# Equivalent SDOF Load Transformation Factors for One-Way Members Subjected to Air Blast Loading

Authors: Yaobei He; Shaobo Geng; Yaxin Wei; Gan Luo; Xin Gong; Huifeng Mao
Journal: Advances in Civil Engineering
 (2021)

---

## Abstract

The equivalent single-degree-of-freedom (SDOF) load transformation factor expressions of the simply supported, fixed end and simply-fixed one-way members subjected to air blast loading are derived by finite series-expansion method or finite point-fitting method. Moreover, figures are used to show the relative relationship between the load transformation factorsKL and the ratio of stand-off distance to structural span (γ). The typical calculation conditions show that the load transformation factors are sensitive to the piecewise function of peak overpressure when γ is less than 0.5. The load transformation factor will be a smooth curve for peak overpressure adopting a single smooth continuous function. For the blast-resistant structural design, if the error of 3% in the elastic stage and 5% in the plastic stage can be allowed, γ = 1.0 is the demarcation point between near field blast and far field blast. In addition, if the error of 0.5% can be allowed, γ ≥ 2.0 can be treated as the far field blast. The calculation results of far field blast show that the values of KL of simply supported, simply-fixed, and fixed-end members are 0.64, 0.58, and 0.54, respectively, which means that the load transformation factor is insensitive to the selection of shape function in vibration mode function or static deflection curve function. The calculation accuracy of KL expression for simply supported beam structure in this paper has been verified through near field blast experiments. This paper provides a clear numerical definition for the distinction between near field and far field explosions. It provides qualification criteria for analyzing explosion categories in blast-resistant structural design and a reference for the selection of theoretical calculation formulas.


---

## Body

## 1. Introduction

Air blast, probably caused by the industrial accident, terrorist attack, and military action, will apply a load of the shock wave to the adjacent buildings. Important structures, such as cross-sea bridge and protective structures, should be designed to withstand a certain degree of blast loading. One-way members, such as beams or one-way slabs, are the main components of blast-resistant structures. Finite-element analysis, experimental test, and equivalent single-degree-of-freedom (SDOF) method are the main three tools to finish structural blast-resistant design [1, 2]. Finite element analysis requires highly specialized software and extensive computing work. The experimental test needs harsh conditions and expensive costs for the safety of engineers and equipments. Therefore, these two methods are mainly used for research. Equivalent SDOF method, which has good calculation and design accuracy, is the simplest tool for designers [3]. As an efficient technique, the SDOF method is also referenced in guidelines such as TM-5-1300 [4]; it provides design methods to resist the impact of blast waves. UFC 3-340-02 [5] provides design methods based on the bending and shear failure modes of the structure subjected to blast loads. GB5009-2012 [6] proposed the basic steps to evaluate the equivalent uniform static loads. Extensive theoretical and experimental researches have been conducted before to investigate the calculation characteristics of the equivalent SDOF method for beam or one-way slab structures subjected to blast loads [7–12]. Stochino and Carta [7] used two SDOF models to approximate the dynamic characteristics. They concluded that the theoretical result of the equivalent SDOF dynamic model is close to the experimental time-histories data, and it is easy to use in practical applications. Bruhl and Varma [8] evaluated the explosion resistance of the small-scale SC wall panel by subjecting it to pressure loads in a blast load simulator. The experimental results are compared with the analysis results of a SDOF model. The results indicate that the SDOF model reasonably estimates the one-way behavior of SC walls subjected to short-duration uniform pressure loads. The updated SDOF design chart including the response to the near field blast was proposed [9]. Based on the acoustic approximation of the rarefaction wave, Rigby et al. [10] conducted a simple method for calculating the clearing relief of blast wave, which is combined with an SDOF model to investigate the effect of clearing on the dynamic response of elastic systems. It is indicated that the SDOF model generally exhibits excellent agreement with the FE model. Nassr et al. [11] employed the SDOF model to analyze the experimental results of 13 full-scale wide flange steel beams subjected to blast loads. The results show that the SDOF model can sufficiently predict the time-dependent deformations, internal forces, and moments of given beams. Carta and Stochino [12] described a continuous beam model aimed at predicting the response of reinforced concrete beams subjected to blast loads. This model can accurately assess the maximum deflection of the beam and capture vertical displacement and curvature profiles at a certain instant of time. All studies show that, compared with other methods for structural engineers, the equivalent SDOF method exhibits cost-effective and simple performance. Wang et al. [13] especially investigated failure modes of the one-way reinforced concrete slab under blast loads, which used the equivalent SDOF method. To observe and define one-way slabs’ failure modes under blast loads, three stand-off distances were considered in the experimental setup. The results indicated that one-way slabs’ failure mode was related to structural span and blast loads value. Nassr et al. [11, 14] developed a new SDOF model which considered the axial load-bending interaction on the dynamic response under blast loads. The concept of the equivalent lateral load was applied to the equivalent SDOF system governing equation to simulate the beam-column interaction. In order to evaluate the new SDOF model, the calculation of finite element and experimental specimens test were carried on under blast loads [15]. Further, Dragos and Wu [16] verified that the equivalent SDOF method for determining pressure-impulse diagrams could predict the failure mechanism and estimate the global instability. Based on a series of experiments with five different two-story structures conducted by Woodson and Baylot [17], Shi et al. [18] proposed a new damage criterion for beam/column subjected to blast loads. Nagata et al. [19] proposed an improved SDOF model considering the blast load distribution under near field explosions. The proposed SDOF model effectively evaluates the displacement response of RC beams subjected to the near field blast loads. However, the column/beam damage levels used the equivalent SDOF method will be overestimated when the blast center is close to the structures. These experiments and numerical simulation studies enrich the civil engineers’ understanding of equivalent SDOF blast-resistant design. For structural buildings, the air blast can be classified into far field blast and near field blast. The typical characteristic of far field blast is that the load acting on the surface of the structure can be simplified to a uniform load type. Not only the above studies but also almost all the research papers adopted far field air blast as the default load type of the equivalent SDOF system [20, 21]. In fact, near-field blast may occur in real building structures. When the distance between the vehicle chassis and the bridge road surface is less than the bridge span, it may be a near-field type for vehicle explosion on the bridge. For example, a vehicle loaded with fireworks exploded on a T-beam bridge in China in early 2013 [22]. This accident caused the southern part of the bridge to collapse. Other experimental studies on beams or columns subjected to close-in air blast loading were finished [23–27]. Zhang et al. [23] investigated the damage characteristics of RC beams subjected to close-in blast loading through experiments. Geng et al. [24] investigated the local failure and overall response characteristics of the steel box girder subjected to close-in air blast loading through experiments. Liu et al. [25] detected the impact of the concrete pavement layer on the blast-resistant performance of the steel box girder deck. It indicated that the concrete pavement layer could effectively block the impact of the air blast loading on the steel box girder deck. Li and Hao [26] proposed a two-step numerical approach, which enormously reduced the amount of modeling and calculation when analyzing the structural response subjected to near field blast loads. To further improve the accuracy of the two-step approach, the influence of brittle shear damage during the blast loading phase was considered. The influence of shear damage was included in the second-step free vibration analysis of the finite element model. The results show that including the damage caused by the blast load in the second-step free vibration analysis can improve the prediction accuracy of the response of beams subjected to blast loads. Fujikura and Bruneau [27] explored the blast-resistant performance of commonly used bridge piers under close-in blast loading. It is obtained through experiments that the steel jacketing known to be effective in providing satisfactory seismic performance was ineffective for the blast loading conditions considered. The American Society of Civil Engineers (ASCE SEI 59-11) [28] defines the blast loads that the scaled distance is less than 1.2 m/kg1/3 as near field explosions. Besides, UFC 3-340-02 [5] regarded a scaled distance of 0.4 m/kg1/3 as the boundary to distinguish near field and far field explosions. Unlike the peak overpressure, explosive mass, and stand-off distance are not the only parameters that distinguish near field and far field blast, the parameter of the structural span should be considered as an important factor because the nonuniform blast loading from the near field blast will ultimately be loaded by the structural building itself. As for the condition that the structural span can be ignored, this is another concern about the boundary between near field and far field blast.The paper aims to derive the equivalent SDOF load expressions and load transformation factors including near field and far field blast and to analyze the difference between near field and far field blast. The objective is to provide a more accurate equivalent load for the practical engineering adopting the equivalent SDOF method. This paper provides a clear numerical definition for the distinction between near field and far field explosions, which considers the structural size of the beam, stand-off distance, and other parameters for one-way members. Simply supported, fixed end, and simply-fixed are the three types of one-way members [29]. It provides qualification criteria for analyzing explosion categories in blast-resistant structural design and a reference for the selection of theoretical calculation formulas. It can also clearly explain why the use of uniform load types in near field blast may overestimate the equivalent loads.


