Idea Transcript
Prestressed Concrete Structures
Dr. Amlan K Sengupta and Prof. Devdas Menon
5.2 Design for Shear (Part I) This section covers the following topics. •
General Comments
•
Limit State of Collapse for Shear
5.2.1 General Comments Calculation of Shear Demand The objective of design is to provide ultimate resistance for shear (VuR) greater than the shear demand under ultimate loads (Vu). For simply supported prestressed beams, the maximum shear near the support is given by the beam theory. For continuous prestressed beams, a rigorous analysis can be done by the moment distribution method. Else, the shear coefficients in Table 13 of IS:456 - 2000 can be used under conditions of uniform cross-section of the beams, uniform loads and similar lengths of span.
Design of Stirrups The design is done for the critical section. The critical section is defined in Clause 22.6.2 of IS:456 - 2000. In general cases, the face of the support is considered as the critical section. When the reaction at the support introduces compression at the end of the beam, the critical section can be selected at a distance effective depth from the face of the support. The effective depth is selected as the greater of dp or ds. dp = depth of CGS from the extreme compression fiber ds = depth of centroid of non-prestressed steel. Since the CGS is at a higher location near the support, the effective depth will be equal to ds. To vary the spacing of stirrups along the span, other sections may be selected for design. Usually the following scheme is selected for beams under uniform load. 1) Close spacing for quarter of the span adjacent to the supports. 2) Wide spacing for half of the span at the middle.
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Prestressed Concrete Structures
Dr. Amlan K Sengupta and Prof. Devdas Menon
For large beams, more variation of spacing may be selected. The following sketch shows the typical variation of spacing of stirrups. The span is represented by L.
L/4
L/2
Figure 5-2.1
L/4
Typical variation of spacing of stirrups
5.2.2 Limit State of Collapse for Shear The shear is studied based on the capacity of a section which is the limit state of collapse. The capacity (or ultimate resistance) of a section (VuR) consists of a concrete contribution (Vc) and the stirrup contribution (VS). VuR = VC + VS
(5-2.1)
Vc includes Vcz (contribution from uncracked concrete), Va (aggregate interlock) and Vd (dowel action). The value of Vc depends on whether the section is cracked due to flexure. Section 22.4 of IS:1343 - 1980 gives two expressions of Vc, one for cracked section and the other for uncracked section. Usually, the expression for the uncracked section will govern near the support. The expression for the cracked section will govern near the mid span. Of course, both the expressions need to be evaluated at a particular section. The lower value obtained from the two expressions is selected. For uncracked sections, Vc = Vco Vc = 0.67bD ft 2 + 0.8fcp ft Vco is the shear causing web shear cracking at CGC. In the above expression, b = breadth of the section = bw, breadth of the web for flanged sections Indian Institute of Technology Madras
(5-2.2)
Prestressed Concrete Structures
Dr. Amlan K Sengupta and Prof. Devdas Menon
D = total depth of the section (h) ft
= tensile strength of concrete = 0.24√fck
fcp = compressive stress in concrete at CGC due to the prestress = Pe/A. The value of fcp is taken as positive (numeric value). Note that, a reduced effective prestress needs to be considered in the transmission length (explained in Section 7.1) region of a pre-tensioned beam.
The previous equation can be derived based on the expression of the principal tensile stress (σ1) at CGC.
v fcp
σ2
fcp
State of stress at CGC
Figure 5-2.2
σ1
(–fcp,v) σ2
Principal stresses
σ1
Mohr’s circle
State of stresses at a point on the neutral axis for a prestressed beam
The principal tensile stress is equated to the direct tensile strength of concrete (ft). σ1 = =-
fcp 2 fcp 2
+ +
fcp2 4 fcp2
+ v2
V Q + ⎛⎜ c 0 ⎞⎟ 4 ⎝ Ib ⎠
2
= ft In the previous equation, I
= gross moment of inertia
Q = At y At = area of section above CGC y = vertical distance of centroid of At from CGC.
