Lec03 Flexural Behavior of RC Beams (Reinforced Concrete Design I & Prof. Abdelhamid Charif)
11 likes3,178 views
The document discusses the behavior and analysis of reinforced concrete beams. It describes three stages that beams undergo as loading increases: 1) the uncracked concrete stage, 2) the cracked-elastic stage, and 3) the ultimate strength stage. It also discusses assumptions made in flexural theory, stress-strain curves for concrete and steel, and methods for calculating stresses in uncracked and cracked beams using the transformed area method. Key points covered include cracking moment, modular ratio, and the three-step transformed area method for cracked sections.
Lec03 Flexural Behavior of RC Beams (Reinforced Concrete Design I & Prof. Abdelhamid Charif)
- 1. 24/2/2013 CE 370 : Prof. Abdelhamid Charif 1 CE370 REINFORCED CONCRETE-I Prof. Abdelhamid CHARIF Flexural Behavior and Analysis of RC Beams Behavior stages in a RC beam • A small transverse load placed on a simply supported RC beam is gradually increased until beam failure. • This loading causes positive bending moment (tension, and steel in bottom) and shear force • Perfect bond is assumed between steel bars and concrete • The beam will go through three distinct stages before collapse occurs. These are: • The uncracked concrete stage • The concrete cracked – elastic strength stage • The ultimate – strength stage. 2
- 2. 24/2/2013 CE 370 : Prof. Abdelhamid Charif 2 3 Simply supported beam under uniform load 4 Basic Assumptions in Flexure theory of RC Members 1. Plane sections before bending remain plane after bending. This assumption has been experimentally verified for all normal beams except deep beams (length to thickness ratio less than 3). 2. The strain in reinforcement equals to the strain in concrete at the same level (perfect bond, no slip). 1 & 2 imply that strain in reinforcement and concrete is directly proportional to the distance from N.A. 3. Tensile strength of concrete is neglected in flexural strength calculations (after cracking). 4. Stresses in concrete and reinforcement are calculated from strains using material stress-strain curves. Other assumptions will be made at ultimate state
- 3. 24/2/2013 CE 370 : Prof. Abdelhamid Charif 3 5 Recall Concrete Stress-Strain Curve Compression: For a stress less or equal to 50% of fc’ , the curve is almost linear. Tension: Strength very small and curve assumed linear. 6 Recall Steel Stress-Strain Curve (Large plastic deformations after yielding) ysy ysss s f E f when when gradesallfor 200000MPaEs
- 4. 24/2/2013 CE 370 : Prof. Abdelhamid Charif 4 Uncracked Concrete Stage • At small loads when the tensile stresses are less than the modulus of rupture, the entire cross section of the beam resists bending. • Standard bending theory can be used. • Compression develops on top side and tension on bottom (for positive moments. 7 • Neutral axis passes through centroid • Moment of inertia of gross section is used 8 Uncracked Concrete Stage Strains and stresses
- 5. 24/2/2013 CE 370 : Prof. Abdelhamid Charif 5 Concrete-Cracked – Elastic Stage • As load is increased, the modulus of rupture of the concrete is exceeded, and cracks begin to develop in bottom of the beam. • Cracking Moment: The moment at which tensile stress in the bottom of the beam equals the modulus of rupture (i.e. when cracks begin to form) is referred to as the cracking moment, Mcr • As load is further increased, cracks quickly spread closer to the neutral axis, which then begins to move upward. • Note: The cracks occur at all places along the beam where the actual moment is greater than the cracking moment. 9 Concrete-Cracked – Elastic Stage (contd.) • As concrete in the cracked zone cannot resist tensile stresses – the steel must do it. Perfect bond is still maintained. • This stage will continue as long as the compression stress in the top fibers is less than about one-half of the concrete compressive strength fc’ and as long as steel stress is less than its yield stress • In this stage the compressive stresses vary linearly with the distance from the neutral axis (straight line). 10 ratioModular with :pointsameat thestrainsEqual n E E nnff E E ff E f E f c s cs c s cs s s c c sc
- 6. 24/2/2013 CE 370 : Prof. Abdelhamid Charif 6 • Service or Working Loads: Loads that actually occur when a structure is in normal service use. • Under working loads, moments develop which are considerably larger than the cracking moments. Obviously the tensile side of the beam will be cracked. • Under working loads, concrete compression stress is less than 50% of its strength and linear stress-strain variation is assumed. • Service loads and serviceability limit state correspond thus to the cracked elastic stage • To compute concrete and steel stresses in this range, the transformed-area method is used. 11 Cracked – Elastic Stage and Service Loads Beam Failure- Ultimate Strength Stage • As load is increased further, compressive stresses are greater than 0.5fc’, cracks and neutral axis move further upward . • Concrete compressive stresses begin to change appreciably from linear to curved, and reinforcing bars yield. 12
