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.

1. SEMINAR
N I T C | C E 4 0 9 2

2. A UNIFIED APPROACH FOR SHEAR
DESIGN OF PRESTRESSED
CONCRETE
2

3.  SHEAR STRENGTH
STRENGTH TO RESIST SHEAR
3

4. MODES OF SHEAR FAILURE IN
PRESTRESSED CONCRETE BEAM
Web-Shear cracking
Flexure-Shear cracking
4

5.  Web-Shear strength (Vco):
According to IS:1343 (2012) : 0. 𝟔𝟕𝐛𝑫 𝒇 𝒕
𝟐
+ 𝟎. 𝟖𝒇 𝒄𝒑 𝒇 𝒕 + 𝑷 𝒗
According to ACI 318 (2005) : (0.29 𝒇 𝒄𝒌 + 𝟎. 𝟑𝒇 𝒄𝒑)𝒃𝑫 + 𝑷 𝒗
5

6.  Flexure-Shear strength (Vcr):
According to IS:1343 (2012) : 𝟏 − 𝟎. 𝟓𝟓
𝒇 𝒑𝒆
𝒇 𝒑
𝝉𝒃𝒅 + 𝑴 𝟎
𝑽 𝒖
𝑴 𝒖
According to ACI 318 (2005) ∶ (𝟎. 𝟓𝟓 𝒇 𝒄𝒌 𝒃𝒅 + 𝑽 𝒅 + 𝑴 𝟎
𝑽 𝒖
𝑴 𝒖
)
6

7. EFFECTIVE SHEAR AREA:
7

8. MOHR’S CIRCLE FOR AN ELEMENT
TO FIND PRINCIPLE STRESSES
8

9. Equation of Mohr’s Circle
−𝑓𝑡 =
𝜎
2
−
𝜎
4
2
+ 𝜏2
At critical case where tensile strength of concrete reaches its limiting
value, evaluating for 𝜏 𝑐𝑟
𝜏 𝑐𝑟 =
𝜎
2
+ 𝑓𝑐𝑡
2
−
𝜎2
4
The flexural strength σ, along the depth is calculated and substituted
in above equation to get 𝜏 𝑐𝑟
If 𝜏 at section is greater than 𝜏 𝑐𝑟 crack initiation in compression
zone takes place and indicated as flexure shear crack
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10. COMPARISON WITH TEST RESULTS
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11. SIMPLIFIED METHOD USING A COEFFICIENT K
Above method is an iterative method
Instead an average shear strength of flexure shear is considered
coefficients of 𝑓𝑐𝑘 defined as K
Average shear strength of rectangular beam can be defined as
𝑽 𝒄𝒓= 𝑲 𝒇 𝒄𝒌 𝒃𝒙 𝒖
where 𝑥 𝑢 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)
12

13. 𝐾 = 0.42 + 𝛼(𝜐 − 1)
𝛼 is a variable and 𝜐 is the ratio of Mo and Mcr
However value of zero to 𝛼 is found satisfactory with
many test results.
Therefore 𝑽 𝒄𝒓= 𝟎. 𝟒𝟐 𝒇 𝒄𝒌 𝒃𝒙 𝒖
13

14. Above equation holds good for rectangular sections
Cannot be applied directly for Flanged sections
Tests on T beams are carried out to find flange contribution for
shear
Then b is changed to beff , beff= bw+tf .
𝑽 𝒄𝒓= 𝟎. 𝟒𝟐 𝒇 𝒄𝒌 𝑨 𝒆𝒇𝒇
14

15. Example Question: (Question is in such a way that Vcr is less than Vco)
A 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.
Using IS method of analysis:
Substituting values in equation 1 − 0.55
𝑓𝑝𝑒
𝑓𝑝
𝜏𝑏𝑑 + 𝑀0
𝑉𝑢
𝑀 𝑢
, flexure
shear strength is 132.17 kN
 Using the method described:
finding neutral axis of cracked section we get 121.53 mm substituting
this value in 𝑉𝑐𝑟= 0.42 𝑓𝑐𝑘 𝑏𝑥 𝑢, flexure- shear strength of 129.126
kN.
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16. COMPARISON & CONCLUSIONS:
Code considers calculating both Vcr and Vco over the length of the
beam and taking minimum of either.
Which means both types of shear cracks can occur even at places
with moment less than cracking moment.
However the method described above considers flexure shear
cracks to occur only at places where flexure cracks occur.
At sections with moment greater than cracking moment the lower
of flexure-shear and web-shear controls
At other sections shear cracks are controlled by web shear cracks.
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17. COMPARISON & CONCLUSIONS
Following 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
𝑽 𝒄𝒓= 𝟎. 𝟒𝟐 𝒇 𝒄𝒌 𝑨 𝒆𝒇𝒇
Wherever 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) 𝑓𝑡- Tensile strength of concrete
2) 𝑓𝑐𝑝- Compressive stress at centroidal axis due to prestressing
3) 𝑃𝑣- Vertical component of prestress
4) 𝑓𝑐𝑘- Characteristic strength of concrete
5) 𝑓𝑝𝑒- Effective Prestress
6) Mo –Moment required to produce zero stress in concrete at
the level of steel
7) Mu –Ultimate bending moment at the section considered due
to applied loads
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19. THANK YOU ALL
FOR LISTENING
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