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1、Lecture 5 - Flexure,June 11, 2003 CVEN 444,Lecture Goals,Rectangular Beams Safety factors Loading and Resistance Balanced Beams,Flexural Stress,The compressive zone is modeled with a equivalent stress block.,Flexural Stress,The equivalent rectangular concrete stress distribution has what is known as
2、 a b1 coefficient is proportion of average stress distribution covers.,Flexural Stress,Requirements for analysis of reinforced concrete beams,1 Stress-Strain Compatibility Stress at a point in member must correspond to strain at a point.,2 Equilibrium Internal forces balances with external forces,Fl
3、exural Stress,Example of rectangular reinforced concrete beam.,(1) Setup equilibrium.,Flexural Stress,Example of rectangular reinforced concrete beam.,(2) Find flexural capacity.,Flexural Stress,Example of rectangular reinforced concrete beam.,(2) Find flexural capacity.,Flexural Stress,Example of r
4、ectangular reinforced concrete beam.,(3) Need to confirm es ey,Flexural Stress Rectangular Example,Example of rectangular reinforced concrete beam.,Given a rectangular beam fc = 4000 psi fy = 60 ksi (4 #7 bars) b = 12 in. d = 15.5 in. h= 18 in. Find the neutral axis. Find the moment capacity of the
5、beam.,Flexural Stress Rectangular Example,Determine the area of steel, #7 bar has 0.6 in2. The b value is b1 = 0.85 because the concrete has a fc =4000 psi.,Flexural Stress Rectangular Example,From equilibrium (assume the steel has yielded),The neutral axis is,Flexural Stress Rectangular Example,Che
6、ck to see whether or not the steel has yielded.,Check the strain in the steel,Steel yielded!,Flexural Stress Rectangular Example,Compute moment capacity of the beam.,Flexural Stress Non-Rectangular Example,For the given beam with concrete rated at fc = 6 ksi and the steel is rated at fs = 60 ksi. d
7、= 12.5 in.,For a non-rectangular beam,(a) Determine the area of the steel for a balanced system for shown area of concrete. (b) Determine the moment capacity of the beam. Mn (c) Determine the NA.,Flexural Stress Non-Rectangular Example,The area of the concrete section is,For a non-rectangular beam,T
8、he force due to concrete forces.,Flexural Stress Non-Rectangular Example,Using equilibrium, the area of the steel can be found,Flexural Stress Non-Rectangular Example,Find the center of the area of concrete area,Flexural Stress Non-Rectangular Example,The moment capacity of the beam is,Flexural Stre
9、ss Non-Rectangular Example,Compute the b1 value,Flexural Stress Non-Rectangular Example,Find the neutral axis,Safety Provisions,Structures and structural members must always be designed to carry some reserve load above what is expected under normal use.,Safety Provisions,There are three main reasons
10、 why some sort of safety factor are necessary in structural design. 1 Consequences of failure. 2 Variability in loading. 3 Variability in resistance.,Consequences of Failure,Potential loss of life. Cost of clearing the debris and replacement of the structure and its contents. Cost to society. Type o
11、f failure warning of failure, existence of alternative load paths.,A number of subjective factors must be considered in determining an acceptable level of safety.,Variability in Loading,Frequency distribution of sustained component of live loads in offices.,Variability in Resistance,Variability of t
12、he strengths of concrete and reinforcement. Differences between the as-built dimensions and those found in structural drawings. Effects of simplification made in the derivation of the members resistance.,Variability in Resistance,Comparison of measured and computed failure moments based on all data
13、for reinforced concrete beams with fc 2000 psi.,Margin of Safety,The distributions of the resistance and the loading are used to get a probability of failure of the structure.,Margin of Safety,The term Y = R - S is called the safety margin. The probability of failure is defined as: and the safety in
14、dex is,Loading,SPECIFICATIONS Cities in the U.S. generally base their building code on one of the three model codes: Uniform Building Code Basic Building Code (BOCA) Standard Building Code,Loading,These codes have been consolidated in the 2000 International Building Code. Loadings in these codes are
15、 mainly based on ASCE Minimum Design Loads for Buildings and Other Structures has been updated to ASCE 7-02.,Loading,The loading variations are taken into consideration by using a series of “load factors” to determine the ultimate load, U.,Loading,The equations come from ACI code 9.2 on loading (4.6
16、 in your book), D Dead LoadW Wind Load L Live LoadLr Roof Load F Fluid Pressure R Rain Load E Earthquake LoadT Temperature Load S Snow LoadH Soil Load,Loading,The most general equation for the ultimate load, U (Mu) that you will see is going to be:,Resistance,The load factors will generate the ultimate load, which is used in the design and analysis of the structural member. Mu Ultimate Moment Mn Nominal Moment f Strength Reduction Factor,Resistance,The strength reduction factor, f, varies from
