This analysis was done with the LS-DYNA Finite Element Code. This code is a very powerful tool for analyzing the dynamic and static loading of structures.
The loading condition is 50,000 psi on the face of the bolt. This it the equivalent force that would be applied to the bolt face from a 223 Rem caliber with a case head separation. The components in the calculation remain in the elastic state. There is no yielding of the 4140 steel, so the stress levels may be linearly scaled as long as the maximum effective stress level remains below the 120,000 psi yield stress. The dimensions of the model were taken from my Stolle Panda rifle in 23/40 caliber. This caliber is essentially a long neck 223 Ackley Improved with the neck to shoulder junction set back about 0.1". In the model, the Z-axis is inline with the bullet's path, see the triad in the lower left corner of each view. There is info on how could actually measure a representative chamber pressure here: Recreational Software, Inc. Software & Instrumentation Technology for the Shooting Sports.
The material properties for the 4140 steel were implemented with a power law hardening model that allows for plastic flow if the effective stress (Huber-Hencky-von-Mises stress) exceeds the 120 ksi yield stress.
The maximum effective stress level of 97,390 psi is located at
the surface of the bolt lugs where they contact the receiver. For
50,000 psi in an intact 223 Rem case, the maximum effective
stress level would be about 63,500 psi and well below the yield
stress. This view of the bolt, lugs, and receiver does not
clearly show the location of the high stresses. In the analysis,
the top plane of the receiver, shown in the view, is fixed in the
Z-direction and the pressure is applied to the bolt face. For
simplicity, the firing pin and hole was not included in the mesh.
The contact surface between the bolt lugs and the receiver was
modeled with a sliding surface including a 0.3 friction
coefficient.
This view of the bolt and lugs, shows more clearly that the high
stress levels are located at the lug contact surfaces.
This view of the receiver shelf shows the high contact stress
levels where the bolt lugs rest.
This it the actual mesh used in the calculation. An X-Z plane of
symmetry and Y-Z plane of symmetry were used in the model. Using
these two planes of symmetry requires a much smaller mesh and
gives results identical to what a full 3-D mesh would give.
Here are the fringes of axial displacement in the Z direction.
The bolt face moves rearward about 0.0077". There is about
0.0002" of slop in the model before the contact surfaces from the
bolt lugs to the receiver shelf start to load each other. This
was intentional so that when the mesh was being made, certain
nodes wouldn't be removed. This displacement with a 50,000 psi
pressure on the bolt face would be equivalent to a non ruptured
223 Rem brass case with about 76,600 psi internal pressure. For
an intact 223 brass case head with 50,000 psi internal pressure,
the bolts rearward deflection would be about 0.005". This amount
of bolt deflection is twice that calculated by Dan Lilja of Lilja Precision
Barrels, but his calculation was only for the bolt lugs
and did not include the deflections of the lug contact points
and a slight amount of stretch in the receiver from the barrel
threads rearward. The magnitude of the action's axial
deflections is one more reason that the chamber should be low
friction and allow the brass case head to contact the bolt
face early in the firing sequence.
Finally, this plot shows the fringes of maximum shear in the
bolt. However, in this view, one can not view the shear stresses
at the junction of the bolt body to bolt lug.
Here the bolt lug has been removed so the shear stresses can be
viewed at the base of the bolt lugs. Dan Lilja assumes, in his
analysis, that the shear stresses are uniform over the area of
the bolt body to lug. Assuming a uniform shear is necessary for a
simple stress calculation and gives reasonable results. However
with the more detailed calculation, the shear stress are shown to
have a maximum near the base of the lugs and decreases to as
little as fifth of maximum near the forward end of the bolt. The
use of the LS-DYNA code allows one to view and understand more
clearly the complicated stress and deflection conditions that
structures undergo when resisting service loading.
WHAT IS EFFECTIVE STRESS?.... Here is some information on Effective Stress or often referred to as Von Mises Stress. In structures under load, the stress state is not a simple uniaxial state of stress but a more complex triaxial state of stress. Plastic deformations (also called plastic flow) will occur in a material when the effective stress, seff exceeds the uniaxial yield stress. Here is the equation for the effective stress when a triaxial state of stress exists and s1, s2, and s3 are the principal stresses in the three orthogonal directions. The effective strain is eeff.
seff= (Φ2/2) [(s1s2)2+ (s2s3)2+ (s3s1)2]½
eeff= (Φ2/3) [(e1e2)2+ (e2e3)2+ (e3e1)2]½
1. When all three orthogonal stresses are equal
(hydrostatic stress state), the effective stress is zero. There
is no yielding.
2. When s2 and s3 are zero, then the effective stress
is the uniaxial stress. This is the stress condition of an axial
pull test specimen.
Looking at the equation, one can see some interesting concepts.
If there is no tension, but sufficient biaxial compression in the
other two directions, then there could be axial yielding in
extension in the third direction. (This is like squeezing a
cylinder of putty in your hand and having it squirt out each
end).
WHEN YIELDING OCCURS.... One can visualize yielding as a cylinder in 3-D space. The axis of the cylinder is the line of the hydrostat. It is a line where s1= s2= s3. When the triaxial stress state deviates from the hydrostat line but stays within the cylinder, there is no yielding. When the triaxial stress state deviates from the hydrostat outside of the cylinder then yielding occurs.
NO YIELDING IN HYDROSTATIC COMPRESSION.... When a metal
is in hydrostatic compression (immersed in a fluid and the
pressure increased), there is no yielding, no matter how high the
pressure. With a real material, micro voids would be closed and
at extremely high pressures, the crystal structure could be
changed, such as converting carbon to a diamond. These pressures
are beyond those normally considered in engineering analysis of
the strength of structures.
NO YIELDING IN HYDROSTATIC TENSION.... When a metal is in
hydrostatic tension, in theory, there is no yielding. I can't
think of any realistic way to apply extremely high hydrostatic
tensions. The limiting point would be the bond strength of the
atoms in the crystals.
Finally... The rifle bolt and lugs are in a complicated triaxial
stress state at each location and the FEA codes determine this
for each tiny element. The back-of-the-envelope p/a type
calculations are of little value in accurately predicting and
understanding what is happening during loading such structures
where a triaxial state of stress exists.
Good Hunting... from Varmint Al
For the serious reader: How to Check Another Engineer's Calculation.
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Last Updated: 01/31/2008
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