- 7. 24/2/2013 CE 370 : Prof. Abdelhamid Charif 7 Moment Curvature Diagram • From the measured compressive strain in concrete and tensile strain in steel, the relationship between the bending moment and the curvature can be tracked up to failure. 13 dy y sc ntaanglessmallVery NAfromdistancesomeatfiberbeamainstrain curvatureorsectionbeamtheofchangeAngle y s c d • Bending moments cause curvatures (rotations). • Curvature more objective than deflection as bending deformation. 14 Typical Moment Curvature Diagram for a Reinforced Concrete Beam There are three different stages in the curve : OC – CY - YF
- 8. 24/2/2013 CE 370 : Prof. Abdelhamid Charif 8 15 Moment Curvature Diagram 1. First stage is for moments less than the cracking moment Mcr . The entire beam cross section is available to resist bending. The relationship is quasi-linear. 2. When the moment is increased beyond Mcr, the slope of the curve decreases as the cracked beam is not as stiff as in the initial stage. The diagram will still follow a straight line from Mcr to the point where steel is stressed to its yield point. 15 3. After steel yields, the beam has very little additional moment capacity, and only a small additional load is required to substantially increase rotations and deflections. The slope of the diagram is now almost flat. Bending stress in an Uncracked Beam • Steel area is small (usually 2% or less), and its effect on beam properties is negligible as long as the beam is uncracked • The bending stress in such a beam can be obtained based on the gross properties of the beam’s cross section. • The neutral axis coincides with the centroid • The stress in the concrete at a distance y from the neutral axis of the cross section can be determined from the following flexure formula: 16 section.crossgrosstheofinertiaofMoment sectiontheofmomentBending g cr g I MM I My f
- 9. 24/2/2013 CE 370 : Prof. Abdelhamid Charif 9 Cracking Moment 17 section.crosstheofinertiaofmomentGross momentBending g g I M I My f t gr cr g tcr r cr t y If M I yM f fff yy :Thus 9.5.2.3)sectionSBC/(ACI0.7ruptureofModulus fibertensionextremetoaxiscentroidalfromdistance Substitute ' Example 1: Uncracked section • Assuming concrete is uncracked, compute bending stresses in extreme fibers of the section below for a bending moment of 34 kN.m. fc’ = 30 MPa. • To check that the section is uncracked, first determine the cracking moment of the section. 18 375 450 300 283 All dimensions in mm
- 10. 24/2/2013 CE 370 : Prof. Abdelhamid Charif 10 19 Solution of Example 1 uncrackedissectionthekN.m,34.0At kN.m8.38 N.mm108.38 225 1028.283.3 :)(MomentCracking mm1028.2450300 12 1 12 1 MPa83.3307.07.0 :ruptureofModulus 6 9 4933 ' M M y If M M bhI ff cr t gr cr cr g cr 19 375 450 300 283 Solution of Example 1 (continued) 20 uncracked.wellissectionThe MPa83.3307.07.0ruptureofModulus MPa35.3 1028.2 225)1034( 225 2 :fibersextremetwoAt the mm1028.2450300 12 1 12 1 :StressesBending ' 9 6 4933 r cr g g g ff ff I My f mm h y bhI I My f An uncracked beam is assumed to be homogeneous with neutral axis passing through the centroid of the beam section. Bending stress is given by the standard flexural equation. 375 450 300 283
- 11. 24/2/2013 CE 370 : Prof. Abdelhamid Charif 11 Calculation of elastics stresses in cracked sections • When bending moment is sufficiently large to cause tensile stress in the extreme fibers , greater than the modulus of rupture, it is assumed that all of the concrete on the tensile side of the beam is cracked and must be neglected in the flexure calculations. • Cracking of the beam does not necessarily mean that the beam is going to fail. The reinforcing bars on the tension side begin to pick up the tension caused by the applied moment. • Assumption of perfect bond is made between reinforcing bars and concrete. Thus the strain in the concrete and in the steel will be the same at equal distances from the neutral axis. • Although the strains in the two materials at a particular point are the same, their stresses can not be the same as they have different values of elastic modulus. 21 Modular Ratio (n) 22 steelofthattimesbewillconcreterequiredofArea required?bewilltwotheofareaswhatconcreteorsteelbyresistedbetoisforceIf with nnAA n A A A F n A F nff F E E nnfff E E f E f E f sc s c cs cs c s csc c s s s s c c sc Definition: The ratio of the steel modulus to the concrete modulus is called the modular ratio. If the modular ratio for a beam is 10, stress in steel will be 10 times the stress in concrete at the same distance from the neutral axis. Otherwise: when n = 10, 500 mm2 of steel will carry the same total force as 5000 mm2 of concrete. At the same point, steel and concrete strains are equal and the stresses are proportional to their modulus of elasticity.
