Bolted Joint Stiffness and Separation Calculator

Bolt stiffness with the shank and thread in series, member stiffness from the Rotscher cone at a half-angle you can change, the load factor Φ they give — typically 0.15 to 0.25 for a steel joint, so the bolt feels a fifth of an external load — and the external load at which the joint separates and stops protecting it.

Last updated 24 Sep 2026

Joint stiffness and separation

Bolt, grip and members → Φ and separation

Thread sizeBoth stiffnesses depend on it: the bolt through its areas and the members through the hole and the bearing face the compression cone starts from.

Property class or gradeThe class sets the preload, which sets the separation load. It does not change the load factor at all — that is geometry and modulus only.

Preload, % of the proof stress75% removable, 90% permanent. This is the number the separation load is proportional to, and it is the reason preload is worth getting right.

Grip length — total thickness clamped (mm)From the bearing face of the head to the bearing face of the nut, including washers. A long grip makes the bolt softer and the members softer too, but not in the same proportion.

Of that, how much is thread inside the grip (mm)A partially threaded bolt has a stiff shank and a softer threaded length in series. The two are computed separately here, on the shank area and the tensile stress area.

Member materialOnly the modulus matters for stiffness, and it varies by a factor of twenty across this list. A soft member is what destroys the protection preload gives.

Modulus of the members (N/mm²)205,000 for bolt steel, 190,000 for structural steel, 180,000 for grey iron, 100,000 for brass, 69,000 for aluminium, 46,000 for magnesium, 9,500 for 30% glass filled nylon. A gasket is far softer still.

Member tensile strength (N/mm²)Not used for stiffness; carried so the bearing pressure under the head can be checked.

Member shear strength (N/mm²)Not used on this page; carried for consistency with the thread engagement calculator.

External tensile load on the joint, per bolt (N)The load trying to pull the joint apart, on top of the preload. The whole point of this page is how little of it the bolt actually feels.

Cone half-angle for the member stiffness (degrees)Sandia’s guideline: between 25 and 33 degrees, 30 recommended. It is the model’s biggest free parameter and it moves the member stiffness by about a quarter across that range — which is why this page shows you the number rather than hiding it.

Not a circuit: the classic joint diagram, and the drawing this batch worked hardest on. The horizontal axis is deflection and the vertical axis is force. The line rising from the left is the BOLT stretching; the line falling to the right is the MEMBERS being compressed; they meet at the preload, which is the horizontal line across the figure. The apex is placed at 1 − Φ of the way along, which makes the two slopes the real stiffness ratio — a steel joint's members are three or four times stiffer than its bolt, so the bolt line is shallow and long and the member line is steep and short. That is the whole reason an external load barely reaches the bolt: the load moves both parts the same DISTANCE, and a shallow slope times a small distance is a small force. The vertical line is the external load you entered, placed the same fraction of the way from the apex to the right-hand end as your load is of the separation load; read the bolt line where it crosses that vertical and you have the bolt's tension. The double line at the right-hand end is SEPARATION, where the member line reaches zero. Past it there is no member line left, the bolt's line is the only one, and the joint is a hinge.

33.13kNExample

M10 class 8.8 at 75% of proof, 30 mm of grip with 10 mm of that threaded, structural steel members, 10 kN of external tensile load per bolt

Two springs, one load, and the fraction the bolt feels

1/k_b = L_shank/(A_shank E) + L_thread/(A_s E) · k_frustum = π E d_h tanα / ln{[(2t tanα + d_w − d_h)(d_w + d_h)] / [(2t tanα + d_w + d_h)(d_w − d_h)]} · k_m = k_frustum/2 · Φ = k_b/(k_b + k_m) · F_bolt = max(F_i + ΦF_A, F_A) · F_separation = F_i/(1 − Φ)

k_b: bolt stiffness. The shank and the threaded length inside the grip are two springs in series on two different areas, and doing it properly matters on a short grip
k_m: member stiffness. Two compression cones, one from each bearing face, meeting at mid-thickness — so two frusta of half the grip each, in series
α: cone half-angle. 30° by convention, 25 to 33 in the literature. The model’s biggest free parameter
d_w: bearing-face diameter, where the cone starts. A washer widens it and stiffens the members, which is a real effect and a reason to use one
Φ: the load factor. Typically 0.15 to 0.25 for a steel joint, meaning the bolt feels a fifth of an external load and the members give up the rest. The whole point of preload, and almost nobody states it
F_i: preload. Note that it does not appear in Φ at all — Φ is geometry and modulus. Preload sets where the joint separates, not how the load divides before it
F_separation: the external load at which the members reach zero compression. Past it the bolt takes the whole load and the joint is a hinge

