2.1.3.1 Introduction
Three different design codes are presented: The Swedish handbook on concrete structures (BBK04), the Eurocode 2 (EC2) from Europe and the American Concrete Institute building code (ACI 318-08).
In order to be clearer for the reader, the variable names were harmonised on the basis of Eurocode 2 notations.
For each code, a presentation of the fundamental equations for shear, also called one- way shear, design is made. A comparison is then made between the different approaches. In section 7.2, the predictions of EC2 and BBK04 are compared with experimental failure loads of 4-pile caps without shear reinforcement. Some additional comments on the efficiency of the design codes are also made in that part.
2.1.3.2 ACI
Reference is made to ACI 318-08 (ACI318-08) in this part.
The design approach of the ACI building code is cross-sectional which means that the sectional capacity is compared to the sectional shear force.
The design approach of the ACI building code is based on the following three equations:
Ed
n V
V ≥
ϕ (2.1)
The design shear capacity should be higher than the actual shear force and is determined as a nominal shear capacity multiplied with φ, the strength reduction factor equal to 0.75 for shear. The nominal shear capacity can be expressed as:
d b
Vn =τn w (2.2)
The contribution of concrete and shear reinforcement are added to define the nominal shear capacity of the section.
s c
n V V
V = + (2.3)
Members not requiring design shear reinforcement
The contribution to the shear capacity of concrete, Vc, is set equal to the shear force required to cause significant inclined cracking and is, in the ACI code, considered to be the same for beams with and without shear reinforcement.
d b f
Vc =2 c w (2.4)
Or, with a more detailed equation:
d b f d
M b d f V
V w c w
Ed Ed c
c 1.9 2500 ≤3.5
+
= λ ρ (2.5)
The maximum nominal shear stress is proportional to the tensile strength of concrete, which is defined as proportional to the square root of the concrete compressive strength. The shear capacity is also directly influenced by the amount of flexural reinforcement, the more the flexural steel ratio ρ=As/bwd is high, the more the propagation of a critical crack in the web is reduced. The term VEd d/MEd limits the concrete shear capacity near inflexion points.
Another set of formulas also allows modifying the shear capacity of a member depending on with axial compression/tension. This case in encountered mainly in prestressed and post-tensioned members and is not relevant for pile caps.
Members requiring design shear reinforcement
According to the ACI code a minimum amount of shear reinforcement should be provided as soon as VEd exceeds 0.5φVc.
c
Ed V
V ≤0.5ϕ (2.6)
This limitation reduces the risk of brittle failure in the web and allows crack width control.
A minimum area of shear reinforcement is required:
y w c
sw f
s f b
A ,min =0.75 (2.7)
This area is chosen bigger for higher concrete strengths in order to prevent brittle failure.
Where shear reinforcement perpendicular to the axis of the beam are provided, the steel contribution to the shear capacity is:
s d f
Vs = Aw y (2.8)
Where Aw is the area of shear reinforcement within spacing s. Vs is calculated as the capacity provided by vertical stirrups in a 45 degrees truss model, see Figure 2.13.
The ACI code considers a modified truss analogy including both the tensile capacity of the stirrups, Vs, and the tensile capacity provided by the concrete, Vc. The nominal shear capacity of the flexural element is then calculated using Equation 2.3.
Load applied close to a support
Figure 2.13 Free body diagrams of the end of a beam
The closest inclined crack at the support will extend in the web and meet the compression zone at approximately a distance d from the face of the support. The loads applied at a distance less than d from the column face are transferred directly by compression in the web above the crack; they do not enter in the calculation of the applied shear force V and do not increase the need for shear capacity. Accordingly, the ACI code states that sections located less than d from the support face are allowed to be designed for the applied shear force V at a distance d from the support face as well. However, this can only be applied if the shear force VEd at d is not radically different from the one applied at the support face. For instance, when a major part of the load is applied within d from the support face, the web might fail in a combination of splitting and crushing. This is the kind of failure that may occur in stocky pile caps and that are not well treated by design codes.
2.1.3.3 Eurocode 2
Reference is made to EN 1992-1-1:2004 (EN 1992-1-1:2004) in this part.
Members not requiring design shear reinforcement
The design shear capacity of a beam without shear reinforcement is:
( f ) b d
k C
VRdc Rdc ck w
= , 31
, 100ρ (2.9)
In order to avoid the shear capacity of the beam to be null when the amount of flexural reinforcement goes to zero, the capacity of the beam should always be taken higher than:
d b v
VRd,c = min w (2.10)
This last expression is often preferred to the Equation 2.9 for the calculation of the shear capacity of a pile cap. Indeed, as pile caps often have low reinforcement ratios, Equation 3.10 gives a higher capacity.
The maximum nominal shear strength is proportional to the cubic root of the concrete compressive strength, fck and to the cubic root of the amount of flexural reinforcement, ρ. C,Rd,c and vmin are found in the respective national annex, the recommended values are C,Rd,c=0,18/γc and:
2 2 1 3
min k fck
v = (2.11)
The size effect factor, k = 1+√200/d, traduces the shear transfer capacity reduction occurring in deep flexural members.
The shear capacity of the section is the product of the nominal shear strength and the cross sectional area, bwd
Members requiring design shear reinforcement
The design of members requiring shear reinforcement is based on a truss model with a variable inclination θ between the struts and the direction of the bea, see Figure 2.14.
Figure 2.14 Truss model of the shear force transfer in a web (EN 1992-1-1:2004).
The variable inclination method assumes that the inclination θ of the average principal strains direction varies when the load increases and is finally (in the ultimate limit state) controlled by the reinforcement arrangement. Force redistribution results in an inclination smaller than 45 degrees, see Figure 2.15. Cracking and force redistribution processes are explained in section 2.1.2.4.
Figure 2.15 Variable inclination method (Walraven 2002)
The angle of inclination of the struts must be restricted due to the limited plastic strain redistribution capacity of concrete and steel. However the allowable value of θ is a national parameter stated in the respective national appendices. These are the recommended limits: