chapter 3 reinforced concrete beams(混凝土结构设计原理英文课件)
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• For purposes of analysis and design, the ACI Code, section 8.10, has established limits on the effective flange width as follows: • 1. the effective flange width must not exceed one-fourth of the span length of the beam, and the effective overhanging flange width on each side of the web must not exceed eight times the thickness of the slab nor onehalf of the clear distance to the next beam. In other words, the effective flange width must not exceed: • A. one – fourth of the span length • B. b 16h
As ,min 3 f c' fy bw d 200 bw d fy
• Note that for T-beams, bw represents the width of the web. Also note that the first expression controls only is • f c' 4440psi . For negative moment (flange in tension) ' 6 f c • As ,min the smaller of bw d or 3 f bd
• Because of the T-shaped compressive area. A variation of the rectangular beam approach for 0.75 b may be observed in section 3-3. • The procedure for determining the minimum steel for a T-beam is the same as for rectangular beam when the Tbeam flange is in compression. Where tensile reinforcement is required by analysis, the steel area As shall not be less than that given by
• One such system, called a beam(横梁) and girder （纵梁） system, is composed of a slab on supporting reinforced concrete beams and girders. The beam and girder framework is, in turn, supported by columns. In such a system, the beams and girders are commonly placed monolithically with the slab. Systems other than the monolithic system do exist, and these may make use of some precast and some cast – in – place concrete. These are generally of a proprietary nature. The typical monolithic system is shown in figure 3-1. the beams are commonly spaced so that they intersect the girders at the midpoint, third points, or quarter points, as shown in figure 3-2.
w f
• C. center- to – center spacing of beams • 2. for beams having a flange on one side only, the effective overhanging flange width must not exceed one – twelfth of the span length of the beam nor six times the slab thickness, nor one-half of the clear distance to the next beam. • 3. for isolated beams in which the T-shapes is used only for the purpose of providing additional compressive area, the flange thickness must not be less than one half of the width of the wed, and total flange width must not be more than fore times the web width. • The ductility requirement for T-beams are similar to those for rectangular beams. The maximum steel ratio • Shall not exceed 0.75 b . This 0.75 b , however, is not the same value as that tabulated for rectangular beams
• In the analysis and design of such floor and roof systems, it is common practice to assume that the monolithically placed slab and supporting beam interact as a unit in resisting positive bending moment. As shown in figure 3-3, the slab becomes the compression flange, and the supporting beam becomes the web or stem . The interacting flange and stem produce the cross section having the typical T-shape from which the Tbeam gets its name. it should be noted that the slab, which comprises the T-beam flange, must itself be designed to span across the supporting beams. Therefore, the slab behaves as a bending member acting in two directions. It should also be noted that should the T-beam cross section be subjected to negative bending moment the slab as the top of the stem
• Will be in tension while the bottom of the stem is in compression. It will be seen that this situation will occur at interior supports of continuous beams, which are discussed later. • To simplify the complex two-way behavior of the flange, the ACI Code, for design and analysis purposes, has established criteria whereby the flange , when acting together with the web, will have a limited width that may be considered affective in resisting applied moment. This effective flange width for symmetrical shapes will always be equal to or less than the beam spacing (see figure 33)
fy
' c
fy
• Because of the relatively large compression area available in the flange of the T-beam, the moment strength is usually limited by the yielding of the tensile steel. Therefore, it is both customary and safe to assume that the tensile steel will yield before the concrete reaches its ultimate strain and crushes. The total tensile force N T , at the ultimate condition, may then be found by • NT As f y • To proceed with the analysis, the shape of the compressive stress block must be defined. As in our previous analyses, the total compressive force N must be equal to the tensile force N T . The shape of the stress block must be compatible with the area in the compressive. Two conditions may exist: the stress block may be completely within the flange, or it may cover the
3-1
T – BEAMS: INTRODUCTION
T – BEAMS: INTRODUCTION
• Floors and roofs in reinforced concrete buildings may be composed of slabs that are supported so that loads are carried to columns and then to the building foundation. As previously discussed, these are termed flat slabs or flat plates. The span of such a slab cannot become vary large before its own dead weight causes it to become uneconomical. Many types of systems have been devised to allow greater spans without the problem of excessive weight.