Worked example

M10 class 8.8 at 75% of proof, 30 mm of grip with 10 mm of that threaded, structural steel members, 10 kN of external tensile load per bolt
Bolt: 20 mm of shank on 78.54 mm² and 10 mm of thread on 57.99 mm², both at 205,000 N/mm². The compliances add: 1.2422 + 0.8412 µm/N, so k_b = 480 kN/mm
Members: two cones, each 15 mm thick, starting from the 14.63 mm washer face over an 11 mm hole, at 30°. One frustum is 3.065 MN/mm and two in series give k_m = 1.533 MN/mm — 3.2 times the bolt
Φ = k_b/(k_b + k_m) = 0.2385, which is just inside the 0.15 to 0.25 band the literature gives for a steel joint and close to Shigley's “good design target … around 0.2”
THE RESULT NOBODY STATES. The 10 kN external load raises the bolt's tension by Φ × 10 kN = 2.385 kN. Not 10 kN. The other 7.615 kN is paid for by the members relaxing, and it costs the bolt nothing. That is what preload buys, and it is why a preloaded joint under a fluctuating load has a fatigue life a preload-free one does not
Preload = 0.75 × 33.63 kN = 25.23 kN. Separation is at F_i/(1 − Φ) = 33.13 kN, so the 10 kN load has a margin of 3.31 on it
Past 33.13 kN the members have no compression left, and the bolt's tension stops being F_i + ΦF_A and becomes F_A. At 60 kN external the bolt would carry 60 kN, not 39.53 kN — which is 78% above its proof load. The transition is a corner, not a curve, and it is where joints fail
Change one thing: make the members 6061-T6 aluminium at 69,000 N/mm². k_m falls to 556.6 kN/mm, Φ rises to 0.463, and the bolt's share of that same 10 kN load roughly doubles. Nothing about the bolt changed

The same M10 joint in every member material — what a soft member costs

This is the table that explains why preload works and when it stops working. In structural steel the members are six times stiffer than the bolt, so Φ is 0.24 and a 10 kN external load raises the bolt tension by 2.4 kN — the members simply relax by that much and the bolt hardly notices. Swap the members for aluminium and Φ rises to 0.47; the bolt now feels nearly half the load. In 30% glass-filled nylon Φ is over 0.9 and the preload is doing almost nothing: the bolt is a spring in series with something softer than itself, which is the definition of a joint with no protection. Note the last column: the separation load falls as Φ rises, so a soft member loses you the fatigue protection AND the sealing margin at the same time. A gasket is softer than anything on this list, which is why VDI 2230 handles gasketed joints separately and why this page will not pretend to. This is a first-pass calculation on an idealised part — uniform section, static load, no stress raisers beyond those stated, and room-temperature material properties. Real parts have fillets, keyways, surface finish and duty cycles that a closed-form answer cannot see.
Member material	E (kN/mm²)	k_m (kN/mm)	k_m ÷ k_b	Load factor Φ	Bolt’s share of a 10 kN load (kN)	Separation load (kN)
Structural steel, ASTM A36 / S275	190.0	1,533	3.19	0.238	2.38	33.1
Grey cast iron, ASTM class 35 / EN-GJL-250	180.0	1,452	3.03	0.248	2.48	33.6
Aluminium 6061-T6	69.0	557	1.16	0.463	4.63	47.0
Magnesium AZ91D (die cast)	46.0	371	0.77	0.564	5.64	57.9
Free-cutting brass C36000	100.0	807	1.68	0.373	3.73	40.2
PA66 with 30% glass fibre, dry	9.5	77	0.16	0.862	8.62	183.2
Steel of a class 8 nut, for reference	205.0	1,654	3.45	0.225	2.25	32.5