## 2. Idealized Equivalent SDOF Load System

The blast loads generated by explosives on the one-way members can be namedp(x)f (t). Unlike the uniform load used in other literatures, p (x) means that the peak overpressure of the shock wave varies depending on the location x. Assuming that the peak overpressure reaches all the locations simultaneously and the overpressure reflection and superposition effect are ignored, f (t) is the peak overpressure decay function along the structural span. If the explosive fixed at the same specific height h, the explosive should be set at the location where the maximum deflection of the structure occurs for structural conservative design or safety [30, 31]. According to structural symmetrical constraints, this location will be at the middle span for simply-supported or fixed-end members. For simply-fixed members, this location will be off-center and close to the simply supported end because the structural constraint of the simple end is weaker than that of the fixed end. The significant point on the whole structural system is selected as a deflection location of the equivalent SDOF system. Since the time variable of both systems is same, the response of the equivalent system y (t) is the same as the significant point on the real system. In order to simplify the calculation, this significant point is also used as an original point for coordination. The displacement of other points in the actual structural system can be obtained from the shape function X (x), which is generally determined by the first-order vibration mode and is often replaced by the curve function of static deflection subjected to uniform loads [3]. Figure 1 shows the correspondence between the parameters of the two systems. For the equivalent SDOF system, the concentrated equivalent load Pe(t) can be gotten [29, 30]:(1)Pet=∫0lpxftXxdx.Figure 1
(a) Real structural system; (b) equivalent SDOF system.
(a)(b)Combined with the superposition principle and illustrated by Figure1, equation (1) is the superimposed sum of the forces along the structural span multiplied by the corresponding displacements. Therefore, the equivalent load Pe (t) in the SDOF system derived from the concept of equivalent work is not only related to the mode function X (x) but also related to the overpressure distribution p (x)f (t) along the structural span. The difference caused by these simplifications has not been thoroughly studied in the previous literature.The actual total force on the structure can be named asPtf (t). The load transformation factor KL, as a scalar factor, is defined as the ratio of the equivalent concentrated force Pe (t) to the actual total force Ptf(t) [29, 30]:(2)KL=PetPtft=∫0lpxftXxdx∫0lpxftdx=∫0lpxXxdx∫0lpxdx.The load transformation factorKL will be a constant value when peak overpressure p (x) is a uniform load type in far field blast. The shape function X (x) used in most literature adopts the displacement function under uniform load.The peak overpressure functionp (x) is a composite function of scaled distance Z, given by [8, 9](3)Z=RW3,where W is the TNT equivalent mass in kilograms and R is the distance to the center of the explosive in meters. From Figure 1, R can be easily expressed as(4)R=h2+x2.According to experimental tests and theoretical analyses of previous researchers, the peak overpressure function can be easily expressed as a cubic term of the scaled distanceZ [29]:(5)px=aZ+bZ2+cZ3.Various researchers have slightly different values for parametersa, b, and c based on their experimental conditions and test methods. The peak overpressure expression of blast loading in free air given by Mills is [31, 32](6)px=0.108Z−0.114Z2+1.772Z3.The expression given by Baker is [33](7)px=2.006Z+0.194Z2−0.004Z3,0.05≤Z<0.5,0.067Z+0.301Z2+0.431Z3,0.5≤Z<10.The expression given by Henrych and Rudolf is [30](8)px=1.4072Z+0.5540Z2−0.0357Z3,0.05≤Z<0.3,0.6194Z−0.0326Z2+0.2132Z3,0.3≤Z<1,0.062Z+0.405Z2+0.3288Z3,1≤Z<10.Many popular expressions could be known from the papers of other researchers, such as Wu’s and Hao formula [34], Brode’s formula [35], and Sadovskyi’s formula [36]. This paper only takes the above expressions for the typical and comparative analysis. To facilitate the derivation and programming, equations (3) and (4) are substituted into equation (5). The general concise form is as follows:(9)px=a∗W1/3h2+x2+b∗W2/3h2+x22+c∗Wh2+x23=Ah2+x2+Bh2+x22+Ch2+x23,where A, B, and C are newly defined parameters of a, b, c, and W, respectively.Peak overpressurep (x) is an external force, which is independent of the structures. To determine the equivalent load transformation factor, another parameter of shape function X (x) is related to the structural type and supported condition. Most literature [3–6, 29] adopted the displacement function of various members subjected to static uniform load as shape function X (x). Because the mode function is a function that reflects structural vibration itself and boundary condition, which is independent of the applied external load and is more suitable for shape function.


