University of British Columbia | Okanagan School of Engineering
CHAPTER 7 – SHEAR IN REINFORCED CONCRETE 7.1 INTRODUCTION
(Fig. Ref. Brzev and Pao 2006)
Brittle failure undesirable Shear in reinforced concrete is complex: • • • •
Non-linear material Non-homogeneous material Cracking Presence of reinforcement
A realistic description of the shear distribution is shown as:
ENGR 327 Reinforced Concrete Design I Dr. Solomon Tesfamariam
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University of British Columbia | Okanagan School of Engineering
7.2 MECHANICS OF SEAR IN BEAMS
(Fig. Ref. Brzev and Pao 2006)
ENGR 327 Reinforced Concrete Design I Dr. Solomon Tesfamariam
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University of British Columbia | Okanagan School of Engineering
, Principal tension , Principal compression
Shear cracks develop when principal tensile stresses σ1 exceeds the tensile strength of the concrete Cracking is perpendicular to principal tension stress A convenient way of determining the principal stresses at a point is to use a Mohr’s circle form stress. There are no shear stresses acting on the plane of maximum and minimum principal stress
(Fig. Ref. Brzev and Pao 2006)
ENGR 327 Reinforced Concrete Design I Dr. Solomon Tesfamariam
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University of British Columbia | Okanagan School of Engineering
7.3 SHEAR REINFORCEMENT CSA A23.3 Clause 11.2.4
(Fig. Ref. Brzev and Pao 2006)
ENGR 327 Reinforced Concrete Design I Dr. Solomon Tesfamariam
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University of British Columbia | Okanagan School of Engineering
7.4 CSA A23.3 DESIGN FOR SHEAR
The CSA A23.3 provisions for shear design were changed substantially in the 2004 version Previously (1994), three distinct approaches were permitted: 11.3 Simplified Method 11.4 General Method 11.5 Strut and Tie Models
The Simplified and General Methods have now been “combined” in the 2004 Code to provide a common approach with two variations: •
A modified simplified method
(Clause 11.3.6.3)
•
Revised general method
(Clause 11.3.6.4)
The Strut and Tie Method is included in the 2004 Code in Clause 11.4.
ENGR 327 Reinforced Concrete Design I Dr. Solomon Tesfamariam
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University of British Columbia | Okanagan School of Engineering
7.4.1 CSA A23.3 SHEAR REQUIREMENTS (CHAPTER 11)
= Factored shear resistance = Factored shear load The factored shear resistance
(Clause 11.3.1)
is supported by concrete and steel :
(Clause 11.3.3)
Additional provisions for:
Minimum Shear Reinforcement (location and amount) Maximum Spacing of Shear Reinforcement Maximum Shear Resistance Critical cross-section for shear design near support
The factored shear resistance
(Clause 11.3.3)
value is Maximum allowable ′
(Clause 11.3.3)
Where,
= 0.65 = effective shear depth = 0.9d or 0.72h, whichever is greater = web width
ENGR 327 Reinforced Concrete Design I Dr. Solomon Tesfamariam
7-6
University of British Columbia | Okanagan School of Engineering
7.4.2 CONCRETE RESISTANCE IN SHEAR, VC
CSA A33-4 Cluse 1134
Three contributions: Vcz shear in compression zone Va aggregate interlock Vd dowel action Combined empirically in V c
′
(Clause 11.3.4)
Where,
= 0.65 = facto to account for low-density concrete = 1 for normal density concrete = factor accounting for shear resistance of cracked concrete, determined in Clause 11.3.6 = effective shear depth = 0.9d or 0.72h, whichever is greater = web width Note that, in the determination of , 8 MPa ′
ENGR 327 Reinforced Concrete Design I Dr. Solomon Tesfamariam
7-7
University of British Columbia | Okanagan School of Engineering
7.4.3 STEEL RESISTANCE IN SHEAR, VS
(Fig. Ref. Brzev and Pao 2006)
CSA A33-4 Cluse 113
Clause 11.3.5
Where, = 0.65 = facto to account for low-density concrete = 1 for normal density concrete = area of shear reinforcement within distance “s” = Ab no. legs in stirrup = stirrup spacing = angle of inclination of compression stresses, determined in Clause 11.3.6 angle of inclined cracks due to shear
For design:
Therefore, the required spacing, is ′
′
′
ENGR 327 Reinforced Concrete Design I Dr. Solomon Tesfamariam
7-8
University of British Columbia | Okanagan School of Engineering
CSA A33-4 Cluse 1136 = 0.21 = 42 7.4.4 DETERMINATION OF
AND
o
For,
• Slabs with thickness < 350 mm • Beams with overall thickness < 250 mm • Concrete joist construction (Clause 10.4) • Beams cast integrally with slabs where the depth of the beam below the slab is not greater than one-half of the web width or 350 mm Clause 11.3.6.3 – Simplified Method
