Chapter 1

1. Chapter 1 Materials and Mechanics of Bending

2. Concrete Concrete is a mixture of cement along with fine and coarse aggregates. • Aggregates: sand, gravel, crushed rock, and other materials. • Water is added for the chemical reaction of curing. Concrete strength and durability depend on the proportions of the mix, along with the placing, finishing, and curing of the concrete. The compressive strength of concrete is high; the tensile strength of concrete is low. • Concrete is a brittle material. • Reinforcing steel (in the form of bars and mesh) is used to resist tension. Reinforced concrete is a combination of steel and concrete.

3. The ACI Building Code The design and construction of reinforced concrete buildings is controlled by the Building Code Requirements for Structural Concrete (ACI 318-08) of the American Concrete Institute. • The term “code” generally refers to the ACI Code. • The code is revised, updated, and reissued on a 3-year cycle. • The code has no legal status, but the ACI Code is incorporated into the building codes of almost all states and municipalities throughout the United States. • When incorporated into a state or local building code, the ACI Code has official sanction, becomes a legal document, and is part of the law controlling reinforced concrete design and construction in that area.

4. Cement and Water Structural concrete generally uses hydraulic cement. • Water is required for the chemical reaction of hydration. • During hydration, the concrete hardens into a solid mass. • Portland cement, which originated in England, is the most common form of cement. • Cement is marketed in bulk or in 94 lb (1 cubic foot) bags. In concrete, the ratio of the amount of water to the amount of cement, by weight, is termed the water/cement ratio. • This ratio is often expressed in terms of gallons of water per bag of cement. • For complete hydration of the cement in a concrete mix, a water/cement ratio of 0.35 to 0.40 (4 to 4½ gal/bag) is required. • To increase the workability of concrete (i.e. the ease with which it can be mixed, handled, and placed), higher water/cement ratios are normally used.

5. Aggregates In ordinary concretes, aggregates occupy approximately 70% to 75% of the volume of the hardened mass. • Gradation of aggregate size to produce close packing is desirable since the denser the concrete the better the strength and durability. Aggregates are classified as fine and coarse. • Fine aggregate is generally sand. • Fine aggregate is classified as particles that pass a No. 4 sieve (four openings per linear inch). • Coarse aggregates consist of particles (such as gravel) that are retained on a No. 4 sieve. • The maximum size of coarse aggregate in reinforced concrete is governed by ACI Code requirements. • Maximum size requirements for aggregate are established to assure that the concrete can be placed into forms without any danger of large particles lodging between adjacent bars or between bars and the sides of the forms.

6. Concrete in Compression The compressive strength of concrete is denoted fc’ with units of pounds per square inch (or kips per square inch). A test standardized by the American Society for Testing and Materials (ASTM C39) is used to determine the compressive strength (fc’) of concrete. • Figure 1-1 (p. 4 of the textbook) shows the results of compression tests on 28-day standard cylinders for varying concrete design mixes. • A specimen cylinder of concrete is loaded in compression to failure. • The compressive strength of the concrete is the highest compressive stress to which the specimen is subjected. • The compressive strength is not the stress that exists in the specimen at failure but the stress that occurs at a strain of approximately 0.002. • Currently, 28-day concrete strengths (fc’) range from 2500 to 9000 psi. • 3000 to 4000 psi is common for reinforced concrete structures. • 5000 to 6000 psi is common for pre-stressed concrete members. • Higher strength concretes have been achieved under laboratory conditions.

7. Concrete in Compression (continued) Note the following observations from the stress-strain curves. • The maximum compressive strength is generally achieved at a unit strain of approximately 0.002. • Higher-strength concretes are more brittle and fracture at a lower maximum strain than the low-strength concretes. • The initial slope of the curve varies (unlike that of steel) and only approximates a straight line. • For steel (that behaves elastically up to a yield point), the stress-strain plot is a straight line below the yield point. • For concrete, the straight-line portion of the plot is very short (if it exists at all). • For concrete, there is no constant value of modulus of elasticity for a given concrete strength. • If a straight-line portion is assumed, the modulus of elasticity is different for concretes of different strengths. • At low and moderate stresses (up to about 0.5 fc’), concrete is assumed to behave elastically.