## 3. Idealized Equivalent Load and Transformation Factor in Elastic Stage

### 3.1. Simply Supported One-Way Members

The mode functionX (x) in the elastic stage for simply supported one-way members can be obtained from the structural dynamics literature. Mostly, the theoretical solution of the mode function X (x) is built on the left end of the structure and is a combination of infinite terms. Generally, the first-order mode function has good accuracy. Therefore, moving the origin of coordinates to midspan position, the first-order mode function can be expressed as [37](10)Xx=cosπxl.Equations (10) and (9) are even functions; they are substituted into equation (1), and (1) becomes(11)Pet=2ft∫−l/20Ah2+x2+Bh2+x22+Ch2+x23cosπxldx.Equation (11) cannot be integrated in mathematics. In order to get its theoretical solution, there are two solutions. The first method of dealing with nonintegrable function is to use the finite point fitting function, and the second method is to use finite series expansion. After analyzing equation (11), we can conclude that it is easier to use the finite series expansion instead of the cosine function. After comparing the calculation results, the error of using trinomial and quadrinomial is less than 0.1%. To simplify the space, this paper only lists the expansion result of the trinomial [38]:(12)cosπxl≈1−π22l2x2+π424l2x4.Substituting equation (12) into equation (11), the dimensionless parameter γ = h/l, and completed integral computation. The result is shown in Formula 1 (A.1) in the appendix.The actual structural total forcePtf (t), that is, the denominator part of equation (2), which is integrated as follows:(13)Ptft=∫−l/2l/2pxftdx=2ftAln12γ+12γ2+1+Bharctan12γ+Ch2sinarctan12γ.Substituting equation (13) and (A.1) into equation (2), the expression of load transformation factor KL is finally determined. It can be seen from equation (13) and (A.1) that the equivalent load transformation factor is not only a function of explosive equivalent mass W and stand-off distance h but also a function of the structural span l. If the function curve of the equivalent load transformation factor can be drawn, the demarcation position between the near field and far field blast will be determined. When using uniform load as the type of explosive load on a member, it is necessary to determine the accuracy and appropriateness of this assumption. Various researchers have proposed different peak overpressure functions for load transformation factors. The parameters a, b, and c in different peak overpressure functions also need to be compared and analyzed. Select the typical structural span l, the peak overpressure parameters a, b, and c proposed by various researchers and the explosive mass W. The equivalent load transformation factor KL with the dimensionless parameter γ as a variable can be plotted as follows. When γ ≥ 2.0, the absolute differences between all curves are below the value of 10^−6 in the same calculation condition, and basically all curves coincide with a line parallel to the abscissa. In order to show the changing part of the curves, especially the difference of near blast field and far blast field, the part of all curves with γ ≥ 2.0 is discarded. In addition to the simply supported one-way members, fixed-end and simply-fixed one-way members are also applicable to the above description.Since the peak overpressure expression proposed by Mills is a continuous smooth function, the function ofKL on γ for various calculation conditions is a smooth curve which can be observed from Figure 2. When γ approaches 0, KL approaches 1. When γ ≥ 2, KL is basically constant at 0.64, which is the calculated value of the equivalent load transformation factor using the uniform load type as the structural blast load. This can be verified from Biggs [3] or Fang and Liu [29]. On the premise of the same explosive mass calculation condition, the smaller the structural span is, the greater the KL value will be. If considering the allowable error in the structural design calculation, when γ ≥ 1, the error between the KL value for various calculation conditions and the value 0.64 is less than 3%. This means that if a calculation error of 3% is allowed, the boundary position between the near field and far field blast is when the stand-off distance h is equal to the structural span length l.Figure 2
Equivalent load transformation factors of simply supported one-way members using Mills’ formula parameters: (a)W = 1 kg. (b)W = 10 kg. (c)W = 100 kg.
(a)(b)(c)Mills’ peak overpressure expression is a two-stage piecewise function. The corresponding figures ofKL for typical calculation conditions have a transition point, which can be observed from Figure 3. For the same explosive mass, the smaller the structural span is, the greater the value γ of the transition point position will be. In the smoothing interval of the piecewise function before or after the transition, the smaller the structural span is, the higher the KL value will be. As the explosive mass increases, the transition point value increases. When γ is close to 0, the KL is not close to 1. This indicates that KL is greatly affected by the parameters a, b, and c in the near field blast. When γ ≥ 2, the values of equivalent load transformation factor using the Baker parameter and Mills parameter are basically constant at 0.64.Figure 3
Equivalent load transformation factors of simply supported one-way members using Baker’s formula parameters: (a)W = 1 kg. (b) W = 10 kg. (c) W = 100 kg.
(a)(b)(c)Henrych’s peak overpressure expression is a three-stage piecewise function, and the corresponding figures ofKL for typical calculation conditions have two transition points for some calculation conditions. According to the definition of γ, with the same γ as the premise, the greater the structural span l is, the greater the stand-off distance h will be. The peak overpressure is related to the stand-off distance h and the explosive mass W. Therefore, on the premise of the same explosive mass calculation condition, the larger the structural span l is, the larger the scaled distance Z will be. Furthermore, when the explosive mass W is a minimal value, the segmentation interval related to the small-scaled distance Z may be skipped and never adopted by the larger structural span l. For example, the curve of KL on γ with a structural span of 50 m has no transition point in Figure 4(a), while it only has one transition point in Figure 4(c).Figure 4
Equivalent load transformation factors of simply supported one-way members using Henrych’s formula parameters. (a)W = 1 kg. (b) W = 10 kg. (c) W = 100 kg.
(a)(b)(c)Whenγ is close to 0.5, the KL curve with a structural span of 10 m has the last transition position. It is known from the KL figures that it is still a near field blast. When γ ≥ 2, the values of equivalent load transformation factor for various calculation conditions are also basically constant at 0.64. When γ ≥ 1, the KL curves decrease slightly and the relative error between the KL value and 0.64 is also less than 3.0%.


### 3.2. Fixed-End One-Way Members

When the coordinate origin is at the members’ end, the mode functionX (x) of the fixed end one-way members in elastic stage can be expressed as [37](14)Xx′=0.62sinβ0x′−shβ0x′+sinβ0l−shβ0lchβ0l−cosβ0lcosβ0x′−chβ0x′,where 0.62 is the adjustment factor of the mode function and β0l = 3π/2.After moving the coordinate origin to the middle span position, which is the maximum and symmetrical position of the mode function, the first-order mode function forx∈[−0.5 l, 0] can be expressed as(15)Xx≈0.662692cosβ0x+e−β0x+3π/4,where the initial conditions X (−0.5 l) = 0 at x = 0 and X (0) = 1 at x = 0 are met.Substituting equations (15) and (9) into equation (1), and equation (1) becomes(16)Pet=2ft∫−l/200.66269Ah2+x2+Bh2+x22+Ch2+x232cosβ0x+e−β0x+3π/4dx.Equation (16) also cannot be integrated in mathematics. The integral result contains too many terms for the finite series expansion method. Therefore, the finite point fitting function method is adopted. Let the dimensionless parameter i = x/h, an exponential function with an accuracy of 99.9% on the interval (−100, 0) is fitted instead of the peak overpressure function equation (9):(17)px=pi=Ah0.9123e0.5387i+0.1708e0.05872i+Bh21.107e0.8694i+0.03201e0.1157i+Ch31.172e1.178i+0.0001033e−0.01216i.Substituting equations (17) and (15) into equation (1), letting the dimensionless parameter δ = β0h and integrating. The result is shown in Appendix (B.1).The actual total forcePtf(t) is independent of the structural supported type. Therefore, equation (13) is still the calculation expression of fixed end one-way members for Ptf(t). Substituting (B.1) and equation (13) into equation (2), the expression of load transformation factor KL is finally determined.Because the parameters are selected for the calculation conditions of the simply supported one-way members, the same parameters are selected for fixed end one-way members.It can be observed from Figures4 to 7 that the KL curves of fixed end one-way members have a similar downtrend and transition positions with the simply supported one-way members. One noticeable difference is that when γ ≥ 2, KL is basically constant at 0.54, which is the result of the uniform load type in far field blast and can be verified from Biggs [3] or Fang and Liu [29]. For the KL calculation conditions using piecewise functions, both the transition value using Baker’s parameters and the second transition value using Henrych’s parameters are significant, and the relative error is approximately 10%.Figure 5
Equivalent load transformation factors of fixed end one-way members using Mills’ formula parameters: (a)W = 1 kg. (b) W = 10 kg. (c) W = 100 kg.
(a)(b)(c)Figure 6
Equivalent load transformation factors of fixed end one-way members using Baker’s formula parameters: (a)W = 1 kg. (b) W = 10 kg. (c) W = 100 kg.
(a)(b)(c)Figure 7
Equivalent load transformation factors of fixed end one-way members using Henrych’s formula parameters: (a)W = 1 kg. (b) W = 10 kg. (c) W = 100 kg.
(a)(b)(c)