Grip length changes both stiffnesses, and not by the same amount

A longer bolt is a softer bolt, which sounds bad and is usually good. Both stiffnesses fall as the grip grows, but the bolt’s falls almost as 1/L while the members’ falls more slowly at first and then faster, because the compression cone keeps widening and adds area. The net effect over the range that matters is that Φ drifts down as the grip grows — a long bolt through a thick flange is better protected from an external load than a short one, and it also tolerates embedment better because the same few micrometres of settling is a smaller fraction of a longer bolt’s stretch. That is the real reason aerospace practice prefers a long thin bolt to a short fat one at the same preload. The cone diameter column is the warning: if the member is not at least that wide, or if another bolt’s cone overlaps this one, the frustum model has run out of material and the real stiffness is lower. This is a first-pass calculation on an idealised part — uniform section, static load, no stress raisers beyond those stated, and room-temperature material properties. Real parts have fillets, keyways, surface finish and duty cycles that a closed-form answer cannot see.
Grip (mm)	k_b (kN/mm)	k_m (kN/mm)	k_m ÷ k_b	Φ	Cone diameter at mid-thickness (mm)	Separation load (kN)
10	1,189	2,532	2.13	0.3195	20.4	37.1
20	684	1,790	2.62	0.2764	26.2	34.9
30	480	1,533	3.19	0.2385	32.0	33.1
50	301	1,319	4.38	0.1857	43.5	31.0
80	193	1,193	6.19	0.1391	60.8	29.3
120	130	1,121	8.60	0.1041	83.9	28.2
200	79	1,062	13.42	0.0693	130.1	27.1

What the cone model leaves out

The Rotscher cone is an idealisation that happens to work well. Sandia’s guideline records the half-angle as “between 25 and 33 degrees” with 30 recommended, and this page prints Φ at all three so you can see the spread — which is about a quarter on the member stiffness and a few per cent on Φ. That is the model’s own uncertainty, and it is smaller than the scatter in the preload you will actually achieve. This is a first-pass calculation on an idealised part — uniform section, static load, no stress raisers beyond those stated, and room-temperature material properties. Real parts have fillets, keyways, surface finish and duty cycles that a closed-form answer cannot see.
Left out	Effect	What to do about it
The head and nut are treated as rigid	They are not. VDI 2230 adds a compliance for the head and for the engaged thread, typically 0.4 d and 0.5 d of extra effective bolt length between them	Expect the real bolt to be a little softer than computed here, which makes Φ slightly LOWER — so this page is conservative on that count. If Φ is load-bearing in your design, use VDI 2230’s full R-sequence.
Whether the member is wide enough for the cone	The frustum grows to the diameter reported above. If the plate ends before that, or a neighbouring bolt’s cone overlaps, there is less material in compression than the model assumes	Check the cone diameter against the real plate. If the joint is narrow, the member stiffness is lower and Φ is higher than computed.
Two different member materials, or a stack of layers	Real joints are often steel on aluminium, or metal with a gasket between	Compute each layer’s frustum separately and add the compliances. This page takes one material for the whole grip and says so.
Gaskets	A gasket is far softer than any metal and it also creeps. It raises Φ towards 1 and destroys the protection preload gives	Not modelled here. VDI 2230 treats gasketed joints separately and ASME PCC-1 governs pressure-boundary flanges. This page will not pretend a gasket is a linear spring.
Bending, eccentric loading and prying	An external load that does not act on the bolt axis adds bending to the bolt and lifts one side of the joint first	VDI 2230 has an eccentric-loading treatment. If the load path is eccentric, Φ is not the whole story and separation starts at one edge rather than everywhere at once.

Why the bolt only feels a fifth of the load, and what takes that away

A properly preloaded bolt barely feels an external load, and this is the page that says why. The bolt is a spring in tension and the clamped members are a much stiffer spring in compression. When an external tensile load arrives, it does not go into the bolt: it is shared, and the split is set by the ratio of the two stiffnesses. Φ = k_b/(k_b + k_m) is the bolt’s share. For a steel joint it is typically 0.15 to 0.25, which means a 10 kN external load raises the bolt’s tension by about 2 kN. The other 8 kN is paid for by the members relaxing, and it costs the bolt nothing.

