VDI 2230-2 Multi-Bolt Joint Calculator: User Manual

The MechCalc VDI 2230-2 multi-bolt joint calculator uses the rigid-body method of VDI 2230 Part 2:2014-12, §6.3, to split the external loads on a bolt array over its bolts. It gives the axial force $F_{Ai}$ and the transverse force $F_{qi}$ of each bolt, flags the candidate critical bolts and can hand the selected bolt over to the VDI 2230-1 single-bolt calculator for the check. VDI 2230 Part 1 applies to single-bolt joints, so the most highly loaded bolted joint has to be determined and separated out first (VDI 2230 Part 2:2014-12, Introduction). Related articles in the Engineering section: Scope and positioning of the standard , When a bolted joint needs the systematic VDI 2230 calculation and the VDI 2230 series .

Scope

What it can calculate

• Bolt pattern: any planar pattern from a coordinate table (2 to 200 bolts), plus a rectangular-array generator and a bolt-circle generator. All bolts have the same nominal diameter.
• External loads: transverse loads $F_{QBx}$ and $F_{QBz}$, axial load $F_{AB}$ (tension positive, compression negative), bending moments $M_x$ and $M_z$, and the torque $M_Y$ about the bolt axis. The load point can be off the centroid, and the height $h$ of the load point above the interface can be given (VDI 2230 Part 2:2014-12, §6.3.1).
• Bending rotation model: free rotation about the principal axes through the centroid (VDI 2230 Part 2:2014-12, §6.3.1.3, Eq. (18), (19); §6.3.1.4, Figure 12; §6.3.1.5, Figure 13), or rotation about a fixed axis, for a pure bending moment with transverse load or torque only (VDI 2230 Part 2:2014-12, §6.3.1.4, Eq. (23), Figure 11).
• Internal pressure: only for a bolt array on one evenly spaced bolt circle; the prying load follows a circular plate clamped all round (VDI 2230 Part 2:2014-12, §6.3.2.4, Figure 20).
• Global slip check: with a coefficient of friction given, the sum of the preloads that friction needs to carry the transverse load (VDI 2230 Part 2:2014-12, §6.3.1.2, Eq. (11)).
• Results: $F_{Ai}$, $F_{qxi}$, $F_{qzi}$ and $F_{qi}$ of each bolt; the centroid and the moments of inertia; the candidate critical bolts; the $F_A$ and $F_Q$ handed over to VDI 2230-1.

What it cannot calculate

• Strength, fatigue, opening and slip checks of a single bolt: VDI 2230-1 does these; this calculator only distributes the loads (VDI 2230 Part 2:2014-12, Scope).
• Not implemented in this calculator: the area-weighted centroid for bolts of different sizes (VDI 2230 Part 2:2014-12, §6.3.1.1, Eq. (3), (4)), the elastomechanical methods (§6.4), the quantitative treatment of unevenly loaded bolt rows (§6.6), finite elements (§7) and the corrected calculations of Annex A1 and A2.
• Fatigue verification under cyclic loading: the rigid-body model is only of very limited application for it (VDI 2230 Part 2:2014-12, §5.2); the calculator does not hand over a minimum axial load $F_{A,min}$.
• Coordinate axes that are not the principal axes of the bolt array: the calculator refuses when a bending moment acts.
• A fixed axis of rotation together with an axial load or internal pressure: the calculator refuses; Eq. (23) is given for a pure bending moment (VDI 2230 Part 2:2014-12, §6.3.1.4, Eq. (23)).
• A flange whose main connection has gaskets between the plates: the calculation is to follow DIN EN 1591 and DIN EN 13445, not VDI 2230 (VDI 2230 Part 2:2014-12, §3.2, Flange, Note).

Input cards at a glance

Inputs

Bolt coordinates (x_i^*, z_i^*) [mm]

Coordinates $x_i^*$ and $z_i^*$ of bolt $i$ in the $x^*$-$z^*$ coordinate system (VDI 2230 Part 2:2014-12, §6.3.1.1, Figure 3).

Where to get the value

• The calculator takes the coordinates seen from the bolt-head side (looking along $-y$): $x^*$ is positive upwards and $z^*$ is positive to the right, in the same directions as the axes $x$ and $z$.
• Figure 3 draws $x^*$ pointing down (VDI 2230 Part 2:2014-12, §6.3.1.1, Figure 3); the calculator does not mirror or convert the values entered.
• The origin can be anywhere: the calculator moves all loads to the centroid $S$ (VDI 2230 Part 2:2014-12, §6.3.1.1, steps 1 and 4).
• In the free-rotation model with a bending moment acting, the calculator requires the axes of the frame to be parallel to the principal axes of the bolt array and refuses otherwise.
• Table: 2 to 200 bolts; two bolts at the same place are refused; two centres closer than the nominal diameter $d$ give a warning that the holes may overlap; the optional label (up to 16 characters) replaces the number in the results.
• Paste: one bolt per line as “x z” or “x z label”, separated by space, comma, semicolon or tab; if a line is not two numbers, the calculator names the row and applies nothing.
• The generators only write rows into the table (replace or append); coordinates are rounded to $10^{-4}$ mm.
• Rectangular array: $n_x$ rows and $n_z$ columns with spacings $l_x$ and $l_z$; the bolt in row $j$ and column $k$ (counted from 0) is at $x^*=j\,l_x$, $z^*=k\,l_z$; “Perimeter bolts only” keeps the outer ring when there are more than 2 rows and more than 2 columns.
• Bolt circle: $n$ bolts evenly spaced on a circle of pitch diameter $d_t$ about $(x_0^*, z_0^*)$; bolt $k$ is at $\varphi_k=\varphi_0+k\cdot 360°/n$, with $x^*=x_0^*+\dfrac{d_t}{2}\sin\varphi_k$ and $z^*=z_0^*+\dfrac{d_t}{2}\cos\varphi_k$; $\varphi_0$ is measured from $+z$ towards $+x$; $\varphi_0=180°/n$ places the bolts symmetrically about the axis $z$-$z$ (VDI 2230 Part 2:2014-12, §6.3.2.3, Figure 19).

