This document discusses approaches for shear design of prestressed concrete beams. It describes two modes of shear failure: web-shear cracking and flexure-shear cracking. Formulas are presented from codes like IS and ACI for calculating web-shear and flexure-shear strength. Mohr's circle analysis is used to derive an expression for flexure-shear cracking. Test results are compared. A simplified method is proposed using a coefficient K to calculate average flexure-shear strength. Values of K are plotted against initial prestressing. The document concludes by recommending an equation that can be used to calculate flexure-shear strength for both prestressed and non-prestressed concrete members.
9. Equation of Mohrβs Circle
βππ‘ =
π
2
β
π
4
2
+ π2
At critical case where tensile strength of concrete reaches its limiting
value, evaluating for π ππ
π ππ =
π
2
+ πππ‘
2
β
π2
4
οThe flexural strength Ο, along the depth is calculated and substituted
in above equation to get π ππ
οIf π at section is greater than π ππ crack initiation in compression
zone takes place and indicated as flexure shear crack
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11. SIMPLIFIED METHOD USING A COEFFICIENT K
οAbove method is an iterative method
οInstead an average shear strength of flexure shear is considered
οcoefficients of πππ defined as K
οAverage shear strength of rectangular beam can be defined as
π½ ππ= π² π ππ ππ π
where π₯ π’ is the distance of neutral axis from topmost fiber
of the cracked section.
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12. VALUES OF K PLOTTED AGAINST
INITIAL PRESTRESSING (PO)
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13. οπΎ = 0.42 + πΌ(π β 1)
οπΌ is a variable and π is the ratio of Mo and Mcr
οHowever value of zero to πΌ is found satisfactory with
many test results.
Therefore π½ ππ= π. ππ π ππ ππ π
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14. οAbove equation holds good for rectangular sections
οCannot be applied directly for Flanged sections
οTests on T beams are carried out to find flange contribution for
shear
οThen b is changed to beff , beff= bw+tf .
π½ ππ= π. ππ π ππ π¨ πππ
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15. οExample Question: (Question is in such a way that Vcr is less than Vco)
οA post tensioned of 400 mm wide and 550 mm deep.
Factored shear is 150kN,
Factored bending moment is 375 kN-m,
Effective prestress is 700 kN,
fck is 40N/mm2
Area of prestressing steel is 700mm2.
οUsing IS method of analysis:
Substituting values in equation 1 β 0.55
πππ
ππ
πππ + π0
ππ’
π π’
, flexure
shear strength is 132.17 kN
ο Using the method described:
finding neutral axis of cracked section we get 121.53 mm substituting
this value in πππ= 0.42 πππ ππ₯ π’, flexure- shear strength of 129.126
kN.
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16. COMPARISON & CONCLUSIONS:
οCode considers calculating both Vcr and Vco over the length of the
beam and taking minimum of either.
οWhich means both types of shear cracks can occur even at places
with moment less than cracking moment.
οHowever the method described above considers flexure shear
cracks to occur only at places where flexure cracks occur.
οAt sections with moment greater than cracking moment the lower
of flexure-shear and web-shear controls
οAt other sections shear cracks are controlled by web shear cracks.
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17. COMPARISON & CONCLUSIONS
οFollowing the methods that are discussed so far, following
equation can be used to compute flexure- shear strength of both
Prestressed and non Prestressed concrete members
π½ ππ= π. ππ π ππ π¨ πππ
οWherever the applied moment is greater than the cracking
moment, Vcr can be calculated at each section along the depth of
section, also to simplify for calculating minimum shear strength
section having maximum bending moment can be considered.
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18. APPENDICES:
1) ππ‘- Tensile strength of concrete
2) πππ- Compressive stress at centroidal axis due to prestressing
3) ππ£- Vertical component of prestress
4) πππ- Characteristic strength of concrete
5) πππ- Effective Prestress
6) Mo βMoment required to produce zero stress in concrete at
the level of steel
7) Mu βUltimate bending moment at the section considered due
to applied loads
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