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FIGURE 1410 Assumptions in ultimate-strength design
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Rectangular Beams with Tensile Reinforcing Only This type of beam includes slabs [for which b 12 in (3048 mm) when the moment and shear are expressed per foot of width] The ultimate resisting moment is Mu where As a b d c y Mu p pb q k [bd 2 c q(1 059q)] As y d a 2 (14104)
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area of tensile reinforcement, in2 (mm2) depth of rectangular stress block As y / 085 cb, in (mm) width of compressive face of exural member, in (mm) distance from extreme compression ber to centroid of tension reinforcement, in (mm) compressive strength of concrete, lb / in2 (kPa) yield strength of reinforcement, lb / in2 (kPa) ultimate resisting moment, in lb (N m) As / bd, 075pb reinforcement ratio producing balanced conditions* As y / bd c p y / c capacity reduction factor, in concrete code concrete stress intensity factor 085 when c 4000 lb / in2 (276 MPa)
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T Beams with Tensile Reinforcing Only For preliminary designs, the formulas given above for rectangular beams with tensile reinforcing only can be used, since the neutral axis is usually in or near the ange The area of tensile reinforcing will usually be critical
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* Balanced conditions occur when the tension reinforcement is at its yield strength y and the concrete in compression is at its assumed ultimate strain of 0003; balanced conditions exist when pb 085k c y 87,000 87,000 y (14104a)
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CHAPTER FOURTEEN
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Beams with Tensile and Compressive Reinforcing Beams with compression reinforcing are generally used when the size of the beam is limited, or where p 075pb The ultimate resisting moment is then Mu where As Asc a b d d y Mu p p pb (As Asc) y d a 2 Asc y(d d ) (14105)
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area of tensile reinforcement, in2 (mm2) area of compressive reinforcement, in2 (mm2) depth of equivalent rectangular stress block ( As Asc) y / 085 c b, in (mm) width of compressive face of exural member, in (mm) distance from extreme compression ber to centroid of tension reinforcement, in (mm) distance from extreme compression ber to centroid of compression steel, in (mm) yield strength of reinforcement, lb / in2 (kPa) ultimate resisting moment, in lb (N m) As / bd Asc / bd ( p p 075pb) reinforcement ratio producing balanced conditions, Eq (14104a) capacity reduction factor
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Shear and Diagonal Tension in Beams The ultimate shearing unit stress, as a measure of diagonal tension, in a reinforced-concrete beam is
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Vu bd
(14106)
where
Vu b d
ultimate shearing unit stress, lb / in2 (kPa) ultimate total shear, lb (N) width of beam (for T beam use width of stem), in (mm) effective depth of beam, in (mm)
For design, the maximum shear is considered to occur at a distance d from the face of the support The shear stress carried by concrete c should not exceed c Wherever the ultimate shear stress c , exceeds the shear stress c , web 2 reinforcement is mandatory Such reinforcement will usually consist of stirrups The cross-sectional area required for a stirrup placed perpendicular to the longitudinal reinforcement is Au where Au d y s Vu Vus y d (14107)
total area of web reinforcement in tension within a distance measured in a direction parallel to the longitudinal reinforcements, in2 (mm2) effective depth, in (mm) yield strength of reinforcement, lb / in2 (kPa) spacing of stirrups, in (mm) total ultimate shear, lb (N) capacity reduction factor, in design codes
The shear stress u should not exceed 10 c and y should not exceed 60,000 lb / in2 (4136 MPa) Stirrups should be anchored at both ends to be considered
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