Effect on the results

• ↑ Increase: all bolts farther from the centroid: the share from a bending moment ($\propto x_i/\sum x_j^2$) and the share from a torque ($\propto r_i/\sum r_j^2$) get smaller; the shares $F_{AB}/n_S$ and $F_{QB}/n_S$ do not change (VDI 2230 Part 2:2014-12, §6.3.1.3, Eq. (16), (19); §6.3.1.2, Eq. (12); §6.3.2.1, Eq. (34)).

Standard reference: VDI 2230 Part 2:2014-12, §6.3.1.1, Eq. (1), (2), Figure 3; §6.3.1.2, Eq. (12); §6.3.1.3, Eq. (19); §6.3.2.3, Figure 19

Bolt nominal diameter (d) [mm]

Bolt diameter = outside diameter of thread (nominal diameter) (VDI 2230 Part 2:2014-12, §4, Symbols). The calculation assumes only one nominal bolt diameter in a bolt array (VDI 2230 Part 2:2014-12, §6.3.1.1).

Where to get the value

• The calculator has one field for $d$; bolt arrays of different diameters (Eq. (3), (4)) are not implemented in this calculator (VDI 2230 Part 2:2014-12, §6.3.1.1, Eq. (3), (4)).

• The calculator uses $d$ for $A=\dfrac{\pi d^2}{4}$ and for the moments of inertia $I_{xx}=A\sum z_i^2$, $I_{zz}=A\sum x_i^2$ (VDI 2230 Part 2:2014-12, §6.3.1.1, Eq. (9), (10)).

• Two bolt centres closer than $d$ give a warning that the holes may overlap.

• The hand-over to VDI 2230-1 carries only $F_A$ and $F_Q$.

• The calculator accepts only $0.1 \le d \le 200$ mm.

Effect on the results

• ↑ Increase: only $A$, $I_{xx}$, $I_{zz}$ and $I_p$ in the results grow; $A$ cancels in Eq. (17) to (19), so the axial and transverse forces of the bolts do not depend on $d$ (VDI 2230 Part 2:2014-12, §6.3.1.3, Eq. (17) to (19)).

Standard reference: VDI 2230 Part 2:2014-12, §4; §6.3.1.1, Eq. (3), (4), (7) to (10); §6.3.1.3, Eq. (17) to (19)

Transverse load, x component (F_QBx) [N]

Component along $x$ of the transverse working load $F_{QB}$ of the bolt array; it acts perpendicularly to the bolt axes and can also result from a working load $F_B$ (VDI 2230 Part 2:2014-12, §4, Symbols; §6.3.1.2, Figure 5). The calculator takes it positive along $+x$, which is up in the plan view seen from the bolt-head side.

Where to get the value

• The calculator takes the load on the whole bolt array and distributes it over the bolts; unit N.
• Transverse forces are to be transmitted by friction in the contact surfaces of the clamped parts; the bolt is to be kept free of shear loading, apart from close-fitting bolts (VDI 2230 Part 2:2014-12, §6.3.1.2).
• An oblique force $F_B$ at the angle $\alpha$ is split into $F_{AB}=F_B\sin\alpha$ and $F_{QB}=F_B\cos\alpha$ (VDI 2230 Part 2:2014-12, §6.3.1.5, Eq. (26.1), (26.2), Figure 14); the calculator takes the axial part in the axial load and the transverse parts in $F_{QBx}$ and $F_{QBz}$.

Effect on the results

• ↑ Increase: $\lvert F_{QBx}\rvert$ gives every bolt the share $F_{QBx}/n_S$ of the transverse force (VDI 2230 Part 2:2014-12, §6.3.1.2, Eq. (13); §6.3.2.1, Eq. (34)); at a height $h\ne 0$ it also changes the moment $M_z=F_{QB}\cdot h-F_{AB}\cdot x_S$ (VDI 2230 Part 2:2014-12, §6.3.1.5, Eq. (28)).

Standard reference: VDI 2230 Part 2:2014-12, §6.3.1.2, Eq. (13), Figure 5; §6.3.1.5, Eq. (26.1), (26.2), (28), Figure 14; §6.3.2.1, Eq. (34)

Axial load (F_AB) [N]

Axial load of the bolt array arising from the working load application $F_B$ (VDI 2230 Part 2:2014-12, §4, Symbols). The $y$ axis is the bolt axis and points towards you (bolt-head side); the calculator takes $F_{AB}>0$ as tension (it pulls the clamped part away from its base) and $F_{AB}<0$ as compression.

