A bolted joint: preload, torque, and what the external load does

A bolted joint is not a bolt carrying a load. It is a spring in tension clamping springs in compression, and almost all of the external load is taken by the members relaxing rather than by the bolt stretching further. That is why a joint is designed around its preload and only then checked against the load it carries — and why a torque figure without a preload behind it says nothing. The joint: two 15 mm steel plates, one M12 × 1.75 bolt of property class 8.8, carrying 10 kN of external tension per bolt.

use steel

Four significant figures. A preload is not known to six, and a page of six digit numbers invites a reader to believe the last two.

digits 4

The bolt

The tensile stress area is the area of a circle whose diameter is the mean of the pitch and minor diameters — a definition, not a table lookup, so it is written out (ISO 724).

d

12 mm

pitch

1.75 mm

A_{s}

\frac{π}{4} \cdot {(d - 0.9382 \cdot pitch)}^{2}

\frac{π}{4} \cdot {(12 mm - 0.9382 \cdot 1.75 mm)}^{2}

84.27 {mm}^{2}

Property class 8.8 states its own strengths: the first digit is the tensile strength in hundreds of MPa, and the second is ten times the ratio of yield to tensile (ISO 898-1). So the class is the data, and nothing here is copied from a table of them.

R_{m}

800 MPa

R_{e}

0.8 \cdot R_{m}

0.8 \cdot 800 MPa

640 MPa

The proof stress is where the standard's permanent-set limit sits, close to nine tenths of yield. ISO 898-1 tabulates 580 MPa for this class up to M16; this is that number, arrived at rather than quoted.

S_{p}

0.9 \cdot R_{e}

0.9 \cdot 640 MPa

576 MPa

Preload

Seventy-five per cent of the proof load is the usual target for a reusable joint: high enough that the members stay in compression under load, low enough that scatter in the tightening method does not take the bolt past proof.

F_{i}

0.75 \cdot A_{s} \cdot S_{p}

0.75 \cdot 84.27 {mm}^{2} \cdot 576 MPa

36.4 kN

The torque that produces it, through the nut factor K — which is not a friction coefficient but an empirical lumping of thread and face friction and the thread geometry. 0.2 is the conventional dry-steel value, and it is the least certain number on this page: K varies by ±25% with lubrication, plating and reuse, and the preload varies with it.

K_{nut}

0.2

T

K_{nut} \cdot F_{i} \cdot d

0.2 \cdot 36.4 kN \cdot 12 mm

87.37 N*m

Stiffness, and how the external load is shared

The bolt is a bar in tension over its grip length. The tensile stress area is used over the whole of it, which understates the stiffness of a bolt with a plain shank in the grip — the shank is the full 12 mm — and so overstates the bolt's share of the external load. That is the safe direction for this check and the wrong one for a fatigue calculation, which wants the two lengths in series.

The members are the compressed cone around the hole, for which the standard frustum approximation is used (Shigley, for two members of the same material with a 30° cone half-angle).

L_{g}

30 mm

k_{b}

\frac{A_{s} \cdot E_{steel}}{L_{g}}

\frac{84.27 {mm}^{2} \cdot 200 GPa}{30 mm}

561.8 MN/m

k_{m}

\frac{0.5774 \cdot π \cdot E_{steel} \cdot d}{2 \cdot \ln (\frac{5 \cdot (0.5774 \cdot L_{g} + 0.5 \cdot d)}{0.5774 \cdot L_{g} + 2.5 \cdot d})}

\frac{0.5774 \cdot π \cdot 200 GPa \cdot 12 mm}{2 \cdot \ln (\frac{5 \cdot (0.5774 \cdot 30 mm + 0.5 \cdot 12 mm)}{0.5774 \cdot 30 mm + 2.5 \cdot 12 mm})}

2414 MN/m

The joint constant is the fraction of an external load that reaches the bolt. It is small — the members are several times stiffer than the bolt — and that is the whole reason a preloaded joint survives a fatigue load that would break the same bolt loose.

C

\frac{k_{b}}{k_{b} + k_{m}}

\frac{561.8 MN/m}{561.8 MN/m + 2414 MN/m}

0.1888

Under the external load

P_{ext}

10 kN

F_{bolt}

F_{i} + C \cdot P_{ext}

36.4 kN + 0.1888 \cdot 10 kN

38.29 kN

σ_{bolt}

\frac{F_{bolt}}{A_{s}}

\frac{38.29 kN}{84.27 {mm}^{2}}

454.4 MPa

The bolt must stay below yield with the external load applied.

check

σ_{bolt} \leq R_{e}

454.4 MPa \leq 640 MPa

pass

Separation

The joint opens when the external load has relieved all of the clamp. Past that point the bolt carries the load directly, the members no longer share it, and the fatigue argument above stops being true.

P_{sep}

\frac{F_{i}}{1 - C}

\frac{36.4 kN}{1 - 0.1888}

44.88 kN

check

P_{ext} \leq 0.75 \cdot P_{sep}

10 kN \leq 0.75 \cdot 44.88 kN

pass

Bolt force against external load

Two straight lines with a corner between them, and the corner is separation. Below it the bolt sees only its share of the load and the line is nearly flat — which is the point of the whole arrangement. Above it the members have let go and the bolt carries everything, so the line turns and follows the load.

The horizontal axis is the external load in kN and the vertical is the bolt force in the same unit.

fn bolt_force defined

plot ({bolt}_{force}, 0, 60)