This page provides Codes and equations related to shear. Although researchers have taken considerable effort to understand shear behavior, due to the lack of an universally accepted model for shear behavior, shear design provisions still generally consist of empirical relationships that differ from code-to-code to such an extent that the shear strength of a particular member as calculated by one code-of-practice may be a few  times as much as the strength predicted by another design code.  Some of the differences in the relationships for the concrete contribution to shear resistance are illustrated in Table 1 followed by description of various equations by Codes or researchers.

Table 1: Summary of Major Code Expressions for the Concrete Contribution to Shear Resistance

Codes or Researcher

Equations

*Factors Accounted

ACI 318-95 (1995)

 (N) or (N) 

f'c, (a/d), r

AASHTO LRFD 1996

f'c, (d), (a/d), (r), agg

Canadian Standard

CSA A23.3-94 (1994)

 (N) if  or  (N) if ,

f'c, d, a/d

Eurocode EC2, Part 1 (1990)

 
   (N) where , ,

f'c, d, a/d, r

British Standard BS 8110 (1985)

 (N)  for

f'c, d, r

CEB-FIP Model Code (1990)

  (N)

f'c, d, a/d, r

Japanese JSCE Code (1986)

  ( kgf )  for  
 where , ,

f'c, d, a/d, r

Australian Standard

 AS 3600 (1994)

  (N)  
 where , 1.0 (when no axial force exists),

f'c, d, a/d, r

New Zealand Standards NZS 3101 (1995)

 
 , (N)   
 with concentrated load,    (N)   
 with distributed load,   (N) 

f'c, a/d, r

Zsutty¡¯s equation (1968)

 (N) for

f'c, a/d, r

Bazant¡¯s equation (1987)

  (N)

f'c, d, a/d, r , agg

Collins & Kuchma¡¯s equation (1999)

 (N) where 0.9d

f'c, (d), agg

*f'c: compressive strength of concrete, d: size effect, a/d: effect of shear span to depth ratio, r: effect of longitudinal reinforcement, agg: effect of aggregate size, ( ) : accounted indirectly

  ACI 318-95  
1. Reinforced Concrete Members 
        (limit < 70 MPa)  
1) Equation 11-3  
        (N)  
2) Equation 11-5  

  (N)  when where  
  AASHTO LRFD 1996  
     (limit 16 < < 70 MPa)  
       

or  : distance between the resultants of the tensile and compressive forces due to flexure

factor indicating ability of diagonally cracked concrete to transmit tension
 ( , )

      - Culverts under 600 mm or more of fill  

                 where  
         For single cell box culverts only,
                Vc for slabs monolithic with walls 
                Vc for slabs simply supported  
  AASHTO Standard Specifications for Highway Bridges 1996  
  Prestressed Concrete Members  
  The lesser of Vci and Vcw
         
         
  where       d 0.8h

: moment causing flexural cracking at section due to externally applied loads (Article 9.20)  

 = compressive strength of concrete at 28 days

*   = width of a web of a flanged member

*   = distance from extreme compressive fiber to centroid of the prestressing force, or to centroid of negative moment reinforcing for precast girder bridges made continuous

= shear force at section due to unfactored dead load (Article 9.20)

= factored shear force at section due to externally applied loads occurring simultaneously with Mmax (Article 9.20)

Mmax = maximum factored moment at section due to externally applied loads (Article 9.20)  

= compressive stress in concrete (after allowance for all prestress losses) at centroid of cross section resisting externally applied loads or at junction of web and flange when the centroid lies within the flange (In a composite member,  is resultant compressive stress at centorid of composite section, or at junction of web and flange when the centroid lies within the flange, due to both prestress and moments resisted by precast member acting alone.) (Article 9.20)  

= compressive stress in concrete due to effective prestress forces only (after allowance for all prestress losses) at extreme fiber of section where tensile stress is caused by externally applied loads (Article 9.20)

= stress due to unfactored dead load, at extreme fiber of section where tensile stress is caused by externally applied loads (Article 9.20)

= distance from centroidal axis of gross section, neglecting reinforcement, to extreme fiber in tension (Article 9.20)

I = moment of inertia about the centroid of the cross section (Article 9.20)

  Canadian Standard CSA A23.3-94, Simplified Method  
       (N)  when  or

       (N) when  and  
  Eurocode EC2, Part 1 (1990)
        (limit 12 < < 50 MPa)  

Consist of 3 parts, , , and .

where   : The shear strength for members without shear reinforcement
: The upper limit of the shear strength to prevent web crushing failures.
: The shear strength for members with shear reinforcement
  
1)
       (N)
,   : an enhancement factor can be applied if the member is loaded by a concentrated load situated at a distance from the face of the support
 = Basic design shear strength
the lower 5% fractile characteristic tensile strength
 mean value of the tensile concrete strength
 
= characteristic cylinder compressive strength of concrete  
 = effective web width
*   = effective depth
Thus, the above equation can be simplified to the following equation.
  (N)
2)
       (N)
where

: the factored design strength, may be taken as

       (For analysis purpose is considered to be appropriate)
* : the effectiveness factor  (may taken as 0.6)
3)
- two method : standard method and Variable-angle truss mehtod

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Under construction below!!!!

 

4. Eurocode EC2, Part 1

                        (limit range 12 £  £ 50 MPa)

            Consist of 3 parts, , , and .

where

            : The shear strength for members without shear reinforcement

            : The upper limit of the shear strength to prevent web crushing failures.

            : The shear strength for members with shear reinforcement

 

1)

              (N)

 

,   : an enhancement factor can be applied if the member is loaded by a concentrated load situated at a distance x£2.5d from the face of the support

 = Basic design shear strength