Where to get the value

• Unit N. The calculator takes the end thrust $p\cdot A_I$ of an internal pressure from $p$ in card 5, not from this field (VDI 2230 Part 2:2014-12, §6.3.2.4, Eq. (45)).
• If the line of action of the working load does not coincide with the centroid $S$, the loading is eccentric and the bolt array is additionally loaded with a bending moment (VDI 2230 Part 2:2014-12, §6.3.1.3, Figure 10); the calculator takes the force as entered and adds this moment from the load point of card 3.
• For an oblique force the part along the bolt axis is $F_{AB}=F_B\sin\alpha$ (VDI 2230 Part 2:2014-12, §6.3.1.5, Eq. (26.1)).
• With a fixed axis of rotation (card 4) the calculator requires $F_{AB}=0$: Eq. (23) is given for a pure bending moment (VDI 2230 Part 2:2014-12, §6.3.1.4, Eq. (23), Figure 11).

Effect on the results

• ↑ Increase: every bolt takes the same additional axial load $F_{AB}/n_S$ (VDI 2230 Part 2:2014-12, §6.3.1.3, Eq. (16); §6.3.1.5, Eq. (27)); with the load point off the centroid the moment $M_z=F_{QB}\cdot h-F_{AB}\cdot x_S$ also changes the bending shares (VDI 2230 Part 2:2014-12, §6.3.1.5, Eq. (28)).

Standard reference: VDI 2230 Part 2:2014-12, §6.3.1.3, Eq. (16), Figure 10; §6.3.1.4, Eq. (23), Figure 11; §6.3.1.5, Eq. (26.1), (27), (28); §6.3.2.4, Eq. (45)

Transverse load, z component (F_QBz) [N]

Component along $z$ of the transverse working load $F_{QB}$ of the bolt array; it acts perpendicularly to the bolt axes and can also result from a working load $F_B$ (VDI 2230 Part 2:2014-12, §4, Symbols; §6.3.1.2, Figure 6). The calculator takes it positive along $+z$, which is to the right in the plan view seen from the bolt-head side.

Where to get the value

• The calculator takes the load on the whole bolt array and distributes it over the bolts; unit N.
• An oblique force $F_B$ at the angle $\alpha$ has the transverse part $F_{QB}=F_B\cos\alpha$ (VDI 2230 Part 2:2014-12, §6.3.1.5, Eq. (26.2)).

Effect on the results

• ↑ Increase: $\lvert F_{QBz}\rvert$ gives every bolt the share $F_{QBz}/n_S$ of the transverse force (VDI 2230 Part 2:2014-12, §6.3.1.2, Eq. (14); §6.3.2.1, Eq. (34)); at a height $h\ne 0$ the calculator also forms $M_{x,S}=M_x+h\,F_{QBz}-z_S\,F_{AB}$, analogous to Eq. (28) (VDI 2230 Part 2:2014-12, §6.3.1.5, Eq. (28)).

Standard reference: VDI 2230 Part 2:2014-12, §6.3.1.2, Eq. (14), Figure 6; §6.3.1.5, Eq. (26.2), (28); §6.3.2.1, Eq. (34)

Bending moment about x (M_x) [N·mm]

Component about the $x$ axis of the working moment (bending moment) $M_B$ acting on a bolting point, $M_B=M_x$ (VDI 2230 Part 2:2014-12, §4, Symbols; §6.3.1.3, Eq. (17)). The calculator takes it positive by the right-hand rule about $+x$; a positive $M_x$ loads the bolts with $z_i<0$ (VDI 2230 Part 2:2014-12, §6.3.1.3, Eq. (18)).

Where to get the value

• Unit N·mm. The calculator takes the moment acting on the bolt array as such and adds the moments of off-centre forces and of the height $h$: $M_{x,S}=M_x+h\,F_{QBz}-z_S\,F_{AB}$.
• If all bolts lie on the axis of the moment ($\sum z_j^2=0$ in Eq. (19)), the calculator refuses and refers to the fixed axis of rotation of card 4 (VDI 2230 Part 2:2014-12, §6.3.1.3, Eq. (19); §6.3.1.4, Eq. (23)).

Effect on the results

• ↑ Increase: $\lvert M_x\rvert$ raises the axial load of a bolt in proportion to its distance from the $x$ axis ($F_{A(M)i}\propto M_x\,z_i/\sum z_j^2$); bolts on the centroidal axis are not loaded (VDI 2230 Part 2:2014-12, §6.3.1.3, Eq. (18), (19); §6.3.1.4, Figure 12).

Standard reference: VDI 2230 Part 2:2014-12, §6.3.1.3, Eq. (17) to (19); §6.3.1.4, Eq. (23), Figure 12

Torque about the bolt axis (M_Y) [N·mm]

Torque about the bolt axis or the bolt array (VDI 2230 Part 2:2014-12, §4, Symbols). A torque $M_Y$ about the axis $y$ perpendicular to the interface rotates the bolt array about the centroid $S$ (VDI 2230 Part 2:2014-12, §6.3.1.2, Figure 7). The calculator takes it positive by the right-hand rule about $+y$, counter-clockwise seen from the bolt-head side.

