Dynamic coefficient analysis of enhanced plastic resistance beams under Friedlander-type blast loading

Abstract

Compared to alternative blast loading formulations, the Friedlander-type loading function effectively captures the attenuation characteristics inherent in explosive pressure-time histories, making it extensively adopted for validating blast loading effects on structures. This empirical model also demonstrates significant potential for enabling refined blast-resistant design methodologies in structural engineering. Furthermore, beam components have gained widespread application in protective structural systems due to their inherent energy absorption capacity and load redistribution mechanisms under blast-induced dynamic loading conditions. Considering the enhanced plastic resistance characteristics of beam components, this study investigates the variation of the dynamic coefficient of beam components under Friedlander-type blast loading. The motion differential equations for both flexible and rigid beam components at different stages are derived using the equivalent single-degree-of-freedom (SDOF) method. The results of dynamic coefficient calculations under 27 typical loading conditions are presented. The displacement response of I-shaped steel beams under Friedlander-type blast loading for different conditions is simulated using the finite element software LS-DYNA. The theoretical results are then compared with finite element and code-based calculations, validating the feasibility of the proposed theory through multiple methods. The research findings indicate that the finite element method results align well with the theoretical results derived from the Friedlander-type loading model. The current blast-resistant design code is more applicable to flexible beam components, while it shows significant discrepancies for rigid beams, which could hinder blast-resistant design. Both the resistance enhancement factor and the Friedlander-type loading shape parameter reduce the dynamic coefficient, with the latter exerting a stronger influence on the dynamic coefficient than the resistance enhancement factor.

Keywords

blast loading dynamic response dynamic coefficient beam component protective structures

Introduction

Blast loading, characterized by a high overpressure peak and short duration, are often simplified as linear decay loads in various national engineering blast-resistance design codes (Department of the Army, 1990; Biggs, 1964; Canadian Standards Association, 2012; China Architecture and Building Press, 2006; Department of the ; China Architecture and Building Press, 2012). After introducing a dynamic coefficient, blast loading is equivalently treated as static loading. Enhancing the accuracy of dynamic response calculations for beam components under blast loading has been a key research focus in structural blast resistance. For instance, Wei et al. (2023), Liao et al. (2019), Chen et al. (2020), Ma et al. (2023), and Nagata et al. (2018) conducted field explosion tests on reinforced concrete beam components. Using the equivalent single-degree-of-freedom (SDOF) method, they calculated the dynamic response of beam components under blast loading and compared these results with experimental data, indicating that the SDOF method is an effective tool for calculating dynamic responses under blast loading. Chen et al. (2021) and Luo et al. (2020) developed blast analysis models for beam components based on the equivalent SDOF method, and performed explosion resistance tests on eight hybrid fiber lightweight aggregate concrete beams. Their results showed that the SDOF model provided a good match with experimental data, further validating the reliability of the SDOF theoretical model. Liu et al. (2021) theoretically derived the dynamic coefficient under series-type explosive loading curves, suggesting a high correlation between the dynamic coefficient and the shape parameter. Geng et al. (2019) used the Friedlander-type loading to study the effect of the shape parameter on the dynamic coefficient of beam components under blast loading, demonstrating that an increase in the shape parameter leads to a reduction in the dynamic coefficient. Cheng et al. (2009), Lin et al. (2001), and Xin et al. (2018) found that Friedlander-type blast loading could accurately model the propagation of shock waves over time. Zhang et al. (2019) analyzed TNT implosion test loads using Friedlander-type loading and showed that this model accurately described the changes in blast loading. Nassr et al. (2012) conducted static tests on six I-shaped steel beams, revealing that all beams exhibited resistance enhancement in the plastic phase and indicating that the plastic resistance enhancement model could accurately describe the relationship between resistance and deformation in components. Hu et al. (2023) suggested that the resistance enhancement effect in the plastic phase significantly influences the overall displacement of steel beams. Chen et al. (2010) pointed out that the resistance enhancement effect notably reduces the dynamic coefficient. Fallah and Louca, (2007) and Cui et al. (2021) used an elastic-plastic enhancement model based on the SDOF system to generate pressure-impulse (P-I) diagrams with the resistance enhancement coefficient as a variable, and concluded that considering the resistance enhancement coefficient provides more accurate predictions of component dynamic responses. Lan et al. (2018) studied the explosion response of components based on the rigid-plastic model both theoretically and numerically. Geng et al. (2021) found that for structures with higher ductility ratios, the dynamic coefficient is smaller, leading to better blast resistance performance. Xu et al. (2018) analyzed the impact of blast loading duration on the dynamic coefficient using both theoretical and finite element simulations.

In summary, current research on dynamic coefficients of beam components under blast loading has generally employed simple load functions and resistance models, without conducting in-depth studies on the dynamic coefficient under blast loading effects. By adopting the more precise Friedlander-type blast loading in conjunction with the plastic resistance enhancement model, more accurate conclusions regarding the influencing factors of dynamic coefficients for beam components can be obtained.

Based on this, considering the classification of flexible and rigid beam components (Geng et al., 2023) the motion displacement solutions for different stages of beam components were solved. Twenty-seven typical working conditions were designed, and feasibility verification was conducted by comparing finite element simulations, current blast-resistant design codes, and theoretical calculations from this study. The effects of different shape parameters and resistance enhancement coefficients on the dynamic coefficient of beam components were analyzed, providing a theoretical basis for blast-resistant design of beam components.

Friedlander-type blast loading and differential equations of plastic resistance enhancement in the SDOF system of beam components

Friedlander-type blast loading and plastic resistance enhancement model

In this study, the dynamic response of beam components under Friedlander-type blast loading is accurately analyzed using the plastic resistance enhancement model based on the single-degree-of-freedom (SDOF) system. The Friedlander-type blast loading curve is shown in Figure 1 below.The relationship between the resistance R of the component during positive motion and its deformation y is shown in Figure 2. The specific expressions for the Friedlander-type blast loading and the plastic resistance enhancement model are as follows Cheng et al. (2009):

R (y) = {\begin{cases} K_{e} y & (0 \leq y \leq y_{T}) \\ r_{m} + α K_{e} (y - y_{T}) & (y_{T} < y \leq y_{m}) \\ R_{m} & (y = y_{m}) \end{cases}

(1)

Δ p_{e} (t) = {\begin{cases} Δ p_{m} l k_{L (l)} (1 - t / t_{z}) e^{- a t / t_{z}}, & 0 ⩽ t ⩽ t_{z} \\ 0, & t > t_{z} \end{cases}

(2)

Where, K_e represents the equivalent stiffness during the elastic phase, while r_m denotes the maximum resistance of the beam component in the elastic phase and R_m corresponds to the maximum resistance in the plastic phase. The peak displacement during the elastic phase is denoted by y_T, and α refers to the plastic resistance enhancement coefficient of the beam component. The peak displacement in the elasto-plastic phase is represented by y_m. The term Δp_e(t) indicates the time-dependent blast loading applied to the beam component, with Δp_m being the overpressure peak of the shock wave. The span length of the beam component is represented by l, while k_L(1) is the load transformation coefficient for either the elastic phase k_L or the plastic phase k₁. The duration of the Friedlander-type blast loading is denoted by t_z, and a represents the shape parameter of the blast loading.

Figure 1.

Schematic diagram of Friedlander-type blast loading.

Figure 2.

Plastic resistance enhancement model.

