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Lecture chapter 3 copyu

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Lecture chapter 3 copyu

1. 1. By: Edgar B. Manubag, CE, PhD
2. 2. Reinforced concrete floors usually consist ofslabs and beams, which are placed or pouredmonolithically. In this effect, the beam willhave an extra width at the top- (which isusually under compression) called flanges,and the resulting section is called a T -beam.The beam may also be L-shaped if it is locatedat the end of a slab.
3. 3. The analysis of T-beams is similar to rectangularbeams, but in unlike rectangular beams wherein wealways check for maximum steel ratio (ρmax), thislimiting ratio is very seldom reached in T -beamsbecause the compression side of the beam is solarge which makes the neutral axis so far away fromAs, and hence one would almost never use anamount of steel greater than ρmax · Thus in T -beamswhere the flange is in compression, it is very oftenthat the steel will yield.
4. 4. The compression block of a T -beam can fall eitherwithin the flange only or partly in the web. If it fallswithin the flange as shown in Figure 3.1 (a), therectangular beam formulas (in Chapter 2) apply sincethe concrete below neutral axis is assumed to becracked and its shape has no effect on the flexurecalculations (other than weight). If however it coversthe web as shown in Figure 3,1 (b), the compressionconcrete no longer consist of a single rectangle andthus the rectangular beam formulas do not apply.
5. 5. In analysis of beams (whatever shape it is), once the value ofc is known, the actual stress in tension steel can becomputed using Eq. 3-1. From the strain diagram shown:
6. 6. THE c/d RATIOOne can actually detect (without furthercomputation) when steel will yield once the valueof c is known. Note that the strain in concrete istaken as 0.003 and the strain in steel is /s/ Es. ForFy = 415 MPa, the maximum strain Es= 415/200,000= 0.0021, and for Fy = 276 MPa, Es = 0.0014.
7. 7. THE c/d RATIOAs shown in Figure 3.2 (a), the grade 415 steel will not yield ifc/d is greater than 0.59 and will yield if c/d is less than 0.59.The grade 276 steel as shown in Figure 3.2 (b) will yield if c/d isless than 0.7. Since the maximum steel strength usually usedin construction is the grade 415 (fy = 415 MPa), we cantherefore conclude that if c/d is less than 0.59, the tension steelwill yield.
8. 8. THE c/d RATIOIn T-beams where the flange is incompression, the c/d ratio is usually thatshown in Figure 3.2 (c), which easily lead us toa conclusion that the steel yields.
9. 9. If a is less than the slab thickness t, the balanced steelratio is computed using the Eg. 2 - 11. However, if a isgreater than t, the following-formula will be used .
10. 10. Substituting c for a:
11. 11. Section 5.10.5.1 of NSCP provides that the minimum steel ratiobe 1.4/fy. It also states that in T-beams where the web is intension, the ratio p shall be computed for this purpose usingwidth of web.
12. 12. 1. In T-beam construction, the flange and web shall be built integrally orotherwise effectively bonded together.2. The width of slab effective as a T -beam shall not exceed 1/4 of the.span of the beam, and the effective overhanging flange on each side ofthe web shall not exceed:(a) 8 times the slab thickness, and(b) 1/2 the clear distance to the next web.3. For beams with slab on one side only, the effective overhangingflange shall not exceed:(a) 1/12 the span length of the beam,(b) 6 times the slab thickness, and(c) 1/2 the clear distance to the next web.
13. 13. Design Problem, a< t
14. 14. PROBLEM 3.2A reinforced concrete T -beam with bf=813 mm, d = 300 mm, bw = 200 mm, t=102 mm, fc = 20.7 MPa, and fy = 414 MPais to be designed to carry a factoredmoment of 221 kN-m. Determine therequired steel area As.