Where to get the value

• Unit N·mm. The calculator takes the torque acting on the bolt array as such and adds the torque of an off-centre transverse load: $M_{Y,S}=M_Y+z_S\,F_{QBx}-x_S\,F_{QBz}$.
• The calculator divides the torque in proportion to the distance from the centroid, which needs a slight relative displacement (slip) in the interface; if such displacements are not permissible, only the bolting points with the shortest distance to the load introduction are to be included (VDI 2230 Part 2:2014-12, §6.3.1.2, Figure 8).
• A warning appears when the bolts have more than one distance from the centroid.
• When the moment is introduced and exits via a structure inside the bolt array, the inner bolts (distance $r_{min}$) are more highly loaded; Annex A1 is not implemented in this calculator (VDI 2230 Part 2:2014-12, Annex A1, Eq. (A1), Figure A1).
• The torque is not part of the global slip check of card 6 (VDI 2230 Part 2:2014-12, §6.3.1.2, Eq. (11)).

Effect on the results

• ↑ Increase: $\lvert M_Y\rvert$ raises the transverse force of every bolt in proportion to its distance from the centroid ($F_{qM}\propto M_Y\,r_i/\sum r_j^2$); it is largest at the bolt with the greatest distance $r_{max}$ (VDI 2230 Part 2:2014-12, §6.3.1.2, Eq. (12)).

Standard reference: VDI 2230 Part 2:2014-12, §6.3.1.2, Eq. (11) to (14), Figures 7, 8; Annex A1, Eq. (A1), Figure A1

Bending moment about z (M_z) [N·mm]

Component about the $z$ axis of the working moment (bending moment) $M_B$ acting on a bolting point, $M_B=M_z$ (VDI 2230 Part 2:2014-12, §4, Symbols; §6.3.1.4). The calculator takes it positive by the right-hand rule about $+z$; a positive $M_z$ loads the bolts with $x_i>0$ (VDI 2230 Part 2:2014-12, §6.3.1.3, Eq. (18)).

Where to get the value

• Unit N·mm. The calculator takes the moment acting on the bolt array as such and adds the moments of off-centre forces and of the height $h$: $M_{z,S}=M_z+x_S\,F_{AB}-h\,F_{QBx}$.
• If all bolts lie on the axis of the moment ($\sum x_j^2=0$ in Eq. (19)), the calculator refuses and refers to the fixed axis of rotation of card 4 (VDI 2230 Part 2:2014-12, §6.3.1.3, Eq. (19); §6.3.1.4, Eq. (23)).

Effect on the results

• ↑ Increase: $\lvert M_z\rvert$ raises the axial load of a bolt in proportion to its distance from the $z$ axis ($F_{A(M)i}\propto M_z\,x_i/\sum x_j^2$); bolts on the centroidal axis are not loaded (VDI 2230 Part 2:2014-12, §6.3.1.3, Eq. (18), (19); §6.3.1.4, Figure 12).

Standard reference: VDI 2230 Part 2:2014-12, §6.3.1.3, Eq. (18), (19); §6.3.1.4, Eq. (23), Figure 12; §6.3.1.5, Eq. (25), Figure 13

Load application point

Position where the line of action of the external force passes the interface in the plan view. If the point of application or the line of action of the working load does not coincide with the centroid $S$, the loading is eccentric and the bolt array is additionally loaded with a bending moment (VDI 2230 Part 2:2014-12, §6.3.1.3, Figure 10).

Where to get the value

• Through the centroid S (default): the calculator sets $x_S=z_S=0$.
• Custom point: the calculator asks for $x_F^*$ and $z_F^*$ and takes $x_S=x_F^*-x_S^*$ and $z_S=z_F^*-z_S^*$ as the distances of the force application from the centroid along $x$ and $z$ (VDI 2230 Part 2:2014-12, §4, Symbols); it adds the moments of this offset to the moments of card 2.

Standard reference: VDI 2230 Part 2:2014-12, §4; §6.3.1.1, step 4; §6.3.1.3, Figure 10; §6.3.1.5, Eq. (28), Figure 15

Load point x_F^* [mm]

Coordinate $x^*$ of the load application point in the frame of the bolt coordinates. The calculator takes $x_S=x_F^*-x_S^*$ as the distance of the force application from the centroid along $x$ (VDI 2230 Part 2:2014-12, §4, Symbols). The field appears only for the custom load point.

Where to get the value

Required for the custom load point. The calculator takes it in the frame of the bolt coordinates: seen from the bolt-head side, $x^*$ is positive upwards.

Effect on the results

• ↑ Increase: a larger distance $x_S$ raises the moment of an axial load about $S$, $M_z=F_{QB}\cdot h-F_{AB}\cdot x_S$ (VDI 2230 Part 2:2014-12, §6.3.1.5, Eq. (28)).

Standard reference: VDI 2230 Part 2:2014-12, §4; §6.3.1.1, Figure 3; §6.3.1.5, Eq. (28)

Load point z_F^* [mm]

Coordinate $z^*$ of the load application point in the frame of the bolt coordinates. The calculator takes $z_S=z_F^*-z_S^*$ as the distance of the force application from the centroid along $z$ (VDI 2230 Part 2:2014-12, §4, Symbols). The field appears only for the custom load point.

