4 Eurocode provisions
4.7 Ultimate limit states
Flexural limit state
The ultimate bending moment demand is calculated based on the ultimate limit state load combination.
The ultimate moment of resistance is determined according to part 3.3 of the thesis then checked to be beyond the ultimate bending moment demand.
Shear limit state
4.7.2.1 Shear design of the beam
According to clause 6.2.1, for the verification of the shear resistance the following symbols are defined:
VRd, c is the design shear resistance of the member without shear reinforcement.
VRd, s is the design value of the shear force which can be sustained by the yielding shear reinforcement.
In regions of the member where VEd ≤VRd, c no calculated shear reinforcement is necessary.
VEd is the design shear force in the section considered resulting from external loading and prestressing (bonded or unbonded). The critical section is at d from the face of the support in case of uniformly distributed load.
In members where the design shear force is less than the shear capacity of the concrete section alone, no design shear reinforcement is required, although the minimum shear reinforcement in accordance with clause 9.2.2 should be provided.
The calculation of shear capacity according to EC2 is divided into two sections:
Regions uncracked in flexure, which occurs near to the simply supported beam ends where moment is normally low and shear is normally high. In this case shear capacity is limited by the maximum tensile stress in the web, which is the principle tensile stress due to the interaction between the compressive stress due to prestressing force (neglecting flexure stress which is small) and the shear stress.
Regions cracked in flexure, which occurs away from the supports in the simply supported beam or at the internal support of continuous beam where moment and shear are expected to be high. The interaction between moment and shear is very complex and cannot be simply modelled. Therefore, empirical formula is developed on the basis of tests to calculate the shear capacity.
In regions uncracked in bending (where the flexural tensile stress is smaller than fctk,0,05/ γc), the shear resistance is given by:
𝑉𝑅𝑑,𝑐 =𝑏𝑤𝐼
𝐴 𝑦√𝑓𝑐𝑡𝑑2 + 𝛼1𝜎𝑐𝑝𝑓𝑐𝑡𝑑 where:
Ι is the second moment of area
bw is the width of the cross-section at the centroidal axis, allowing for the presence of ducts S is the first moment of area above and about the centroidal axis
αI = lx/lpt2 ≤ 1,0 for pre-tensioned tendons = 1,0 for other types of prestressing
lx is the distance of section considered from the starting point of the transmission length lpt2 is the upper bound value of the transmission length of the prestressing element
σcp is the concrete compressive stress at the centroidal axis due to axial loading or prestressing (σcp = NEd /Ac in MPa, NEd > 0 in compression)
In prestressed single span members without shear reinforcement, the shear resistance of the regions cracked in bending may be calculated using the design value for the shear resistance VRd, c:
𝑉𝑅𝑑,𝑐 = [𝐶𝑅𝑑,𝑐𝑘(100𝜌1𝑓𝑐𝑘)13+ 𝑘1𝜎𝑐𝑝 ] 𝑏𝑤𝑑 With minimum value of:
𝑉𝑅𝑑,𝑐 = [𝑣𝑚𝑖𝑛+ 𝑘1𝜎𝑐𝑝 ]𝑏𝑤𝑑 where:
fck is in MPa
𝑘 = 1 + √200
𝑑 ≤ 0.02 with d in mm 𝜌𝑙 = 𝐴𝑠𝑙
𝑏𝑤 𝑑 ≤ 0.02
Asl is the area of the tensile reinforcement, which extends ≥ (lbd + d) beyond the section considered.
bw is the smallest width of the cross-section in the tensile area [mm]
σcp = NEd/Ac < 0.2 fcd [MPa]
NEd is the axial force in the cross-section due to loading or prestressing in newtons (NEd>0 for compression). The influence of imposed deformations on NE may be ignored.
AC is the area of concrete cross section [mm2] VRd,c is in newtons
Note: The values of CRd,c, vmin and k1 for use in a Country may be found in its National Annex. The recommended value for CRd,c is 0,18/ γc and that for k1 is 0.15.
𝑣min = 0.035 𝑘32 ⋅ 𝑓𝑐𝑘
1 2
The equation is similar to that of reinforced concrete except for the term k1σcp which indicates enhancement of the shear capacity by 15% of the longitudinal stress due to prestress forces.
However, EC2 assumes if VEd > VRd,c that the entire shear demand is resisted by the shear reinforcement only.