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(5-2.3)
Prestressed Concrete Structures
Dr. Amlan K Sengupta and Prof. Devdas Menon
At + CGC
Figure 5-2.3
y
Cross-section of a beam showing the variables for calculating shear stress in the web
Transposing the terms, Vc 0 =
Ib 2 ft + fcp ft Q
→ 0.67bD ft 2 + 0.8fcp ft
(5-2.4)
The term 0.67bD represents Ib/Q for the section. It is exact for a rectangular section and conservative for other sections. To be conservative, only 80% of the prestressing force is considered in the term 0.8fcp. For a flanged section, when the CGC is in the flange, the intersection of web and flange is considered to be the critical location. The expression of Vc0 is modified by substituting 0.8fcp with 0.8 × (the stress in concrete at the level of the intersection of web and flange). In presence of inclined tendons or vertical prestress, the vertical component of the prestressing force (Vp) can be added to Vc0.
Vc → Vc 0 +Vp = 0.67bD ft 2 + 0.8fcp ft +Vp
(5-2.5)
For cracked sections, Vc = Vcr ⎛ f ⎞ V Vc = ⎜ 1- 0.55 pe ⎟ τc bd + M0 u fpk ⎠ Mu ⎝ ≥ 0.1bd fck
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(5-2.6)
Prestressed Concrete Structures
Dr. Amlan K Sengupta and Prof. Devdas Menon
Vcr is the shear corresponding to flexure shear cracking. The term (1 – 0.55fpe /fpk)τcbd is the additional shear that changes a flexural crack to a flexure shear crack. The notations in the previous equation are as follows. fpe = effective prestress in the tendon after all losses ≤ 0.6fpk fpk = characteristic strength of prestressing steel
τc = ultimate shear stress capacity of concrete, obtained from Table 6 of IS:1343 - 1980. It is given for values of Ap / bd, where d is the depth of
CGS. The values are plotted in the next figure. b = breadth of the section = bw , breadth of the web for flanged sections d = distance from the extreme compression fibre to the centroid of the tendons at the section considered M0 = moment initiating a flexural crack Mu = moment due to ultimate loads at the design section Vu = shear due to ultimate loads at the design section.
τ c (N/mm2)
1.2 0.8 0.4 0 0
1
2 A p /bd x 100
M30
Figure 5-2.4
3
4
M40
Variation of shear strength of concrete
The term (M0/Mu)Vu is the shear corresponding to the moment M0, that decompresses (nullifies the effect of prestress) the tension face and initiates a flexural crack. The expression of M0 is given below. M0 = 0.8fpt
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I y
(5-2.7)
Prestressed Concrete Structures
Dr. Amlan K Sengupta and Prof. Devdas Menon
In the above expression, fpt = magnitude of the compressive stress in concrete at the level of CGS due to prestress only. An equal amount of tensile stress is required to decompress the concrete at the level of CGS. The corresponding moment is fptI / y. In the expression of M0, I
= gross moment of inertia
y = depth of the CGS from CGC. The factor 0.8 implies that M0 is estimated to be 80% of the moment that decompresses the concrete at the level of CGS. Since the concrete is cracked and the inclination of tendon is small away from the supports, any vertical component of the prestressing force is not added to Vcr. Maximum Permissible Shear Stress
To check the crushing of concrete in shear compression failure, the shear stress is limited to a maximum value (τc,max).
The value of τc,max depends on the grade of
concrete and is given in Table 7 of IS:1343 - 1980. Vu ≤ τ c,max bdt
In the previous expression, dt = greater of dp or ds dp = depth of CGS from the extreme compression fiber ds = depth of centroid of regular steel Vu = shear force at a section due to ultimate loads.
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(5-2.8)
Prestressed Concrete Structures
Dr. Amlan K Sengupta and Prof. Devdas Menon
2
τ c, max (N/mm )
6
4
2
0 30
40
50
60
2
f ck (N/mm )
Figure 5-2.5
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Variation of maximum permissible shear stress in concrete