8. Concrete in Compression (continued) The ACI Code (Section 8.5.1) provides the accepted empirical expression for the modulus of elasticity. Ec = wc1.5 33√fc’ where Ec = modulus of elasticity of concrete in compression (psi) wc = unit weight of concrete (lb/ft3) fc’ = compressive strength of concrete (psi) The ACI Code expression for modulus of elasticity is valid for concretes having wc between 90 and 160 lb/ft3. • If the unit weight for concrete is 144 lb/ft3, the expression for modulus of elasticity becomes Ec = 57,000 √fc’ • Table A-6 (p. 486 of the textbook) lists values for Ec using this expression.

9. Concrete in Compression (continued) Concrete strength varies with time and rates of loading. • The specified concrete strength is usually the strength that occurs 28 days after the concrete is placed. • Concrete attains approximately 70% of its 28-day strength in 7 days. • Concrete attains approximately 85% to 90% of its 28-day strength in 14 days.

10. Concrete in Tension The tensile and compressive strengths of concrete are not equal or proportional. • The tensile strength of normal-weight concrete is approximately 10% to 15% of the compressive strength (per ACI Code Commentary). • An increase in compressive strength is accompanied by an appreciably smaller percentage increase in tensile strength. The true tensile strength of concrete is difficult to determine. • The split-cylinder test (ASTM C496) is used to determine the tensile strength of lightweight aggregate concrete. • Splitting tensile strength, fct, may be calculated by the following expression (derived from the theory of elasticity): fct = 2P/πLD where fct = splitting tensile strength of lightweight aggregate concrete (psi) P = applied load at splitting (lb) L = length of cylinder (inch) D = diameter of cylinder (inch)

11. Concrete in Tension (continued) Another common approach to measure tensile strength is to use the modulus of rupture, fr. • The modulus of rupture is the maximum tensile bending stress in a plain concrete test beam at failure (ASTM C78). • The moment that produces a tensile stress just equal to the modulus of rupture is termed the cracking moment, Mcr. • The ACI Code recommends that the modulus of rupture fr (psi) be taken as fr = 7.5 λ √fc’ where λ (Greek lower case “lambda”) = modification factor reflecting the lower tensile strength of lightweight concrete relative to normal-weight concrete λ = 1.0 for normal-weight concrete λ = 0.85 for sand-lightweight concrete λ = 0.75 for all-lightweight concrete

12. Reinforcing Steel Most concrete is reinforced in some way to resist tensile forces. Tensile reinforcement is embedded in concrete to withstand the tensile stress. • Reinforcement is in the form of steel reinforcing bars or welded wire reinforcing (often called “mesh”) composed of steel wire. • Reinforcing in the form of structural steel shapes, steel pipe, steel tubing, and high-strength steel tendons is also permitted by the ACI Code. • Other economical reinforcement includes fiber-reinforced concrete (using short fibers of steel or fiberglass). Specifications for steel reinforcement are published by ASTM (American Society for Testing and Materials). • The specifications are generally accepted for the steel used in reinforced concrete construction in the United States. • The ASTM specifications are identified in the ACI Code (Section 3.5).

13. Reinforcing Steel (continued) Steel bars used for reinforcing are generally round deformed bars. • Reinforcing bars feature patterned ribbed projections rolled onto their surfaces conforming to ASTM specifications. • Steel reinforcing bars are available in straight lengths of 60 feet. • Smaller bar sizes are available in coil stock for use in automatic bending machines. • Bars designations vary from No. 3 to No. 11, along with No. 14 and No. 18. • The designation for the No. 3 to the No. 8 bars represents the bar diameter in eighths of an inch (e.g. No. 5 bar is 5/8” in diameter). • The No. 9, No. 10, and No. 11 bars have diameters that provide areas equal to 1” square bars, 1.125” square bars, and 1.25” square bars, respectively. • The No. 14 and No. 18 bars have diameters that provide areas equal to 1.5” and 2” square bars, respectively, and are commonly available by special order.

14. Reinforcing Steel (continued) Reinforcing bars are usually made from newly manufactured steel (billet steel per ASTM A615). • Billet steel is available in grades 40, 60, and 75 (with minimum yield strengths of 40,000, 60,000, and 75,000 psi, respectively). Note the useful references and tables in Appendix A of the textbook. • Table A-1 (p. 483 of the textbook) lists the steel grades that are available for each bar size. • Table A-1 also lists the weight, diameter, and cross-sectional area for each bar size. • Table A-2 (p. 484 of the textbook) provides cross-sectional areas of multiple reinforcing bars. • Table A-3 (p. 484 of the textbook) provides the minimum required beam width based on the number of bars in one layer.

