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2.1.2 Shear Force

Shear Force

The check of shear force resistance is to be performed only in the ultimate limit state (ULS). The actions and resistances are considered with their design values. The general design requirement according to EN 1992-1-1, clause 6.2.1 is the following:

  • VEd ≤ VRd

where

    • VEd : design value of applied shear force
    • VRd : design value of shear force resistance

Depending on the failure mechanism, the design value of the shear force resistance is determined by one of the following three values.

    • VRd,c : design shear resistance of a structural component without shear reinforcement
    • V Rd,s : design shear resistance of a structural component with shear reinforcement, limited by the yield point of shear reinforcement (failure of tie)
    • VRd, max : design shear resistance limited by strength of concrete compression strut

If the acting shear force VEd remains below the value of VRd,c, no calculated shear reinforcement is necessary and the check is verified.

If the applied shear force VEd is higher than the value of VRd,c, a shear reinforcement must be designed. The shear reinforcement must resist the entire shear force. In addition, the bearing capacity of the concrete compression strut must be analyzed.

  • VEd ≤ VRd,s and VEd ≤ VRd,max

The various types of shear force resistance are determined according to EN 1992-1-1 as follows.

Design shear resistance without shear reinforcement

The design value for the design shear resistance VRd,c may be determined by:

VRd,c = CRd,c · k 100σl · fck13 - k1 · σcp bw · d 

Equation 2.1 EN 1992-1-1, Eq. (6.2a)

where

CRd,c

: recommended value: 0.18/γc

: scaling factor for considering cross-section depth
d: mean static depth in [mm]

: ratio of longitudinal reinforcement
  Asl : ­ area of tensile reinforcement which extends by at least (lbd + d)
  beyond the considered cross-section

fck

: characteristic value of concrete compressive strength in [N / mm2]

k1

: recommended value: 0.15

bw

: minimum cross-section width within tension zone in [mm]

d

: static effective depth of bending reinforcement in [mm]

: design value of concrete longitudinal stress in [N/mm2]

It is allowed, however, to apply a minimum value of the shear force resistance VRd,c,min.

VRd,c,min = vmin + k1 · σcp · bw · d 

Equation 2.2 EN 1992-1-1, Eq. (6.2b)

where

Design shear resistance with shear reinforcement

The following applies for structural components with shear reinforcement running perpendicular to the component's axis (α = 90°):

VRd,s = Asws · z · fywd · cot θ 

Equation 2.3 EN 1992-1-1, Eq. (6.8)

where

Asw

: cross-sectional area of shear reinforcement

s

: spacing of links

z

: lever arm of internal forces assumed with 0.9 d

fywd

: design yield strength of shear reinforcement

θ

: inclination of concrete compression strut

The inclination of the concrete compression strut θ may be selected within certain limits depending on the loading. This way, the equation can take the fact that a part of the shear force is resisted by crack friction into account. Thus, the virtual truss is less stressed. The following limits are recommended in equation (6.7) of EN 1992-1-1.

  • 1 ≤ cot θ ≤ 2.5

Thus, the compression strut inclination θ can vary between the following values:

Table 2.1 Recommended limits for inclination of compression strut

Minimum inclination Maximum inclination

θ

21.8°

45.0°

cot θ

2.5

1.0

Design shear resistance of concrete compression strut

The following applies for structural components with shear reinforcement running perpendicular to the component's axis (α = 90°):

VRd,max = αcw · bw · z · ν1 · fcdcot θ + tan θ  

Equation 2.4 EN 1992-1-1, Eq. (6.9)

where

Table 2.1

αcw

: coefficient for considering stress state in compression flange

bw

: cross-section width

z

: lever arm of internal forces (exactly calculated in bending design)

ν1

: reduction factor for concrete strength in case of shear cracks

fcd

: design value of concrete strength

θ

: inclination of concrete compression strut