Both stiffnesses have to be computed properly for that number to mean anything. The bolt is two springs in series: the plain shank on the full cross-section, and the threaded length inside the grip on the tensile stress area, which is 68 to 82% of it. Adding the compliances rather than using one area matters on a short grip. The members are the Rotscher cone, which Shigley gives as a closed form and which this batch verified against a numerical integration of dx/(E·A(x)) over three dozen geometries: two frusta, one from each bearing face, each half the grip thick, in series. The cone’s half-angle is the model’s free parameter; Sandia’s bolted joint guideline records it as “between 25 and 33 degrees” with 30 recommended, and this page prints Φ at 25, 30 and 33 so you can see what that costs. VDI 2230 has a much longer method that adds head and nut compliance and handles layered members and eccentric loads; this is the first-pass version of it.

The separation load is where the joint stops being a joint. F_separation = F_i/(1 − Φ). Below it the bolt’s tension is F_i + ΦF_A and rises slowly. At it, the members reach zero compression. Above it the bolt carries the entire external load, the relationship becomes F_bolt = F_A, and the joint is a hinge — no friction to carry shear, no protection against fatigue, and a bolt load that can pass proof in a few kilonewtons. The transition is a corner, not a curve. The chart on this page sweeps the external load through it so the corner is visible.

A soft member destroys all of this, and that is the practical warning. Keep the same M10 bolt and change only what it is clamping: in structural steel Φ is 0.24, in aluminium 0.47, in 30% glass-filled nylon over 0.9. At Φ = 0.9 the bolt takes nine tenths of every external load and the preload is doing almost nothing but setting a separation threshold. A gasket is softer than anything on that list and it creeps as well, which is why VDI 2230 treats gasketed joints separately and why this page refuses to model one. Note also what does NOT help: a stronger bolt. Strength does not change stiffness, and a fatter bolt raises k_b and makes Φ worse. The fixes are a wider bearing face, a longer grip, a harder member, or more bolts.

Where this number goes. Straight into the bolt fatigue under alternating load calculator, where the alternating stress in the bolt is Φ times the external load range divided by twice the stress area — and where the counter-intuitive result lives, that raising preload usually improves fatigue life. The preload itself comes from the bolt torque calculator and the proof load from the proof load and tensile stress area calculator. And the series stiffness reported here is what embedment acts against, which the preload accuracy by tightening method calculator uses to turn a few micrometres of settling into a preload loss.

Frequently asked questions

What is the load factor of a bolted joint?

Φ = k_b/(k_b + k_m), the fraction of an external tensile load that shows up as extra tension in the bolt. For a steel joint it is usually 0.15 to 0.25, and Shigley calls 0.2 a good design target. The consequence is worth stating plainly because hardly anyone does: a 10 kN load on a joint with Φ = 0.2 raises the bolt’s tension by 2 kN, not 10. The members relax by 8 kN and that is where the load goes. It only works while the joint stays closed — past the separation load, Φ stops applying and the bolt takes everything.

Does higher preload make the bolt work harder?

It raises the bolt’s mean stress and it does not raise the alternating stress at all while the joint stays closed, because Φ does not depend on preload. What higher preload does is raise the separation load, and separation is the thing that actually hurts: past it the bolt’s alternating stress jumps by roughly 1/Φ, which is a factor of four or five. That is why raising preload usually IMPROVES fatigue life even though it raises the mean stress, and the fatigue calculator works it through.

What is the separation load and why does it matter?

F_i/(1 − Φ): the external tensile load at which the clamped members reach zero compression. It matters for three separate reasons. The joint stops carrying shear by friction, because there is no normal force. Any seal in the joint leaks. And the bolt’s load stops being cushioned, so both its peak stress and its alternating stress jump. It is a corner in the behaviour, not a gradual change, and it is the load a joint should be designed to stay well below.

How accurate is the Rotscher cone model?

Good enough to design with and not good enough to be precise about. Its closed form is exact for the assumed cone — this batch checked it against a numerical integration of the cone’s own compliance over three dozen geometries and it agrees to twelve figures — so the uncertainty is entirely in the assumption. The half-angle is quoted between 25° and 33°, which moves the member stiffness by about a quarter and Φ by a few per cent. It also assumes an infinite plate, a rigid head and nut, and one material for the whole grip. All of that is a smaller error than the ±25% scatter in the preload you will actually achieve.

Why does a washer stiffen the joint?

Because the compression cone starts at the bearing face, so a larger bearing diameter starts the cone wider and puts more material in compression for the same grip. In the formula, d_w appears in both the numerator bracket and the denominator, and raising it raises k_m. There is a second and usually more important effect: a hardened washer spreads the contact pressure, which reduces the local yielding that causes embedment and therefore the preload lost in the first hours. And a third: a washer under the turned element makes the underhead friction — about half the tightening torque — repeatable.