### 3.3. Simply-Fixed One-Way Members

When the coordinate origin is at the simply supported end of the simply-fixed one-way members, the mode functionX (x′) in the elastic stage can be expressed as [37](18)Xx′=shβx′−shβlsinβlsinβx′,where βl = 5π/4.The mode functionX (x′) is asymmetrical about the middle span due to different boundary support conditions. To shift the coordinate origin to the maximum position of the mode function, letting the derivative of X (x′) be 0, the solution is x0 = 0.42 l. After shifting the coordinate origin to 0.42 l, the first-order mode function can be expressed as(19)Xx=D0D1eβx−D2e−βx+D3sinβx+D4cosβx,where D0 = 0.02613, D1 = 0.5e0.42βl, D2 = 0.5e−0.42βl, D3 = 35.8746cos (0.42βl), and D4 = 35.8746sin (0.42βl) and conditions X (−0.42 l) = X (0.58 l) = 0, and X (0) = 1 are met.Substituting equations (19) and (9) into equation (1), the resulting expression also cannot be integrated in mathematics. Similar to the solving process of fixed end one-way members, it is suitable for peak overpressure function with the high-precision fitting method. Using the symmetry of equation (9) and the result of equation (17), letting i = x/h to fit an exponential function with an accuracy of 99.9% in the interval (0, 100) instead of the peak overpressure function equation (18):(20)px=pi=Ah0.9123e−0.5387i+0.1708e−0.05872i+Bh21.107e−0.8694i+0.03201e−0.1157i+Ch31.172e−1.178i+0.0001033e+0.01216i.Equation (20) is obtained by replacing i in equation (17) with −i. After substituting equations (17) and (20) into the corresponding integral interval in equation (1), changing the integral variable with i = x/h, the expression is shown in Appendix (C.1).It can be seen from (C.1) that there are many integral items. To write concisely, some parameters about the upper and lower limits of integration points are defined. Define the parameters D5 = sin (−0.42βl), D6 = cos (−0.42βl), D7 = sin (−0.42βl), D8 = cos (−0.42βl), and λ = βh. The calculation result can also be expressed as(21)Pet=Pe1t+Pe2t+Pe3t,where Pe1 (t), Pe2 (t), and Pe3 (t) are shown in Appendix.Because the peak overpressure load of the blast shock wave is asymmetric along the structural span, the actual structural total forcePtf(t) can be expressed as(22)ptft=∫−0.42l0.58lftAh2+x2+Bh2+x22+Ch2+x23dx=ftAln0.582+γ2+0.580.422+γ2−0.42+Bharctan0.58γ+arctan0.42γ+Ch20.580.582+γ2+0.420.422+γ2.Substituting equations (21) and (22) into equation (2), the expression of load transformation factor KL is finally determined.As in the previous section, the same parameters of explosive massW and structural span l are selected for the calculation conditions of simply-fixed one-way members.It can be seen from Figures8 to 10 that when the explosive equivalent mass is 1.0 kg to 100 kg, the KL curves of simply-fixed one-way members have a similar downtrend as the KL curves of simply supported or fixed one-way members. When γ < 1, KL is mainly related to the value γ and the peak overpressure parameters. Therefore, the range can basically be classified as the near field blast. When γ ≥ 2, KL is basically constant at 0.58, which can be verified from Biggs [3] or Fang and Liu [29]. With the same structural span and external load conditions, the structural constraint will change the mode function X (x). The stronger the constraint condition is, the smaller the relative value of the mode function X (x) along the structural span will be. Therefore, the fixed end one-way members without hinged constraints have the smallest equivalent load transformation factor KL. With one hinged constraint, the KL of the simply supported one-way members is about 0.04 bigger than that of the fixed end one-way members in a far field blast. With both hinged constraints, the KL of the simply supported one-way members is about 0.10 bigger than that of the fixed end one-way members in a far field blast.Figure 8
Equivalent load transformation factors of simply-fixed one-way members using Mills’ formula parameters: (a)W = 1 kg. (b) W = 10 kg. (c) W = 100 kg.
(a)(b)(c)Figure 9
Equivalent load transformation factors of simply-fixed one-way members using Baker’s formula parameters: (a)W = 1 kg. (b) W = 10 kg. (c) W = 100 kg.
(a)(b)(c)Figure 10
Equivalent load transformation factors of simply-fixed one-way members using Henrych’s formula parameters: (a)W = 1 kg. (b) W = 10 kg. (c) W = 100 kg.
(a)(b)(c)


## 3.1. Simply Supported One-Way Members

The mode functionX (x) in the elastic stage for simply supported one-way members can be obtained from the structural dynamics literature. Mostly, the theoretical solution of the mode function X (x) is built on the left end of the structure and is a combination of infinite terms. Generally, the first-order mode function has good accuracy. Therefore, moving the origin of coordinates to midspan position, the first-order mode function can be expressed as [37](10)Xx=cosπxl.Equations (10) and (9) are even functions; they are substituted into equation (1), and (1) becomes(11)Pet=2ft∫−l/20Ah2+x2+Bh2+x22+Ch2+x23cosπxldx.Equation (11) cannot be integrated in mathematics. In order to get its theoretical solution, there are two solutions. The first method of dealing with nonintegrable function is to use the finite point fitting function, and the second method is to use finite series expansion. After analyzing equation (11), we can conclude that it is easier to use the finite series expansion instead of the cosine function. After comparing the calculation results, the error of using trinomial and quadrinomial is less than 0.1%. To simplify the space, this paper only lists the expansion result of the trinomial [38]:(12)cosπxl≈1−π22l2x2+π424l2x4.Substituting equation (12) into equation (11), the dimensionless parameter γ = h/l, and completed integral computation. The result is shown in Formula 1 (A.1) in the appendix.The actual structural total forcePtf (t), that is, the denominator part of equation (2), which is integrated as follows:(13)Ptft=∫−l/2l/2pxftdx=2ftAln12γ+12γ2+1+Bharctan12γ+Ch2sinarctan12γ.Substituting equation (13) and (A.1) into equation (2), the expression of load transformation factor KL is finally determined. It can be seen from equation (13) and (A.1) that the equivalent load transformation factor is not only a function of explosive equivalent mass W and stand-off distance h but also a function of the structural span l. If the function curve of the equivalent load transformation factor can be drawn, the demarcation position between the near field and far field blast will be determined. When using uniform load as the type of explosive load on a member, it is necessary to determine the accuracy and appropriateness of this assumption. Various researchers have proposed different peak overpressure functions for load transformation factors. The parameters a, b, and c in different peak overpressure functions also need to be compared and analyzed. Select the typical structural span l, the peak overpressure parameters a, b, and c proposed by various researchers and the explosive mass W. The equivalent load transformation factor KL with the dimensionless parameter γ as a variable can be plotted as follows. When γ ≥ 2.0, the absolute differences between all curves are below the value of 10^−6 in the same calculation condition, and basically all curves coincide with a line parallel to the abscissa. In order to show the changing part of the curves, especially the difference of near blast field and far blast field, the part of all curves with γ ≥ 2.0 is discarded. In addition to the simply supported one-way members, fixed-end and simply-fixed one-way members are also applicable to the above description.Since the peak overpressure expression proposed by Mills is a continuous smooth function, the function ofKL on γ for various calculation conditions is a smooth curve which can be observed from Figure 2. When γ approaches 0, KL approaches 1. When γ ≥ 2, KL is basically constant at 0.64, which is the calculated value of the equivalent load transformation factor using the uniform load type as the structural blast load. This can be verified from Biggs [3] or Fang and Liu [29]. On the premise of the same explosive mass calculation condition, the smaller the structural span is, the greater the KL value will be. If considering the allowable error in the structural design calculation, when γ ≥ 1, the error between the KL value for various calculation conditions and the value 0.64 is less than 3%. This means that if a calculation error of 3% is allowed, the boundary position between the near field and far field blast is when the stand-off distance h is equal to the structural span length l.Figure 2
Equivalent load transformation factors of simply supported one-way members using Mills’ formula parameters: (a)W = 1 kg. (b)W = 10 kg. (c)W = 100 kg.
(a)(b)(c)Mills’ peak overpressure expression is a two-stage piecewise function. The corresponding figures ofKL for typical calculation conditions have a transition point, which can be observed from Figure 3. For the same explosive mass, the smaller the structural span is, the greater the value γ of the transition point position will be. In the smoothing interval of the piecewise function before or after the transition, the smaller the structural span is, the higher the KL value will be. As the explosive mass increases, the transition point value increases. When γ is close to 0, the KL is not close to 1. This indicates that KL is greatly affected by the parameters a, b, and c in the near field blast. When γ ≥ 2, the values of equivalent load transformation factor using the Baker parameter and Mills parameter are basically constant at 0.64.Figure 3
Equivalent load transformation factors of simply supported one-way members using Baker’s formula parameters: (a)W = 1 kg. (b) W = 10 kg. (c) W = 100 kg.
(a)(b)(c)Henrych’s peak overpressure expression is a three-stage piecewise function, and the corresponding figures ofKL for typical calculation conditions have two transition points for some calculation conditions. According to the definition of γ, with the same γ as the premise, the greater the structural span l is, the greater the stand-off distance h will be. The peak overpressure is related to the stand-off distance h and the explosive mass W. Therefore, on the premise of the same explosive mass calculation condition, the larger the structural span l is, the larger the scaled distance Z will be. Furthermore, when the explosive mass W is a minimal value, the segmentation interval related to the small-scaled distance Z may be skipped and never adopted by the larger structural span l. For example, the curve of KL on γ with a structural span of 50 m has no transition point in Figure 4(a), while it only has one transition point in Figure 4(c).Figure 4
Equivalent load transformation factors of simply supported one-way members using Henrych’s formula parameters. (a)W = 1 kg. (b) W = 10 kg. (c) W = 100 kg.
(a)(b)(c)Whenγ is close to 0.5, the KL curve with a structural span of 10 m has the last transition position. It is known from the KL figures that it is still a near field blast. When γ ≥ 2, the values of equivalent load transformation factor for various calculation conditions are also basically constant at 0.64. When γ ≥ 1, the KL curves decrease slightly and the relative error between the KL value and 0.64 is also less than 3.0%.


