This section presents the current capabilities of SimEx, starting with the main interface used for computing the dynamic response of SDOF systems subjected to blast loading, and following with the description of the remaining calculation assistants.
2.1. Single-Degree-of-Freedom System Analysis
In many situations of practical interest, the response of structural elements to blast loading can be reduced, in first approximation, to that of an equivalent spring-mass SDOF system. As sketched in
Figure 1, this system is made up of a concentrated mass subject to external forcing and a nonlinear weightless spring representing the resistance of the structure against deformation [
8]. The mass of the equivalent system is based on the component mass, the dynamic load is imposed by the blast wave, and the spring stiffness and yield strain on the component structural stiffness and load capacity. Generally, a small viscous damping is also included to account for all energy dissipated during the dynamic response that is not accounted by the spring-mass system, such as slip and friction at joints and supports, material cracking, or concrete reinforcement bond slip [
31].
If the system properties are properly defined, the deflection of the spring-mass system,
, will reproduce the deflection of a characteristic point on the actual system (e.g., the maximum deflection). The system properties required for the determination of the maximum deflection are the effective mass of the equivalent SDOF system,
, the effective viscous damping,
, the effective resistance function,
, and the effective load history acting on the system,
. To systematize the calculations, the effective properties are obtained using dimensionless transformation factors that multiply the actual properties of the blast-loaded component, respectively,
M,
C,
, and
[
32]. These factors are obtained from energy conservation arguments in order to guarantee that the equivalent SDOF system has the same work, kinetic, and strain energies as the real component for the same deflection when it responds in a given, assumed mode shape, typically the fundamental vibrational mode of the system [
31].
In the analysis of blast-loaded SDOF systems, it is therefore of prime importance to identify the fundamental vibrational mode of the structural element. This procedure is not trivial, since obtaining the fundamental mode can entail certain difficulties, in which case its shape must be approximated in some way [
32]. To determine the equivalent properties of the SDOF system, it is also necessary to determine the type of structure (beam, pillar, frame, etc.) and how the load is applied (typically, a uniform load is assumed). The elastic behavior of the material is often modeled as perfect elasto-plastic, probably the simplest of all nonlinear material models. This assumes that the initial response follows a linear elastic behavior described by an apparent elastic constant
K, but once the yield strain is reached,
, the material behaves as plastic, flowing at a constant stress with an ultimate resistance
, i.e.,
Although more complex models could be used, they are not considered here due to the heavy simplifications introduced in the formulation of the problem.
The mass transformation factor,
, is defined as the ratio between the equivalent mass
and the real mass
M of the blast-loaded component; the load transformation factor
is defined as the ratio between the equivalent load
and the actual load
, and usually coincides with the resistance and damping transformation factors; and finally the load-mass factor
is defined as the ratio between the mass factor and the load factor
Although all these factors are easy to obtain, even through analytical expressions in some cases, most of them can be found tabulated in the UFC-3-340-02 [
16].
The linear momentum equation for the equivalent SDOF system then takes the form [
32]
where, as previously discussed,
C represents the viscous damping constant of the blast-loaded component. This constant is often specified as a small percentage,
z, of the critical viscous damping,
, with a damping coefficient
being a good value when not otherwise known (for further details see [
31]). Note, however, that damping has very little effect on the maximum displacement, which typically occurs during the first cycle of oscillation, so the actual value of
z is not of major relevance. The inhomogeneous term,
, appearing on the right-hand side of Equation (
3) represents the dynamic load associated with the blast wave, to be discussed in
Section 2.1.1 below.
SimEx provides an easy and intuitive GUI environment for the study of the dynamic response to blast loadings of a variety of structural elements that can be modeled as SDOF systems.
Figure 2 shows the main SimEx interface, divided into three calculation assistants for the three basic elements that make up the SDOF system: a module for calculating the properties of the blast wave (forcing term,
), a module for calculating the equivalent mechanical properties (resistance term,
), and a module for the numerical integration of the problem, which includes the post-processing of the results and their graphic representation in the form of displacements, forces, and deformation diagrams (see the bottom plots of
Figure 2) and of CW–S damage charts, to be discussed in
Section 3.3.
As a final remark, it is important to note that, following standard practice, the SDOF analysis carried out by SimEx uses the load defined in terms of pressure, (), so that both the mass M (), the damping coefficient C () and the ultimate resistance () must all be introduced as distributed values per unit surface (p.u.s.) in the different calculation assistants.
2.1.1. Forcing Term
As previously discussed, the blast wave overpressure defined in Equation (
4) below can be used directly in Equation (
3) as forcing term,
, as long as the analysis is formulated per unit surface and uses distributed masses and forces. In order to determine the blast parameters (arrival time, peak overpressure, positive phase duration, impulse per unit area, waveform parameter, etc.), classical correlations [
1,
2,
17,
18,
19,
33,
34] in terms of scaled distance are used together with the scaling laws for spherical or hemispherical blast waves [
1,
17,
35,
36], which allow their evaluation for arbitrary CW–S pairs. It is interesting to note that the standoff distance is defined as the minimum distance from the charge to the structural element under study (e.g., a wall). However, the actual distance to a given point of that element, e.g., the centroid (or geometric center), which may be considered the most representative point of the structure, may be slightly different due to the incidence angle being larger than 0 at that point.