Differential equations of the elasto-plastic equivalent system

According to the theory of the equivalent single-degree-of-freedom (SDOF) system, the differential equation for the elastic response stage of the beam component is given by:

M_{e} \ddot{y} + C_{e} \dot{y} + K_{e} y = Δ p_{e} (t)

(3)

where: M_e is the equivalent mass in the elastic phase, and C_e is the equivalent damping in the elastic phase. The formulas for calculating these equivalent parameters are as follows:

\begin{array}{l} M_{e} = k_{M} m l, \\ C_{e} = 2 ξ \sqrt{k_{M} m l k_{L} K}, \\ K_{e} = k_{L} K \end{array}

(4)

where: m is the mass per unit length of the beam component, ξ is the damping ratio of the beam component, K is the stiffness of the beam component, and k_M is the mass transformation coefficient in the elastic phase.

At the end of the elastic motion phase, the beam component reaches the peak elastic displacement y_T, after which it enters the plastic motion phase. In this phase, the plastic resistance enhancement model is considered, and the equivalent SDOF differential equation is given by:

m_{e} \ddot{y} + c_{e} \dot{y} + K_{e} y_{T} + α K_{e} (y - y_{T}) = Δ p_{e} (t)

(5)

where: m_e is the equivalent mass in the plastic phase, and c_e is the equivalent damping in the plastic phase. The formulas for calculating these equivalent parameters are as follows:

\begin{array}{l} m_{e} = k_{m} m l, \\ c_{e} = 2 ξ \sqrt{k_{m} m l k_{l} K} \end{array}

(6)

where: k_m is the mass transformation coefficient in the plastic phase.

Dynamic response solution for flexible beam components

Elastic forced motion stage

At the initial stage of the blast loading application, the beam component enters the elastic forced motion phase with external loading, where both displacement and velocity are set to zero as initial conditions. The duration of this phase is defined by the relationship 0 < t < t_z. By solving equation (3) and combining equations (2) and (4), the expressions for displacement and velocity in this stage are derived as:

y = e^{- ξ ω t} (C_{1} \cos ω_{d} t + C_{2} \sin ω_{d} t) + \frac{y_{st} θ_{z} e^{- a \frac{t}{t_{z}}}}{a^{2} - 2 ξ a θ_{z} + {θ_{z}}^{2}} (θ_{z} - ω t + \frac{2 ξ {θ_{z}}^{2} - 2 a θ_{z}}{a^{2} - 2 ξ a θ_{z} + {θ_{z}}^{2}})

(7)

v = e^{- ξ ω t} [(ω_{d} C_{2} - ω ξ C_{1}) \cos ω_{d} t - (ω ξ C_{2} + ω_{d} C_{1}) \sin ω_{d} t] - \frac{y_{st} θ_{z} e^{- a \frac{t}{t_{z}}}}{a^{2} - 2 ξ a θ_{z} + {θ_{z}}^{2}} [\frac{a}{t_{z}} (θ_{z} - ω t + \frac{2 ξ {θ_{z}}^{2} - 2 a θ_{z}}{a^{2} - 2 ξ a θ_{z} + {θ_{z}}^{2}}) + ω]

(8)

where ω is the undamped natural frequency, ω_d is the damped natural frequency, y_st is the static displacement of the beam component when the blast loading is considered as a static loading, and θ_z is the time duration parameter of the Friedlander-type blast loading for the beam component. The formulas for these parameters are as follows:

\begin{array}{l} ω = \sqrt{\frac{K_{e}}{M_{e}}}, \\ ω_{d} = ω \sqrt{1 - ξ^{2}}, \\ y_{st} = \frac{Δ p_{m} l}{K}, θ_{z} = ω t_{z} \end{array}

(9)

Substituting the initial conditions of zero displacement and velocity into equations (7) and (8), the constants C₁ and C₂ can be solved as:

C_{1} = - y_{st} \frac{{θ_{z}}^{2}}{a^{2} - 2 ξ a θ_{z} + {θ_{z}}^{2}} (1 + \frac{2 ξ θ_{z} - 2 a}{a^{2} - 2 ξ a θ_{z} + {θ_{z}}^{2}})

(10)

C_{2} = \frac{y_{st}}{γ} [\frac{{θ_{z}}^{2}}{a^{2} - 2 ξ a θ_{z} + {θ_{z}}^{2}} (1 + \frac{2 ξ θ_{z} - 2 a}{a^{2} - 2 ξ a θ_{z} + {θ_{z}}^{2}}) (\frac{a}{θ_{z}} - ξ) + \frac{θ_{z}}{a^{2} - 2 ξ a θ_{z} + {θ_{z}}^{2}}]

(11)

where γ is the total damping reduction coefficient, calculated as:

γ = \sqrt{1 - ξ^{2}}

(12)

Substituting t_z into equations (7) and (8), and using equations (10) and (11), the expressions for displacement y_z and velocity v_z at the end of the elastic forced motion phase can be obtained.

Elastic free motion stage

After the blast loading is removed, the beam component enters the elastic free motion phase, with no external loading and initial conditions of displacement y_z and velocity v_z. The duration of this phase is defined by the relation t_z<t<t_T, where t_T is the time required for the beam component to reach the peak elastic displacement. The equivalent SDOF differential equation for this phase is given by:

M_{e} \ddot{y} + C_{e} \dot{y} + K_{e} y = 0

(13)

By solving equation (13), the expressions for displacement and velocity during this phase are obtained as:

y = e^{- ξ ω (t - t_{z})} [C_{3} \cos ω_{d} (t - t_{z}) + C_{4} \sin ω_{d} (t - t_{z})]

(14)

v = e^{- ξ ω (t - t_{z})} [(ω_{d} C_{4} - ω ξ C_{3}) \cos ω_{d} (t - t_{z}) - (ω ξ C_{4} + ω_{d} C_{3}) \sin ω_{d} (t - t_{z})]

(15)

Substituting the initial conditions y_z and v_z into equations (14) and (15), the constants C₃ and C₄ are solved as:

\begin{array}{l} C_{3} = y_{z}, \\ C_{4} = \frac{v_{z} + ω ξ y_{z}}{ω_{d}} \end{array}

(16)

By substituting t = t_T into equations (14) and (15) and combining with equation (16), the displacement and velocity expressions at this moment can be obtained as:

y_{T} = e^{- ξ (θ_{T} - θ_{z})} [y_{z} \cos γ (θ_{T} - θ_{z}) + \frac{v_{z} + ω ξ y_{z}}{ω_{d}} \sin γ (θ_{T} - θ_{z})]

(17)

v_{T} = e^{- ξ (θ_{T} - θ_{z})} [v_{z} \cos γ (θ_{T} - θ_{z}) - (\frac{v_{z} ξ + ω ξ^{2} y_{z}}{\sqrt{1 - ξ^{2}}} + ω_{d} y_{z}) \sin γ (θ_{T} - θ_{z})]

(18)

where θ_T is the time duration parameter of the beam component. Its expression is as follows:

θ_{T} = ω t_{T}

(19)

Plastic free motion stage

After the elastic free motion phase, the beam component enters the plastic free motion phase, with no external loading and initial conditions of displacement y_T and velocity v_T. The duration of this phase is defined by the relation t_T<t<t_m, where t_m is the time required for the beam component to reach the peak elasto-plastic displacement. The equivalent SDOF differential equation for this phase is given by:

m_{e} \ddot{y} + c_{e} \dot{y} + α K_{e} y + (1 - α) K_{e} y_{T} = 0

(20)

Depending on the relationship between the damping ratio ξ and the resistance enhancement coefficient α, equation (20) needs to be solved in two cases:

(1) When ξ²<α: By solving equation (20), the expressions for displacement and velocity in this case are:

y = e^{- \sqrt{\frac{k_{M - L}}{k_{m - l}}} ξ ω (t - t_{T})} [C_{5} \cos \sqrt{\frac{k_{M - L}}{k_{m - l}}} ω_{e} (t - t_{T}) + C_{6} \sin \sqrt{\frac{k_{M - L}}{k_{m - l}}} ω_{e} (t - t_{T})] + \frac{α - 1}{α} y_{T}

(21)

v = \sqrt{\frac{k_{M - L}}{k_{m - l}}} e^{- \sqrt{\frac{k_{M - L}}{k_{m - l}}} ξ ω (t - t_{T})} [(ω_{e} C_{6} - ξ ω C_{5}) \cos \sqrt{\frac{k_{M - L}}{k_{m - l}}} ω_{e} (t - t_{T}) - (ω_{e} C_{5} + ξ ω C_{6}) \sin \sqrt{\frac{k_{M - L}}{k_{m - l}}} ω_{e} (t - t_{T})]

(22)

where ω_e is the damped natural frequency in the plastic phase with resistance enhancement, and k_M−L and k_m−1 are the ratios of mass transformation coefficients and loading transformation coefficients in the elastic and plastic phases, respectively. The expressions for these parameters are as follows:

\begin{array}{l} ω_{e} = ω \sqrt{α - ξ^{2}}, \\ k_{M - L} = \frac{k_{M}}{k_{L}}, \\ k_{m - 1} = \frac{k_{m}}{k_{1}} \end{array}

(23)

Substituting the initial conditions y_T and v_T into equations (21) and (22), the constants C₅ and C₆ are solved as:

\begin{array}{l} C_{5} = \frac{y_{T}}{α}, \\ C_{6} = \sqrt{\frac{k_{m - l}}{k_{M - L}}} \frac{v_{T}}{ω_{e}} + \frac{ξ ω y_{T}}{α ω_{e}} \end{array}

(24)

Setting equation (22) to zero, the time t_m corresponding to the peak elasto-plastic displacement y_m is obtained as:

t_{m} = \sqrt{\frac{k_{m - l}}{k_{M - L}}} \frac{1}{ω_{e}} \arctan \sqrt{\frac{k_{m - l}}{k_{M - L}}} \frac{α ω_{e} v_{T}}{y_{T} ({ω_{e}}^{2} + ξ^{2} ω^{2}) + α ξ ω v_{T}} + t_{T}

(25)

Substituting equations (24) and (25) into equation (21), the expression for the displacement at the peak elasto-plastic displacement y_m is obtained.

(2) When ξ²≥α: By solving equation (20), the expressions for displacement and velocity in this case are:

y = C_{7} e^{- ω \sqrt{\frac{k_{M - L}}{k_{m - l}}} (ξ - \sqrt{ξ^{2} - α}) (t - t_{T})} + C_{8} e^{- ω \sqrt{\frac{k_{M - L}}{k_{m - l}}} (ξ + \sqrt{ξ^{2} - α}) (t - t_{T})} + \frac{α - 1}{α} y_{T}

(26)

v = - ω \sqrt{\frac{k_{M - L}}{k_{m - l}}} [(ξ - \sqrt{ξ^{2} - α}) C_{7} e^{- ω \sqrt{\frac{k_{M - L}}{k_{m - l}}} (ξ - \sqrt{ξ^{2} - α}) (t - t_{T})} + (ξ + \sqrt{ξ^{2} - α}) C_{8} e^{- ω \sqrt{\frac{k_{M - L}}{k_{m - l}}} (ξ + \sqrt{ξ^{2} - α}) (t - t_{T})}]

(27)

Similarly, the constants C₇ and C₈ are solved, and the time t_m corresponding to the peak elasto-plastic displacement y_m is obtained as:

\begin{array}{l} C_{7} = \frac{1}{2 \sqrt{ξ^{2} - α}} \sqrt{\frac{k_{m - l}}{k_{M - L}}} [\frac{v_{T}}{ω} + \sqrt{\frac{k_{M - L}}{k_{m - l}}} (ξ + \sqrt{ξ^{2} - α}) \frac{y_{T}}{α}], \\ C_{8} = \frac{y_{T}}{α} - C_{7} \end{array}

(28)

t_{m} = \frac{1}{2 ω \sqrt{ξ^{2} - α}} \sqrt{\frac{k_{m - l}}{k_{M - L}}} \ln \frac{(\sqrt{ξ^{2} - α} + ξ) C_{10}}{(\sqrt{ξ^{2} - α} - ξ) C_{9}} + t_{T}

(29)

Substituting equations (28) and (29) into equation (26), the expression for the displacement at the peak elasto-plastic displacement y_m is obtained.

Based on the theoretical model of blast loading for the beam component and the definitions of anti-explosion parameters, the expressions for the equivalent static loading dynamic coefficient k_h and ductility ratio β are given by:

\begin{array}{l} k_{h} = \frac{y_{T}}{y_{st}}, \\ β = \frac{y_{m}}{y_{T}} \end{array}

(30)

Substituting the values of y_T and y_m from both cases into equation (30), the dynamic coefficient and ductility ratio for flexible beam components can be obtained.

Dynamic response solution for rigid beam components

Elastic forced motion stage

For rigid beam components, the elastic forced motion phase occurs within the time interval 0 < t < t_T. The equivalent SDOF equation for this phase is the same as equation (3). By directly substituting t = t_T into equations (7) and (8), the expressions for displacement and velocity at the end of the elastic phase of the rigid beam component are obtained as:

y_{T} = e^{- ξ θ_{T}} (C_{1} \cos γ θ_{T} + C_{2} \sin γ θ_{T}) + \frac{y_{st} θ_{z} e^{- a \frac{θ_{T}}{θ_{z}}}}{a^{2} - 2 ξ a θ_{z} + {θ_{z}}^{2}} (θ_{z} - θ_{T} + \frac{2 ξ {θ_{z}}^{2} - 2 a θ_{z}}{a^{2} - 2 ξ a θ_{z} + {θ_{z}}^{2}})

(31)

\begin{array}{l} v_{T} = e^{- ξ θ_{T}} [(ω_{d} C_{2} - ω ξ C_{1}) \cos γ θ_{T} - (ω ξ C_{2} + ω_{d} C_{1}) \sin γ θ_{T}] \\ - y_{st} e^{- a \frac{θ_{T}}{θ_{z}}} \frac{a}{t_{z}} (\frac{{θ_{z}}^{2} - θ_{z} θ_{T}}{a^{2} - 2 ξ a θ_{z} + {θ_{z}}^{2}} + \frac{2 ξ {θ_{z}}^{3} - 2 a {θ_{z}}^{2}}{{(a^{2} - 2 ξ a θ_{z} + {θ_{z}}^{2})}^{2}}) - y_{st} e^{- a \frac{θ_{T}}{θ_{z}}} \frac{ω θ_{z}}{a^{2} - 2 ξ a θ_{z} + {θ_{z}}^{2}} \end{array}

(32)

Plastic forced motion stage

After the elastic forced motion phase, the blast loading continues to act on the beam component, which enters the plastic forced motion phase with initial conditions of displacement y_T and velocity v_T. The duration of this phase is defined by the relation t_T<t<t_z. The equivalent SDOF differential equation for this phase is given by:

m_{e} \ddot{y} + c_{e} \dot{y} + α K_{e} y + (\begin{array}{l} 1 - \\ α \end{array}) K_{e} y_{T} = Δ p_{m} l k_{1} (\begin{array}{l} 1 - \\ t / t_{z} \end{array}) e^{- a \frac{t}{t_{z}}}

(33)

Similar to the method for solving the plastic free motion stage for flexible beam components, solving equation (33) also requires considering two cases:

(1) When ξ²<α: By solving equation (33), the expressions for displacement and velocity are:

\begin{array}{l} y = e^{- \sqrt{\frac{k_{M - L}}{k_{m - l}}} ξ ω t} [C_{9} \cos \sqrt{\frac{k_{M - L}}{k_{m - l}}} ω_{e} t + C_{10} \sin \sqrt{\frac{k_{M - L}}{k_{m - l}}} ω_{e} t] + \frac{α - 1}{α} y_{T} \\ + y_{st} e^{- a \frac{t}{t_{z}}} (\frac{(k_{M - L} / k_{m - l}) ({θ_{z}}^{2} - θ_{z} ω t)}{a^{2} - 2 \sqrt{k_{M - L} / k_{m - l}} ξ a θ_{z} + (k_{M - L} / k_{m - l}) α {θ_{z}}^{2}} + \frac{2 (k_{M - L} / k_{m - l}) {θ_{z}}^{2} (\sqrt{k_{M - L} / k_{m - l}} ξ θ_{z} - a)}{{(a^{2} - 2 \sqrt{k_{M - L} / k_{m - l}} ξ a θ_{z} + (k_{M - L} / k_{m - l}) α {θ_{z}}^{2})}^{2}}) \end{array}

(34)

\begin{array}{l} v = \sqrt{\frac{k_{M - L}}{k_{m - l}}} e^{- \sqrt{\frac{k_{M - L}}{k_{m - l}}} ξ ω t} [(ω_{e} C_{10} - ξ ω C_{9}) \cos \sqrt{\frac{k_{M - L}}{k_{m - l}}} ω_{e} t - (ω_{e} C_{9} + ξ ω C_{10}) \sin \sqrt{\frac{k_{M - L}}{k_{m - l}}} ω_{e} t] \\ - y_{st} e^{- a \frac{t}{t_{z}}} [\begin{array}{l} \frac{a}{θ_{z}} (\frac{(k_{M - L} / k_{m - l}) ({θ_{z}}^{2} - θ_{z} ω t)}{a^{2} - 2 \sqrt{k_{M - L} / k_{m - l}} ξ a θ_{z} + (k_{M - L} / k_{m - l}) α {θ_{z}}^{2}} + \frac{2 (k_{M - L} / k_{m - l}) {θ_{z}}^{2} (\sqrt{k_{M - L} / k_{m - l}} ξ θ_{z} - a)}{{(a^{2} - 2 \sqrt{k_{M - L} / k_{m - l}} ξ a θ_{z} + (k_{M - L} / k_{m - l}) α {θ_{z}}^{2})}^{2}}) \\ + \frac{(k_{M - L} / k_{m - l}) θ_{z} ω}{a^{2} - 2 \sqrt{k_{M - L} / k_{m - l}} ξ a θ_{z} + (k_{M - L} / k_{m - l}) α {θ_{z}}^{2}} \end{array}] \end{array}

(35)

To simplify the expressions for displacement and velocity after substituting the initial conditions of displacement y_T and velocity v_T into equations (34) and (35), we define:

\begin{array}{l} f_{1}^{*} = \frac{(k_{M - L} / k_{m - l}) {θ_{z}}^{2}}{a^{2} - 2 \sqrt{k_{M - L} / k_{m - l}} ξ a θ_{z} + (k_{M - L} / k_{m - l}) α {θ_{z}}^{2}} (1 + \frac{2 \sqrt{k_{M - L} / k_{m - l}} ξ θ_{z} - 2 a}{a^{2} - 2 \sqrt{k_{M - L} / k_{m - l}} ξ a θ_{z} + (k_{M - L} / k_{m - l}) α {θ_{z}}^{2}}), \\ f_{2}^{*} = - \frac{(k_{M - L} / k_{m - l}) ω θ_{z}}{a^{2} - 2 \sqrt{k_{M - L} / k_{m - l}} ξ a θ_{z} + (k_{M - L} / k_{m - l}) α {θ_{z}}^{2}}, \\ f_{3}^{*} = \sqrt{\frac{k_{M - L}}{k_{m - l}}} ω_{e} t_{T}, \\ f_{4}^{*} = e^{- \sqrt{\frac{k_{M - L}}{k_{m - l}}} ξ ω t_{T}} \end{array}

(36)

Using equation (36), the constants C₉ and C₁₀ are solved as:

C_{9} = \frac{1}{ω_{e} f_{4}^{*}} (ω_{e} \cos f_{3}^{*} - ξ ω \sin f_{3}^{*}) [\frac{y_{T}}{α} - y_{st} e^{- a \frac{θ_{T}}{θ_{z}}} (f_{2}^{*} t_{T} + f_{1}^{*})] - \frac{v_{T} + y_{st} e^{- a \frac{θ_{T}}{θ_{z}}} [\frac{a}{t_{z}} (f_{2}^{*} t_{T} + f_{1}^{*}) - f_{2}^{*}]}{\sqrt{k_{M - L} / k_{m - l}} ω_{e} f_{4}^{*}} \sin f_{3}^{*}

(37)

C_{10} = \frac{\frac{y_{T}}{α} - y_{st} e^{- a \frac{θ_{T}}{θ_{z}}} (f_{2}^{*} t_{T} + f_{1}^{*})}{f_{4}^{*} \sin f_{3}^{*}} - \frac{C_{9}}{\tan f_{3}^{*}}

(38)

(2) When ξ²≥α: By solving equation (33), the expressions for displacement and velocity are:

\begin{array}{l} y = C_{11} e^{- ω \sqrt{\frac{k_{M - L}}{k_{m - l}}} (ξ + \sqrt{ξ^{2} - α}) t} + C_{12} e^{- ω \sqrt{\frac{k_{M - L}}{k_{m - l}}} (ξ - \sqrt{ξ^{2} - α}) t} \\ + y_{st} e^{- a \frac{t}{t_{z}}} (\frac{(k_{M - L} / k_{m - l}) ({θ_{z}}^{2} - θ_{z} ω t)}{a^{2} - 2 \sqrt{k_{M - L} / k_{m - l}} ξ a θ_{z} + (k_{M - L} / k_{m - l}) α {θ_{z}}^{2}} + \frac{2 (k_{M - L} / k_{m - l}) {θ_{z}}^{2} (\sqrt{k_{M - L} / k_{m - l}} ξ θ_{z} - a)}{{(a^{2} - 2 \sqrt{k_{M - L} / k_{m - l}} ξ a θ_{z} + (k_{M - L} / k_{m - l}) α {θ_{z}}^{2})}^{2}}) + \frac{α - 1}{α} y_{T} \end{array}

(39)

\begin{array}{l} v = ω \sqrt{\frac{k_{M - L}}{k_{m - l}}} [- (ξ + \sqrt{ξ^{2} - α}) C_{11} e^{- ω \sqrt{\frac{k_{M - L}}{k_{m - l}}} (ξ + \sqrt{ξ^{2} - α}) t} - (ξ - \sqrt{ξ^{2} - α}) C_{12} e^{- ω \sqrt{\frac{k_{M - L}}{k_{m - l}}} (ξ - \sqrt{ξ^{2} - α}) t}] \\ - y_{st} e^{- a \frac{t}{t_{z}}} [\begin{array}{l} \frac{a}{θ_{z}} (\frac{(k_{M - L} / k_{m - l}) ({θ_{z}}^{2} - θ_{z} ω t)}{a^{2} - 2 \sqrt{k_{M - L} / k_{m - l}} ξ a θ_{z} + (k_{M - L} / k_{m - l}) α {θ_{z}}^{2}} + \frac{2 (k_{M - L} / k_{m - l}) {θ_{z}}^{2} (\sqrt{k_{M - L} / k_{m - l}} ξ θ_{z} - a)}{{(a^{2} - 2 \sqrt{k_{M - L} / k_{m - l}} ξ a θ_{z} + (k_{M - L} / k_{m - l}) α {θ_{z}}^{2})}^{2}}) \\ + \frac{(k_{M - L} / k_{m - l}) θ_{z} ω}{a^{2} - 2 \sqrt{k_{M - L} / k_{m - l}} ξ a θ_{z} + (k_{M - L} / k_{m - l}) α {θ_{z}}^{2}} \end{array}] \end{array}

