By employing high-strength bolts, some of the strength deficiencies can be partially compensated. However, this approach may lead to a reduction in the impact toughness of the bolts. Additionally, due to the structural limitations of the flange, it remains challenging to fully resolve the strength issues. In this study, the combined sections and bolts of the bolted three-stage explosion vessel are simplified into rigid bodies and springs, respectively, forming a three-degree-of-freedom spring-mass system. This model is used to analyze the dynamic behavior of the system and explore ways to improve the stress conditions of the bolts.
The primary forces acting on the coupling bolts come from three main sources of excitation: (1) Axial impact loads, which determine the axial force inside the container caused by the movement of the explosion shock wave. This load is the main cause of deformation. In flanged bolted explosion vessels, the superposition of reflected overpressure from the explosion shock wave along the axial direction of the container wall is given by $ P(t) = \int_{sp} r,H,U,t \cos U_1 ds $, where $ r, H, U $ are the surface coordinates of the container head, and $ U_1 $ is the angle between the normal direction of the head and the axial direction of the container. (2) The influence of the container shell on the bolt. The vibration of the container shell is complex with dense frequency content. Under specific loading conditions, the shell is mainly governed by a few low-frequency vibrations. While the coupling between shell vibration and bolt vibration is weak, torsional bending vibrations from the shell can still cause uneven stress distribution on the bolt, leading to fatigue cracks at the root of the bolt head and the threaded section. (3) The natural vibration frequency of the bolt has minimal impact on its strength. Assuming the bolt behaves like a homogeneous, isotropic one-dimensional linear elastic body, the natural frequencies for a steel bolt with length $ L = 0.58 \, \text{m} $, modulus of elasticity $ E = 206 \, \text{GPa} $, and density $ \rho = 7800 \, \text{kg/m}^3 $ are approximately 5.3 kHz, 8.6 kHz, and 12.9 kHz. These frequencies are significantly higher than the actual impact load response range, thus having little effect on the bolt's stress state.
From the above analysis, it is reasonable to consider only the axial impact load as the main factor affecting the coupling bolt in a flanged explosion vessel. Ignoring damping effects, the explosion vessel can be modeled as a three-degree-of-freedom spring-mass system. The system is simplified without considering initial bolt pre-tension. The equation of motion is expressed as:
$$
\begin{bmatrix}
m_1 & 0 & 0 \\
0 & m_2 & 0 \\
0 & 0 & m_3
\end{bmatrix}
\begin{bmatrix}
\ddot{x}_1 \\
\ddot{x}_2 \\
\ddot{x}_3
\end{bmatrix}
+
\begin{bmatrix}
k_1 & -k_1 & 0 \\
-k_1 & k_1 + k_2 & -k_2 \\
0 & -k_2 & k_2 + k_3
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}
=
\begin{bmatrix}
P(t) \\
0 \\
-P(t)
\end{bmatrix}
$$
The initial conditions are $ x_i(0) = 0 $ and $ \dot{x}_i(0) = 0 $ for $ i = 1, 2, 3 $. This can be rewritten in matrix form as:
$$
M \ddot{X} + K X = P(t)
$$
Where $ M $ is the mass matrix, $ K $ is the stiffness matrix, $ \ddot{X} $ is the acceleration vector, $ X $ is the displacement vector, and $ P(t) $ is the external load vector. Considering the bolt’s preload, the flange joint maintains a constant preload within a certain range. The stiffness $ k_i = k_b + k_m $, where $ k_b $ and $ k_m $ are the bolt and flange stiffnesses, respectively. The explosive load $ P(t) $ is modeled as an exponentially decaying function: $ P(t) = P_0 e^{-At} $, where $ P_0 $ is the peak axial load and $ A $ is the attenuation coefficient.
Due to symmetry and equal preload on both flanges, $ k_1 = k_2 = k $. Thus, the system equation becomes:
$$
\begin{bmatrix}
m_1 & 0 & 0 \\
0 & m_2 & 0 \\
0 & 0 & m_3
\end{bmatrix}
\begin{bmatrix}
\ddot{x}_1 \\
\ddot{x}_2 \\
\ddot{x}_3
\end{bmatrix}
+
\begin{bmatrix}
k & -k & 0 \\
-k & 2k & -k \\
0 & -k & k + k_3
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}
=
\begin{bmatrix}
P(t) \\
0 \\
-P(t)
\end{bmatrix}
$$
To estimate the maximum strain in the three bolts, we assume the solution takes the form:
$$
x_1 = U_e1 \sin(\omega_1 t + \phi_1), \quad x_2 = U_e2 \sin(\omega_2 t + \phi_2), \quad x_3 = U_e3 \sin(\omega_3 t + \phi_3)
$$
Substituting this into the system equation yields an eigenvalue problem, from which the natural frequencies $ \omega_1, \omega_2, \omega_3 $ can be determined.
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