## 3.2. Fixed-End One-Way Members

When the coordinate origin is at the members’ end, the mode functionX (x) of the fixed end one-way members in elastic stage can be expressed as [37](14)Xx′=0.62sinβ0x′−shβ0x′+sinβ0l−shβ0lchβ0l−cosβ0lcosβ0x′−chβ0x′,where 0.62 is the adjustment factor of the mode function and β0l = 3π/2.After moving the coordinate origin to the middle span position, which is the maximum and symmetrical position of the mode function, the first-order mode function forx∈[−0.5 l, 0] can be expressed as(15)Xx≈0.662692cosβ0x+e−β0x+3π/4,where the initial conditions X (−0.5 l) = 0 at x = 0 and X (0) = 1 at x = 0 are met.Substituting equations (15) and (9) into equation (1), and equation (1) becomes(16)Pet=2ft∫−l/200.66269Ah2+x2+Bh2+x22+Ch2+x232cosβ0x+e−β0x+3π/4dx.Equation (16) also cannot be integrated in mathematics. The integral result contains too many terms for the finite series expansion method. Therefore, the finite point fitting function method is adopted. Let the dimensionless parameter i = x/h, an exponential function with an accuracy of 99.9% on the interval (−100, 0) is fitted instead of the peak overpressure function equation (9):(17)px=pi=Ah0.9123e0.5387i+0.1708e0.05872i+Bh21.107e0.8694i+0.03201e0.1157i+Ch31.172e1.178i+0.0001033e−0.01216i.Substituting equations (17) and (15) into equation (1), letting the dimensionless parameter δ = β0h and integrating. The result is shown in Appendix (B.1).The actual total forcePtf(t) is independent of the structural supported type. Therefore, equation (13) is still the calculation expression of fixed end one-way members for Ptf(t). Substituting (B.1) and equation (13) into equation (2), the expression of load transformation factor KL is finally determined.Because the parameters are selected for the calculation conditions of the simply supported one-way members, the same parameters are selected for fixed end one-way members.It can be observed from Figures4 to 7 that the KL curves of fixed end one-way members have a similar downtrend and transition positions with the simply supported one-way members. One noticeable difference is that when γ ≥ 2, KL is basically constant at 0.54, which is the result of the uniform load type in far field blast and can be verified from Biggs [3] or Fang and Liu [29]. For the KL calculation conditions using piecewise functions, both the transition value using Baker’s parameters and the second transition value using Henrych’s parameters are significant, and the relative error is approximately 10%.Figure 5
Equivalent load transformation factors of fixed end one-way members using Mills’ formula parameters: (a)W = 1 kg. (b) W = 10 kg. (c) W = 100 kg.
(a)(b)(c)Figure 6
Equivalent load transformation factors of fixed end one-way members using Baker’s formula parameters: (a)W = 1 kg. (b) W = 10 kg. (c) W = 100 kg.
(a)(b)(c)Figure 7
Equivalent load transformation factors of fixed end one-way members using Henrych’s formula parameters: (a)W = 1 kg. (b) W = 10 kg. (c) W = 100 kg.
(a)(b)(c)


## 3.3. Simply-Fixed One-Way Members

When the coordinate origin is at the simply supported end of the simply-fixed one-way members, the mode functionX (x′) in the elastic stage can be expressed as [37](18)Xx′=shβx′−shβlsinβlsinβx′,where βl = 5π/4.The mode functionX (x′) is asymmetrical about the middle span due to different boundary support conditions. To shift the coordinate origin to the maximum position of the mode function, letting the derivative of X (x′) be 0, the solution is x0 = 0.42 l. After shifting the coordinate origin to 0.42 l, the first-order mode function can be expressed as(19)Xx=D0D1eβx−D2e−βx+D3sinβx+D4cosβx,where D0 = 0.02613, D1 = 0.5e0.42βl, D2 = 0.5e−0.42βl, D3 = 35.8746cos (0.42βl), and D4 = 35.8746sin (0.42βl) and conditions X (−0.42 l) = X (0.58 l) = 0, and X (0) = 1 are met.Substituting equations (19) and (9) into equation (1), the resulting expression also cannot be integrated in mathematics. Similar to the solving process of fixed end one-way members, it is suitable for peak overpressure function with the high-precision fitting method. Using the symmetry of equation (9) and the result of equation (17), letting i = x/h to fit an exponential function with an accuracy of 99.9% in the interval (0, 100) instead of the peak overpressure function equation (18):(20)px=pi=Ah0.9123e−0.5387i+0.1708e−0.05872i+Bh21.107e−0.8694i+0.03201e−0.1157i+Ch31.172e−1.178i+0.0001033e+0.01216i.Equation (20) is obtained by replacing i in equation (17) with −i. After substituting equations (17) and (20) into the corresponding integral interval in equation (1), changing the integral variable with i = x/h, the expression is shown in Appendix (C.1).It can be seen from (C.1) that there are many integral items. To write concisely, some parameters about the upper and lower limits of integration points are defined. Define the parameters D5 = sin (−0.42βl), D6 = cos (−0.42βl), D7 = sin (−0.42βl), D8 = cos (−0.42βl), and λ = βh. The calculation result can also be expressed as(21)Pet=Pe1t+Pe2t+Pe3t,where Pe1 (t), Pe2 (t), and Pe3 (t) are shown in Appendix.Because the peak overpressure load of the blast shock wave is asymmetric along the structural span, the actual structural total forcePtf(t) can be expressed as(22)ptft=∫−0.42l0.58lftAh2+x2+Bh2+x22+Ch2+x23dx=ftAln0.582+γ2+0.580.422+γ2−0.42+Bharctan0.58γ+arctan0.42γ+Ch20.580.582+γ2+0.420.422+γ2.Substituting equations (21) and (22) into equation (2), the expression of load transformation factor KL is finally determined.As in the previous section, the same parameters of explosive massW and structural span l are selected for the calculation conditions of simply-fixed one-way members.It can be seen from Figures8 to 10 that when the explosive equivalent mass is 1.0 kg to 100 kg, the KL curves of simply-fixed one-way members have a similar downtrend as the KL curves of simply supported or fixed one-way members. When γ < 1, KL is mainly related to the value γ and the peak overpressure parameters. Therefore, the range can basically be classified as the near field blast. When γ ≥ 2, KL is basically constant at 0.58, which can be verified from Biggs [3] or Fang and Liu [29]. With the same structural span and external load conditions, the structural constraint will change the mode function X (x). The stronger the constraint condition is, the smaller the relative value of the mode function X (x) along the structural span will be. Therefore, the fixed end one-way members without hinged constraints have the smallest equivalent load transformation factor KL. With one hinged constraint, the KL of the simply supported one-way members is about 0.04 bigger than that of the fixed end one-way members in a far field blast. With both hinged constraints, the KL of the simply supported one-way members is about 0.10 bigger than that of the fixed end one-way members in a far field blast.Figure 8
Equivalent load transformation factors of simply-fixed one-way members using Mills’ formula parameters: (a)W = 1 kg. (b) W = 10 kg. (c) W = 100 kg.
(a)(b)(c)Figure 9
Equivalent load transformation factors of simply-fixed one-way members using Baker’s formula parameters: (a)W = 1 kg. (b) W = 10 kg. (c) W = 100 kg.
(a)(b)(c)Figure 10
Equivalent load transformation factors of simply-fixed one-way members using Henrych’s formula parameters: (a)W = 1 kg. (b) W = 10 kg. (c) W = 100 kg.
(a)(b)(c)