(40)

Similarly, to simplify the expressions, let:

\begin{array}{l} f_{5}^{*} = - \sqrt{\frac{k_{M - L}}{k_{m - l}}} (ξ - \sqrt{ξ^{2} - α}) θ_{T}, \\ f_{6}^{*} = - \sqrt{\frac{k_{M - L}}{k_{m - l}}} (ξ + \sqrt{ξ^{2} - α}) θ_{T} \end{array}

(41)

Substituting the initial conditions of displacement y_T and velocity v_T into equations (39) and (40), and using equation (41), the constants C₁₁ and C₁₂ are solved as:

C_{11} = \frac{v_{T} t_{T} + y_{st} t_{T} e^{- a \frac{θ_{T}}{θ_{z}}} [(\frac{f_{5}^{*}}{t_{T}} + \frac{a}{t_{z}}) (\begin{array}{l} f_{1}^{*} + \\ f_{2}^{*} t_{T} \end{array}) - f_{2}^{*}] - \frac{y_{T} f_{5}^{*}}{α}}{(f_{6}^{*} - f_{5}^{*}) e^{f_{6}^{*}}}

(42)

C_{12} = \frac{y_{T} - y_{st} α e^{- a \frac{θ_{T}}{θ_{z}}} (f_{1}^{*} + f_{2}^{*} t_{T}) - α C_{11} e^{f_{6}^{*}}}{α e^{f_{5}^{*}}}

(43)

Substituting t = t_z into equations (39) and (40), the expressions for displacement y_z and velocity v_z at the end of the blast loading effect are obtained.

Plastic free motion stage

After the blast loading is removed, the beam component enters the plastic free motion stage with initial conditions of displacement y_z and velocity v_z. The duration of this stage is defined by the relation t_z<t<t_m. The equivalent SDOF equation for this stage is the same as equation (20), and the solution must also be divided into two cases:

(1) When ξ²<α: By solving equation (20), the expressions for displacement and velocity are:

y = e^{- \sqrt{\frac{k_{M - L}}{k_{m - l}}} ξ ω (t - t_{z})} [C_{13} \cos \sqrt{\frac{k_{M - L}}{k_{m - l}}} ω_{e} (t - t_{z}) + C_{14} \sin \sqrt{\frac{k_{M - L}}{k_{m - l}}} ω_{e} (t - t_{z})] + \frac{α - 1}{α} y_{T}

(44)

v = \sqrt{\frac{k_{M - L}}{k_{m - l}}} e^{- \sqrt{\frac{k_{M - L}}{k_{m - l}}} ξ ω (t - t_{z})} [(ω_{e} C_{14} - ξ ω C_{13}) \cos \sqrt{\frac{k_{M - L}}{k_{m - l}}} ω_{e} (t - t_{z}) - (ω_{e} C_{13} + ξ ω C_{14}) \sin \sqrt{\frac{k_{M - L}}{k_{m - l}}} ω_{e} (t - t_{z})]

(45)

In this case, the initial conditions are the displacement y_z and velocity v_z obtained from the plastic forced motion stage for ξ²<α. Substituting these initial conditions into equations (44) and (45), the constants C₁₃ and C₁₄ are solved as:

\begin{array}{l} C_{13} = y_{z} + \frac{1 - α}{α} y_{T}, \\ C_{14} = \sqrt{\frac{k_{m - l}}{k_{M - L}}} \frac{v_{z}}{ω_{e}} + \frac{ξ ω}{ω_{e}} C_{13} + \frac{ξ ω y_{T}}{α ω_{e}} \end{array}

(46)

Setting equation (45) to zero, the expression for the time t_m when the beam component reaches the plastic displacement peak y_m is:

t_{m} = \sqrt{\frac{k_{m - l}}{k_{M - L}}} \frac{1}{ω_{e}} \arctan \frac{ω_{e} C_{14} - ξ ω C_{13}}{ω_{e} C_{13} + ξ ω C_{14}} + t_{z}

(47)

Substituting equations (46) and (47) into equation (44), the expression for the displacement at the plastic displacement peak y_m is obtained.

(2) When ξ²≥α: By solving equation (20), the expressions for displacement and velocity are:

y = C_{15} e^{- ω \sqrt{\frac{k_{M - L}}{k_{m - l}}} (ξ - \sqrt{ξ^{2} - α}) (t - t_{z})} + C_{16} e^{- ω \sqrt{\frac{k_{M - L}}{k_{m - l}}} (ξ + \sqrt{ξ^{2} - α}) (t - t_{z})} + \frac{α - 1}{α} y_{T}

(48)

v = ω \sqrt{\frac{k_{M - L}}{k_{m - l}}} [(\sqrt{ξ^{2} - α} - ξ) C_{15} e^{- ω \sqrt{\frac{k_{M - L}}{k_{m - l}}} (ξ - \sqrt{ξ^{2} - α}) (t - t_{z})} - (ξ + \sqrt{ξ^{2} - α}) C_{16} e^{- ω \sqrt{\frac{k_{M - L}}{k_{m - l}}} (ξ + \sqrt{ξ^{2} - α}) (t - t_{z})}]

(49)

In this case, the initial conditions are the displacement y_z and velocity v_z obtained from the plastic forced motion stage for ξ²≥α. Substituting these initial conditions into equations (48) and (49), the constants C₁₅ and C₁₆ are solved as:

\begin{array}{l} C_{15} = \frac{1}{2 \sqrt{ξ^{2} - α}} [\sqrt{\frac{k_{m - l}}{k_{M - L}}} \frac{v_{z}}{ω} + (\begin{array}{c} ξ + \\ \sqrt{ξ^{2} - α} \end{array}) (y_{z} + \frac{1 - α}{α} y_{T})], \\ C_{16} = y_{z} + \frac{1 - α}{α} y_{T} - C_{15} \end{array}

(50)

Setting equation (49) to zero, the expression for the time t_m when the beam component reaches the plastic displacement peak y_m is:

t_{m} = \frac{1}{2 ω \sqrt{ξ^{2} - α}} \sqrt{\frac{k_{m - l}}{k_{M - L}}} \ln \frac{(\sqrt{ξ^{2} - α} + ξ) C_{16}}{(\sqrt{ξ^{2} - α} - ξ) C_{15}} + t_{z}

(51)

Substituting equations (50) and (51) into equation (48), the expression for the displacement at the plastic displacement peak y_m is obtained.

By substituting the displacement y_T and y_m obtained from the above two cases into equation (30), the expressions for the dynamic coefficient and ductility ratio of the rigid beam component are obtained.

Verification of the theoretical accuracy

Experimental verification

To verify the theoretical model, two beam components (2B3 and 5B1) from Nassr’s (Nassr et al., 2012a) field explosion tests on simply supported I-beams were selected as the research objects. Finite element models of these components were established and simulated using LS-DYNA software. The simulation results were compared with the experimental data to validate the reliability of the finite element model. The parameters of the two steel beams are provided in Table 1.

Table 1.

Parameter information for steel beams.

