Bolt Stiffness
Calculate a bolt's stiffness (spring constant), k_b = (A_t·E) ÷ L, from the tensile area A_t (mm²), the material elastic modulus E (MPa) and the grip length L (mm, the effective length under tension between head and nut). When tensioned by the preload, the bolt behaves as a very stiff SPRING: it stretches an amount proportional to the force (Hooke's law), and its stiffness is force per unit elongation. This stiffness is one of two essential ingredients of bolted-joint analysis — the other is the stiffness of the clamped PARTS (members). The ratio between these two stiffnesses (the joint stiffness constant C) determines how an external load splits between the bolt and the parts. Typically the parts (massive, with large effective compression area) are MUCH stiffer than the bolt (thin and long), which is the DESIRED situation: stiff parts absorb most of the external load, protecting the bolt from stress variation and fatigue. Long, thin bolts have low stiffness (good for sharing load), while short, thick bolts are stiff. Knowing k_b is the starting point of fatigue and joint-separation analysis. Enter the tensile area, elastic modulus and grip length.
Result
—
Bolt stiffness
The stiffness (spring rate) of a bolt is k_b = (A_t·E) ÷ L, from the tensile stress area A_t, the modulus of elasticity E and the grip length L (the effective length in tension between the head and the nut). Stretched by the preload, the bolt behaves like a very stiff spring: it elongates an amount proportional to the force (Hooke's law), and its stiffness is force per unit of elongation. That stiffness is one of the two essential ingredients in the analysis of bolted joints — the other one is the stiffness of the clamped members. The ratio between those two stiffnesses (the joint stiffness constant C) governs how an external load splits between the bolt and the members. Typically the members, being solid and having a large effective compression area, are far stiffer than the bolt, which is thin and long, and that is the desirable situation: stiff members take up most of the external load, shielding the bolt from stress swings and from fatigue. Long, slender bolts have low stiffness (good for sharing the load), while short, thick bolts are stiff. Knowing k_b is the starting point for fatigue analysis and for checking joint separation. Enter the tensile stress area, the modulus of elasticity and the grip length.
Related Tools
Bolt Load under External Load
Calculate the total tensile force in the bolt when an external load is applied to the joint, F_b = F_i + C·P, from the preload F_i (N), the joint stiffness constant C and the external tensile load P (N). This is one of the most important — and most surprising to the uninitiated — relations of bolted joints: when you apply an external load P trying to 'separate' the parts, the bolt tension does NOT rise from F_i to F_i + P (as intuition suggests), but only to F_i + C·P, where C is typically 0.2-0.4. That is, the bolt only 'feels' a FRACTION of the external load! The reason: most of the external load (1−C)·P merely RELIEVES the compression between the parts (which were compressed by the preload), rather than stretching the bolt more. This is the genius of the preloaded joint — it 'hides' the external load from the bolt. So a well-tightened joint, under a CYCLIC external load (causing fatigue), exposes the bolt to a very small stress variation (proportional to C·ΔP, not ΔP), making it extremely fatigue-resistant. This formula holds while the joint does NOT separate (P below the separation load); above that, the bolt carries the whole load. Enter the preload, stiffness constant and external load.
Joint Stiffness Constant
Calculate a bolted joint's stiffness constant, C = k_b ÷ (k_b + k_m), from the bolt stiffness k_b (N/mm) and the members' (clamped parts) stiffness k_m (N/mm). The constant C (also called bolt load fraction) is the heart of bolted-joint analysis: it tells what FRACTION of an external tensile load is carried by the BOLT, the rest (1−C) being carried by decompression of the MEMBERS. The value of C reveals the elegant, protective behavior of a preloaded joint: since the (massive) members are usually much stiffer than the (thin) bolt, k_m >> k_b, so C is SMALL (typically 0.2-0.4). This means that when an external load P is applied, only a small portion C·P adds to the bolt tension — most of the load (1−C)·P is absorbed by RELIEF of the compression between the parts. That is why the bolt stress variation is small (good fatigue resistance) and why preload is so beneficial. The smaller C (stiff parts, flexible bolt), the better the bolt protection. The constant C appears in all subsequent formulas: bolt load, residual member force, separation load and fatigue safety factor. Enter the bolt stiffness and the members' stiffness.