The design of members with shear reinforcement is based on a truss model in Figure 11:
Figure 11 Truss model used to design shear reinforcement [3]
VRd,s is limited to VRd, max (the design value of the maximum shear force which can be sustained by the member, limited by crushing of the compression struts) equals:
𝑉𝑅𝑑,𝑠 =𝐴𝑠𝑤
𝑠 𝑧𝑓𝑦𝑤𝑑cot 𝜃 and
𝑉𝑅𝑑,𝑚𝑎𝑥 = 𝛼𝑐𝑤𝑏𝑤𝑧𝜈1𝑓𝑐𝑘 [1.5(cot 𝜃 + tan 𝜃)]
where:
Asw is the cross-sectional area of the shear reinforcement s is the spacing of the stirrups
fywd is the design yield strength of the shear reinforcement 𝑣 = 0.6 [1 − 𝑓𝑐𝑘
250] (𝑓𝑐𝑘 𝑎𝑛𝑑 σcp 𝑖𝑛 𝑀𝑃𝑎)
For reinforced and prestressed members, if the design stress of the shear reinforcement is below 80% of the characteristic yield stress fyk, ν may be taken as:
𝑣 = 0.6 𝑓𝑜𝑟 𝑓𝑐𝑘 ≤ 60 𝑀𝑃𝑎 𝑣 = 0.9 − 𝑓𝑐𝑘
200> 0.5 𝑓𝑜𝑟 𝑓𝑐𝑘 ≥ 60 𝑀𝑃𝑎
Note: The value of tan αc for use in a Country may be found in its National Annex. The recommended value is as follows:
1 for non-prestressed structures (1 + σcp/fcd) for 0 < σcp ≤ 0.25 fcd
1.25 for 0.25 fcd < σcp ≤ 0.5 fcd
2.5 (1 - σcp/fcd) for 0.5 fcd < σcp < 1.0 fcd
where:
σcp is the mean compressive stress, measured positive, in the concrete due to the design axial force
This should be obtained by averaging it over the concrete section taking account of the reinforcement.
The value of σcp need not be calculated at a distance less than 0.5d cot θ from the edge of the support.
However, some considerations must be taken:
If the web of the section contains grouted ducts with diameter greater than one-eighth of the web thickness, in the calculation of VRd, max, the web thickness should be reduced by one-half of the sum of the duct diameter.
For non-grouted ducts, grouted plastic ducts and unbonded tendons the web thickness should be reduced by 1.2 times the sum of the duct diameters.
It can be assumed that z=0.9d although the EC2 restricts this approximation only for the reinforced concrete. But, because calculating z at every section will be tedious and make calculation complex and its consequential effects will likely be small, this approximation can be used.
For the two limiting values of cot θ according to EC2 (2.5 to 1) to ensure that the angle θ does not exceed 45°.
So, the angle can be determined to calculate the required shear reinforcement area.
4.7.2.2 Interface shear design
Ever since composite construction has been employed to create more efficient designs, horizontal shear strength at the interface has been a topic full of challenges and controversies. Figure 12 illustrates the horizontal shear forces required to develop composite action.
Figure 12 Development of shear forces during composite action [19]
According to clause 6.2.5, vEdi is the design value of the shear stress in the interface and is given by:
𝑣𝐸𝑑𝑖= 𝛽𝑉𝐸𝑑/(𝑧𝑏𝑖) where:
β is the ratio of the longitudinal force in the new concrete area and the total longitudinal force either in the compression or tension zone, both calculated for the section considered
VEd is the transverse shear force
z is the lever arm of composite section bi is the width of the interface
The design shear resistance vRdi at the interface is a combination of frictional and cohesional resistance.
𝑣𝑅𝑑𝑖 = 𝑐𝑓𝑐𝑡𝑑+ 𝜇𝜎𝑛+ 𝜌𝑓𝑦𝑑(𝜇 sin(𝛼) + cos(𝛼)) ≤ 0.5ν𝑓𝑐𝑑 where:
C and μ are cohesion and friction factors depending on surface characteristics and equal to 0.45 and 0.7 for exposed aggregate, respectively.
fctd is the design tensile strength of the concrete with the lowest strength with fctd = fctk,0.05/γc.
σn stress per unit area caused by the minimum external normal force across the interface that can act simultaneously with the shear force, positive for compression, such that σn < 0.6 fcd, and negative for tension. When σn is tensile c fctd should be taken as 0.
ρ = As / Ai
As area of reinforcement crossing the interface, including ordinary shear reinforcement (if any), with adequate anchorage at both sides of the interface.
Ai area of the joint
α the angle of reinforcement with the interface and should be limited by 45° ≤ α ≤ 90°
ν Is defined before