The local atmospheric pressure,
, and temperature,
, are determined using the International Civil Aviation Organization (ICAO) Standard Atmosphere (ISA) [
37] with a temperature offset (ISA
). The user must specify the geopotential height, in meters, and the non-standard offset temperature
, although arbitrary ambient temperature and pressure can also be introduced directly [
38]. TNT is used as reference explosive, although the results can be extrapolated to other compositions using either the equivalence tables included in SimEx for selected explosives [
39], or the thermochemical calculation assistant, to be presented in
Section 2.2.1, for less conventional formulations or explosive mixtures.
To estimate the dynamic load exerted by the blast wave, the angle of incidence of the incoming shock wave must be considered, the worst-case conditions being usually those of normal incidence. UFC 3-340-02 [
16] contains scaled magnitude data for both spherical and hemispherical blast waves. It also provides methods to calculate the properties of the blast wave with different incidence angles, including both ordinary and Mach reflections for oblique shocks. The time evolution of the blast wave overpressure
at a fixed distance,
d, sufficiently far from the charge (at least, larger than the fireball scaled distance) is approximated using the modified Friedlander’s equation, which captures also the negative overpressure phase [
1,
17,
40]
where
represents the peak overpressure measured from the undisturbed atmospheric pressure
, with
denoting the peak post-shock pressure,
is time measured from the blast arrival time,
is the positive phase duration, and
is the waveform parameter, closely related to the impulse per unit area of the positive phase
(area under the positive phase of the overpressure-time curve) according to
. SimEx performs by default the complete integration of the Friedlander waveform, but the equivalent triangular pressure pulse can also be used without significant errors [
32]. This simplified waveform has the same maximum peak overpressure,
, but a fictitious positive duration computed in terms of the total positive impulse and the peak over pressure,
.
The “Blast wave” calculation assistant allows the activation of the effects of clearing and confined explosions, which increases the computational capabilities to more realistic situations. The clearing effect takes into account the time required for reflected pressures to clear a solid wall that has received the impact of a blast wave as a result of the propagation of rarefaction waves from the edges of the wall. In the case of confined explosions, SimEx implements the procedure outlined in UFC 3-340-02 [
16] to estimate the gas phase peak overpressure and duration of the equivalent triangular pressure pulse in terms of the chamber’s total vent area and free volume. These effects can be activated on the lower part of the “Blast wave” calculation assistant.
2.1.2. Resistance Term
The “Resistance” calculation assistant provides a means to define the equivalent mechanical properties (i.e., structural mass, damping coefficient, and structural strength) of the SDOF system under study modeled as a perfectly elasto-plastic system with elastic stiffness
K until the yield strain, as given in Equation (
1). The characteristic length,
L, of the structural element must also be provided, as it is required to determine the maximum rotation angle at its boundaries, often referred to as support rotation,
. For the equivalent SDOF system, the assistant computes the fundamental natural period,
, the critical damping,
, and the deflection at which plastic deformation initiates in the system,
. Direct access to calculation assistants that compute the equivalent properties (
M,
K,
,
) required for the calculations is also provided for various types of systems. Currently, standard European wide flange “metal beams” [
41] and reinforced “concrete beams” are included (see
Section 3.2), although it could be possible to incorporate additional assistants for other elements, such as metal panels/plates, open-web steel joists, reinforced concrete slabs, reinforced/unreinforced masonry, or wood panels/beams. The metal beams assistant also provides the possibility of studying custom (i.e., non-normalized) profiles and materials in order to widen the computation capabilities.
2.1.3. Numerical Integration
Once the characteristics of the equivalent SDOF system have been defined, the resulting ordinary differential equation that models the transient nonlinear response of the equivalent structural system (
3) must be integrated numerically. The integration module implements the two numerical methods recommended by UFC-3-340-02 [
16], namely the “Acceleration-Impulse-Extrapolation Method” and the “Average Acceleration Method” [
16], which can be selected from a drop-down menu. Text boxes are also included to set the initial conditions (displacement and initial speed, which are zero by default) as well as the final integration time. Since both numerical methods use constant time steps, a sufficiently short time increment, typically of the order of a few percentage of the natural period or the positive phase duration (usually, fractions of a millisecond), should be used in order to ensure the numerical convergence of the integration.
2.1.4. Post-Processing
After integration, three plots appear in a pop-up window and a summary table is provided at the bottom left corner of the main window. The left plot shows the instantaneous displacement (solid line) and the permanent displacement, or deformation (dashed line). The central plot shows the temporal variation of the forcing term (i.e., the blast pressure wave, solid line) together with the resistance strength of the SDOF system (dashed line). The right plot shows the displacement–resistance graph, in which it is possible to determine more clearly whether permanent deformations occur or not. Finally, the table of results shows the maximum displacement obtained, , along with two damage indicators: the ductility ratio, , defined as the ratio of the peak deflection to the ultimate elastic deflection, and the maximum support rotation, , whose calculation depends on the type of structure under study.
By integrating different combinations of charge weights and standoff distances for the same structural element, damage level diagrams can be rapidly obtained in the CW–S distance space. SimEx has a function for it located in the central part of the integrator module. One can select the range of charge weights and standoff distances, the number of intermediate values and the type of damage in terms of the quantitative indicators
and
[
15]. From the two quantitative indicators, the structural damage level can be classified qualitatively into: superficial, moderate, heavy, hazardous failure, and blowout, with response limit boundaries between these levels denoted respectively by B1 (superficial to moderate), B2 (moderate to heavy), B3 (heavy to hazardous failure), and B4 (hazardous failure to blowout). Convenient limits for the boundaries of component damage levels for common structural components in terms of
and
are provided in [
15]. An example of a damage level diagram for the façade of a conventional building subject to blast loading computed with SimEx will be presented in
Section 3.3.