A new general approach to derive normalised pressure impulse curves

International Journal of Impact Engineering 62 (2013) 1e12

Contents lists available at SciVerse ScienceDirect

International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng

A new general approach to derive normalised pressure impulse curves Jonathon Dragos, Chengqing Wu* School of Civil and Environmental Engineering, The University of Adelaide, Adelaide, SA, Australia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 17 April 2013 Accepted 23 May 2013 Available online 21 June 2013

A pressure impulse (PI) diagram is an important tool used for the preliminary design of structural members against blasts. Normalised PI diagrams can be derived using single degree of freedom (SDOF) theory to quickly determine the PI diagram of a given structural member. In order to use PI diagrams for blasts occurring in various confined environments, characterised by irregular shaped pulse loads, an investigation into the effects of pulse shape on a given point on the normalised PI curve is undertaken. Relying on the concept of the effective pulse load, three parameters which define the shape of the effective pulse load are determined. These parameters are then used to derive a method for determining a point on the normalised PI curve for elastic, rigid plastic and elastic plastic hardening structural members. The overall procedure can be iterated to determine many points, thus forming the entire normalised PI curve. Due to the generality of this new approach, it can be applied to structural members subjected to any arbitrary pulse load as long as its response and failure are controlled by its flexural behaviour. Ó 2013 Elsevier Ltd. All rights reserved.

Keywords: Blasts Effective pulse load Normalised PI diagram SDOF

1. Introduction The SDOF method is a first order method used to model the response of structural members, such as reinforced concrete (RC), steel members and unreinforced masonry walls, against blasts based on an anticipated response mode [1e3]. Its applications can even be extended to that of structures against sonic booms [4] and aircraft structures against blasts [5]. ASCE guidelines [6,7] and the most recent guidelines of UFC-3-340-02 [8] all recommend the use of the SDOF method for such analyses. The SDOF method models the deflection of an important point on the structural member, for example the mid span of a simply supported beam, by simplifying it into a lumped mass on a spring type system. Therefore, the method relies on certain parameters, such as the equivalent mass and the resistance deflection (RD) function of the idealized spring system [9e11]. Using the SDOF model, pressure impulse (PI) curves for structural members can be determined. Provided that the parameters used in the SDOF model accurately predict the response of a given structural member, the PI curve can be used to quickly determine whether the member fails or survives a given blast [12e14]. Therefore, it can be used as a preliminary design tool for structural members against blasts. A normalized PI curve represents a family

* Corresponding author. E-mail address: [email protected] (C. Wu). 0734-743X/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijimpeng.2013.05.005

of PI curves and can be converted quickly into a PI curve for a given structural member. This is done using expressions for the minimum impulse and the minimum peak reflected pressure asymptotes. For this reason, it is recognized as a useful tool for analysis. Krauthammer et al. [15] and Li & Meng [16] undertook a dimensional analysis on PI curves for elastic members subjected to different pulse loads in order to investigate the effects of pulse shape on the normalized PI curve. This analysis was undertaken for a load function which is typically used for external blasts in which the parameters can be manipulated to form three pulse shapes; exponential, triangular, and rectangular. Li & Meng [16] then attempted to eliminate the effects of pulse shape on the normalised PI curve for elastic members. Two parameters, n1 and n2, were introduced which were empirical functions of the geometric centroid of the pulse load. These parameters were then part of a general hyperbolic function which describes the normalised PI curve for any elastic member subjected to any pulse load shape. This empirical approach for eliminating the effects of pulse shape was proven to be very efficient and accurate for the pulse load shapes tested in the study which are typically associated with external blast loads. However, Campidelli & Viola [17] attempted to extend those empirical equations to be more applicable to a wider range of pulse shapes. They found that for some pulse shapes, the errors involved in those empirical equations were quite significant. This suggests that the empirical approach contains limitations, one of which is that the shape of the pulse load was defined by a single parameter which was labelled d. Another limitation of this