## 4. Idealized Equivalent Load and Transformation Factor in Plastic Stage

### 4.1. Idealized Computation Modal in Plastic Stage

It should be pointed out that it is difficult to deal with the plastic behavior of one-way members. From a practical design point of view, generally the fixed end or simply-fixed one-way members go through two plastic stages. The first stage is the combination of elasticity and plasticity, which has the characteristic shape of a simply supported one-way member; the second occurs after the hinge has been formed in the midspan of simply supported one-way members. This paper uses the second plasticity stage as the analysis object.After shifting the coordinate origin to the middle span position and the structure is symmetrical, the shape function can be expressed as [3, 29](23)Xx=2xl+1,where the initial conditions X (−0.5 l) = 0 and X (0) = 1 are met.Substituting equations (23) and (9) into equation (1), equation (1) becomes(24)Pet=2ft∫−l/20Ah2+x2+Bh2+x22+Ch2+x232xl+1dx,And, the expression can be solved as(25)Pet=2ftAln12γ+1+14γ2+Bharctan12γ+Ch2sinarctan12γ+2Aγ1+14γ2−1+Bγhln1+14γ2+Cγh21−cosarctan12γ.The actual structural total forcePtf(t) is the same as the simply supported or fixed end one-way members in the elastic stage and should be expressed by equation (13).Substituting equations (25) and (13) into equation (2), the expression of the load transformation factor KL for one-way members in the plastic stage is finally determined.


### 4.2. Load Transformation Factor Characteristic in Plastic Stage

It can be easily seen from equations (25) and (13) that neither of the expressions contains the parameter l. This means that the load transformation factor KL in the plastic stage is independent of the structural span l. To compare with the load transformation factor in the elastic stage, the structural spans are taken as 10 m∼50 m and the same explosive mass or peak overpressure parameters are used as calculation examples.It can be seen from Figures11 to 13 that when the explosive mass is 1.0 kg to 100 kg, the decreasing rate of KL on γ in the plastic stage is faster than that in the elastic stage. The limit value of KL is 0.5 with γ approaches infinity, which can be verified from Biggs [3] or Fang and Liu [29]. The value 0.5 is lower than any KL value of beam in the elastic stage for various support conditions. When KL is 0.52, γ is 1.0, which means that if the calculation error within 4% is negligible, the boundary position between the far field and near field blast in the plastic stage is when the stand-off distance h is equal to the structural span l.Figure 11
Equivalent load transformation factors of one-way members using Mills’ formula parameters in plastic stage.Figure 12
Equivalent load transformation factors of one-way members using Baker’s formula parameters in plastic stage: (a)W = 1 kg. (b) W = 10 kg. (c) W = 100 kg.
(a)(b)(c)Figure 13
Equivalent load transformation factors of one-way members using Henrych’s formula parameters in plastic stage: (a)W = 1 kg. (b) W = 10 kg. (c) W = 100 kg.
(a)(b)(c)It can be seen from Figure11 that the KL values of different structural spans using Mills’ formula parameters are the same under the same explosive mass, reflecting the characteristics unrelated to the structural span l. When the explosive mass W is changed, the difference between KL values of different structural spans with the same value γ in the plastic stage is smaller than that in the elastic stage. It can be seen from Figures 12 and 13 that the KL values of the peak overpressure parameters described by the piecewise functions are the same for various structural spans in most calculation intervals of γ under the same explosive mass, reflecting the characteristics unrelated to the structural span l.


## 4.1. Idealized Computation Modal in Plastic Stage

It should be pointed out that it is difficult to deal with the plastic behavior of one-way members. From a practical design point of view, generally the fixed end or simply-fixed one-way members go through two plastic stages. The first stage is the combination of elasticity and plasticity, which has the characteristic shape of a simply supported one-way member; the second occurs after the hinge has been formed in the midspan of simply supported one-way members. This paper uses the second plasticity stage as the analysis object.After shifting the coordinate origin to the middle span position and the structure is symmetrical, the shape function can be expressed as [3, 29](23)Xx=2xl+1,where the initial conditions X (−0.5 l) = 0 and X (0) = 1 are met.Substituting equations (23) and (9) into equation (1), equation (1) becomes(24)Pet=2ft∫−l/20Ah2+x2+Bh2+x22+Ch2+x232xl+1dx,And, the expression can be solved as(25)Pet=2ftAln12γ+1+14γ2+Bharctan12γ+Ch2sinarctan12γ+2Aγ1+14γ2−1+Bγhln1+14γ2+Cγh21−cosarctan12γ.The actual structural total forcePtf(t) is the same as the simply supported or fixed end one-way members in the elastic stage and should be expressed by equation (13).Substituting equations (25) and (13) into equation (2), the expression of the load transformation factor KL for one-way members in the plastic stage is finally determined.


## 4.2. Load Transformation Factor Characteristic in Plastic Stage

It can be easily seen from equations (25) and (13) that neither of the expressions contains the parameter l. This means that the load transformation factor KL in the plastic stage is independent of the structural span l. To compare with the load transformation factor in the elastic stage, the structural spans are taken as 10 m∼50 m and the same explosive mass or peak overpressure parameters are used as calculation examples.It can be seen from Figures11 to 13 that when the explosive mass is 1.0 kg to 100 kg, the decreasing rate of KL on γ in the plastic stage is faster than that in the elastic stage. The limit value of KL is 0.5 with γ approaches infinity, which can be verified from Biggs [3] or Fang and Liu [29]. The value 0.5 is lower than any KL value of beam in the elastic stage for various support conditions. When KL is 0.52, γ is 1.0, which means that if the calculation error within 4% is negligible, the boundary position between the far field and near field blast in the plastic stage is when the stand-off distance h is equal to the structural span l.Figure 11
Equivalent load transformation factors of one-way members using Mills’ formula parameters in plastic stage.Figure 12
Equivalent load transformation factors of one-way members using Baker’s formula parameters in plastic stage: (a)W = 1 kg. (b) W = 10 kg. (c) W = 100 kg.
(a)(b)(c)Figure 13
Equivalent load transformation factors of one-way members using Henrych’s formula parameters in plastic stage: (a)W = 1 kg. (b) W = 10 kg. (c) W = 100 kg.
(a)(b)(c)It can be seen from Figure11 that the KL values of different structural spans using Mills’ formula parameters are the same under the same explosive mass, reflecting the characteristics unrelated to the structural span l. When the explosive mass W is changed, the difference between KL values of different structural spans with the same value γ in the plastic stage is smaller than that in the elastic stage. It can be seen from Figures 12 and 13 that the KL values of the peak overpressure parameters described by the piecewise functions are the same for various structural spans in most calculation intervals of γ under the same explosive mass, reflecting the characteristics unrelated to the structural span l.