Can I use this for a gasketed flange?

No, and that is a deliberate refusal. A gasket is far softer than any metal on this page, it is non-linear in compression, and it creeps, so a linear spring is the wrong model and a stiffness computed today is not the stiffness next week. VDI 2230 handles gasketed joints with its own treatment and ASME PCC-1 governs bolted flange assembly on a pressure boundary, including the tightening sequence, which matters as much as the torque. What this page can honestly tell you is the direction: a soft layer raises Φ and lowers the separation load, so it costs you protection at both ends.

Related calculators

References

Sandia National Laboratories, SAND2008-0371, Guideline for Bolted Joint Design and Analysis: Version 1.0, as reproduced by the Engineering Library. Source for the target preload rule, attributed there to Machinery’s Handbook: “use 75% of the proof strength … for removable fasteners and 90% of the proof strength for permanent fasteners”; for the torque-wrench accuracies by lubrication state (unlubricated ±35%, cadmium plated ±30%, lubricated ±25%); and for Shigley’s frustum form of the member stiffness with the statement that the cone half-angle “should be between 25 and 33 degrees and in general recommends 30 degrees”.
R. G. Budynas and J. K. Nisbett, Shigley’s Mechanical Engineering Design, chapter 8. Cited for the frustum member-stiffness model, the stiffness constant C = k_b/(k_b + k_m) with the note that “a good design target is around 0.2”, and Table 8-17’s fully corrected endurance strengths for rolled threads. The one Table 8-17 value this batch could confirm from a second document is 129 N/mm² for an M10 class 8.8, which A. Mosedal’s Three textbook analyses of fatigue in the bolted tension joint reproduces with K_f = 3.0.
VDI 2230 Blatt 1, Systematic calculation of highly stressed bolted joints — Joints with one cylindrical bolt. Cited by clause and table; the guideline is copyrighted and was not reproduced. The tightening-torque terms used here are its own form, M_A = F_M[0.16 P + 0.58 d₂ μ_G + 0.5 D_Km μ_K], where the 0.58 is 0.5/cos 30° written out. Table A8 gives the tightening factor α_A; clause R12 gives the endurance limit; clause R4 gives the embedding amount f_Z.
MakeItFrom.com material property pages for ASTM A36 / S275 structural steel, ASTM class 35 / EN-GJL-250 grey cast iron, 6061-T6 aluminium, AZ91D magnesium, C36000 free-cutting brass and dry 30% glass-fibre PA 6/6. One publisher throughout, so the shear-to-tensile ratios on this page are comparable with each other. The grey iron row is the one worth looking at: 250 N/mm² tensile against 330 N/mm² shear, a ratio of 1.32, where every other material on the list is near 0.6.
ISO 898-1:2013, Mechanical properties of fasteners made of carbon steel and alloy steel — Part 1: Bolts, screws and studs with specified property classes. Cited by clause; the standard is copyrighted and was not fetched. Clause 9.1.6.1 defines the nominal stress area as the circle on the mean of the pitch diameter d₂ and the minor diameter d₃ of the basic profile, which is where the 0.938194 on this page comes from — it is derived here from d₂ = d − 0.649519 P and d₃ = d − 1.226870 P, not copied. Table 3 carries the property classes.
Kova Fasteners. ISO 898 Part 1 – 2013 (Extract). The source for the minimum tensile strength R_m, the minimum yield or 0.2% proof strength, and the stress under proof load S_p for every property class used here. It also carries the note that for classes 4.8, 5.8 and 6.8 “the values for R_pf min are under investigation” — which is why the second digit of those three designations is a label and not a property.
Embedment in a bolted joint: the Wikipedia article Embedment, which cites SAE data for “values of up to 0.0005 inches … at each surface mate” and notes that “most of the embedment occurs during torquing” so only what happens afterwards costs preload, together with VDI 2230’s own worked example, where “for a clamping length ratio of 3.5 there is a total amount of embedding of f_Z = 5 × 10⁻³ mm”, as reported in PCB Load & Torque’s Review of the Application of Design Guideline VDI 2230. This page computes the preload loss from f_Z and the joint’s own stiffness rather than quoting a percentage.