By employing high-strength bolts, some of the strength deficiencies can be partially compensated. However, this approach may reduce the impact toughness of the bolt itself. Moreover, due to the limitations of the flange structure, it is challenging to fully resolve the strength issues. In this study, the combined sections and bolts of a bolted three-stage explosion vessel are simplified as rigid bodies and springs, respectively, forming a three-degree-of-freedom spring-mass system. Through dynamic analysis of this system, we aim to explore ways to improve the stress condition of the bolts. The primary forces acting on the coupling bolts come from three main sources of excitation.
First, the axial impact load, caused by the movement of the explosion shock wave inside the container, determines the axial force and is the main cause of deformation. In flange-connected explosion vessels, the superposition of reflected overpressure from the explosion shock wave along the axial projection of the container wall is given by:
P(t) = ∫ sp(r, H, U, t) cos U₠ds,
where r, H, and U represent the surface coordinates of the container head, and Uâ‚ is the angle between the normal direction of the head and the axial direction of the container.
Second, the effect of the container shell on the bolt. The vibration of the container shell is complex, with a dense frequency spectrum. Under specific loads, the shell is mainly influenced by low-frequency vibrations. The coupling between the shell's vibration and the bolt is weak, and thus has minimal impact on the bolt's stress state. However, torsional bending vibrations from the shell can lead to uneven stress distribution in the bolt, creating a weak point—specifically at the root of the bolt head and the threaded section—potentially causing fatigue cracks.
Third, the influence of the bolt’s natural vibration frequency on its strength. When treated as a homogeneous, isotropic one-dimensional linear elastic body, the natural frequencies of the bolt can be calculated. For a steel bolt with length L = 0.58 m, modulus of elasticity E = 206 GPa, and density Ï = 7800 kg/m³, the natural frequencies are approximately 5.3 kHz, 8.6 kHz, and 12.9 kHz. These frequencies are significantly higher than the actual impact load response range, meaning they do not have a major effect on the bolt’s stress state.
From this analysis, it is reasonable to focus only on the axial impact load when analyzing the forces acting on the coupling bolts in the combined explosive container.
Ignoring damping effects, the explosion vessel can be modeled as a three-degree-of-freedom spring-mass system. Without considering the initial bolt pre-tension, the system simplifies into a basic model. The equation of motion can be written as:
mâ‚ 0 0
0 mâ‚‚ 0
0 0 m₃
ẍâ‚
ẍ₂
ẍ₃
+
kâ‚ -kâ‚ 0
-kâ‚ kâ‚ + kâ‚‚ -kâ‚‚
0 -k₂ k₂ + k₃
xâ‚
xâ‚‚
x₃
=
P(t)
0
-P(t)
With initial conditions x_i(0) = 0 and ẋ_i(0) = 0 for i = 1, 2, 3. This can be expressed in matrix form as:
M Ẍ + K X = P(t)
Where M is the mass matrix, K is the stiffness matrix, Ẍ is the acceleration vector, X is the displacement vector, and P(t) is the load vector. Considering the bolt pre-tension, the flange joint maintains a pre-load within a certain range. The stiffness matrix elements are defined as k_i = k_b + k_m, where k_b and k_m are the bolt and flange stiffnesses, respectively.
The explosive load acting on the container wall can be described as an exponentially decaying function:
P(t) = P₀ e^(-At), for t ≥ 0,
where Pâ‚€ is the peak axial explosive load, and A is the decay coefficient.
Since only axial deformation and bolt strength are analyzed, radial forces are neglected. Due to symmetry and equal preloads, kâ‚ = kâ‚‚ = k. Thus, the equation becomes:
mâ‚ 0 0
0 mâ‚‚ 0
0 0 m₃
ẍâ‚
ẍ₂
ẍ₃
+
k -k 0
-k 2k -k
0 -k k + k₃
xâ‚
xâ‚‚
x₃
=
P(t)
0
-P(t)
To estimate the maximum strain in the bolts, we assume solutions of the form:
xâ‚, xâ‚‚, x₃ = U_e1, U_e2, U_e3 sin(ω_i t + φ)
Substituting into the equation yields an eigenvalue problem, from which the three natural frequencies ωâ‚, ω₂, ω₃, and the zero mode can be determined.
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