- 12. 24/2/2013 CE 370 : Prof. Abdelhamid Charif 12 Transformed Area Method • When the steel area As is replaced with an equivalent area of fictitious concrete (nAs), which supposedly can resist tension, the area is referred to as the transformed area. • As the transformed area is of concrete only, it can be analyzed by the usual methods for elastic homogeneous beams. 23 Tensile concrete is ignored Tensile steel area As replaced by equivalent concrete area nAs N.A. cf sA n fs snA Steps in transformed area method 1. Locate the neutral axis: – Assume it is located a distance x from the compression surface of the beam. – Equate the first moment of the compression area of the beam cross section about the neutral axis to the first moment of the tensile area about the neutral axis. – Solve the resulting quadratic equation. 2. Calculate the moment of inertia of the transformed section wit respect to the neutral axis. 3. Compute the stresses in the concrete and the steel with the flexure formula ( f = My/I ). 24
- 13. 24/2/2013 CE 370 : Prof. Abdelhamid Charif 13 Example 2: Transformed Area Method • (a) Calculate the bending stresses in the beam shown below using the transformed area method • fc’ = 20 MPa, n = 9 and M = 95 kN.m • (b) Determine the allowable resisting moment of the beam, if the allowable stresses are : fac = 10 MPa and fas = 140 MPa 25 425 500 300 283 All dimensions in mm 26 Solution: 425 500 300 )1846( 283 sA All dimensions in mm 300 425 x )425( x 2 mm16614snA 49 2323 2 22 22 mm10571.1 168.5)(425(16614)168.5300 3 1 )(425)(168.5300 3 1 :axis)neutral(aboutInertiaofMoment mm5.168issolutionPositive 84.44778.200559470734)76.110( 04707376.1100706095016614150 166147060950150)425(16614150)425( 2 )(300 :)isfibertopfromdepthaxis(neutralaxisneutralaboutmomentsfirstTaking I xnAI x xxxx xxxxxnA x x x s s
- 14. 24/2/2013 CE 370 : Prof. Abdelhamid Charif 14 27 MPa6.139 10571.1 5.256)1095( 9 :mm)256.5168.5-425(levelSteel MPa2.10 10571.1 5.168)1095( :mm)168.5(ncompressioExtreme :StressesBending 9 6 9 6 I My nf I My n f fy I My fy I My f s s c c kN.m2.95N.mm102.95 5.2569 10571.1140)/( mm256.5168.5-425:levelsteelAt the :steelintensionallowabletoingcorrespondmomentAllowable kN.m2.93N.mm102.93 5.168 10571.110 :concreteinncompressioallowabletoingcorrespondmomentAllowable :momentresistingAllowable(b) 6 9 6 9 ny If y Inf M y y If M y If M I yM f asas as ac ac allowable allowable allowable allowable Solution – Cont. 28 Concrete and steel will reach their permissible stresses at these moments respectively. Discussion: Concrete and steel will not reach their maximum allowable stresses at exactly the same bending moments. The resisting moment of the section is 93.2 kNm (the smallest) because if that value is exceeded, concrete becomes overstressed even though the steel stress is less than its allowable stress. kN.m2.95 kN.m2.93 :momentsresistingAllowable as ac M M Solution – Cont.