Where to get the value

Required for the custom load point. The calculator takes it in the frame of the bolt coordinates: seen from the bolt-head side, $z^*$ is positive to the right.

Effect on the results

• ↑ Increase: a larger distance $z_S$ raises the moment of an axial load about $S$; the calculator forms $M_{x,S}=M_x+h\,F_{QBz}-z_S\,F_{AB}$, analogous to Eq. (28) (VDI 2230 Part 2:2014-12, §6.3.1.5, Eq. (28)).

Standard reference: VDI 2230 Part 2:2014-12, §4; §6.3.1.1, Figure 3; §6.3.1.5, Eq. (28)

Height above the interface (h) [mm]

Height $h$ of the load application point above the interface (VDI 2230 Part 2:2014-12, §4, Symbols; §6.3.1.5, Figure 15). The calculator measures it along the bolt axis towards the bolt-head side, for both load point modes.

Where to get the value

Default 0. The calculator adds the moments $h\,F_{QBz}$ to $M_{x,S}$ and $-h\,F_{QBx}$ to $M_{z,S}$ itself, so the fields $M_x$ and $M_z$ take only the moments acting directly on the bolt array.

Effect on the results

• ↑ Increase: at a larger $\lvert h\rvert$ the transverse load gives a larger moment about $S$, $M_z=F_{QB}\cdot h-F_{AB}\cdot x_S$ (VDI 2230 Part 2:2014-12, §6.3.1.5, Eq. (28)).

Standard reference: VDI 2230 Part 2:2014-12, §4; §6.3.1.5, Eq. (28), Figure 15

Bending rotation model

How the bolt array turns under a bending moment. Free rotation: the bolts share the moment in proportion to their distance from the principal axes through the centroid (VDI 2230 Part 2:2014-12, §6.3.1.3, Eq. (18), (19); §6.3.1.4, Figure 12; §6.3.1.5, Figure 13). Fixed axis of rotation: the greatest tensile force acts on the bolt row farthest from the axis (VDI 2230 Part 2:2014-12, §6.3.1.4, Eq. (23), Figure 11).

Where to get the value

• Free rotation about the centroid (default): the calculator uses Eq. (19) for every combination with an axial load and for internal pressure; bolts on a centroidal axis are not loaded (VDI 2230 Part 2:2014-12, §6.3.1.3, Eq. (19); §6.3.1.4, Figure 12).
• Fixed axis parallel to z or to x: the axis lies at $x^*=x_D^*$ or at $z^*=z_D^*$, given in the next field; the calculator takes $F_{A(M)i}=\dfrac{\lvert M_{axis}\rvert\,e_i}{\sum e_j^2}$ with $e_i$ the distance of bolt $i$ from the axis (VDI 2230 Part 2:2014-12, §6.3.1.4, Eq. (23)).
• With a fixed axis the calculator requires $F_{AB}=0$ and $p=0$, and the moment component perpendicular to the axis must be 0 after the moments are moved to the centroid; transverse load and torque are allowed.
• With a moment about the axis ($\lvert M_{axis}\rvert\ne 0$), all bolts must lie on the side that the moment lifts or on the axis; otherwise the calculator refuses.
• The calculator shows a notice that the fixed axis needs a contact (an edge or a support) able to carry compression, and reports its reaction $R_D=\sum F_{A(M)i}$.

Standard reference: VDI 2230 Part 2:2014-12, §6.3.1.3, Eq. (18), (19); §6.3.1.4, Eq. (23), Figures 11, 12; §6.3.1.5, Figure 13

Fixed axis position (x_D^* or z_D^*) [mm]

Position of the fixed axis of rotation in the frame of the bolt coordinates: $x_D^*$ for an axis parallel to $z$, $z_D^*$ for an axis parallel to $x$ (VDI 2230 Part 2:2014-12, §6.3.1.4, Figure 11). The field appears only for a fixed axis, and its label shows which of the two it is.

Where to get the value

• Required for a fixed axis. The calculator takes it in the frame of the bolt coordinates ($x^*$ up, $z^*$ to the right, seen from the bolt-head side).
• With a moment about the axis the calculator checks, for an axis parallel to $z$: $x_i^*\ge x_D^*$ when $M_{z,S}>0$ and $x_i^*\le x_D^*$ when $M_{z,S}<0$; for an axis parallel to $x$: $z_i^*\le z_D^*$ when $M_{x,S}>0$ and $z_i^*\ge z_D^*$ when $M_{x,S}<0$. Otherwise it refuses with the message that a bolt is on the pressed side.

Effect on the results

• ↑ Farther axis: the farther the axis of rotation is from the bolt array, the smaller the bolt loading; in case 1 of Figure 11 row I carries a bolt load 2.5 times higher than in case 2 (VDI 2230 Part 2:2014-12, §6.3.1.4, Figure 11).

Standard reference: VDI 2230 Part 2:2014-12, §6.3.1.4, Eq. (23), Figure 11

Internal pressure (p) [MPa]

Internal pressure in a flange. A bending moment in the circumferential direction results from the deformation of the flange; the calculation takes a circular plate clamped all round with the pressure-loaded area $A_I$ (VDI 2230 Part 2:2014-12, §6.3.2.4, Figure 20). The calculator applies it only to one evenly spaced bolt circle; with $p=0$ the whole card is ignored.

