Design for Shear

University of British Columbia | Okanagan

School of Engineering

ENGR 327 Reinforced Concrete Design I

Dr. Solomon Tesfamariam

7-1

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:





7-2

7.2 MECHANICS OF SEAR IN BEAMS






7-3

��

� �� , ��

� ��

Principal tension

��

� �� , ��

� ��

Principal compression

tan 2�� /2

� 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






7-4

7.3 SHEAR REINFORCEMENT CSA A23.3 Clause 11.2.4






7-5

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.





7-6

7.4.1 CSA A23.3 SHEAR REQUIREMENTS (CHAPTER 11) �� (Clause 11.3.1)

�� = Factored shear resistance � = Factored shear load

The factored shear resistance �� 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) Maximum allowable �� value is

��,�� 0.25"��′#$%& (Clause 11.3.3)

Where, "�= 0.65 %&= effective shear depth = 0.9d or 0.72h, whichever is greater #$= web width





7-7

7.4.2 CONCRETE RESISTANCE IN SHEAR, VC CSA A23.3-04 Clause 11.3.4

Three contributions:

Vcz shear in compression zone Va aggregate interlock Vd dowel action

� Combined empirically in Vc

�� "�345��′#$%& (Clause 11.3.4) Where, "�= 0.65 3 = facto to account for low-density concrete = 1 for normal density concrete 4 = factor accounting for shear resistance of cracked concrete, determined in Clause 11.3.6 %&= effective shear depth = 0.9d or 0.72h, whichever is greater #$= web width

Note that, in the determination of ��, 5��′ 6 8 MPa





7-8

7.4.3 STEEL RESISTANCE IN SHEAR, VS


CSA A23.3-04 Clause 11.3.5

�� "�7&�8%& cot θ< Clause 11.3.5

Where, "�= 0.65 3 = facto to account for low-density concrete = 1 for normal density concrete 7&= 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:

��?@′A � � � �� "�7&�8%& cot θ<

Therefore, the required spacing, <�?@′A is

<�?@′A � "�7&�8%& cot θ� � ��





7-9

7.4.4 DETERMINATION OF B AND C CSA A23.3-04 Clause 11.3.6 4 = 0.21 � = 42o 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 �8 < 400 MPa

� = 35o 4 = 0.18 for sections containing at least minimum transverse

reinforcement (Clause 11.2.8.2)

4 � 2301000 � %&

for sections containing no transverse reinforcement and having maximum Coarse Aggregate size > 20 mm

4 � 2301000 � <E?

for sections containing no transverse reinforcement and all aggregate sizes

<E? = equivalent crack spacing

<E? � 35<E 15 � FG

� 0.85<E





7-10

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 � non-typical members/structures

7.4.5 ADDITIONAL CODE REQUIREMENTS (CSA A23.3-04)

1. Minimum shear reinforcement Clause 11.2.8.1 **Changed in 2004 A minimum area of shear reinforcement is required in the following regions of flexural members: (a) Where Vf > Vc (b) Beams where h > 750 mm (c) Where torsion, Tf > 0.25 Tcr

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:

7&,�IJ � 0.065��′#$<�8

3. Maximum spacing of shear reinforcement Clause 11.3.8

< 6 K 0.7%&600 MMN for � 6 0.1253"��′#$%&





7-11

< 6 K 0.35%&300 MMN for � O 0.1253"��′#$%&

4. Maximum shear resistance Clause 11.3.3

��,�� 0.25"��′#$%&

Thus,

��,�� 0.25"��′#$%& � ��

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





7-12

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 bw and or d

2. Determine θ and β

3. Compute Vc 4. Design stirrups for critical section, Vf, near support:

(a) If Vf < Vc, then no stirrups are required

(b) If Vf > 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 > Av,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 Vf envelope





7-13

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 fy = 400 MPa h = 820 mm, Max C.A. size = 20 mm





7-14

Example 1…





Example 1… Develop stirrup layout and shear resistance envelope:



Develop stirrup layout and shear resistance envelope:

7-15





7-16

SHEAR DESIGN DIGEST STEP 1: FOR THE GIVEN GEOMETRY, CHECK IF THE MEMBER SIZE IS

ADEQUATE… STEP 2: CHECK IF MINIMUM REINFORCEMENT IS SATISFIED…





7-17

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).

Design for Shear

Documents

shear design

shear stresses

concrete resistance

reinforced concrete

vcz shear

shear distribution

area of shear reinforcement

concrete cracking