sustains inelastic rotation sufficient for plastic design. Class 2 (compact) develops \(M_p\). Class 3 (non-compact) develops \(M_y = F_y S_x\). Class 4 (slender) buckles before reaching \(M_y\) and must be analyzed with effective widths.

6.3 Lateral-Torsional Buckling

A long beam bent about its major axis can buckle sideways and twist. The elastic critical moment is

\[ M_{cr} = \frac{\pi}{L}\sqrt{EI_y GJ}\sqrt{1 + \left(\frac{\pi E}{L}\right)^2\frac{I_y C_w}{GJ I_y}}\cdot \omega_2, \]

where \(C_w\) is warping constant and \(\omega_2\) a moment gradient factor. Simplified design charts relate unbraced length to moment capacity; continuous lateral bracing of the compression flange raises capacity to \(M_p\).

6.4 Design Procedure

Given factored moment \(M_f\), the designer selects a W-shape (or equivalent) of the lightest mass such that \(\phi M_r \ge M_f\), checks shear capacity \(\phi V_r = \phi 0.66F_y A_w\), checks local buckling class, checks lateral-torsional buckling at the actual unbraced length, and verifies deflection under service load (typically span/360 for live load).

Chapter 7: Steel Tension Members and Axially Loaded Members

7.1 Tension Members

Three limit states govern tension members: gross section yielding (\(T_r = \phi A_g F_y\)), net section fracture (\(T_r = \phi_u A_n F_u\) with \(\phi_u = 0.75\)), and block shear along potential tear-out lines. Bolt holes reduce the net area; staggered fastener patterns are handled with the Cochrane (\(s^2/4g\)) adjustment.

7.2 Compression Members

For a concentrically loaded column, the elastic buckling stress is the Euler value

\[ F_e = \frac{\pi^2 E}{(KL/r)^2}, \]

where \(KL\) is the effective length and \(r\) is the radius of gyration. Inelastic behaviour at short slenderness is captured by the CSA-S16 column curve

\[ F_{cr} = F_y(1 + \lambda^{2n})^{-1/n}, \]

with \(\lambda = \sqrt{F_y/F_e}\) and \(n = 1.34\) (hot-rolled sections). The column resistance is \(C_r = \phi A F_{cr}\).

7.3 Effective Length Factor

The effective length factor \(K\) captures end conditions. Standard cases: fixed-fixed \(K = 0.5\), fixed-pinned \(K = 0.7\), pinned-pinned \(K = 1.0\), fixed-free \(K = 2.0\). Sway columns in unbraced frames have \(K \ge 1\); alignment charts (nomographs) determine \(K\) from end restraint ratios.

Chapter 8: Comparing Steel and Concrete Systems

8.1 Material Characteristics

Structural steel offers high strength-to-weight (yield/density around 30 kN·m/kg), ductility, and fast erection. Reinforced concrete offers low material cost, fire resistance without additional protection, acoustic mass, and adaptability to complex geometries. Composite construction (concrete slab on steel beams with shear studs) exploits the strengths of each.

8.2 Span and Depth Trade-offs

For spans below 8 m, flat-plate concrete slabs are typically economical. From 8 to 15 m, concrete one-way or two-way joist systems or steel W-shapes with composite deck dominate. Above 15 m, long-span steel trusses, castellated or cellular beams, and post-tensioned concrete become competitive. Depth-to-span ratios give a quick check: \(L/d\) of 18 to 22 for steel composite, 20 to 25 for post-tensioned concrete, 15 to 18 for reinforced concrete beams.

8.3 Fire, Durability, and Sustainability

Steel loses half its strength near 550 °C, requiring spray-applied cementitious coatings or intumescent paint for fire-rated construction. Concrete retains most of its strength through a one-hour fire without added protection. Concrete durability depends on cover and concrete quality; steel durability depends on paint systems and detailing for water runoff. Embodied carbon of a typical concrete frame is dominated by cement (about 90 percent of the CO\(_2\)); of a steel frame, by iron reduction and rolling. Supplementary cementitious materials (fly ash, slag) and electric-arc-furnace steel with high scrap content lower both.