No.	Cross-sectional dimensions (h × b × t_w × t)/(mm × mm × mm × mm)	Blast distanceS/ m	Scaled distanceR/ m·kg^1/3	Beam length l/ m	Overpressure peakΔp_m/ MPa	Loading duration t_z/ ms	StiffnessK/ kN·m^-1	Moment of inertiaI/ mm⁴
2B3	160.0 × 102.0 × 6.6 × 10.3	10.3	2.22	2.413	0.623	6.00	1955	13092667
5B1	216.0 × 206.0 × 10.0 × 17.0	9.5	1.51	2.413	2.098	7.45	66179	71796697

Note: h is the height of the steel beam cross-section; b is the width of the cross-section; t_w is the width of the web; t is the thickness of the flange.

In LS-DYNA, the material constitutive model for bilinear elasto-plasticity was defined using the *MAT_PLASTIC_KINEMATIC keyword, with material model parameters listed in Table 2. The boundary conditions of the steel beams were defined using *BOUNDARY_SPC_SET with simply supported conditions: one end was constrained in the X, Y, and Z displacement directions, as well as the X and Z rotational directions, while the other end was constrained in the Y and Z displacement directions, along with the X and Z rotational directions. A schematic of the numerical simulation setup is shown in Figure 3. The damping was defined using the *DAMPING_GLOBAL keyword, in this paper, the damping ratio is set to 0.21. and the Friedlander-type blast loading measured in the experiment was defined using *DEFINE_CURVE. The blast loading was then applied to the flange surface of the beam using *LOAD_SEGMENT_SET. Based on the mesh sensitivity analysis, an element size of 5 mm was selected. For the two experimental cases from Nassr’s study, distinct blast loading parameters were configured including overpressure peak, loading duration, and shape parameter as specified in Table 2. The corresponding loading profiles are illustrated in Figure 4(c) and 4(d).The displacement time history curves from the experimental tests and the simulation analysis for both steel beam components, observed at the midpoint of the upper flange at the center span of the I-beam, are shown in Figure 4(e) and 4(f).

Table 2.

Material modeling parameters for two types of steel beams.

No.	Density ρ/ kg·m⁻³	Elastic modulus E/ GPa	Tangent modulus E_t/ GPa	Yield stress σ/ MPa	Poisson’s ratio v	Plastic strain F_s	Overpressure peakΔp_m/ MPa	Loading duration t_z/ ms	Shape parameter a
2B3	7850	27.28	0.74	350.00	0.3	0.2	0.6	6	1.5
5B1	7850	168.62	74.19	487.32	0.3	0.2	2.1	7.5	2

Figure 3.

Schematic of the numerical simulation setup.

Figure 4.

Experimental and simulation analysis of I-beam structures under Friedlander-type blast loading.

As shown in Figure 4, the results of the finite element simulation are in excellent agreement with the experimental results, with the errors in the peak elastic-plastic displacement of the two components being 1.07% (Component 2B3) and 2.27% (Component 5B1), respectively. This indicates that the finite element model can accurately reflect the dynamic response of the beam components under blast loading, providing a solid foundation for further in-depth analysis based on finite element methods.

By comparing the theoretical analysis results with the experimental data, the accuracy of the theoretical model can be further evaluated: For Component 2B3, the peak displacement predicted by the theoretical analysis is 38.83 mm, with an error of 4.83% compared to the experimental result of 40.80 mm; for Component 5B1, the theoretical peak displacement is 60.64 mm, showing a 3.44% error relative to the experimental value of 62.80 mm. The overall trend of the theoretical analysis is in good agreement with the experimental results, and the errors are within a reasonable range.

This study validates the accuracy of numerical simulations through experimental results and further comparison analysis indicates that the theoretical model is highly reliable in terms of both precision and practicality, despite some simplifications. Specifically, in the theoretical analysis, the material nonlinearity of the beam components and the complex dynamic characteristics of the blast loading were moderately simplified, which led to corresponding deviations. However, this deviation has a limited impact on the validity of the theoretical model, demonstrating that it possesses strong engineering applicability. Through comparison with the experimental results, this paper finds that the theoretical analysis is not only in good agreement with the experimental data, but the numerical simulation results also match the experimental data, verifying the credibility and high accuracy of dynamic response analysis based on numerical simulations under blast loading. Overall, the theoretical analysis provides a simplified yet effective prediction method, offering valuable insights for preliminary analyses in engineering practice.

Theoretical method verification

Based on the finite element model described earlier, both flexible and rigid I-beams were designed, and simulation analyses were conducted for three different loading conditions to verify the feasibility of the proposed theoretical method. The cross-sectional dimensions of the two types of beams are shown in Figure 5. The tangent modulus E_t for the three loading conditions were 0.21 GPa, 2.1 GPa, and 21 GPa, respectively, with a constant damping ratio of 0.1. The remaining material model parameters for the steel beams are provided in Table 3.

Figure 5.

Cross-section dimensions of steel beams. (unit: mm).

Table 3.

Parameter information for steel beams.

Component type	Beam length l/ m	Overpressure peakΔp_m/ MPa	Loading duration t_z/ ms	Stiffness K/ kN·m^-1	Moment of inertiaI/ mm⁴	Density ρ/ kg·m^-3	Elastic modulus E/ GPa	Yield stress σ/ MPa	Poisson’s ratio v	Plastic strain F_s
Flexibility	2	1.5	1.52	18157.09	9006493	7850	210	375	0.3	0.2
Rigidity	2	3.0	3.23	44451.53	22049369	7850	210	470	0.3	0.2

The range of the shape parameter a is set from 1.0 to 2.0, which effectively fits the positive and negative overpressure decay curves of blast loading, capturing both the rapid decay in the initial phase and the gradual change in the later phase. Reference Ning et al. (2024) found that this range can accurately fit the blast loading of conventional weapons, ensuring the accuracy and reliability of the model, and avoiding distortions caused by excessively large or small parameter values (Table 4).

Table 4.

Design for calculation cases.

Loading condition	Shape parameter a	Resistance enhancement coefficient α
C1	1.0	0.001
C2	1.5	0.01
C3	2.0	0.1

The resistance enhancement coefficient α in the range of 0.001 to 0.1, with values of 0.001, 0.01, and 0.1, represents the transition from ideal elasto-plastic behavior to low-level strengthening. Specifically, α = 0.001 corresponds to nearly ideal elasto-plastic behavior with minimal strengthening effects, suitable for cases with no significant material enhancement; α = 0.01 represents a weak resistance enhancement, applicable to materials with minor strengthening effects; and α = 0.1 indicates moderate strengthening, reflecting the material’s response under higher stress conditions. These values provide a clear description of material resistance enhancement under blast loading, as discussed in reference Geng et al. (2024).

Displacement time history curves for both types of beams under the three conditions, measured at the midpoint of the upper flange at the center span, are presented in Figure 6.

Figure 6.

Midspan displacement-time curves for two types of beam components.

Since the theoretical model in this study treats the displacement (y) and duration parameter (θ) as dimensionless by using the static displacement (y_st) and motion frequency (ω) of the blast loading when simplified as a static loading, the units of both the theoretical and simulation results need to be normalized. The calculation of y_st and ω is explained below. By substituting the overpressure peak value (Δp_m), beam length (l), and stiffness (K) from Table 3 for the flexible and rigid beam components into equation (9), the static displacement (y_st) for the two beam types is calculated to be 165.22 mm and 134.98 mm, respectively. Based on the formula for the first natural frequency of motion for I-beams from the literature (China Architecture and Building Press, 2006), and substituting the relevant parameters, the motion frequencies ( $ω = π^{2} \sqrt{E I / ρ A} / l^{2}$ ) for the two types of beams are calculated as 264 m/s and 310 m/s, respectively.