Applicable to cases other than Clause 11.3.6.2 and members not subject to significant axial tension Limitations: < 60 MPa < 400 MPa
′
= 35 = 0.18 13 13 o
for sections containing at least minimum transverse reinforcement (Clause 11.2.8.2) for sections containing no transverse reinforcement and having maximum Coarse Aggregate size > 20 mm for sections containing no transverse reinforcement and all aggregate sizes
= equivalent crack spacing 13
ENGR 327 Reinforced Concrete Design I Dr. Solomon Tesfamariam
7-9
University of British Columbia | Okanagan School of Engineering
Clause 11.3.6.4 – General Method
Based on Modified Compression Field Theory Use for: • > 60 MPa • Members subject to significant tension • Situations where designer wants a more rigorous approach typical members/structures
′
non-
7.4.5 ADDITIONAL CODE REQUIREMENTS (CSA A23.3-04) 1. Minimum shear reinforcement **Changed in 2004
Clause 11.2.8.1
A minimum area of shear reinforcement is required in the following regions of flexural members: (a) Where V f > Vc (b) Beams where h > 750 mm (c) Where torsion, T f > 0.25 T cr 2. Minimum area of shear reinforcement
Clause 11.2.8.2
Where shear reinforcement is required by Clause 11.2.8.1 or by calculation, a minimum area of shear reinforcement shall be provided:
6 ′
3. Maximum spacing of shear reinforcement
6
for
Clause 11.3.8
1
ENGR 327 Reinforced Concrete Design I Dr. Solomon Tesfamariam
′
7-10
University of British Columbia | Okanagan School of Engineering
3 3
for
1 ′
4. Maximum shear resistance
Clause 11.3.3
′
Thus,
′
If too may stirrups are provided ( is too large), then the concrete web may crush before the stirrups yields. If < , then the cross-section dimension need to be increased
5. Sections near supports
Clause 11.3.2
Critical section for shear design: • Compute at a distance from support where support reaction introduces compression • Compute as support where support reaction introduces tension
ENGR 327 Reinforced Concrete Design I Dr. Solomon Tesfamariam
7-11
University of British Columbia | Okanagan School of Engineering
7.4.6 SHEAR DESIGN PROCEDURE Must satisfy
> along length of member.
1. Determine if size of cross-section is adequate: Check
< If not, increase b
w
and or d
2. Determine θ and β 3. Compute V c 4. Design stirrups for critical section, V f, near support: (a) If V f < Vc, then no stirrups are required (b) If V f > Vc, then:
• Choose Av • Compute required spacing, s • Choose a reasonable value for s
round to nearest multiple of 10 mm or 25 mm less than or equal to calculated s
• Check Av > A v,min • Check s < smax 5. Design stirrups for selected other sections along length of beam following Step 4 procedure 6. Determine stirrup layout along beam length. Draw Vr diagram for beam and compare to V f envelope
ENGR 327 Reinforced Concrete Design I Dr. Solomon Tesfamariam
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University of British Columbia | Okanagan School of Engineering
Example 1: The factored shear force envelope for a continuous interior beam is shown below. Design the shear reinforcement for the beam.
= 25 MPa and f = 400 MPa ′
y
h = 820 mm, Max C.A. size = 20 mm
ENGR 327 Reinforced Concrete Design I Dr. Solomon Tesfamariam
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University of British Columbia | Okanagan School of Engineering
Example 1…
ENGR 327 Reinforced Concrete Design I Dr. Solomon Tesfamariam
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University of British Columbia | Okanagan School of Engineering
Example 1… Develop stirrup lay ut and shear resistance envelope:
ENGR 327 Reinforced Con rete Design I Dr. Solomon Tesfamariam
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University of British Columbia | Okanagan School of Engineering
SHEAR DESIGN DIGEST STEP 1: FOR THE GIVEN GEOMETRY , CHECK IF THE MEMBER SIZE IS ADEQUATE … STEP 2: CHECK IF MINIMUM REINFORCEMENT IS SATISFIED …
ENGR 327 Reinforced Concrete Design I Dr. Solomon Tesfamariam
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University of British Columbia | Okanagan School of Engineering
7.7 REFERENCES 1) Brzev, S. and Pao, J. 2006. Reinforced Concrete Design-A Practical Approach, Prentice Hall. 2) MacGregor, J.G. and Bartlett, F.M. 2000. Reinforced Concrete – Mechanics and Design, Prentice Hall, 1st Canadian Edition. 3) Canadian Portland Cement Association 2005. Concrete Design Handbook. Third edition. (Contains the 2004 edition of the design standard for reinforced concrete structures, CSA A23.3-04).
ENGR 327 Reinforced Concrete Design I Dr. Solomon Tesfamariam
7-17
