Saturday, September 28, 2019

Vehicle-buggy accidents, Part 2: Some math

Fortunately, there is a great deal of research if you know where to look.

Head Injury Criteria

For instance, the Federal Motor Vehicle Safety Standards define the Head Injury Criteria as a moving window with the average acceleration (in Gs) raised to the 2.5th power. A Head Injury Criteria over a thousand is not good. The moving window is usually 15ms as the structural properties of the human head function like a low-pass filter. Jiggly acceleration signals in the 5ms range and shorter duration are noise.

The HIC (Head Injury Criteria) was developed for impacts that drive the human head rearward. There are significant structural differences in how the human head responds to frontal hit and to rear hits, but we will use the FMVSS HIC number as a starting point.

Crush Space
Not the easiest table read. A seven G pulse (which equates to HIC of 130) requires 206 inches...17 feet... to accelerate a mass to sixty miles per hour. A fifteen G pulse (which equates to a HIC of 870) requires 24 inches to accelerate a mass to thirty miles per hour.

Simply equating kinetic energy (1/2*mass*velocity^2) and force through a distance (accel rate X distance) and doing a bit of math shows us that an object can be accelerated to 30 miles per hour in a space of 51.5 inches when subjected to an even acceleration of 7 times the rate of gravity.

Said another way, the chart tells me that if I want to limit the HIC to 499, then I need to limit the maximum acceleration to 12 Gs. If I limit the acceleration to 12 Gs, then I need to find 30 inches of crush space and design a structure that will generate the proper force pulse to generate 12 G.

Magee and Thornton (1978)

Magee and Thornton published formulas for approximating the average force it takes to crush round and square pipe in the axial direction.

As a convenience, rectangular sections are approximated using the formula for the square section by taking the average of the height and width of the section.

Magee and Thornton's approximation is:
Average axial crush force for a rectangular section = 17 * (metal thickness in millimeters)^1.8 * (Material's Ultimate Tensile Strength) * ((Height + Width)/2)^0.2

This is where reality rears its head. Steel tubing is not available in an infinite range of exterior dimensions and metal thicknesses. Furthermore, axial crush force is exquisitely sensitive to metal thickness and surprisingly insensitive to the height and width.

2" by 2" and 4" and 6". Metal thickness  in inches listed in second row from top. 60ksi ultimate tensile strength assumed.

Assuming four fat men and two hundred pounds of seats and framing, we are looking at a max load of 1000 pounds. Assuming we have one longitudinal rail along each side of the buggy, the table shown above translates rectangular tubing dimensions into the accelerations in Gs for maximum ballast condition.

Functionally, this would perform similar to a 2"-by-6" section if the flanges were one inch, each.
The longitudinal rails for the occupant protection structure for the buggy would optimally be 16 gauge steel and 2" wide by 4" high in order to provide sufficient energy capacity for maximum ballast conditions.

Additional "crush distance" for less than maximum ballast conditions must be engineered into the individual seats to avoid excessive HIC numbers for the occupants.

Part 3

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