Where to get the value

• Unit MPa (N/mm²).

• With $p>0$ the calculator requires: free rotation as the bending model; at least 3 bolts, all on one pitch circle and evenly spaced (it allows a radius deviation of 0.1 % of the mean radius and an angular deviation of 0.1° from $360°/n_S$); and $D_I$ given. Otherwise it refuses.

• The calculator takes the end thrust $p\cdot A_I$ from this field, not from the axial load (VDI 2230 Part 2:2014-12, §6.3.2.4, Eq. (45)).

• If gaskets are arranged between the plates in the main connection, the calculation is to follow DIN EN 1591 and DIN EN 13445, not VDI 2230 (VDI 2230 Part 2:2014-12, §3.2, Flange, Note).

• The calculator accepts only $p \ge 0$ MPa.

Effect on the results

• ↑ Increase: the maximum axial load of every bolt rises in proportion to $p$: $F_{A\max}=F_A+F_{AM}=1.25\,\dfrac{p\,A_I}{n_S}$ (VDI 2230 Part 2:2014-12, §6.3.2.4, Eq. (44) to (46)).

Standard reference: VDI 2230 Part 2:2014-12, §3.2, Flange; §6.3.2.4, Eq. (43) to (46), Figure 20; §6.3.2.5

Pressurised plate diameter (D_I) [mm]

Internal diameter (VDI 2230 Part 2:2014-12, §4, Symbols); here the diameter of the circular plate under internal pressure (VDI 2230 Part 2:2014-12, §6.3.2.4, Figure 20). The calculator takes the pressure-loaded area as $A_I=\dfrac{\pi D_I^2}{4}$. The field appears, and is required, only when $p>0$.

Where to get the value

The calculator requires $D_I$ larger than 0 and smaller than the pitch diameter $d_t$ of the bolt circle, taking $d_t=2\bar r$ (twice the mean distance of the bolts from the centroid), and refuses otherwise.

Effect on the results

• ↑ Increase: the area $A_I$ grows with $D_I^2$, and so does $F_{A\max}=1.25\,p\,A_I/n_S$ (VDI 2230 Part 2:2014-12, §6.3.2.4, Eq. (43) to (46)).

Standard reference: VDI 2230 Part 2:2014-12, §4; §6.3.2.4, Eq. (43) to (46), Figure 20

Minimum coefficient of friction (μ_T,min)

Coefficient of friction in the interface, minimum value (VDI 2230 Part 2:2014-12, §6.3.1.2, Eq. (11)). As long as contact of the bolts with the hole wall can be ruled out (fundamental requirement), Eq. (11) is an approximation for the transverse force carried by friction (VDI 2230 Part 2:2014-12, §6.3.1.2, Eq. (11)). The field is optional: if it is empty, the calculator skips the global slip check.

Where to get the value

• Allowed range: more than 0 and at most 1.
• The printed Eq. (11) has $\mu_{T,min}$ in the denominator; the calculator evaluates $\sum F_V\cdot\mu_{T,min}\cdot q_F>F_{QBmax}$ with $F_{QBmax}=\sqrt{F_{QBx}^2+F_{QBz}^2}$ (VDI 2230 Part 2:2014-12, §6.3.1.2, Eq. (11)). Eq. (94) of the same part has $\mu_{TH}$ in the numerator (VDI 2230 Part 2:2014-12, §7.4, R12, Eq. (94)).
• The results give the required sum of preloads $\sum F_{V,erf}$ and its mean per bolt.
• Eq. (11) counts the transverse load only, not the torque (VDI 2230 Part 2:2014-12, §6.3.1.2, Eq. (11)).

Effect on the results

• ↓ Decrease: a lower $\mu_{T,min}$ raises the required sum of preloads in inverse proportion: $\sum F_{V,erf}=\dfrac{F_{QBmax}}{\mu_{T,min}\,q_F}$ as evaluated by the calculator (VDI 2230 Part 2:2014-12, §6.3.1.2, Eq. (11)).

Standard reference: VDI 2230 Part 2:2014-12, §6.3.1.2, Eq. (11); §7.4, R12, Eq. (94)

Number of force-transmitting internal interfaces (q_F)

Number of force-transmitting ($F_{QB}$) internal interfaces which are involved in possible slipping in the joint and/or a shearing load on the bolts (VDI 2230 Part 2:2014-12, §4, Symbols; §6.3.1.2, Eq. (11)). The calculator uses it only when $\mu_{T,min}$ is given.

Where to get the value

Default 1; a whole number of at least 1.

• The calculator accepts only $q_F \ge 1$.

Effect on the results

• ↑ Increase: each additional interface lowers the required sum of preloads in inverse proportion, $\sum F_{V,erf}=\dfrac{F_{QBmax}}{\mu_{T,min}\,q_F}$ as evaluated by the calculator (VDI 2230 Part 2:2014-12, §6.3.1.2, Eq. (11)).

Standard reference: VDI 2230 Part 2:2014-12, §4; §6.3.1.2, Eq. (11)

Calculation steps

The “Steps” page of the calculator lists the whole calculation chain step by step: each step gives the formula, the substituted values, the result and the clause of the standard.

The fixed axis replaces step 7, and pressure and slip each insert one step; the numbers of the later steps shift accordingly. When a bending moment acts and the equilibrium check finds that the axes are not principal axes, the calculation is refused; see the FAQ .

