The anchor bolt resistances are evaluated according to STO 36554501-048-2016 for headed and post-installed anchors. STO often uses tabulated values in Annex, in which case the formulas from SP 43, SP16 or EN 1992-4 are used because of their general validity. Pull-out failure of straight anchors, combined pull-out and concrete failure of bonded anchors, and concrete splitting failure are not checked due to missing information available only for the particular anchor and glue type from the anchor manufacturer.
In the Code setup, concrete can be set as cracked or uncracked. The resistances of uncracked concrete are higher.
Tensile steel resistance (SP 43 - Annex G):
Tensile steel resistance of anchors in STO uses tabulated values in Annex A. Therefore, a general formula from SP 43 - Annex G is used.
\[ N_{ult,s} = \frac{A_{sa} \cdot R_{ba} \cdot \gamma_c}{k_0} \]
where:
- Rba = 0.8 ⋅ Rbyn – anchor bolt design yield strength
- Rbyn – characteristic yield strength of anchor steel
- Asa – net cross-sectional area of a bolt
- k0 – factor for loading type; editable in Code setup; k0 = 1.05 for static loading and k0 = 1.35 for dynamic loading; for the portable anchors with anchor plates, installed freely in the tubes, k0 is taken equal to 1.15 for dynamic loads (SP 43 – G.9)
- γc – service factor – SP 16, Table 1, editable in Code setup
Pull-out resistance (EN 1992-4, Cl. 7.2.1.5)
Pull-out resistance of anchors in STO uses tabulated values in Annex A. Therefore, the general formula from EN 1992-4, Cl. 7.2.1.5 is used for anchors with washer plates:
\[ N_{ult,p}=\frac{N_{n,p} \cdot \psi_c}{\gamma_{bt} \gamma_{Np}} \]
where:
- Nn,p \(\cdot \psi_c\) = k2 ∙ Ah ∙ Rbn – characteristic resistance in case of pull-out failure
- k2 – coefficient dependent on concrete condition, k2 = 7.5 for cracked concrete, k2 = 10.5 for non-cracked concrete
- Ah – bearing area of head of anchor; for circular washer plate \(A_h = \frac{\pi}{4} \left ( d_h^2 - d^2 \right )\), for rectangular washer plate \(A_h = a_{wp}^2 - \frac{\pi}{4} d^2\)
- dh ≤ 6 th + d – diameter of the head of the fastener
- th – thickness of the head of the headed fastener
- d – diameter of the shank of the fastener
- Rbn – characteristic concrete compressive cylinder strength
- γbt – partial safety factor for concrete (editable in Code setup)
- γNp – partial safety factor taking account of the installation safety of an anchor system (editable in Code setup)
The pullout resistance of other types of anchors is not checked and must be guaranteed by manufacturer or determined by STO, Annex A.
Concrete cone failure resistance of anchor or group of anchors (STO - Cl. 6.1.3):
\[N_{ult,c}=\frac{N_{n,c}^0}{\gamma_{bt} \cdot \gamma_{Nc}} \cdot \frac{A_{c,N}}{A_{c,N}^0} \cdot \psi_{s,N} \cdot \psi_{re,N} \cdot \psi_{ec,N}\]
where:
- \(N_{n,c}^0 = k_1 \sqrt{R_{b,n}} h_{ef}^{1.5}\) – characteristic resistance of a single fastener placed in concrete and not influenced by adjacent fasteners or edges of the concrete member
- k1 – factor taking into account concrete condition; k1 = 8.4 for cracked concrete and k1 = 11.8 for non-cracked concrete
- Rb,n – characteristic concrete compressive cylinder strength
- hef – embedment depth of the anchor in concrete; for three or four close edges, effective \(h'_{ef} = \max \left \{ \frac{c_{max}}{c_{cr,N}} \cdot h_{ef}, \, \frac{s_{max}}{s_{cr,N}} \cdot h_{ef} \right \}\) is used instead in formulas for Nn,c0, ccr,N, scr,N, Ac,N, Ac,N0, ψs,N, and ψec,N
- Ac,N – actual projected area, limited by overlapping concrete cones of adjacent fasteners as well as by edges of the concrete member
- Ac,N0 = scr,N2 – reference projected area, i.e. area of concrete of an individual anchor with large spacing and edge distance at the concrete surface
- \(\psi_{s,N}=0.7+0.3 \cdot \frac{c}{c_{cr,N}} \le 1\) – factor taking into account disturbance of the distribution of stresses in the concrete due to the proximity of an edge of the concrete member
- c – smallest edge distance
- ccr,N = 1.5 ∙ hef – characteristic edge distance for ensuring the transmission of the characteristic resistance of an anchor in case of concrete break-out under tension loading
- \(\psi_{re,N}=0.5+\frac{h_{ef}}{200} \le 1\) – shell spalling factor
- \(\psi_{ec,N}=\frac{1}{1+2 \cdot (e_N / s_{cr,N})} \le 1\) – factor taking into account group effect when different tension loads are acting on the individual fasteners of a group; ψec,N is determined separately for each direction and the product of both factors is used
- eN – eccentricity of resultant tension force of tensioned fasteners in respect to the centre of gravity of the tensioned fasteners
- scr,N = 2 ∙ ccr,N – characteristic spacing of anchors to ensure the characteristic resistance of the anchors in case of concrete cone failure under tension load
- γbt – partial safety factor for concrete (editable in Code setup)
- γNc – partial safety factor taking account of the installation safety of an anchor system (editable in Code setup)
The concrete breakout cone area for group of anchors loaded by tension that create common concrete cone, Ac,N, is shown by red dashed line.
