爆風荷重応答解析
Theory and Physics
What is Blast Load?
Professor, what is blast load?
It is the pressure load acting on a structure caused by the shock wave (blast wave) generated by an explosion (chemical explosive, gas explosion, etc.). It has a characteristic time waveform: peak pressure → exponential decay → negative pressure.
Friedlander Waveform
Standard approximation formula for blast pressure (modified Friedlander equation):
- $P_s$ — Peak overpressure
- $t_d$ — Positive phase duration
- $b$ — Decay parameter
- $p_0$ — Atmospheric pressure
So it decays exponentially and even becomes negative pressure, right?
There is a positive pressure phase (pushing force) followed by a negative pressure phase (pulling force). For a building wall, it's a repetition of positive pressure (outward) → negative pressure (inward).
Estimating Blast Parameters
Estimate blast parameters using the Kingery-Bulmash equations from ConWep (Conventional Weapons Effects Program):
$Z$ is the scaled distance (m/kg^{1/3}), $R$ is the distance from the source, $W$ is the TNT equivalent mass. Estimate $P_s, t_d, I$ from $Z$.
So the parameters are determined just by the explosive mass and distance?
It's the "Scaling Law" (Hopkinson-Cranz law). The same $Z$ yields the same pressure waveform. TNT 1 kg at 10 m and TNT 1000 kg at 100 m have the same $Z$ and thus the same overpressure.
FEM Setup
Two approaches:
1. Direct input of pressure time history — Apply the pressure from the Friedlander equation as a time function onto structural surfaces. The simplest method.
2. ALE method (Arbitrary Lagrangian-Eulerian) — Solve blast wave propagation using Eulerian mesh and structural deformation using Lagrangian mesh simultaneously. Includes reflection and diffraction.
Is the ALE method more accurate?
The ALE method automatically calculates blast reflection, diffraction, and superposition, so it's more accurate. However, it requires a 3D air mesh and has a high computational cost. For simple shapes, direct pressure input is sufficient.
Summary
Key points:
- Friedlander waveform — Peak overpressure → exponential decay → negative pressure
- Parameter estimation with ConWep (Kingery-Bulmash equations) — $Z = R/W^{1/3}$
- Direct pressure input or ALE method — Choose based on complexity
- LS-DYNA's *LOAD_BLAST_ENHANCED — Automatically calculates ConWep
Explosion Pressure Decays with the Cube of Distance
According to the TNT equivalent scaling law (Hopkinson-Cranz law), the maximum overpressure from an explosion is organized by the ratio Z = R/W^(1/3) of distance R and charge mass W to the 1/3 power. At Z = 1, the maximum overpressure reaches about 0.3 MPa. Brode's equation from 1944 was later refined into the Kingery-Bulmash database (US Army TM5-855-1), which is still adopted in ANSYS Autodyn's ATBLAST function.
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward when a car brakes suddenly? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, which assumes "forces are applied slowly enough that acceleration can be ignored". It absolutely cannot be omitted in impact loads or vibration problems.
- Stiffness term (elastic restoring force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it", right? That's Hooke's law $F=kx$, the essence of the stiffness term. Now a question — an iron rod and a rubber band, which stretches more when pulled with the same force? Obviously the rubber band. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "high stiffness = strong" is incorrect. Stiffness is "resistance to deformation", strength is "resistance to failure" — they are different concepts.
- External force term (load term): Body force $f_b$ (gravity, etc.) and surface force $f_s$ (pressure, contact force, etc.). Think of it this way — the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road surface is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common mistake here: getting the load direction wrong. Intending "tension" but it becomes "compression" — sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades away. That's because the vibration energy is converted into heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle — they intentionally absorb vibration energy to improve ride comfort. What if damping were zero? Buildings would keep swaying forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity.
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and stress-strain relationship is linear.
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only equilibrium between external and internal forces.
- Non-applicable cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity or creep, constitutive law extensions are needed.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify load and elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformation) |
| Elastic modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel) |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system |
Numerical Methods and Implementation
Blast Analysis in LS-DYNA
LS-DYNA has dedicated functions for blast:
```
*LOAD_BLAST_ENHANCED
1, 100., 10., 0., 0., 0., 1, 1.0 $ W=100kg TNT, R=10m
```
Automatically calculates and applies ConWep blast pressure to each surface of the structure. Also automatically considers reflection angles.
It's convenient that ConWep is built-in.
*LOAD_BLAST_ENHANCED automatically calculates reflected pressure from explosive mass, source location, and structural surface orientation. Much lighter than ALE method and sufficiently accurate for many problems.
ALE Method Setup
1. Place Eulerian mesh for air around the structure
2. Define the explosion source with *INITIAL_DETONATION etc.
3. Define the structure with Lagrangian mesh
4. Connect fluid and structure with ALE coupling (*CONSTRAINED_LAGRANGE_IN_SOLID)
Summary
ALE Method Solves Blast-Structure Coupling
In explosion analysis, the ALE (Arbitrary Lagrangian-Eulerian) method is mainstream, treating air/explosives with Eulerian grids and structures with Lagrangian elements. LS-DYNA's ALE multi-material code was developed at LLNL (Lawrence Livermore) in the 1990s; simulating a 1 kg TNT free-field explosion for 5 ms with 1 million nodes takes about 30 minutes on current HPC.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated with reduced integration or B-bar method).
Quadratic Elements (with Mid-side Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended when stress evaluation is important.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates tangent stiffness matrix every iteration. Achieves quadratic convergence within convergence radius, but computationally expensive.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial value or every few iterations. Cost per iteration is low, but convergence speed is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Applies total load not all at once, but in small increments. The arc-length method (Riks method) can trace beyond extremum points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper" — reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer" — starts with a rough answer but improves accuracy with each iteration. It's the same principle as looking up a word in a dictionary: opening to an estimated page and adjusting forward/backward (iterative method) is more efficient than searching sequentially from the first page (direct method).
Relationship Between Mesh Order and Accuracy
1st order elements are like "approximating a curve with a ruler" — represented by straight line segments, so accuracy is limited. 2nd order elements are like a "flexible curve" — can represent curved changes, dramatically improving accuracy even with the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
Blast Analysis in Practice
Used in structural design for defense, petrochemical plants, and counter-terrorism measures.
Application Examples
| Application | Purpose |
|---|---|
| Blast wall design | Shielding from blast. Keep deformation within allowable limits. |
| Building blast-resistant design | Glass breakage, structural response. |
| Vehicle blast resistance | Resistance to IEDs (Improvised Explosive Devices). |
| Plant safety distance | Impact on structures during explosion accidents. |
Practical Checklist
What is UFC 3-340-02?
The US Department of Defense's Structural Design Manual for Blast Loads. Specifies peak pressure, duration, and allowable structural deformation. The world standard for blast-resistant design.
Protection Design Standard is UFC 4-010-01
The US Department of Defense UFC (Unified Facilities Criteria) 4-010-01 is the blast protection design standard for government facilities, specifying standoff distances and structural response limits. The 2012 edition revised after 9/11/2001 requires the Ductility Ratio (μ) of RC slabs to be kept below 10 and explicitly mentions the use of dynamic analysis (SDOF or FEM) for verification.
Analogy of Analysis Flow
The analysis flow is actually very similar to cooking. First, you buy ingredients (prepare CAD model), do the prep work (Related Topics
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