Anchor Bolt Design Theory (EN 1992-4)

The theory behind this anchor-bolt calculator: how a group of cast-in or post-installed anchors fixing a steel base plate to concrete is verified to EN 1992-4. We cover each failure mode in tension (steel, concrete cone breakout, pull-out or combined pull-out and cone, and splitting) and in shear (steel, concrete pry-out and concrete edge breakout), the projected-area method that scales a single-anchor resistance to a group near edges, the ψ modification factors, the combined tension-shear interaction, and how EN 1992-4 differs from the older ETAG 001 / CEN-TR 1992-4 it replaced.

An anchor connection transfers tension and shear from a steel base plate into a concrete member through a group of anchors. EN 1992-4:2018 governs the design of these fastenings and requires that every credible failure mode - in the steel and in the concrete - is verified, because the governing mode is rarely obvious in advance. This page explains the mechanics and the formulas behind each check this calculator performs.

The verification framework

For each failure mode the design action effect must not exceed the design resistance. In tension $N_{Ed} \le N_{Rd}$ and in shear $V_{Ed} \le V_{Rd}$, with the utilisation $\beta = E_{d}/R_{d} \le 1$. Design resistances come from a characteristic resistance divided by a partial factor: steel modes use $\gamma_{Ms}$, and all concrete modes use $\gamma_{Mc} = \gamma_c\,\gamma_{inst}$.

The full set of modes and their governing equations:

Tension - steel failure

The simplest mode: the anchor shank yields and fractures in tension. The characteristic resistance is the stress area times the ultimate strength, and the design value divides by the steel partial factor.

Tension - concrete cone breakout

Under tension a cone of concrete can break away from the anchor head, idealised as a pyramid spreading at roughly 35 degrees from the embedded head to the surface. The characteristic resistance of a single anchor far from any edge is

where $k_1 = 8.9$ for cast-in headed anchors in cracked concrete ($12.7$ uncracked), $f_{ck}$ is the characteristic cylinder strength and $h_{ef}$ the effective embedment. This is scaled to the real group and edge condition by the projected-area method and three modification factors:

The projected-area method

A single anchor projects a reference area $A_{c,N}^{0} = s_{cr,N}^{2}$ with$s_{cr,N} = 3\,h_{ef}$ and $c_{cr,N} = 1.5\,h_{ef}$. When anchors are close together their cones overlap, and when they sit near an edge the cone is cut off, so the actual area $A_{c,N}$ is smaller. The ratio $A_{c,N}/A_{c,N}^{0}$therefore captures the group and edge effect directly, instead of multiplying a single resistance by the number of anchors.

• ψ_s,N = 0.7 + 0.3·c/c_cr,N ≤ 1 - the edge disturbance factor.
• ψ_re,N = 0.5 + h_ef/200 ≤ 1 - shell spalling for shallow embedment.
• ψ_ec,N = 1/(1 + 2e_N/s_cr,N) ≤ 1 - eccentric tension on the group.

Tension - pull-out and bonded combined failure

A headed anchor can pull through the concrete bearing on its head before a cone forms. The characteristic pull-out resistance is

where $A_h$ is the net bearing area of the head. For a bonded(adhesive) anchor there is no head; instead the bond can fail together with a concrete cone, so the single-anchor bond resistance $N_{Rk,p}^{0} = \pi\,d\,h_{ef}\,\tau_{Rk}$ is scaled by the same projected-area and ψ-factor structure as the cone, using the bond spacing$s_{cr,Np}$ from the anchor ETA.

Tension - concrete splitting

In thin members or close to an edge the concrete can split along the anchor axis. EN 1992-4 takes the cone resistance and multiplies it by a member-thickness factor:

Because $\psi_{h,sp} \ge 1$, splitting is never more critical than the cone when the member is thick ($h \ge 2h_{ef}$); it only governs for thin slabs. It need not be checked when the edge distance and member depth both exceed the characteristic splitting values.

Shear - steel failure

Without a lever arm the anchor simply shears: $V_{Rk,s} = k_6\,A_s\,f_{uk}$ with$k_6 = 0.5$. If the base plate stands off the concrete (a grout gap or levelling nuts), the shear also bends the shank and the resistance drops to

with lever arm $l = e_1 + 0.5\,d$ and restraint factor $\alpha_M$ = 1.0 (free to rotate) or 2.0 (fully restrained). A stand-off can cut the shear capacity by half or more, so a flush grouted plate is preferable when shear governs.

Shear - concrete pry-out

A short, stiff anchor can lever a cone of concrete out on the side away from the load. The resistance is tied directly to the cone tension resistance:

Shear - concrete edge breakout

When shear pushes an anchor towards a free edge, a half-cone of concrete can break out from the edge. This is often the governing shear mode near an edge. The single-anchor resistance is

with $\alpha = 0.1\,(l_f/c_1)^{0.5}$, $\beta = 0.1\,(d_{nom}/c_1)^{0.2}$,$k_9 = 1.7$ (cracked) or $2.4$ (uncracked), edge distance$c_1$ and load-transfer length $l_f = \min(h_{ef},\,12\,d_{nom})$. It is then scaled by $A_{c,V}/A_{c,V}^{0}$ (with $A_{c,V}^{0} = 4.5\,c_1^{2}$) and the factors $\psi_{s,V}$, $\psi_{h,V}$, $\psi_{ec,V}$,$\psi_{\alpha,V}$ and $\psi_{re,V}$.

Combined tension and shear

Finally the interaction is checked separately for each material. For steel failure the demands combine as squares; for all concrete modes as the power 1.5:

where the concrete terms use the governing (smallest) concrete resistance in tension and in shear. Both equations must hold in addition to every individual mode.

Differences from ETAG 001 / CEN-TR 1992-4

EN 1992-4 superseded ETAG 001 Annex C and CEN/TS 1992-4 in 2018. It keeps the same projected-area framework but updates several coefficients. The most consequential for hand calculations is the concrete-cone factor: EN 1992-4 uses $k_1 = 8.9$ (cracked) /$12.7$ (uncracked) for cast-in headed anchors, whereas the old documents used$7.7$ / $11.0$, and the cone and edge strengths are written consistently in terms of the cylinder strength $f_{ck}$. Legacy spreadsheets built on the old values - and some still in circulation - can return non-conservative cone resistances; this calculator uses the current EN 1992-4 values throughout.

Note also that EN 1992-4 does not cover every detail (for example anchors with supplementary reinforcement to carry the tension or shear directly into the member need the additional provisions of §7.2.1.8 / §7.2.2.6), and product-specific values such as the bond strength$\tau_{Rk}$, the head bearing area and some k-factors come from the anchor's European Technical Assessment (ETA) - always check those against the product you specify.

Frequently asked questions

Ready to check your anchorage? Run the full EN 1992-4 verification for a cast-in or bonded anchor group with step-by-step derivations for every failure mode.