Shear and links

How a beam carries shear to EN 1992-1-1 6.2, on the Eurocode 2 route of the STATIX RC beam calculator: the concrete resistance V_Rd,c, the truss with a variable strut angle, designed links V_Rd,s, strut crushing V_Rd,max, and the minimum links and spacing.

1. Learning outcomes

After this lesson you can:

find the design shear at d from the face of a support;
work out V_Rd,c and decide whether designed links are needed;
explain the truss model and the limits on the strut angle;
size links for V_Rd,s and check the struts against V_Rd,max;
apply the minimum link ratio and the spacing limits.

2. Before you start

Lesson 1: the beam, its loads and its bars.
The RC beam calculator, set to Eurocode 2 (EN 1992-1-1).
A pocket calculator and about 50 minutes.
The same lesson on the SA code: RC Design 1, lesson 3 (SANS 10100-1).

The values this lesson uses are those of the calculator’s Eurocode 2 route: the recommended C_Rd,c = 0.18/γ_c, v_min, ν₁ and the cot θ range 1 to 2.5, with cot θ = 2.5 used for the links; α_cc 1.0 for shear; z = 0.9d; the minimum link ratio 9.5N and the spacing 9.6N and 9.8N. The links are taken as B500 (f_ywk = 500 MPa).

3. How a beam carries shear

Where the shear is. Under a uniform load the shear force falls in a straight line from the supports to zero at mid-span. The design shear is taken at a distance d from the face of the support, where a direct load is near the support (6.2.1(8)).

Shear cracks. Near the supports the beam cracks on a slope. The concrete between the cracks acts as inclined struts, and the links that cross a crack act as ties.

The truss model. EN 1992-1-1 6.2.3 treats a beam with links as a truss: compression struts at an angle θ, the links as vertical ties, and the top and bottom chords a lever arm z = 0.9d apart.

4. The resistances

The shear the concrete carries. Without designed links, a beam carries V_Rd,c: more with more tension steel, less in deep members (the size factor k). The calculator uses the recommended C_Rd,c 0.18/γ_c and v_min.

The decision. If the design shear is above V_Rd,c, the links must carry the whole of it: the Eurocode does not add the concrete contribution to the links, as some other codes do. Beams always get at least the minimum links.

The strut angle. The designer may choose θ within limits: a flatter strut (cot θ up to 2.5) crosses more links, so fewer are needed, but it squeezes the concrete harder. The calculator starts from cot θ = 2.5, the flattest the code allows.

Designed links. The links that cross the strut zone over a length z cot θ carry the shear. A two-leg Y10 link gives A_sw = 157 mm²; with B500 links f_ywd = 434.8 MPa.

Crushing the struts. However many links there are, the concrete struts can crush. V_Rd,max is largest at θ = 45°. The calculator’s checks take f_cd with α_cc 1.0 for shear (bending uses 0.85).

5. Minimum links, spacing and zones

Minimum links and spacing. Every beam has links at least at the minimum ratio, and no further apart than 0.75d along the beam, so that every possible shear crack crosses at least one link.

Link zones. Since the shear falls towards mid-span, a beam usually has closer links near the supports and the minimum in the middle. The calculator’s detailing lays the links in zones like this.

Shear adds tension. The truss pulls the bottom bars harder than the bending moment alone suggests. EN 1992-1-1 allows for this by shifting the moment curve by a_l = z cot θ/2 when the bars are curtailed and anchored (lesson 8).