Anchor shear steel resistance (SP16 - Cl. 14.2.9 and STO - Cl. 6.2.1)
According to STO - Cl. 6.2.1, two scenarios are investigated:
- Shear without lever arm (Stand-off: Direct)
- Shear with lever arm (Stand-off: Mortar joint)
Shear without lever arm
Shear steel resistance of anchors in STO uses tabulated values in Annex A. Therefore, a general formula from SP16 is used. It is assumed that anchors are threaded rods. Friction is not taken into account.
A bolt subject to a design shear force is designed according to SP16 - Cl. 14.2.9 and shall satisfy:
\[ V_{ult,s} = R_{bs} A_b \gamma_b \gamma_c \]
where:
- Rbs – design shear strength of a bolt – SP 16, Table 5
- Ab – bolt gross section area
- γb – service factor of bolt joint – SP 16, Table 41 – γb = 1.0 for single bolting and multibolting with accuracy class A, γb = 0.9 for multibolting and accuracy class B and high strength bolts (Rbun ≥ 800 MPa)
- γc – service factor – SP 16, Table 1, editable in Code setup
| Rbyn [MPa] | Rbs [MPa] |
| \(R_{byn} \le 300 \) | \(0.42 \cdot R_{bun} \) |
| \(300 < R_{byn} \le 400 \) | \(0.41 \cdot R_{bun} \) |
| \(400 < R_{byn} \le 936 \) | \(0.40 \cdot R_{bun} \) |
| \(936 > R_{byn} \) | \(0.35 \cdot R_{bun} \) |
Shear with lever arm (STO - Cl. 6.2.1.5)
\[ V_{ult,s} = \frac{M_{n,s}}{l_s} \gamma_b \gamma_c \]
where:
- \(M_{n,s} = M_{n,s}^0 \left ( 1- \frac{N_{an}}{N_{ult,s}} \right ) \) – characteristic bending resistance of the anchor decreased by the tensile force in the anchor
- Mn,s0 = 1.2 Wel Rbun – characteristic bending resistance of the anchor (ETAG 001, Annex C – Equation (5.5b))
- \( W_{el} = \frac{\pi d^3}{32}\) – section modulus of the anchor
- d – anchor bolt diameter; if shear plane in thread is selected, the diameter reduced by threads is used; otherwise, nominal diameter, dnom, is used
- Rbun – ultimate tensile strength of the anchor
- Nan – tensile force in the anchor
- Nult,s – tensile resistance of the anchor
- ls = (0.5 dnom + tmortar + 0.5 tbp) / \(\alpha_M \) – lever arm
- αM = 2 – full restraint is assumed
- tmortar – thickness of mortar (grout)
- tbp – thickness of the base plate
- γb – service factor of bolt joint – SP 16, Table 41 – γb = 1.0 for single bolting and multibolting with accuracy class A, γb = 0.9 for multibolting and accuracy class B and high strength bolts (Rbun ≥ 800 MPa)
- γc – service factor – SP 16, Table 1, editable in Code setup
Concrete pry-out failure (STO - Cl. 6.2.2):
\[ V_{ult,cp}= k \cdot \frac{N_{ult,c}}{\gamma_{V,cp}} \]
where:
- k – factor for concrete pryout failure (STO 36554501-048-2016 - Cl. 6.2.2.3) taken as k = 2 as default (ETAG 001, Annex C – Cl. 5.2.3.3) editable in Code setup
- Nult,c – resistance of a fastener or a group of fasteners in case of concrete cone failure; all anchors are assumed to be in tension and γNc = 1.0
- γV,cp – partial safety factor taking account of the installation safety of an anchor system for concrete pryout failure editable in Code setup
Concrete edge failure (STO - Cl. 6.2.3):
Concrete edge failure is a brittle failure and the worst possible case is checked, i.e. only the anchors located near the edge transfer the full shear load acting on a whole base plate. If anchors are positioned in a rectangular pattern, the row of anchors at the investigated edge transfers the shear load. If anchors are positioned irregularly, the two anchors nearest to the investigated edge transfer the shear load. Two edges in the direction of the shear load are investigated and the worst case is shown in results.