Reading the results

Where to look first

1. The candidate critical bolts card: lists the numbers of the candidate bolts with the reasons as badges. Click a number to select that bolt for the hand-over.
2. The no-tension or no-axial-load notice: when no bolt is in tension, a banner appears on the candidate critical bolts card, and the same notice (H-notension) is also listed in the standing card. With bolts in compression it says “No bolt is in tension”; when all $F_{Ai}=0$ (only transverse force or torque) it says “No axial load”. The candidates are still listed.
3. The main figure: figure 1 is the plan view of the bolt pattern, figure 2 is a bar chart of the forces of each bolt.
4. Bolt array — global quantities and Per-bolt results.
5. Standing card and pop-up: the standing card is shown after every calculation; the pop-up codes that this load case triggers are shown after the calculation and listed in the “What these warnings mean” card.

The calculator gives no verdict of pass or fail; it only distributes the loads. The checks of each candidate bolt are made in VDI 2230-1 (VDI 2230 Part 2:2014-12, §6.2).

How the candidates are chosen

The critical single-bolt joint is the first one in which a failure is expected from exceeding the permissible operating load or the permissible fatigue strength, or from joint opening; several bolts may have to be investigated (VDI 2230 Part 2:2014-12, §6.2). The calculator lists every bolt and flags the candidates by these rules:

• max $F_A$: the bolts with the largest axial force $F_{Ai}$, all tied bolts included; only if the maximum is above zero.
• max $F_Q$: the bolts with the largest transverse force $F_{qi}$, all tied bolts included; only if the maximum is above zero.
• max resultant: when both of the above exist and have no bolt in common, the bolts with the largest resultant $F_{res,i}=\sqrt{\max(F_{Ai},0)^2+F_{qi}^2}$ are added.
• A bolt can belong to several groups; the card and the table show all its reasons.
• Default bolt for the hand-over: the bolt with the largest $F_A$ (the lowest number if tied); with no bolt in tension, the bolt with the largest $F_Q$.
• The calculator forms no envelope: it does not pair the largest $F_A$ with the largest $F_Q$ of another bolt.

Per-bolt results

One row per bolt, in the order of input. Click a row or the radio button at its start to select the bolt; the bolt gets an extra highlight ring in the figure. Candidate rows are highlighted.

How to read the figures

• Figure 1, plan view of the bolt pattern: the interface seen from the bolt-head side, $z$ to the right, $x$ up, $y$ towards you. A bolt in tension ($F_A>0$) is filled, the darker the larger the force, with ⊙ at its centre; a bolt in compression ($F_A<0$) is drawn as an outline only, with ⊗ at its centre; a bolt with $F_A=0$ has neither fill nor symbol. Transverse forces are arrows whose length is proportional to $F_{qi}$. The principal axes are thin dashed lines, the centroid $S$ is a solid cross, and the origin $O$ of the reference frame is a small square. The external loads are drawn at the load point $F$ (arrows for transverse forces, ⊙ / ⊗ for $F_{AB}$, an arc arrow for $M_Y$, double arrows for $M_x$ and $M_z$), and their values are written in the “External loads” box of the figure; a corner shows the directions $x$, $y$, $z$. Candidate bolts have a bold outline in the warning colour and carry the labels $F_A$ and $F_q$. The bolt you select gets an extra highlight ring and the same labels. The fixed-axis model draws the axis line (warning colour, dashed); with internal pressure the pitch circle and the $D_I$ circle are drawn (dashed). With no bolt in tension the figure says “No bolt in tension”, and with all $F_A=0$ it says “No axial load”. Ctrl + wheel zooms and dragging pans; clicking a bolt in the figure also selects it.
• Figure 2, bar chart: one group of bars per bolt, “axial force” and “transverse force”; below the zero line is compression. A pin marks the axial-force bar of a candidate bolt.
• The Schematic tab draws the bolt pattern from the current input before any calculation.

Bolt array — global quantities

Warnings and notices

The notices in the results area use two channels: the standing card (the “Assumptions and notes” card, shown after every calculation and included in the PDF report) and the pop-up (shown after the calculation and also listed in the “What these warnings mean” card).

Standing card

Pop-up

Hand-over to VDI 2230-1

What is handed over is the own pair of forces of the selected bolt; no envelope is formed:

• $F_A^{EV}=\max(F_{Ai},0)$. If the selected bolt is in compression, 0 is handed over, with the notice “The selected bolt is in compression under this load case; the hand-over axial force is 0”.
• $F_Q^{EV}=F_{qi}$, the resulting transverse force, torque share included.
• The eccentricities $a_{load}$ and $s_{sym}$ are not handed over.
• The load introduction factor $n$ is not handed over. Rigid body models give $a=0$ and $n=1$; for the residual clamp load, $n=0$ should be used (VDI 2230 Part 2:2014-12, §6.2, Figure 2, note a). The VDI 2230-1 calculator requires $0\lt n\le 1$.
• The minimum axial load $F_{A,min}$, the torque $M_Y$, the pressure $p_i$ and the sealing area $A_D$ are not handed over.

