Lec 6-flexural analysis and design of beamns
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This document discusses the cracked transformed section method for analyzing reinforced concrete beams. It explains that when cracking occurs in the tension side of the beam, the concrete is no longer effective and the steel takes the tensile stresses. The effective cracked transformed section is used, where the depth is reduced by a factor k. Equations are provided to calculate k in terms of the steel ratio ρ. The depth and neutral axis location for the cracked transformed section model are then used to calculate the resisting moment capacity of the beam.
![Plain & Reinforced Concrete-1
Cracked Transformed Section (contd…)
Calculations for“k”, taking top face as reference
kd = [b x kd x kd/2 + nAsd]/ [b x kd + nAs]
b x (kd)2
+ nAskd = b x (kd)2
/2 + nAsd
b x (kd)2
/2 = nAsd - nAskd
b x (kd)2
/2 = (1-k) nAsd
bd2
/2 k2
= (1-k) nρbd2
k2
= 2(1-k) nρ
Cracked
Transformed Section
kd
nAs
b
d
ρ = As / bd](https://image.slidesharecdn.com/lec-6-flexuralanalysisanddesignofbeamns-181114184456/85/Lec-6-flexural-analysis-and-design-of-beamns-4-320.jpg)
![Plain & Reinforced Concrete-1
Cracked Transformed Section (contd…)
k2
+ 2nρ k - 2nρ = 0
k = [ -2nρ ± √ (4n2
ρ2
+ 8nρ) ] /2
k = [ -nρ ± √ (n2
ρ2
+ 2nρ) ]
k = √ (n2
ρ2
+ 2nρ) -nρ
Taking only positive value as distance can’t be negative
Cracked
Transformed Section
kd
nAs
b
d](https://image.slidesharecdn.com/lec-6-flexuralanalysisanddesignofbeamns-181114184456/85/Lec-6-flexural-analysis-and-design-of-beamns-5-320.jpg)
Lec 6-flexural analysis and design of beamns
- 1. Plain & Reinforced Concrete-1 CE3601 Lecture # 6 21rd Feb 2012 Flexural Analysis and Design of Beams
- 2. Plain & Reinforced Concrete-1 Cracked Transformed Section When tensile stress cross fr, crack appear on the tension face of beam. Tensile strength of concrete is neglected. All the concrete on the tension side of N.A. is neglected. Only the Shaded area of cross-section is considered effective. N.A Concrete Steel
- 3. Plain & Reinforced Concrete-1 Cracked Transformed Section (contd…) Cc T = Asfs la = jd N.A. fc εc fs Strain Diagram Stress Diagram Internal Force Diagram When the tension side is cracked the concrete becomes ineffective but the strains goes on increasing. The steel comes in to action to take the tension. εs Cracked Transformed Section h kd d nAs b
- 4. Plain & Reinforced Concrete-1 Cracked Transformed Section (contd…) Calculations for“k”, taking top face as reference kd = [b x kd x kd/2 + nAsd]/ [b x kd + nAs] b x (kd)2 + nAskd = b x (kd)2 /2 + nAsd b x (kd)2 /2 = nAsd - nAskd b x (kd)2 /2 = (1-k) nAsd bd2 /2 k2 = (1-k) nρbd2 k2 = 2(1-k) nρ Cracked Transformed Section kd nAs b d ρ = As / bd
- 5. Plain & Reinforced Concrete-1 Cracked Transformed Section (contd…) k2 + 2nρ k - 2nρ = 0 k = [ -2nρ ± √ (4n2 ρ2 + 8nρ) ] /2 k = [ -nρ ± √ (n2 ρ2 + 2nρ) ] k = √ (n2 ρ2 + 2nρ) -nρ Taking only positive value as distance can’t be negative Cracked Transformed Section kd nAs b d
- 6. Plain & Reinforced Concrete-1 Cracked Transformed Section (contd…) Value of “j” jd = d – kd/3 J = 1 – k/3 Cc = volume of stress diagram Cc = ½ kd x fc x b Cc = fc/2 x kdb T = Asfs = nfcAs b kd fc T = Asfs Cc jdN.A d
- 7. Plain & Reinforced Concrete-1 Cracked Transformed Section (contd…) Resisting Moment Capacity For longitudinal equilibrium T = Cc Mr = T x jd = Cc x jd Mr = Asfs x jd = fc/2 x kdb x jd ------------- 1 FOR MAXIMUM VALUE RESISTING MOMENT Put in eq-1 fs = 0.5 fy for grade 300 steel fs = 0.4 fy for grade 420 steel fs = Maximum allowable tensile stress in steel fc= 0.45 fc’ fc = Maximum allowable concrete stress
- 8. Concluded