Investigated edges in dependence on the direction of the shear force resultant
Resistance of a fastener or a group of fasteners loaded towards the edge:
\[ V_{ult,c}= \frac{V_{n,c}^0}{\gamma_{bt} \cdot \gamma_{Vc}} \cdot \frac{A_{c,V}}{A_{c,V}^0} \cdot \psi_{s,V} \cdot \psi_{h,V} \cdot \psi_{\alpha,V} \cdot \psi_{ec,V} \cdot \psi_{re,V} \]
where:
- \( V_{n,c}^0 = k_3 \cdot d_{nom}^\alpha \cdot l_f^\beta \cdot \sqrt{R_{b,n}} \cdot c_1^{1.5}\) – initial value of the characteristic resistance of a fastener loaded perpendicular to the edge
- k3 – factor taking into account concrete condition; k3 = 2.0 for cracked concrete, k3 = 2.8 for non-cracked concrete
- \( \alpha = 0.1 \left ( \frac{l_f}{c_1} \right ) ^{0.5} \)
- \( \beta = 0.1 \left ( \frac{d_{nom}}{c_1} \right ) ^{0.2} \)
- lf = min (hef, 12 dnom) for dnom ≤ 24 mm; lf = min [hef, max (8 dnom, 300 mm)] for dnom > 24 mm – effective length of the anchor in shear - taken from EN 1992-4 - Cl. 7.2.2.5
- hef – embedment depth of the anchor in concrete
- c1 – distance from the anchor to the investigated edge; for fastenings in a narrow, thin member, the effective distance \( c'_1=\max \left \{ \frac{c_{2,max}}{1.5}, \, \frac{h}{1.5}, \, \frac{s_{2,max}}{3} \right \} \) is used instead
- c2 – smaller distance to the concrete edge perpendicular to the distance c1
- dnom – nominal anchor diameter
- Ac,V0 = 4.5 c12 – area of concrete cone of an individual anchor at the lateral concrete surface not affected by edges
- Ac,V – actual area of the concrete cone of anchorage at the lateral concrete surface
- \(\psi_{s,V} = 0.7+0.3 \frac{c_2}{1.5 c_1} \le 1.0 \) – factor which takes account of the disturbance of the distribution of stresses in the concrete due to further edges of the concrete member on the shear resistance
- \( \psi_{h,V} = \left ( \frac{1.5 c_1}{h} \right ) ^ {0.5} \ge 1.0 \) – factor which takes account of the fact that the shear resistance does not decrease proportionally to the member thickness as assumed by the ratio Ac,V / Ac,V0
- \( \psi_{\alpha,V} = \sqrt{\frac{1}{(\cos \alpha_V)^2 + (0.4 \sin \alpha_V)^2}} \ge 1 \) – takes account of the angle αV between the load applied, V, and the direction perpendicular to the free edge of the concrete member
- \( \psi_{ec,V} = \frac{1}{1+e_V / (1.5 c_1)} \le 1 \) – factor which takes account of a group effect when different shear loads are acting on the individual anchors of a group
- ψre,V = 1.0 – factor takes account of the effect of the type of reinforcement used in cracked concrete
- h – concrete block height
- γbt – partial safety factor for concrete (editable in Code setup)
- γVc – partial safety factor taking account of the installation safety of an anchor system (editable in Code setup)
Interaction of tensile and shear forces (STO - Cl. 6.3):
Interaction of tensile and shear forces is determined according to STO - Cl. 6.3., Equation (6.55):
\[ \beta_N^{1.5} + \beta_V^{1.5} \le 1.0 \]
where:
- \(\beta_N = \max \left \{ \frac{N_{an}}{N_{ult,s}}; \, \frac{N_{an}}{N_{ult,p}}; \, \frac{N_{an}}{N_{ult,c}} \right \} \) – coefficient defined as the largest value of the ratio of the design tensile forces to the value of the ultimate tensile resistances for each of the failure mechanisms
- \(\beta_V = \max \left \{ \frac{V_{an}}{V_{ult,s}}; \, \frac{V_{an}}{V_{ult,cp}}; \, \frac{V_{an}}{V_{ult,c}} \right \} \) – coefficient defined as the largest value of the ratio of the design shear forces to the value of the ultimate shear resistances for each of the failure mechanisms
Anchors with stand-off
Anchor with stand-off is designed as a bar element loaded by shear force, bending moment and compressive or tensile force. These internal forces are determined by finite element model. The anchor is fixed on both sides, one side is 0.5×d below the concrete level, the other side is in the middle of the thickness of the plate. The buckling length is conservatively assumed as twice the length of the bar element. Plastic section modulus is used. The bar element is designed according to SP 16. The shear force may decrease the yield strength of the steel but the minimum length of the anchor to fit the nut under the base plate ensures that the anchor fails in bending before the shear force reaches half the shear resistance. The reduction is therefore not necessary. Interaction of bending moment and compressive or tensile strength is assumed linear.