## 5. Experimental Verification for Near-Field Blast Situation

### 5.1. Near-Field Blast Experiment of the Simply Supported Beam

At present, there are few verification experiments on the blast load of the one-way beam structures. This paper quotes the experiment of simply supported beam structure subjected to near field blast load designed by Nagata et al. [19], which is used to verify the theory proposed. The experimental device tested by Nagata et al. [19] is shown in Figure 14.Figure 14
Experiment of simply supported beam structure subjected to near-field blast load [19].The cross-sectional width and height of the experimental RC beam are 120 mm, and the calculated span is 1100 mm. When measuring the overpressure distribution on the RC beam, the pressure sensors were arranged at intervals of 100 mm on a steel plate with a thickness of 12 mm, a width of 120 mm, and a length of 500 mm. Table1 lists the basic parameters of the test. Two sets of experiments were carried out under the same blast condition. The blast center was located directly above the midspan of the simply supported beam. The spatial distribution of the peak overpressure on the RC beam was measured through the experiment. The theoretical fitting formula is shown as follows:(26)Px=Pme−2αp/Lx,0≤x≤L2,αp>0,where x is the distance from underneath the explosive, Pm is the peak overpressure underneath the explosive, αp is the pressure distribution factor, and L is the length of the RC beam.Table 1
Experimental cases and parameters [19].
No.C-4 Mass (g)TNT equivalent mass:WTNT (g)Stand-off distance:h (mm)αp1-21101311045.33-41601901184.65-62502981364.0The pressure distribution factorαp is determined through the function fitting. The fitting value is shown in the last column of Table 1. Figure 15 shows the spatial distribution of peak overpressure on the RC beam tested by Nagata et al. [19].Figure 15
Peak overpressure data at different distances from the underneath explosive [19]: (a) nos. 1-2 (110 g), (b) nos. 3-4 (160 g), and (c) nos. 5-6 (250 g).
(a)(b)(c)Nagata et al. proposed the expression of equivalent load transformation factor of the simply supported beam based on the experiment [19]:(27)KL=KMKLM=αp⋅B⋅C41−e−αp⋅AA=3435αp7+6120e−αpαp6+12e−2αp−25αp5−15e−αpαp4+6−2e−2αp+3αp3+54e−αpαp2+183e−2αp−4αp+63e−2αp−144e−αp+81,B=2αp3+3e−αpαp2−6αp−6e−αp+6,C=2αp3+6e−αpαp2+6e−2αp−2αp+9e−2αp−24e−αp+15.Combined with equation (27) and αp, the equivalent load transformation factors KL of the test cases are obtained, as shown in Figure 16.Figure 16
Comparison ofKL between the theoretical results and the experimental results.


### 5.2. Comparison with Theoretical Result

The relevant parameters of the experiment done by Nagata et al. [19] are substituted into the theoretical method of calculating the equivalent load transformation factor in this paper, which is suitable for the simply supported one-way members. The calculation results of the equivalent load transformation factor KL of the proposed theoretical method under different blast load parameters are obtained, which are compared with the calculation results of the literature [19] based on experiments. The summary is shown in Figure 16, and the relative error is shown in Figure 17.Figure 17
The relative error ofKL between the theoretical results and the experimental results.It can be seen from Figure16 that the calculation results of the overpressure parameters proposed by various representative scholars in this paper are very close to the calculation results based on the experiment by Nagata et al. The theoretical method proposed in this paper can well predict the variation trend of the equivalent load transformation factor under different stand-off distances. Figure 17 shows that the minimum error of KL between the theoretical results obtained from different parameters of various scholars and the experimental results tested by Nagata et al. is 0.011%, the maximum error is no more than 4% and the average error is 1.4%.


## 5.1. Near-Field Blast Experiment of the Simply Supported Beam

At present, there are few verification experiments on the blast load of the one-way beam structures. This paper quotes the experiment of simply supported beam structure subjected to near field blast load designed by Nagata et al. [19], which is used to verify the theory proposed. The experimental device tested by Nagata et al. [19] is shown in Figure 14.Figure 14
Experiment of simply supported beam structure subjected to near-field blast load [19].The cross-sectional width and height of the experimental RC beam are 120 mm, and the calculated span is 1100 mm. When measuring the overpressure distribution on the RC beam, the pressure sensors were arranged at intervals of 100 mm on a steel plate with a thickness of 12 mm, a width of 120 mm, and a length of 500 mm. Table1 lists the basic parameters of the test. Two sets of experiments were carried out under the same blast condition. The blast center was located directly above the midspan of the simply supported beam. The spatial distribution of the peak overpressure on the RC beam was measured through the experiment. The theoretical fitting formula is shown as follows:(26)Px=Pme−2αp/Lx,0≤x≤L2,αp>0,where x is the distance from underneath the explosive, Pm is the peak overpressure underneath the explosive, αp is the pressure distribution factor, and L is the length of the RC beam.Table 1
Experimental cases and parameters [19].
No.C-4 Mass (g)TNT equivalent mass:WTNT (g)Stand-off distance:h (mm)αp1-21101311045.33-41601901184.65-62502981364.0The pressure distribution factorαp is determined through the function fitting. The fitting value is shown in the last column of Table 1. Figure 15 shows the spatial distribution of peak overpressure on the RC beam tested by Nagata et al. [19].Figure 15
Peak overpressure data at different distances from the underneath explosive [19]: (a) nos. 1-2 (110 g), (b) nos. 3-4 (160 g), and (c) nos. 5-6 (250 g).
(a)(b)(c)Nagata et al. proposed the expression of equivalent load transformation factor of the simply supported beam based on the experiment [19]:(27)KL=KMKLM=αp⋅B⋅C41−e−αp⋅AA=3435αp7+6120e−αpαp6+12e−2αp−25αp5−15e−αpαp4+6−2e−2αp+3αp3+54e−αpαp2+183e−2αp−4αp+63e−2αp−144e−αp+81,B=2αp3+3e−αpαp2−6αp−6e−αp+6,C=2αp3+6e−αpαp2+6e−2αp−2αp+9e−2αp−24e−αp+15.Combined with equation (27) and αp, the equivalent load transformation factors KL of the test cases are obtained, as shown in Figure 16.Figure 16
Comparison ofKL between the theoretical results and the experimental results.


## 5.2. Comparison with Theoretical Result

The relevant parameters of the experiment done by Nagata et al. [19] are substituted into the theoretical method of calculating the equivalent load transformation factor in this paper, which is suitable for the simply supported one-way members. The calculation results of the equivalent load transformation factor KL of the proposed theoretical method under different blast load parameters are obtained, which are compared with the calculation results of the literature [19] based on experiments. The summary is shown in Figure 16, and the relative error is shown in Figure 17.Figure 17
The relative error ofKL between the theoretical results and the experimental results.It can be seen from Figure16 that the calculation results of the overpressure parameters proposed by various representative scholars in this paper are very close to the calculation results based on the experiment by Nagata et al. The theoretical method proposed in this paper can well predict the variation trend of the equivalent load transformation factor under different stand-off distances. Figure 17 shows that the minimum error of KL between the theoretical results obtained from different parameters of various scholars and the experimental results tested by Nagata et al. is 0.011%, the maximum error is no more than 4% and the average error is 1.4%.