Procedure:

1. Select a bolt in “Per-bolt results” (click the row, or click the number on the candidate card); the “Hand-over to VDI 2230-1” card shows $F_A^{EV}$ and $F_Q^{EV}$ of this bolt.
2. Click “Check this bolt in VDI 2230-1 →” on the hand-over card, or “VDI 2230-1 →” at the end of a candidate row. The page jumps to VDI 2230-1 in the same tab and overwrites only $F_A$ and $F_Q$; the other inputs stay. The values are rounded to $10^{-4}$ N, and the hand-over is valid for 60 seconds.
3. VDI 2230-1 opens the matching input card and shows a notice; the “Input source” on its “Steps” page names the source bolt and can expand the steps of this calculator.
4. Bolt size, material and tightening method are entered separately in VDI 2230-1.

In card 3 of VDI 2230-1, the “Multi-bolt (VDI 2230-2)” entry below $F_A$ also takes $F_A$ and $F_Q$ of the selected bolt from this calculator.

Limits and cautions

• Rigid-body assumption: the clamped components are assumed to be considerably stiffer than the bolts, and the bolts are replaced by point bearings in the interface; only axial and transverse forces can be calculated, bending moments in the bolts and interfaces are ignored, and the result can be considerably larger and also smaller than reality (VDI 2230 Part 2:2014-12, §5.2).
• Fatigue verification: the rigid-body model is only of very limited application for check calculations under vibrational loading (VDI 2230 Part 2:2014-12, §5.2); the calculator does not hand over a minimum axial load $F_{A,min}$.
• Transverse load carried by friction: transverse forces are to be transmitted by friction in the contact surfaces of the clamped parts, and the bolt is to be kept free of shear loading, apart from close-fitting bolts (VDI 2230 Part 2:2014-12, §6.3.1.2).
• Torque share: the share in proportion to $r_i$ needs a slight relative displacement (slip) in the interface; if it is not permissible, only the bolting points with the shortest distance to the load introduction are included (VDI 2230 Part 2:2014-12, §6.3.1.2, Figure 8).
• Bolt row parallel to the transverse load: when the bolt row runs parallel to the direction of force, the bolts are actually loaded unevenly and the two outer bolts bear the greatest load, while the rigid-body approach gives all bolts an equal share (VDI 2230 Part 2:2014-12, §6.3.1.2, Figure 6).
• Linear superposition: for combined loads the superposition principle applies to rigid bodies; when the load case is unclear, the case of the greatest possible bolt loading is assumed (VDI 2230 Part 2:2014-12, §6.3.1.2; §6.3.1.5).
• No opening: Eq. (39) to (42) assume that no partial opening of the joint occurs (VDI 2230 Part 2:2014-12, §6.3.2.3).
• Axes must be principal axes: with a bending moment and axes that are not principal axes, the calculator refuses (VDI 2230 Part 2:2014-12, §6.3.1.1, step 2).
• Internal pressure: only one evenly spaced bolt circle and free rotation; the pressurised plate must be smaller than the pitch diameter.
• Fixed axis: only a pure bending moment with transverse load or torque; with a moment about the axis ($\lvert M_{axis}\rvert\ne 0$) all bolts must lie on the lifted side or on the axis, and with only a transverse load or torque this is not checked.
• One bolt size only: the area-weighted equations for bolts of different sizes (Eq. (3), (4)) are not implemented in this calculator (VDI 2230 Part 2:2014-12, §6.3.1.1).
• Global slip check: Eq. (11) counts the transverse load only, not the torque (VDI 2230 Part 2:2014-12, §6.3.1.2, Eq. (11)).

Frequently asked questions

How can I try it quickly? The “Verification examples” list above the calculator has three presets: example 1 is a rectangular array under transverse force and torque, example 2 is eccentric tension on a 2x2 array, example 3 is a circular flange under bending moment, axial force and internal pressure. Select one and click “Run calculation”.

The calculation is refused: the axes are not the principal axes of the bolt array. With a bending moment, Eq. (18), (19) presuppose the principal axis system (VDI 2230 Part 2:2014-12, §6.3.1.1, steps 2 and 3). Without a bending moment the calculator does not make this check.

The calculation is refused: the rigid-body model cannot resist this moment. When all bolts lie on the axis of the moment (for example, all bolts have the same $x$ coordinate under $M_z$), the denominator of Eq. (19) is zero (VDI 2230 Part 2:2014-12, §6.3.1.3, Eq. (19)); the calculator refers to the fixed-axis model of card 4.

The fixed axis is refused: a bolt is on the pressed side. With a moment about the axis, the calculator requires all bolts to lie on the side that the moment lifts or on the axis; with only a transverse load or torque this is not checked. The conditions are in Fixed axis position .

Internal pressure is refused. Internal pressure requires one evenly spaced bolt circle (at least 3 bolts on one pitch circle), the free-rotation model, a given pressurised plate diameter $D_I$, and $D_I$ smaller than the pitch diameter.

More than one candidate bolt. Several bolts may have to be investigated (VDI 2230 Part 2:2014-12, §6.2); the calculator hands over each bolt separately.

Which $n$ after the hand-over to VDI 2230-1? Rigid body models give $a=0$ and $n=1$; for the residual clamp load, $n=0$ should be used (VDI 2230 Part 2:2014-12, §6.2, Figure 2, note a). The VDI 2230-1 calculator requires $0\lt n\le 1$.