Shear resistance:
A bolt subject to a design shear force is designed according to SP16 - Cl. 14.2.9 and shall satisfy:
\[ V_{ult,s} = R_{bs} A_{bn} \gamma_b \gamma_c \]
where:
- Rbs – design shear strength of a bolt – SP 16, Table 5
- Abn – bolt gross section area
- γb – service factor of bolt joint – SP 16, Table 41 – γb = 1.0 for single bolting and multibolting with accuracy class A, γb = 0.9 for multibolting and accuracy class B and high strength bolts (Rbun ≥ 800 MPa)
- γc – service factor – SP 16, Table 1, editable in Code setup
| Rbyn [MPa] | Rbs [MPa] |
| \(R_{byn} < 300 \) | \(0.42 \cdot R_{bun} \) |
| \(300 \le R_{byn} < 400 \) | \(0.41 \cdot R_{bun} \) |
| \(400 \le R_{byn} < 936 \) | \(0.40 \cdot R_{bun} \) |
| \(936 < R_{byn} \) | \(0.35 \cdot R_{bun} \) |
Tensile and compressive resistance:
Steel resistance of anchors in STO uses tabulated values in Annex A. Therefore, a general formula from SP 43 - Annex G is used.
\[ N_{ult,s} = \frac{A_{sa} \cdot R_{ba} \cdot \gamma_c }{k_0} \]
where:
- Rba = 0.8 ⋅ Rbyn – anchor bolt design yield strength
- Rbyn – characteristic yield strength of anchor steel
- Asa – net cross-sectional area of a bolt
- γc – service factor – SP 16, Table 1, editable in Code setup
- k0 – factor for loading type; editable in Code setup; k0 = 1.05 for static loading and k0 = 1.35 for dynamic loading; for the portable anchors with anchor plates, installed freely in the tubes, k0 is taken equal to 1.15 for dynamic loads (SP 43 – G.9)
Bending resistance:
\[ M_{ult,s} = W_n R_{ba} \gamma_c \]
- \( W_{n}= \frac{d_s^3}{6} \) – section modulus of the bolt
- \(d_s = \sqrt{\frac{4A_{bn}}{\pi}}\) – anchor bolt diameter reduced by threads
- Rba = 0.8 ⋅ Rbyn – anchor bolt design yield strength
- Rbyn – characteristic yield strength of anchor steel
- γc – service factor – SP 16, Table 1, editable in Code setup
Stand-off anchor steel utilization
Linear interaction is used:
\[ \frac{N}{N_{ult,s}} + \frac{M}{M_{ult,s}} \le 1 \]
Stand-off anchor Concrete utilization
All concrete checks are also performed and following interaction for concrete failure modes is provided:
\[ \beta_N^{1.5} + \beta_V^{1.5} \le 1.0 \]
where:
- \(\beta_N = \max \left \{ \frac{N_{an}}{N_{ult,p}}; \, \frac{N_{an}}{N_{ult,c}} \right \} \) – coefficient defined as the largest value of the ratio of the design tensile forces to the value of the ultimate tensile resistances for each of the failure mechanisms
- \(\beta_V = \max \left \{ \frac{V_{an}}{V_{ult,cp}}; \, \frac{V_{an}}{V_{ult,c}} \right \} \) – coefficient defined as the largest value of the ratio of the design shear forces to the value of the ultimate shear resistances for each of the failure mechanisms