What is buckling?
It is a sudden sideways bow of a slender rod under push.
Buckling is a sudden, large and unstable lateral deflection. It can start with only a small increase in compressive load above a critical level, called the buckling load. The stress at that point can be far below the yield strength of the rod material. Once it happens the rod collapses, which is why buckling is potentially highly dangerous.
Buckling happens on the push stroke. A pulling rod is in tension and does not buckle.
What decides column strength?
Column strength is dictated mainly by rod length, rod diameter and axial alignment, and also by the rod's yield strength. A tape measure shows the effect: the more of it you extend, the less stable it becomes, until even a light touch makes it buckle.
A rod's resistance to buckling is described by its slenderness ratio, λ = K × L ÷ r. L is the length, K is a factor for the end conditions (so K × L is the effective length) and r is the radius of gyration of the section, r = √(I ÷ A). For a solid round rod r = d ÷ 4, so λ = 4 × K × L ÷ d. This is the same definition used in the other articles on push-pull rods and tubes in this knowledge centre (K06-Q1-03, K07-Q1-10). A simpler index, effective length divided by diameter, is sometimes used as a quick comparison; do not set that number against a threshold meant for λ.
In a hydraulic cylinder, the effective length takes in the centre of the rod end mounting point, whether a cross tube, a clevis or a fixed mounting, and the point where the head bearing supports the rod. The buckling length is normally taken pin to pin of the whole cylinder at full extension, not the rod length alone. The distance from the mounting point to the rod bearing matters: shorter is stronger. A cylinder is strongest in compression when retracted, because the supported column is then shortest.
How does the mounting change the result?
The end supports change the effective length of the rod. Euler's formula for an ideal slender rod gives the critical stress as:
σcr = π² × E ÷ λ²
and the critical load is σcr × A. E is the stiffness of the steel (about 200 GPa). Doubling the effective length cuts the critical load to a quarter, and diameter counts to the fourth power.
Euler only applies to slender rods. Below a transition slenderness λt = √(2π² × E ÷ σy), where σy is the yield strength, the rod would reach yield before it could buckle elastically, and Euler overstates the capacity. In that range use the Johnson formula: σcr = σy − σy² × λ² ÷ (4π² × E). This is the same Euler/Johnson split used in K07-Q1-10.
An illustrative example: a 45 mm solid rod (A = 1,590 mm², r = 11.25 mm) of steel with E = 200 GPa and an assumed yield of 400 MPa, so λt = 99. The pin-to-pin length is taken as 1.2 m, treating the whole length as the rod, a conservative simplification. The push force of an 80 mm bore cylinder at 160 bar is about 80.4 kN. The same rod gives very different results depending on the ideal end conditions:
| Ideal end condition | K | Effective length | λ | Regime | Critical stress | Critical load | Ratio to 80.4 kN |
|---|---|---|---|---|---|---|---|
| Fixed at both ends | 0.5 | 0.60 m | 53 | Johnson | 342 MPa | about 544 kN | 6.8 |
| Fixed at one end, pinned at the other | 0.7 | 0.84 m | 75 | Johnson | 287 MPa | about 456 kN | 5.7 |
| Pinned at both ends | 1.0 | 1.20 m | 107 | Euler | 173 MPa | about 276 kN | 3.4 |
| Fixed at one end, free at the other | 2.0 | 2.40 m | 213 | Euler | 43 MPa | about 69 kN | 0.9 |
For the first two rows, Euler alone would give 1,104 kN and 563 kN, which overstates the capacity (about twice for the first row and 1.2 times for the second), because those stresses are above half the yield, where Euler no longer holds. Only the pinned–pinned and fixed–free rows are in Euler's range. The fixed–free row falls below the push force, so that mounting would not work for this load. Even the pinned figure, with a margin of about 3.4, should not be approved as safe from a table like this, because these are ideal figures. A real cylinder is not an ideal column: the piston, rod bearing, clevises, spherical bearings, frame stiffness, mounting clearance, stop tube, load guidance and stroke position all change the boundary condition, and shock, misalignment, weak mounting and the rod's straightness can use the margin up. A cylinder maker's buckling guide (Blince) makes the same warning.
Does a pivoting rod eye lower the column strength?
Usually, yes, compared with a fixed mounting. A cylinder that pivots at both the rod and the clevis allows for possible axial misalignment, especially when fully extended, whereas a fixed head mount pushing a rigidly guided load gives a superior combination of axial stability and high column strength.
The pivoting end does not cause the buckling by itself. It sets the boundary condition. A pivoting cylinder is also more exposed to side load if its pivots are not in line (see Q1-10).
Looseness in the joint adds to the problem. In one service case, the rear clevis pivots and the rod-end eye had noticeable side clearance, and the previous rod bent toward the operator side. Many buckling failures begin outside the rod: a worn clevis pin that lets the load line move, a base plate that twists, a trunnion mounted off centre, or a guided carriage that binds near the end of travel.
Which mounting helps?
For force transfer, a cap mount is the usual choice for push loads (Q1-04); for column strength, a head-end mount is better because the supports are closer together, which means a shorter effective length. On a long-stroke push cylinder, buckling can decide in favour of the head mount. Each mounting style loads the rod end differently.
How do you improve the column strength?
- A larger rod diameter. The effect is large because diameter counts to the fourth power. The drawbacks are weight and cost.
- A shorter effective length. For example, a mounting that puts the supports closer together.
- A stop tube. A stop tube is extra support for an extended rod. It increases the distance between the rod bearing and the piston, which improves the inherent column strength.
- A stiffer, higher-yield rod. Raising the yield strength of the rod steel can often improve security against buckling, and Euler overestimates buckling resistance for stocky rods, where yielding comes into play.
- A guide for the load. A rigidly guided load keeps the force along the rod axis.
A bigger rod end does not help. The buckling limit comes from the rod, its length and its supports, not from the size of the rod end (Q1-06).
What does your cylinder supplier need to know?
Do not approve a long-stroke push cylinder from bore, stroke and pressure alone. Give the supplier the actual push load, the maximum extension, the mount type, the rod-end connection, how the load is guided, the operating angle, the speed, any shock, and the safety consequence of failure. A typical maker's method (Hänchen's, for one) is an Euler calculation that needs the buckling length, the installation factor for the way the cylinder and rod are mounted, and a safety factor of 3 to 5.
Does the check change for a steering or tipper-body cylinder on a vehicle?
Vehicle cylinders (steering, levelling, tipper-body hoist) are usually pinned at both ends, often with spherical bearings, because the structure around them moves. That is the K = 1.0 row of the table, so use it rather than the fixed-end rows. What separates one vehicle cylinder from another is mostly the stroke. An illustration, with the same assumptions as above (E = 200 GPa, yield assumed 400 MPa, λt = 99) and the whole pin-to-pin length taken as rod: a 32 mm rod under the push of a 63 mm bore at 100 bar, about 31 kN.
| Pin-to-pin length | λ = 4KL ÷ d | Regime | Critical load | Ratio to 31 kN |
|---|---|---|---|---|
| 750 mm | 94 | Johnson | about 178 kN | 5.7 |
| 1,500 mm | 188 | Euler | about 45 kN | 1.4 |
Here, doubling the length cuts the capacity to about a quarter. So buckling rarely decides a short steering-type cylinder, but it can decide a long-stroke body or boom cylinder. These are ideal-column figures, not approvals. A worn pin, a loose spherical bearing or side force from a flexing frame all use up margin. The extended length, mounting and load case go to the cylinder maker.