• For purposes of analysis and design, the ACI Code, section 8.10, has established limits on the effective flange width as follows: • 1. the effective flange width must not exceed one-fourth of the span length of the beam, and the effective overhanging flange width on each side of the web must not exceed eight times the thickness of the slab nor onehalf of the clear distance to the next beam. In other words, the effective flange width must not exceed: • A. one – fourth of the span length • B. b 16h
As ,min 3 f c' fy bw d 200 bw d fy
• Note that for T-beams, bw represents the width of the web. Also note that the first expression controls only is • f c' 4440psi . For negative moment (flange in tension) ' 6 f c • As ,min the smaller of bw d or 3 f bd
• Because of the T-shaped compressive area. A variation of the rectangular beam approach for 0.75 b may be observed in section 3-3. • The procedure for determining the minimum steel for a T-beam is the same as for rectangular beam when the Tbeam flange is in compression. Where tensile reinforcement is required by analysis, the steel area As shall not be less than that given by
• One such system, called a beam(横梁) and girder （纵梁） system, is composed of a slab on supporting reinforced concrete beams and girders. The beam and girder framework is, in turn, supported by columns. In such a system, the beams and girders are commonly placed monolithically with the slab. Systems other than the monolithic system do exist, and these may make use of some precast and some cast – in – place concrete. These are generally of a proprietary nature. The typical monolithic system is shown in figure 3-1. the beams are commonly spaced so that they intersect the girders at the midpoint, third points, or quarter points, as shown in figure 3-2.
w f
• C. center- to – center spacing of beams • 2. for beams having a flange on one side only, the effective overhanging flange width must not exceed one – twelfth of the span length of the beam nor six times the slab thickness, nor one-half of the clear distance to the next beam. • 3. for isolated beams in which the T-shapes is used only for the purpose of providing additional compressive area, the flange thickness must not be less than one half of the width of the wed, and total flange width must not be more than fore times the web width. • The ductility requirement for T-beams are similar to those for rectangular beams. The maximum steel ratio • Shall not exceed 0.75 b . This 0.75 b , however, is not the same value as that tabulated for rectangular beams
• In the analysis and design of such floor and roof systems, it is common practice to assume that the monolithically placed slab and supporting beam interact as a unit in resisting positive bending moment. As shown in figure 3-3, the slab becomes the compression flange, and the supporting beam becomes the web or stem . The interacting flange and stem produce the cross section having the typical T-shape from which the Tbeam gets its name. it should be noted that the slab, which comprises the T-beam flange, must itself be designed to span across the supporting beams. Therefore, the slab behaves as a bending member acting in two directions. It should also be noted that should the T-beam cross section be subjected to negative bending moment the slab as the top of the stem
• Will be in tension while the bottom of the stem is in compression. It will be seen that this situation will occur at interior supports of continuous beams, which are discussed later. • To simplify the complex two-way behavior of the flange, the ACI Code, for design and analysis purposes, has established criteria whereby the flange , when acting together with the web, will have a limited width that may be considered affective in resisting applied moment. This effective flange width for symmetrical shapes will always be equal to or less than the beam spacing (see figure 33)
fy
' c
fy
• Because of the relatively large compression area available in the flange of the T-beam, the moment strength is usually limited by the yielding of the tensile steel. Therefore, it is both customary and safe to assume that the tensile steel will yield before the concrete reaches its ultimate strain and crushes. The total tensile force N T , at the ultimate condition, may then be found by • NT As f y • To proceed with the analysis, the shape of the compressive stress block must be defined. As in our previous analyses, the total compressive force N must be equal to the tensile force N T . The shape of the stress block must be compatible with the area in the compressive. Two conditions may exist: the stress block may be completely within the flange, or it may cover the
3-1
T – BEAMS: INTRODUCTION
T – BEAMS: INTRODUCTION
• Floors and roofs in reinforced concrete buildings may be composed of slabs that are supported so that loads are carried to columns and then to the building foundation. As previously discussed, these are termed flat slabs or flat plates. The span of such a slab cannot become vary large before its own dead weight causes it to become uneconomical. Many types of systems have been devised to allow greater spans without the problem of excessive weight.