First, we’ll examine the horizontal force created by the fluid pressure. The magnitude is calculated the same as before, but with one difference. The important diameter is not the braid mean diameter, Dm, but, rather, the end fitting nipple’s outside diameter, Dn. The horizontal force is developed from the pressure acting directly on the nipple; the braid must support the horizontal force (and the vertical force).

The horizontal pressure force, H, always pushes directly against the fitting’s nipple. If the reinforcement wasn’t secured to the connector’s shell or socket, the pressure would push the nipple out of the hose. The horizontal force’s magnitude is the same, regardless of tube’s wall thickness does.

This is where the calculation of the reinforcement requirement changes from that suggested by the Evans model. The magnitude of the reaction force carried by the reinforcement diameter does not change even if the tube wall thickness does. The reaction force is related to the nipple’s outside diameter, and it is carried by the reinforcement!

Now let’s look at the vertical force, V, created by the fluid pressure. Similar to the Evans analysis, we show the vertical force acting at the inside diameter of the tube; it does not act directly on the reinforcement. The reinforcement provides the reaction force to the pressure driven forces applied to the inner surface of the tube.

Justification
The mechanics to justify moving the vertical (radial) force, V, to the inside diameter of the tube is two fold. First, the tube is trapped between the fluid and the reinforcement. The fluid pushes radially outward, and the reinforcement pushes back with equal force. Second, the lay of the reinforcement moves to the neutral angle.

The Evans solution for the direction of the neutral angle as 54° 44´ has been verified by decades of experience. Experience also tells us that the braid will move toward the neutral angle unless it is somehow blocked.

This is because if the horizontal force is greater than the vertical force, the hose reinforcement will elongate until the horizontal and vertical forces are equal. That only happens when the reinforcement braid angle is 54° 44´. The reverse is also true: if the horizontal force is less than the vertical force, the hose reinforcement will shrink until it reaches the neutral angle. This is why hose swells and shortens when pressurized.

To explain, we know from experience that when under pressure, the reinforcement always aligns at the neutral angle. So where does the vertical force act? It must be adjacent to the horizontal force so that the resulting force created by the vector sum of the horizontal and vertical forces lies at the neutral angle.

A suggested improvement
My suggestion is for hose manufacturers to change from using the mean diameter of the reinforcement, Dm, to the connector’s nipple diameter, Dn. Any geometric change to the tube or braid thickness has no impact on the magnitude of the force the reinforcement must support.

In summary, the hydrostatic forces, H and V, act on the inner surface of the tube; that is, at the nipple diameter and along the related wetted pitch inside tube. The resultant force acts along the neutral angle and the magnitude of the resultant force is:

FR = P × 1.36 × Dn2.

Following Evans’ analysis, we can conclude that:

R × N = P × 1.36 × Dn2.

We now have created a model in which we know that the optimum design strength of the reinforcement must be equal to the connector’s nipple diameter instead of the mean diameter of the reinforcement. This means the Evans approach is too conservative for OEMs and users that want a lighter weight, yet equally strong and reliable hose assembly that is designed using the lower forces based on the nipple OD. Adopting this suggested approach may lead to the next generation hose design.

Design theory for optimization
We calculate the hydrostatic force based on the nipple’s diameter, which reduces the hose reinforcement design to N × R. We know the braid angle, so the designer simply selects the solution for N × R.

The minimum number of reinforcement ends, N, must provide sufficient coverage of the outer surface of the hose tube to prevent it from extruding between the reinforcement strands. Furthermore, the product of N and R must exceed the minimum burst pressure. This means that the geometric solution for N × R may require two or more layers of reinforcement to keep the hose tube from failing.

Conclusions
The Evans analysis is successful because it is conservative. It is based on using the mean reinforcement (braid or spiral) diameter as the boundary for generating the hydrostatic force. However, the free-body-diagram at the wetted surfaces indicates the horizontal force is generated by the pressure acting directly on the nipple. The result is that the magnitude of the horizontal and vertical forces is reduced from that calculated by using the mean reinforcement diameter. These reductions will allow manufacturers to design hose that may be lighter and more flexible than hose currently based on the Evans analysis model.

Peter Stroempl is retired after working 33 years in the hydraulics industry.