15. Reinforcing Steel (continued) The most useful physical properties of reinforcing steel for reinforced concrete design calculations are the yield stress (fy) and the modulus of elasticity (Es). • Table A-1 (p. 483 of the textbook) lists the available steel grades and the associated yield stress. • The modulus of elasticity is taken as 29,000,000 psi (per ACI Code, Section 8.5.2). Corrosion of reinforcing steel leads to cracking and spalling of the concrete in which it is embedded. • Quality concrete, with adequate cover, provides good protection against corrosion. • Protective coatings may be used to minimize the corrosion of the reinforcing steel. • Non-metallic materials: epoxy coated, complying with ASTM A775 or ASTM A934 (per ACI Code). • Metallic materials: zinc (galvanizing), complying with ASTM A767 (per ACI Code).

16. Reinforcing Steel (continued) Welded wire reinforcing (WWR) (mesh) is another type of reinforcing. • Welded wire reinforcing consists of cold-drawn wire in square or rectangular patterns, welded at all intersections. • Welded wire reinforcing may be supplied in rolls or sheets depending on the wire size. • Wire sizes are designated by the symbol WWR, followed by the spacing of the longitudinal wires, then the spacing of the transverse wires, by the size of the longitudinal wire, and by the size of the transverse wires. For example: WWR6 x 12 – W16 x W8 This is the designation for a plain WWR with 6” longitudinal spacing, 12” transverse spacing, and a cross sectional area equal to 0.16 in2 for the longitudinal wires and 0.08 in2 for the transverse wires.

17. Beams: Mechanics of Bending Review The expression for the maximum bending stress in a beam (i.e. the flexure formula) is fb = Mc/I where • fb = the calculated bending stress at the outer fiber of the cross section • M = the applied maximum moment • c = the distance from the neutral axis to the outside tension or compression fiber of the beam • I = the moment of inertia of the cross section about the neutral axis The flexure formula represents the relationship between bending stress, bending moment, and the geometric properties of the beam cross section. Assumptions used in developing the flexure formula include: • Beams (such as steel and wood) are composed of homogeneous material. • Beams (such as steel and wood) exhibit elastic behavior up to a certain limit. • The stress distribution developed at any cross section is linear, varying from zero at the neutral axis to a maximum at the outer fibers.

18. Beams: Mechanics of Bending Review (continued) The maximum moment that may be applied to the beam cross section, called the resisting moment (MR), may be found by rearranging the terms of the flexure formula. MR = FbI/c where Fb = the allowable bending stress The use of the flexure formula presents some complications when applied to concrete. • Reinforced concrete is not a homogeneous material. • Concrete does not behave elastically over its full range of strength (stress).

19. Beams: Mechanics of Bending Review (continued) A different approach, called the internal couple method, is used for the design and analysis of concrete beams. • The internal couple (moment) is composed of a compressive force “C” above the neutral axis and a parallel tensile force “T” below the neutral axis forming a couple. • A single span, simply supported beam with a positive bending moment is assumed. • Forces acting at the cross section must be equal and opposite in direction to satisfy equilibrium requirements (i.e. ∑H = 0), thus C = T • The couple formed by C and T must be equal and opposite to the bending moment at the same location produced by the external loads.

20. Beams: Mechanics of Bending Review (continued) The first example that follows demonstrates the concept of cracking moment Mcr and the modulus of rupture (fr) for a plain (non-reinforced) concrete beam. • The cracking moment Mcr causes the maximum tensile stress just to reach the modulus of rupture (fr). • The cracking moment Mcr causes the cross sectional area to be on the verge of cracking. • The first example compares the modulus of rupture developed from the cracking moment with the ACI-recommended modulus of rupture.

21. Beams: Mechanics of Bending Review (continued) The second example of a plain (non-reinforced) concrete beam demonstrates the internal couple method. • Non-reinforced concrete beams are considered homogeneous and elastic (valid for small tensile stresses). • The analysis for bending stresses in the “uncracked” beam is based on the properties of the gross cross sectional area using the elastic-based flexure formula. • The use of the flexure formula is valid as long as the maximum tensile stress does not exceed the modulus of rupture (fr). • The second example compares the internal couple method and the flexure formula approach.