As shown in Figure 6, through finite element simulations, the elastic displacement peak (y_T) of the flexible beam components under the three working conditions are 15.006 mm (t_T = 1.60 ms), 14.899 mm (t_T = 1.65 ms), and 14.748 mm (t_T = 1.84 ms), respectively. For the rigid beam components, the elastic displacement peak (y_T) is 22.092 mm (t_T = 1.75 ms), 20.738 mm (t_T = 1.77 ms), and 19.613 mm (t_T = 1.78 ms), respectively. Using equation (30), the dynamic coefficients (k_h) for both types of beams under the three working conditions are calculated based on the finite element simulation. Substituting the parameters also yields the k_h values for the three working conditions based on the theoretical model presented in this study. A summary of the k_h values obtained from both methods is provided in Table 5.

Table 5.

Comparison of results between theory and finite element simulation.

Loading condition	β	Flexible beam component (θ_{i = 0.40})			β	Rigid beam component (θ_{i = 1.0})
Loading condition	β	Theoretical k_h	Simulation k_h	Error	β	Theoretical k_h	Simulation k_h	Error
C1	1.52	0.0883	0.0908	−2.75%	2.35	0.1577	0.1637	−3.67%
C2	1.36	0.0854	0.0902	−5.32%	2.22	0.1432	0.1536	−6.77%
C3	1.19	0.0831	0.0893	−6.94%	2.02	0.1338	0.1453	−7.91%

As shown in Table 5, the maximum difference between the dynamic coefficient (k_h) calculated using the proposed theoretical method and the finite element simulation results is only 7.91%. This indicates that the theoretical analysis, which simplifies the blast loading to the Friedlander-type loading based on the plastic resistance enhancement model, provides a reliable foundation for simulating and analyzing the dynamic response of beam components under blast loading.

Verification against standards

To further verify the accuracy of the theoretical solutions presented in this study, the calculation method for the equivalent single-degree-of-freedom dynamic coefficient recommended by the current explosion-resistant design code in China is used as a comparative benchmark. The calculation formula provided in the code is (China Architecture and Building Press, 2006):

k_{h} = {[\frac{2}{θ_{i}} \sqrt{2 β - 1} + \frac{2 β - 1}{2 β (1 + 4 / θ_{i})}]}^{- 1}

(52)

where θ_i is the linear decay loading explosion duration parameter. The relationship between this parameter and the Friedlander-type blast loading duration parameter θ_z for the beam component is given by Geng et al. (2019):

θ_{i} = 2 θ_{z} [\frac{1}{a} + \frac{1}{a^{2}} (e^{- a} - 1)]

(53)

In this study, a simply supported beam is selected as the type of beam subjected to loading. The ratio of the elasto-plastic mass transformation coefficient to the loading transformation coefficient (k_M-L/k_m-1) is taken as 1.18 (Fang and Liu, 2010). The range of the beam component’s duration parameter θ_z is considered from 0.2 to 2.2, with the damping ratio ξ set to 0.1. The shape parameter a takes values of 1,1.3, 1.5, 1.8, and 2.0, and the resistance enhancement coefficient α is taken as 0.001, 0.01, and 0.1. The ductility ratio β is considered as 1.0, 3.0, and 5.0. In total, 27 typical working conditions are investigated under the action of the Friedlander-type blast loading and the plastic resistance enhancement model. For the 27 cases, the results are classified according to the values of the shape parameter a. The results are compared with the formula in the code (China Architecture and Building Press, 2006) for ductility ratios of 1.0, 3.0, and 5.0, as shown in Figures 7 –11.

Figure 7.

Comparison results for calculation cases and code of a = 1.0.

Figure 8.

Comparison results for calculation cases and code of a = 1.3.

Figure 9.

Comparison results for calculation cases and code of a = 1.5.

Figure 10.

Comparison results for calculation cases and code of a = 1.8.

Figure 11.

Comparison results for calculation cases and code of a = 2.0.

As shown in Figures 7 –11, the theoretical calculations of k_h in this study are consistently lower than the values obtained from the explosion-resistant design code, with deviations ranging from approximately 13.36% to 26.75%. A smaller k_h indicates a more economical explosion-resistant design. For all working conditions, k_h increases as θ_z increases, and for the elasto-plastic design with β = 1, k_h is always higher than for designs with β > 1. In this scenario, the elasto-plastic explosion-resistant design can be treated as an elastic design, with no plastic deformation of the beam component. Under the same values of a and β, as α increases, k_h decreases; under the same values of α and β, as a increases, k_h decreases; and under the same values of α and a, as β increases, k_h decreases. These results indicate that increasing α, a, or β leads to a reduction in k_h. Moreover, the larger the values of α, a, and β, the greater the relative error between the theoretical results and the code values. The design code formula provides higher accuracy in the explosion-resistant design of rigid beam components compared to flexible beam components.

To further investigate the individual effects of α and a on k_h, a baseline condition with a = 1.0 is taken, and the reduction in k_h for a = 1.3, 1.5, 1.8 and 2.0 relative to this baseline is calculated to study the effect of a, respectively, as shown in Figure 10. Similarly, for α = 0.001 as the baseline, the reduction in k_h for α > 0.001 relative to this baseline is calculated to study the effect of α. The reduction in k_h is denoted as δ_h, as shown in Figure 11.

As shown in Figures 12 –14, the greater the value of a, the larger the reduction in k_h. For a = 1.3, the reduction in k_h ranges from 7.31% to 7.91%, for a = 1.5, the reduction in k_h ranges from 11.68% to 12.63%, for a = 1.8, the reduction in k_h ranges from 17.62% to 20.60%, and for a = 2.0, it ranges from 21.18% to 22.83%. As β decreases, the effect of a on k_h becomes more pronounced. For the same value of β, there is almost no difference in k_h for different values of α, as the effect of α on k_h has already been isolated before examining the effect of a alone. This suggests that α and a have a linear coupling effect.

Figure 12.

The reduction rate curves for dynamic factor of a at β = 1.

Figure 13.

The reduction rate curves for dynamic factor of α at β = 3.

Figure 14.

The reduction rate curves for dynamic factor of a at β = 5.

As shown in Figures 15–16, the larger the value of α, the greater the reduction in k_h. For α = 0.01 and 0.1, the maximum reduction in k_h is 0.69% and 6.83%, respectively. When β = 1, the effect of α on k_h is negligible, with a range of 0% to 0.69%. This indicates that in these cases, the effect of α can be ignored. When β > 1 and α > 0.01, the reduction in k_h ranges from 3.34% to 6.83%, suggesting that a larger value of α has a more significant impact on reducing k_h. In comparison to the effect of a on k_h, the effect of α is less significant when considered alone.

Figure 15.

The reduction rate curves for dynamic factor of a = 0.01.

Figure 16.

The reduction rate curves for dynamic factor of α = 0.1.

Conclusions

This paper theoretically derives the analytical expressions for the dynamic coefficients of beam components under the influence of Friedlander-type blast loading and the plastic resistance enhancement model based on the SDOF system. The results are compared and verified against finite element simulation results and current design code calculations. The influence of shape parameters and resistance enhancement coefficients on the dynamic coefficient is analyzed, leading to the following conclusions:

(1) Under identical parameters related to blast loading, increasing the shape parameter a of the Friedlander-type blast loading consistently reduces the dynamic coefficient k_h. Compared to the baseline case with a = 1.0, the value of k_h decreases by 7.31%–7.91% when a = 1.3, by 11.68%–12.63% when a = 1.5, by 17.62%–20.60% when a = 1.8, and reaches a maximum reduction of 22.83% when a = 2.0. These results highlight the significant influence of the shape parameter on the dynamic response of beams, indicating that the mathematical expressions of blast loading should be carefully considered in blast-resistant design.