## 6. Results and Discussions

### 6.1. Parametric Selection Study and Experimental Verification

The equivalent SDOF load transformation factor expression of the explosion shock wave for the one-way member is a function of the peak overpressure distributionp (x) and the structural shape function X (x). Most of the peak overpressure functions p (x) proposed by different scholars are functions of the scaled distance Z. The coefficients a, b, and c in the function are different. Some functions take the form of piecewise function. From the perspective of peak overpressure, the inconspicuous difference is caused by the different coefficients. When calculating the load transformation factor KL, the difference will be increased by the multiplication. This amplification effect shows some significant value transitions in the KL figures. The three hypothetical explosive mass parameters 1 kg, 10 kg, and 100 kg represent different levels of explosives. The five typical structural span parameters 10 m, 20 m, 30 m, 40 m, and 50 m can basically contain the design dimensions of the single structural member. After combining three different typical peak overpressure parameters, the various load transformation factor values show that it will be insensitive to overpressure parameters and shape functions when γ ≥ 1. In this paper, the first-order vibration mode function (i.e., the main mode function) of the structure is used as the shape function X (x). Other scholars or guidelines often choose the static deflection shape corresponding to the uniform load distribution. This choice has little effect on the simply supported and simply-fixed one-way members and has little effect on the fixed end one-way member in the elastic stage. For the fixed end member, when γ approaches infinity, the calculation result of KL in this paper is 0.54 and other literatures [3, 29] are 0.53. These conclusions are based on the explosive mass and structural span assumed in this paper. It can be determined and unnecessary to increase the calculation values of explosive mass and structural span, and the same or similar conclusions can be drawn.The load type caused by the far field blast is a uniform load. The load transformation factor is a fixed constant for the beam structure, which is unnecessary for verification. Therefore, verifying the formula proposed in this paper is mainly reflected in the near field blast. In this paper, we use a typical test of the simply supported RC beams subjected to near field blast as the verification means. Different equivalent load transformation factors are calculated by using the test parameters proposed by various scholars, which is compared with the equivalent load transformation factors calculated by the experiment. The relative error ofKL between the theoretical results and the experimental results is low, and the maximum error does not exceed 4%. Accordingly, the theoretical formula derived in this paper has good accuracy.


### 6.2. Demarcation between Near- and Far-Field Blast

The important indicator to consider the demarcation between near field and far field blast is whether the blast load can be simplified as a uniform load type. From the current calculation results, if the calculation errors of 3% in elasticity and 4% in plasticity are acceptable, the ratioγ = 1.0 is the demarcation point. If the ratio γ rises to 2.0, the calculation error of KL corresponding to the uniform load will be reduced to 0.5%. This provides a suggestion for the design of blast-resistant structure: when the ratio γ ≤ 1.0, if the blast loading is still considered to the uniform load form and the peak overpressure corresponding to the minimum blast distance (i.e., stand-off distance) is applied to calculate the blast load of equivalent SDOF, the calculation result will overestimate the equivalent load value. Meanwhile, the peak overpressure corresponding to the maximum blast distance, such as the distance from the center of the explosive to the end of the one-way member, is taken as the uniform load value. It will underestimate the equivalent load value. The suggested method is to treat the blast load distribution as a nonuniform load form and use the formula provided in this paper to calculate the corresponding equivalent blast load or load transformation factor.


## 6.1. Parametric Selection Study and Experimental Verification

The equivalent SDOF load transformation factor expression of the explosion shock wave for the one-way member is a function of the peak overpressure distributionp (x) and the structural shape function X (x). Most of the peak overpressure functions p (x) proposed by different scholars are functions of the scaled distance Z. The coefficients a, b, and c in the function are different. Some functions take the form of piecewise function. From the perspective of peak overpressure, the inconspicuous difference is caused by the different coefficients. When calculating the load transformation factor KL, the difference will be increased by the multiplication. This amplification effect shows some significant value transitions in the KL figures. The three hypothetical explosive mass parameters 1 kg, 10 kg, and 100 kg represent different levels of explosives. The five typical structural span parameters 10 m, 20 m, 30 m, 40 m, and 50 m can basically contain the design dimensions of the single structural member. After combining three different typical peak overpressure parameters, the various load transformation factor values show that it will be insensitive to overpressure parameters and shape functions when γ ≥ 1. In this paper, the first-order vibration mode function (i.e., the main mode function) of the structure is used as the shape function X (x). Other scholars or guidelines often choose the static deflection shape corresponding to the uniform load distribution. This choice has little effect on the simply supported and simply-fixed one-way members and has little effect on the fixed end one-way member in the elastic stage. For the fixed end member, when γ approaches infinity, the calculation result of KL in this paper is 0.54 and other literatures [3, 29] are 0.53. These conclusions are based on the explosive mass and structural span assumed in this paper. It can be determined and unnecessary to increase the calculation values of explosive mass and structural span, and the same or similar conclusions can be drawn.The load type caused by the far field blast is a uniform load. The load transformation factor is a fixed constant for the beam structure, which is unnecessary for verification. Therefore, verifying the formula proposed in this paper is mainly reflected in the near field blast. In this paper, we use a typical test of the simply supported RC beams subjected to near field blast as the verification means. Different equivalent load transformation factors are calculated by using the test parameters proposed by various scholars, which is compared with the equivalent load transformation factors calculated by the experiment. The relative error ofKL between the theoretical results and the experimental results is low, and the maximum error does not exceed 4%. Accordingly, the theoretical formula derived in this paper has good accuracy.


## 6.2. Demarcation between Near- and Far-Field Blast

The important indicator to consider the demarcation between near field and far field blast is whether the blast load can be simplified as a uniform load type. From the current calculation results, if the calculation errors of 3% in elasticity and 4% in plasticity are acceptable, the ratioγ = 1.0 is the demarcation point. If the ratio γ rises to 2.0, the calculation error of KL corresponding to the uniform load will be reduced to 0.5%. This provides a suggestion for the design of blast-resistant structure: when the ratio γ ≤ 1.0, if the blast loading is still considered to the uniform load form and the peak overpressure corresponding to the minimum blast distance (i.e., stand-off distance) is applied to calculate the blast load of equivalent SDOF, the calculation result will overestimate the equivalent load value. Meanwhile, the peak overpressure corresponding to the maximum blast distance, such as the distance from the center of the explosive to the end of the one-way member, is taken as the uniform load value. It will underestimate the equivalent load value. The suggested method is to treat the blast load distribution as a nonuniform load form and use the formula provided in this paper to calculate the corresponding equivalent blast load or load transformation factor.


## 7. Conclusions

The equivalent SDOF load transformation factor expressions of the simply-supported, fixed-end, and simply-fixed one-way members subjected to air blast loading in the elastic and plastic stage are solved by the series expansion method or the approximate fitting method. To visually display load transformation factors, the typical peak overpressure, explosive mass and structural span parameters are selected for calculation and drawing. The results show that theKL calculation curves using a single function are smooth and continuous for the peak overpressure function, while for KL calculation curves of the peak overpressure using a piecewise function, there are some massive transition at transition points. However, the above load transformation factors based on SDOF methods are based on some approximations. For instance, assume that the shape of the structural member is consistent with its first vibration mode; the equivalent parameters of the equivalent system (mass, stiffness, and load) and the length of the plastic hinge are estimated by approximate formulas.The calculation results also indicate that allKL values show a downtrend as the variable γ increases. The fastest decline is in the plastic stage. Under different support conditions, the descending speed of KL in the elastic stage is fixed end > simply-fixed end > simply supported. When γ > 1.0, the downtrend becomes slow. If the calculation error of 4% is acceptable, the ratio γ = 1.0 will be the demarcation point for near blast and far blast fields. When γ ≥ 2.0 and the calculation error of 0.5% is acceptable, it can be considered that γ approaches infinity. The KL formula expression and calculation results show that the KL is independent of structural span l in the plastic stage. It is related to the peak overpressure parameters and the ratio of stand-off distance to structural span (γ).Combined with the experiment of the simply supported RC beams subjected to near field blast, the load transformation factor data is obtained. Moreover, using the parameters of the overpressure distribution function proposed by various scholars and substituting them into the theoretical formula proposed in this paper for calculating the load transformation factor, the average error between the theoretical results and the experimental results is 1.4%, and the maximum error is no more than 4%.



---

*Source: 6754381-2021-03-04.xml*