- 15. 24/2/2013 CE 370 : Prof. Abdelhamid Charif 15 2929 d h b sA b h x )( xd snA I xdM nfxdy I Mx fxy xnA bx Ix xnAdnA bx xdnA bx xdnA x x x s c s ssss )( :)(stresstensionSteel :)(stressncompressioconcreteTop )(d)( 3 :NA)(aboutinertiaofMomentforSolve 0 2 )( 2 )( 2 )(b :)isfibertopfromdepthaxis(neutralaxisneutralaboutmomentsFirst 2 3 22 Steps for a General Rectangular Section Ultimate State and Nominal Flexural Moment • After concrete compression stresses exceed about 50% of fc’, they no longer vary linearly. The variation is rather nonlinear as shown in Figure (b) 30 d yf b sA c (a) Beam (b) Actual compression stress variation Neutral axis depth is now noted c
- 16. 24/2/2013 CE 370 : Prof. Abdelhamid Charif 16 31 The ultimate flexural moment capacity is called “Nominal” moment and is noted Mn. It is derived using this assumption: The curved compression diagram can be replaced by a rectangular one with a constant stress of 0.85 fc’ (Whitney). The Whitney rectangular stress block of depth “a” is assumed to have the same area and same centre of gravity as the curved diagram (generating the same force and the same moment). d yf b sA c ca 1 abfC c ' 85.0 ' 85.0 cf (a) Beam (b) Actual block of compression stress (c) Whitney block of compression stress ys fAT Ultimate State and Nominal Flexural Moment 32 Apart from the previous standard assumptions (sections remain plane after bending, perfect bond between concrete and steel bars, concrete tensile strength ignored), two other assumptions are added in the ultimate stage: • Concrete is assumed to fail (crush) when its compressive strain at the extreme fiber reaches its ultimate value εcu = 0.003 • The nonlinear compressive stress block for concrete is replaced by an equivalent rectangular one with constant stress of 0.85 fc’ and a depth a = 1 c Additional assumptions at ultimate state MPafforf MPaffor cc c 3065.0,008.009.1Max 3085.0 1 1 shall not be smaller than 0.65
- 17. 24/2/2013 CE 370 : Prof. Abdelhamid Charif 17 33 Factor 1 to account for skewed shape of compression stress block Question: Why does 1 vary (reduced for fc’ > 30 MPa) ? MPafforf MPaffor cc c 3065.0,008.009.1Max 3085.0 1 34 Variation of coefficient 1 Observe skewed shape of compression stress-strain curves for higher strength The shapes of the stress- strain curves for high strength concrete are skewed. This shifts the centroid of the area (below the curve) closer to the origin, and thus reduces the length of the equivalent rectangle. Thus 1 must be smaller for high strength concrete.
- 18. 24/2/2013 CE 370 : Prof. Abdelhamid Charif 18 35 There are three types of flexural failure of a structural member: 1. Steel reaches its yield strength before concrete reaches its ultimate strain of 0.003 (Under-reinforced section) 2. Steel reaches yield limit at the same time as concrete reaches its ultimate strain (Balanced section). 3. Concrete fails before the yield of steel, in the case of presence of a high percentage of steel in the section (Over-reinforced section). Types of ultimate flexural failure 36 Steel reaches its yield strength before concrete reaches its ultimate strain. Large deformations occur before concrete crushing. Failure is ductile. After yielding, the steel tensile force remains constant. To maintain an equal equilibrating concrete compression force, the neutral axis is shifted up. Cracks will also progress upwards. Under-reinforced section
- 19. 24/2/2013 CE 370 : Prof. Abdelhamid Charif 19 37 Steel reaches its yield limit at the same time as concrete reaches the ultimate strain. Failure is sudden and brittle Balanced section 38 Concrete crushes before the yield of steel, due to the presence of a high percentage of steel in the section. Failure is very sudden and brittle Over-reinforced section
- 20. 24/2/2013 CE 370 : Prof. Abdelhamid Charif 20 39 Moment-Curvature for various steel areas (Increasing steel area reduces ductility) Types of flexural failure • Brittle failures are very dangerous and must be avoided. • All codes of practice (ACI, SBC, …) require beams to be designed as under-reinforced to ensure ductile behavior. • In fact SBC and ACI codes require more ductility than just under reinforced. 40