(2) The effect of the resistance enhancement coefficient α on the dynamic coefficient k_h becomes more pronounced as α increases. When α ≤ 0.01, the plastic resistance enhancement effect is negligible, and the beam component behaves nearly as an ideal elasto-plastic system. In this case, the reduction in k_h does not exceed 0.69%, suggesting that the use of ideal elasto-plastic assumptions offers both computational efficiency and sufficient accuracy. When α > 0.01, the reduction in k_h becomes significant; for instance, when α = 0.1, k_h decreases by 6.83%, demonstrating a notable influence of resistance enhancement. However, comparative analysis indicates that the influence of the shape parameter a on k_h is still more significant than that of α.

(3) The ductility ratio β, which characterizes the allowable extent of plastic deformation in beam components, also markedly affects the dynamic coefficient k_h. When β = 1, the maximum deviation between the present theoretical k_h and the code-recommended value is 13.36%. Under high ductility conditions with β = 5, the beam enters a stage characterized by a high degree of plasticity, where the code predictions of k_h become overly conservative, with a maximum deviation reaching 26.75%. These findings suggest that current design codes are more suitable for low-ductility beam components in blast-resistant applications. For detailed designs involving high-ductility elements, a multi-parameter model such as the one developed in this study should be adopted to achieve greater accuracy and reliability.

Footnotes

ORCID iD

Shaobo Geng

Author Contributions

Shaobo Geng: Conceptualization, methodology, investigation, formal analysis, writing – original draft, and supervision. Haojia Li: Data curation, software implementation, and validation, Writing–original draft. Xin Hong: Data curation and Writing–original draft. Rui Feng: Methodology, formal analysis, and contribution to numerical simulation. Yanwei Niu: Review and editing, project administration, and validation of experimental results.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to acknowledge the Fundamental Research Program of Shanxi Province (Grant No. 202203021211099), the National Natural Science Foundation of China (Grant No. 51408558), the Fundamental Research Funds for the Central Universities, CHD (Grant No. 300102215519, 300102212907), and the Hunan Traffic Science and Technology Project (No. 202510) to provide fund for conducting experiments.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

Data is available and will be shared upon request.*

References

Biggs

(1964) Introduction to Structural Dynamics. McGraw-Hill.

Canada Standards Association (2012) CSA/S 850-12: Design and Assessment of Buildings Subject to Blast Loads. Canadian Standards Association.

Chen

Song

Wang

(2010) Analytical solution of RC beam system under explosion loading. Engineering Mechanics 27(4): 73–78.

Chen

Luo

Guo

, et al. (2020) Responses of HFR-LWC beams under close-range blast loadings accompanying membrane action. Defence Technology 16(6): 1167–1187.

Chen

Guo

Luo

(2021) Theoretical model for HFR-LWC beam under blast loading accompanying membrane action and its experimental validation. Engineering Mechanics 38(2): 77–91.

Cheng

Yang

Guo

(2009) Analysis on an exponential attenuation factor in the modified friedlander equation by overpressure tests. Journal of Explosion and Shock Waves 29(4): 425–428.

China Architecture & Building Press (2006) GB 50038-2005: Code for Design of Civil Air Defence Basement. Beijing: China Architecture & Building Press.

China Architecture & Building Press (2012) GB 50009-2012: Load Code for the Design of Building Structures. China Architecture & Building Press.

Cui

Zhang

Hao

(2021) Improved analysis method for structural members subjected to blast loads considering strain hardening and softening effects. Advances in Structural Engineering 24(12): 2622–2636.

10.

Department of the Army (1990) TM5-1300: Structures to Resist the Effects of Accidental Explosions. Washington: Department of the Army.

11.

Fallah

Louca

(2007) Pressure–impulse diagrams for elastic-plastic-hardening and softening single-degree-of-freedom models subjected to blast loading. International Journal of Impact Engineering 34(4): 823–842. doi: 10.1016/j.ijimpeng.2006.01.007, In press.

12.

Fang

Liu

(2010) Underground Protective Structure. China Water & Power Press.

13.

Geng

(2019) Equivalent static load dynamic coefficient for blast load. Acta Armamentarii 40(10): 2088–2095.

14.

Geng

Niu

Wang

(2021) Dynamic increase factor of an equivalent SDOF structural system for beams with different support conditions under conventional blast loading. Journal of Engineering Mechanics 147(4): 06021002.

15.

Geng

Han

Niu

(2023) Study on dynamic coefficient of beam members with linear hardening resistance model under blast load. Chinese Journal of Computational Mechanics , (Online first)10.7511/jslx20230314002.

16.

Geng

Han

Niu

, et al. (2024) Study on dynamic coefficient of beam members with linear hardening resistance model under blast load. Chinese Journal of Computational Mechanics/Jisuan Lixue Xuebao 41(5): 886–893.

17.

Yang

Wang

(2023) Improved single degree of freedom method considering flange deformation of steel beam under blast loading. Journal of Hebei University of Engineering (Natural Science Edition) 40(2): 44–51.

18.

Lan

Feng

Huang

, et al. (2018) Blast response of continuous-density graded cellular material based on the 3D Voronoi model. Defence Technology 14(5): 433–440.

19.

Liao

Tang

, et al. (2019) Study on explosion resistance performance experiment and damage assessment model of high-strength reinforcement concrete beams. International Journal of Impact Engineering 133: 103362.

20.

Lin

Bai

Zhang

(2001) Overpressure functions of blast waves for explosions in air. Explosion and Shock Waves 21(1): 41–46.

21.

Liu

Geng

(2021) Dynamic coefficient of series chemical blast load in building structure. Journal of Vibration and Shock 40(23): 51–57.

22.

Luo

Chen

Guo

(2020) Experimental study on blast-resistances of HFR-LWC beam with membrane effect. Engineering Mechanics 37(5): 237–248.

23.

Fang

(2023) A unified performance-based blast-resistant design approach for RC beams/columns. International Journal of Impact Engineering 173: 104459.

24.

Nagata

Beppu

Ichino

, et al. (2018) Method for evaluating the displacement response of RC beams subjected to close-in explosion using modified SDOF model. Engineering Structures 157: 105–118.

25.

Nassr

Razaqpur

Tait

, et al. (2012a) Experimental performance of steel beams under blast loading. Journal of Performance of Constructed Facilities 26(5): 600–619.

26.

Nassr

Razaqpur

Tait

, et al. (2012b) Single and multi degree of freedom analysis of steel beams under blast loading. Nuclear Engineering and Design 242: 63–77.

27.

Ning

Hong

Geng

, et al. (2024) Dynamic factor of damped beam member under exponential blast load. Journal of Vibration and Shock 43(21): 310–318.

28.

Wei

Zhang

, et al. (2023) Modification of SDOF model for reinforced concrete beams under close-in explosion. Defence Technology 20: 162–186.

29.

Xin

Wang

, et al. (2018) Empirical formula and numerical simulation of TNT explosion shock wave in free air. Missiles and Space Vehicles 3(3): 98–102. doi: 10.7654/j.issn.1004-7182.20180319.

30.

Tang

Wang

(2018) Analysis of dynamic effects on a single-story pre-stressed cable-stayed steel frame caused by sudden cable failure. Steel Construction 33(1): 6–10.

31.

Zhang

Wang

(2019) Quasi-static pressure experiment and numerical simulation of TNT internal explosion. Journal of Ordnance Equipment Engineering 40(5): 195–199.