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Context

1. Module 2: Flexure and Analysis of Beams Prepared by: Engr. Adrian D. Viloria, MSc.

2. TOPIC: Flexure and Shear Design of Beams Introduction

3. Flexure and Shear Design of Beams Flexure - is associated with lateral deformation of a member under a transversely applied load. Flexural strength - is a measure of the tensile strength of concrete beams or slabs. - identifies the amount of stress and force an unreinforced concrete slab, beam or other structure can withstand such that it resists any bending failures.

4. Flexure and Shear Design of Beams Analysis VS. Design 1. Analysis . Given a cross section, concrete strength, reinforcement size and location, and yield strength, compute the resistance or strength. In analysis there should be one unique answer. 2. Design . Given a factored design moment, normally designated as Mu , select a suitable cross section, including dimensions, concrete strength, reinforcement, and so on. In design there are many possible solutions. Mu? Beam dimension? Area of steel?

5. Flexure and Shear Design of Beams Required Strength and Design Strength The basic strength requirement from the ACI Code [4-2] is: Or for flexure: where  M u  is the moment due to the factored loads, which commonly is referred to as the  factored design moment . The term  M n  refers to the  nominal moment strength  of a cross section, computed from the nominal dimensions and specified material strengths. 

6. TOPIC: Flexure of Beams - Analysis

7. Flexure of Beams - Analysis Objective for analysis Identify the cross-section area allotted for compression and “equate” it to the steel area allotted for tension to identify the capacity of the beam against flexure. Compression Tension

8. Flexure of Beams - Analysis Working Stress Design (WSD) - is a method used for the reinforced concrete design where concrete is assumed as elastic, steel and concrete act together elastically where the relation ship between loads and stresses is linear .

9. Flexure of Beams - Analysis Working Stress Design (WSD) Effective depth, “d” Concrete cover, “Cc” Sometimes with respect to the centroid of the rebar Neutral Axis, “N.A.” Compression block f’c = compressive strength of concrete after 28 days of curing (usually as input), MPa fc = compressive stress occurring in the concrete structure (usually as output), MPa fy = yield strength of steel, MPa fs = yield strength occurring in the steel reinforcement, MPa n = modular ratio, Es / Ec Area steel, A s

10. Flexure of Beams - Analysis Working Stress Design (WSD) Steps you can follow: Determine the Maximum Bending Moment occurring on the beam, Mu Given, or solved through shear & moment diagram

11. Flexure of Beams - Analysis Working Stress Design (WSD) Steps you can follow: Note: Beam cross-section can be analyzed in negative moment (at fixed supports), or positive moment ( midspan )

12. Flexure of Beams - Analysis Working Stress Design (WSD) Steps you can follow: Determine the modular ratio, n Determine the location of the neutral axis M above = M below b(x)(x/2) = nA s (d – x) Hence, x = _____ Effective depth, “d” x It’s like taking a moment at the N.A. x

13. Flexure of Beams - Analysis Working Stress Design (WSD) Steps you can follow: Determine the 2 nd moment of area about the neutral axis, I NA   x Effective depth, “d” The second moment of area is a measure of the ' efficiency ' of a cross-sectional shape to resist bending caused by loading

14. Flexure of Beams - Analysis Working Stress Design (WSD) Steps you can follow: Determine the stress on concrete, fc Determine the stress on steel, fs  

15. Flexure of Beams - Analysis Working Stress Design (WSD) Steps you can follow: Check for allowable values: fc allowable = 0.45 fc’, in MPa fs allowable = 160 MPa

16. Flexure of Beams - Analysis Working Stress Design (WSD) Example: A reinforced concrete beam experiences a flexural load of 100 kNm at the midspan , and has a base of 300 mm and an effective depth of 500 mm, with a concrete cover of 75 mm. The RC beam has reinforcements of 3 – 25 mm diameter bars. The concrete has a compressive strength of 21 MPa . Determine if the cross section, and steel area, are sufficient or not. Steps you can follow: Determine Mu 100 kNm Determine the modular ratio, n 9.28  9 Determine the location of the neutral axis 170.60 mm Determine the 2 nd moment of area about the neutral axis, I NA 1.94 x 10^9 mm^4 Determine the stress on concrete, fc 8.8185 MPa Determine the stress on steel, fs 153.24 MPa Check for allowable values: Ad & Ad

17. Flexure of Beams - Analysis Ultimate Stress Design (USD) - is used extensively and almost exclusively in many countries for structural design practice . - utilizes reserves of strength resulting from a more efficient distribution of stresses allowed by plastic strains in the concrete and reinforcing steel. x Effective depth, “d” c Equivalent stress block, a WSD USD

18. Flexure of Beams - Analysis Ultimate Stress Design (USD) Whitney’s Constant, β β = 0.85 for fc’ = 17 to 28 MPa = 0.85 – (fc - 28) for 28 < fc’ < 55 Mpa = 0.65 for fc’ ≥ 55 MPa Depth of neutral axis, c c = Equivalent stress block, a a = β c Strength Reduction Factor, ϕ for bending, ϕ is equal to 0.9 (ACI, 0.005 mm/mm) Mu = ϕ Mn Mu = Ultimate moment capacity, kNm Mn = Nominal moment capacity, kNm = usually through empirical formulas   0.9 0.65 Fy /Es 0.004 NSCP 0.005 ACI Strain ϕ ?