Bolt Preload
Calculate the recommended preload (initial clamping force) of a bolt, F_i = 0.75·A_t·S_p, from the bolt tensile stress area A_t (mm²) and the material proof strength S_p (MPa); the result is the tensile force installed in the bolt on tightening. Preload is perhaps the MOST important and most misunderstood concept in bolted joints: a well-designed bolt is tightened to be strongly TENSIONED (stretched), clamping the joined parts together. This clamping force keeps the joint tight and, counterintuitively, PROTECTS the bolt from fatigue. The 0.75·A_t·S_p value (75% of proof load) is the classic recommendation for NON-permanent (reusable) joints; permanent joints use 0.90·A_t·S_p. A HIGH preload is desirable because it: keeps the joint together under varying external load; prevents loosening from vibration; and, mainly, makes an external tensile load be absorbed mostly by DECOMPRESSION of the (stiff) parts rather than additional bolt stretch — so the bolt stress variation (which causes fatigue) is very small. That is why well-tightened bolts rarely fail by fatigue, and loose bolts fail. Enter the tensile area and the proof strength.
Peak Shear Stress in a Bonded Lap Joint (Volkersen)
Computes the peak shear stress in the adhesive layer of a single lap joint using the Volkersen model, which treats the adherends as elastic membranes in tension and the adhesive in pure shear: τ_max = τ_avg·(λ/2)·coth(λ/2), with τ_avg = F/(b·L) and λ = L·√(2·G_a/(E·t·t_a)). Because the adherends stretch unevenly along the overlap, the adhesive does not work uniformly: load piles up at both ends while the middle stays almost unloaded, so the peak stress can be several times the average — 3.35 times in the default example. Hence the model most useful and counter-intuitive conclusion: lengthening the overlap pays less and less, because the extra length carries no load; doubling L from 25 to 50 mm halves the AVERAGE stress but cuts the PEAK stress by only 0.25 %, and it is the peak that breaks the joint. The model assumes a balanced joint, with both adherends of the same material and thickness — that is where the 2 inside the root comes from — and since adherend and adhesive thickness enter only as a product, thickening the adherend buys exactly what thickening the glue line does. Far more is gained by thickening the adhesive or choosing a less rigid one, which is what lowers λ. Enter the load, the overlap width and length, the adherend thickness and modulus, and the adhesive thickness and shear modulus.
Equivalent Stiffness (Springs in Parallel)
Calculate the equivalent stiffness of two springs in parallel, k_eq = k₁ + k₂, by adding the individual stiffnesses. The result, in the same unit (N/m), is always larger than the largest stiffness — springs in parallel are stiffer, since they share the load under the same displacement and the forces add. This is the case of mounts, isolators and supports placed side by side carrying the same component. Reducing spring assemblies to an equivalent stiffness is the first step to compute a vibrating system's natural frequency. Enter the two stiffnesses.
Bolt Shear Stress
Calculate the shear stress in transversely loaded bolts, τ = F ÷ (n·A), from the total shear force F (N), the number of bolts (or shear planes) n and each bolt's area A (mm²). Unlike tensioned joints (where the bolt is tightened and the load is axial), in SHEAR joints the bolts resist a transverse force tending to slide one part over another (as in steel structural connections, splice plates, flanges under lateral load). The force is distributed among the bolts and each works in shear — hence the stress is force divided by the number of bolts times the area. There can be SINGLE shear (one shear plane) or DOUBLE shear (two planes, when the bolt passes through three plates), doubling capacity. The area used depends on whether the shear plane passes through the threaded part (use the tensile area) or the smooth shank (nominal-diameter area). Shear stress is compared with the bolt material's shear strength (typically ~0.6 of tensile strength). In structures, bearing-type (bolt in shear/bearing) and slip-critical (preload friction transmits load without bolt shear) connections are distinguished — this formula covers shear resistance. Enter the shear force, the number of bolts and the area.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.