Bolted Joint with Gasket (External Tensile Load)

stress

Tighten a bolt through two flanges and it stretches like a stiff spring, squeezing the members between its head and nut with a preload $F_i$. Now hang an external tensile load $P$ on the joint — internal pressure trying to lift a vessel head, a pipe flange fighting the line pressure behind it. The question every fastener course turns on is deceptively simple: how much of $P$ does the bolt actually feel?

The naïve answer — all of it — is wrong, and usefully so. The bolt and the clamped members are two springs in parallel, both anchored between the same head and nut. When $P$ tries to pull them apart, the bolt stretches a little more and the members, already squeezed, relax a little. They must move together as long as they stay in contact, so the external load splits between them by stiffness. A well-preloaded joint routes most of $P$ into unloading the members and only a small fraction into more bolt tension — which is the entire reason bolts are preloaded at all.

You meet this joint wherever a seal has to hold pressure: cylinder heads, pressure-vessel flanges, pump and valve bonnets, pipe flange pairs, the head of a hydraulic cylinder. The gasket is what makes it its own THING.

The load splits by stiffness

Give the bolt stiffness $k_b$ and the combined member-plus-gasket stiffness $k_m$. The fraction of the external load the bolt takes is the joint stiffness constant:

$C = \frac{k_b}{k_b + k_m}$

— exactly how two parallel springs share a force. The bolt tension and the residual clamping force then follow from equilibrium and compatibility together:

$F_b = F_i + C\,P, \qquad F_m = F_i - (1 - C)\,P$

The bolt sees its preload plus only its share $C\,P$ of the external load; the members shed the rest. In a bare metal joint the members are far stiffer than the bolt, so $C$ is small — typically $0.15$ to $0.25$, meaning the bolt feels only $15$–$25\%$ of $P$. This is the whole payoff of preload: the load the bolt fatigues under is a fraction of the applied load, and the members do the rest of the work by un-squeezing.

The build does not hand you these formulas to trust. It certifies that equilibrium and compatibility form a system linear in $\{F_b, F_m\}$, solves that $2\times2$ system exactly at build time, and checks the solution back through both relations — the same total verification an ordinary closed form gets. This is the solveLinear capability shared with the propped cantilever and the composite bar; what the machine proves and what still rests on a book is on the verification page.

The gasket is the whole point

A gasket is a soft layer in series with the metal members, and softness lowers $k_m$. Look at what that does to $C = k_b/(k_b + k_m)$: as $k_m$ falls, $C$ rises, so the bolt is forced to absorb a larger fraction of every external load. The default state here is already a gasketed joint — the members-plus-gasket stiffness $k_m$ has been brought down to equal the bolt’s own $k_b$, so $C = 0.5$ and the bolt takes half of every external load, well above the $0.15$–$0.25$ of a bare metal joint. Soften the gasket further in the widget (drop $k_m$ below $k_b$) and watch $C$ climb past $0.5$ — the bolt force $F_b$ swings ever more for the same $P$. That is why a soft-gasketed joint is harder on its bolts than a metal-to-metal one, and why gasket choice is a fatigue decision, not just a sealing one. (Here $k_m$ is a direct input — the members and gasket already combined into one series stiffness. Computing $k_m$ from the members’ pressure-cone frusta and the gasket’s own rate is the standard next step, and named future work.)

Separation — where the model stops

Watch $F_m = F_i - (1 - C)\,P$ as you crank $P$ up. The residual clamping force falls, and at

$P_0 = \frac{F_i}{1 - C}$

it reaches zero: the members go slack and the joint separates. Past $P_0$ the bolt carries the entire external load, the two-spring split no longer describes anything, and — for a gasketed joint — the seal has already leaked. Because the linear model is void everywhere beyond separation, the widget refuses the whole evaluation there rather than draw a confident, wrong picture: every readout blanks and the diagram is replaced by a refusal. Separation is not a warning to design near; it is the cliff the preload exists to keep you away from.

What sets the numbers, and the honest scope

Two inputs deserve a note:

• Proof strength $S_p$ is a direct knob, not a seeded material. A bolt’s strength comes from its property class — SAE grades, ISO metric classes like 8.8 or 10.9 — which are specification grades, not the alloys in this site’s materials database. The default $600\ \mathrm{MPa}$ is the ISO 898-1 Class 8.8 proof stress (Shigley Table 8-11). The tensile stress area $A_t$ is a separate knob set by thread size (an M16 bolt is $\approx 157\ \mathrm{mm^2}$, an M20 $\approx 245\ \mathrm{mm^2}$): the two are chosen independently, and the safety factor $n_p = S_p A_t / F_b$ guards the bolt against reaching proof at full external load.