19. Flexure of Beams - Analysis Ultimate Stress Design (USD) Nominal Moment Capacity, in which,  

20. Flexure of Beams - Analysis Ultimate Stress Design (USD) Steps you can follow: Use the “working equation” to determine the location of N.A., c , and the depth of compression block, a . Compression = Tension Therefore, a = ____ & c = ______ Note: c = a/ β   a c Effective depth, “d” COMPRESSION TENSION N.A.

21. Flexure of Beams - Analysis Ultimate Stress Design (USD) Steps you can follow: Check if the steel yields . Note: if the solved is ≥ , “ STEEL YIELDS ”, hence, ok! Proceed to step 3a. if the solved is < , “ STEEL DOES NOT YIELD ”, hence, resolving is needed! Proceed to step 3b.   fy  “capacity” ≥   <   fs  “actual”

22. Flexure of Beams - Analysis Ultimate Stress Design (USD) Steps you can follow: a.) Solve for . or Note: It’s like taking a moment about the centroid of the compression block or centroid of the rebar   a c Effective depth, “d” b COMPRESSION TENSION N.A.

23. Flexure of Beams - Analysis Ultimate Stress Design (USD) Steps you can follow: b.) Resolve for the new c or a . Same with step 1, use the “working equation”, but the difference, change the following: Compression = Tension Therefore, new c = ______ & new a = ____   a c Effective depth, “d” COMPRESSION TENSION N.A.

24. Flexure of Beams - Analysis Ultimate Stress Design (USD) Steps you can follow: c.) Re-check if the steel does not yield . Note: This time, the solved must only be < , or “ STEEL DOES NOT YIELD ”, If not, the obtained new value of c is incorrect..   fy  “capacity” <   fs  “actual”

25. Flexure of Beams - Analysis Ultimate Stress Design (USD) Steps you can follow: d.) Solve for . or Note: It’s like taking a moment about the centroid of the compression block or centroid of the rebar   a new c Effective depth, “d” b COMPRESSION TENSION N.A.

26. Flexure of Beams - Analysis Ultimate Stress Design (USD) Steps you can follow: Solve for .   Reduction factors “limit” the capacity of the beam; act as safety factors so that the beam won’t “operate” at its maximum

27. Flexure of Beams - Analysis Ultimate Stress Design (USD) Example: Determine the ultimate moment capacity of the beam shown using f’c = 21 MPa and fy = 414 MPa . Use NSCP 2001 (for the mean time, use β = 0.85) Use the “working equation” to determine the location of N.A., c, and the depth of compression block, a. a = 113.85 mm Check if the steel yields. fs = 1639.81 MPa Steel Yields Steel Does Not Yield 3-a.) Solve for 𝑀 N 3-b.) Resolve for the new c or a. Mn = 270.13 kNm 3-c.) Re-check if the steel does not yield 3-d.) Solve for 𝑀 N 4.) Solve for Mu. 243.11 kNm 500 mm 300 mm 3 – 25 mm

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29. Flexure of Beams - Analysis Required Strength and Design Strength USD(Ultimate Stress Design)  Method WSD(Working Stress Design) Method 1 Ultimate Stress Design(USD) Method is define  as the method of design structural members by considering the ultimate Strength of steel and concrete is called USD Method. Working Stress Design method is define  as the method of design structural members by considering the allowable Strength of steel and concrete is called WSD Method. 2 It’s primarily based on the strength concept of Concrete. It’s based on the linear theory or elastic theory. 3 Factor of Safety is considered in this design.  Factor of Safety is not considered in this design. 4 It’s Consider to Design  Critical Combination of load It’s Consider to Design  carrying load 5 It’s designing for elastic behavior of materials. It’s designing to plastic behavior of materials 6 Materials strength to be used for member design Modular Ratio used for member design 7 Stability of Structure  is more than WSD Stability of Structure is less than USD 8 It’s a modern design method. It’s a pretty old design method. 9 This method is used for mostly the biggest structures and developed nations. This type of method is used for mostly small structures and undeveloped nations. 10 Economical Design method. Less economical Design method.