• There is no material dropdown, and that is honest. Young’s modulus never appears here — the stiffnesses $k_b$ and $k_m$ are supplied directly by design, so the usual “stiffer material deflects more” cascade has nothing to act on. Bolt strength comes from the grade table via $S_p$, which no seeded metal represents. A token material picker that changed nothing would be dishonest, so there isn’t one. (A frusta-based $k_m$ and a bolt-grade lookup table would each re-introduce a real axis; both are future work.)

Sign convention

$F_i$ is the assembly preload — bolt tension equals member compression before any external load. $F_b$ is the total bolt tension and is always positive. $F_m$ is the residual clamping force, taken positive in compression: it starts at $F_i$ and falls as $P$ grows, and separation is $F_m \le 0$. (Shigley writes the member force as $(1-C)P - F_i$ with compression negative; the sign is flipped here so that a positive $F_m$ reads directly as “the joint is still clamped.”)

Try it

Governing relations

Derivation

Steps marked modeling step are where physics enters by citation — every other line is machine-proven to follow from them, and the cited models are independently re-derived in the test pipeline where possible. See what is and isn't machine-verified.

How it fails

The widget’s safety factor $n_p$ guards one thing — the bolt reaching its proof strength under the maximum static load. Real bolted joints rarely die that way. The interesting failures are the ones the static picture cannot see, and for a preloaded joint the first of them is the real killer.

• Fatigue at the thread root — and why $C$ decides it. Most external loads fluctuate: pressure cycles, road inputs, thermal breathing. If $P$ swings between $P_{min}$ and $P_{max}$, the bolt force swings with it — but only by its share. The alternating bolt stress is

  $\sigma_a = \frac{C\,(P_{max} - P_{min})}{2\,A_t}$

  and there it is: the fatigue amplitude scales with the joint constant $C$. A stiff joint (hard gasket, high $k_m$, small $C$) barely feels a fluctuating load; a soft-gasketed joint (high $C$) hands the bolt a large alternating stress and fatigues it quickly. This is the deep reason preload and gasket choice are fatigue decisions: a high preload keeps the joint from separating so the spring split stays valid, and a stiff member path keeps $C$ — and therefore $\sigma_a$ — small. Fatigue strength sits far below static yield and concentrates at the first engaged thread root, a sharp notch this page’s nominal $\sigma_b = F_b/A_t$ deliberately does not resolve. The endurance limit that turns $\sigma_a$ into a life is a material property this catalog will carry in a later phase; until then, read $C$ as the knob that decides whether this joint has a fatigue problem at all.

• Separation and leakage. The one hard refusal on this page. When $P$ reaches $P_0 = F_i/(1-C)$ the members go slack; for a gasketed joint the seal has usually opened before that, as soon as the clamping pressure on the gasket drops below the line pressure it is holding. A joint that separates under load also slams shut when the load releases — impact the static model knows nothing about, and a fast route to fatigue at the seating faces.

• Preload loss — the number that quietly drifts down. Everything here assumes $F_i$ is the preload that is actually there, but preload leaks away with time. Soft gaskets creep and relax under sustained compression; the seating faces embed as micro-asperities flatten in the first load cycles; thermal cycling ratchets it. Every lost newton of preload lowers $P_0$ and raises the risk of separation and leakage. This is why gasketed joints are re-torqued, and why a stiffer joint (which loses a smaller fraction of its preload for a given embedment) is the more forgiving one.

• Self-loosening under vibration. Transverse vibration can walk a nut loose (the Junker mechanism) entirely independently of tension fatigue — the thread slips in tiny increments and preload bleeds to zero. Prevailing-torque nuts, thread lockers, and correct preload fight it; the static analysis here cannot represent it.

• Overtightening. Push past $S_p$ during assembly — the region the proof warning marks — and the bolt yields, so it no longer returns the preload you dialed in; push further and the threads strip or the bolt necks. The proof strength is a ceiling on assembly, not just on service.

• The thread root and the material. The nominal stress on $A_t$ omits the stress concentration at the thread root (rolled threads help by cold-working it), and high-strength bolts in the wrong environment are prone to hydrogen embrittlement and stress-corrosion cracking — brittle failures with no warning that a yield-based margin does not anticipate.

Sources

• Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — ch. 8 (Screws, Fasteners, and the Design of Nonpermanent Joints): §8-4 the joint constant C = k_b/(k_b + k_m); §8-5 bolt and member forces F_b = F_i + CP, F_m = F_i − (1 − C)P and the separation condition; Table 8-11 metric property classes (proof strength).