2

J. Dragos, C. Wu / International Journal of Impact Engineering 62 (2013) 1e12

empirical approach is that within different regions of a PI curve the shape of the pulse load acting on the structural member, from the time at which the shockwave reaches the member until the time at which the member reaches its maximum deflection, changes. This is due to the relative relationship between the load function and the structural response which is unique for each point along the normalised PI curve [15]. Therefore, by assuming that the shape of the elastic normalised PI curve can be described by a single parameter, d, which is calculated from the pulse load shape, does not take the changing loaderesponse relationships into consideration. Fallah & Louca [2] attempted to extend the studies conducted by Krauthammer et al. [15] and Li & Meng [16] by deriving normalised PI curves, using a dimensional analysis, for an SDOF system with a bilinear resistance deflection (RD) function. The bilinear RD functions investigated were elastic plastic hardening and elastic plastic softening. Idealising the response of an RC or steel structural member using a bilinear RD function is much more appropriate as the yielding and ductility of such a member are taken into account. As done by previous studies, normalised PI curves for such members subjected to pulse loads typically associated with external blasts, that is, exponential, triangular and rectangular shapes, were derived, as they corresponded with that of external blasts. However, for more abstract pulse shapes, the entire approach needs to be repeated and new differential equations need to be solved to derive their associated normalised PI curves. To date, all studies conducted on normalized PI curves have only investigated pulse load shapes corresponding to blasts in an external environment. However, blasts can also occur in a confined or partially confined environment in which shockwave reflections off surrounding walls can occur. This can cause pressure time histories with abstract pulse shapes to act on surrounding structural members. Although there is a high level of variability associated with such blast load scenarios, the most recent guidelines of UFC-3340-02 [8] claim that it is appropriate to simplify a confined blast pressure time history to a bilinear pulse load. Dragos et al. [18] also support this claim but provide a different technique for obtaining such a simplified bilinear pulse load. Therefore, normalised PI curves derived for these simplified bilinear pulse load shapes will facilitate the quick determination of the response of structural members against confined blasts. However, due to the abstract shapes associated with the simplified bilinear pulse load, a thorough understanding of the effects of pulse shape on the normalized PI curve should be established. In the current study, a new approach for deriving a normalised PI curve for any pulse load shape and any bilinear elastic plastic hardening RD function shape is provided. Firstly, the effective pulse load (defined differently to that of Youngdahl [19]), as opposed to the actual pulse load, is investigated and discussed. This is because the concept of the effective pulse load is the foundation of this new approach. Then, by undertaking an approach similar to Li & Meng [16] and Fallah & Louca [2], a database of many points on the normalised PI diagram is determined for various different RD functions and effective pulse shapes. Then, parameters which define the shape of an effective pulse load are derived. These parameters, in conjunction with the database of values, are then used as the basis to derive the formulae so as to determine a single point on the elastic and rigid plastic normalised PI curves. Furthermore, using the database, the formulae for determining a single point on the normalised PI curve for a member with a bilinear RD curve are also derived. Even though the current study provides a methodology for determining a single point on the normalised PI curve, it can be repeated to find many points, thus forming the entire normalised PI curve. Also, due to the generality of this approach, it can be applied to any pulse load shape. Finally, the overall approach to determine a normalised PI curve utilising a spreadsheet tool with the formulae

derived in the current study, is outlined. Such a spreadsheet can be created rather quickly leading to the efficient analysis of structural members against blast load scenarios which generate more abstract pulse load shapes, such as that predicted by UFC guidelines [8] for confined blasts. This approach is more suitable to practicing engineers who do not wish to use a full finite element approach due to its long simulation times and specialised knowledge required. 2. Normalised pressure impulse curves For the purposes of this study, a normalised PI curve, as shown in Fig. 1, is defined as a PI curve with horizontal and vertical asymptotes equal to unity, as seen by the horizontal and vertical dashed lines, respectively. Studies conducted on PI curves, such as Krauthammer et al. [15] and Li & Meng [16], have shown nondimensional PI curves for elastic members in which the magnitude of the horizontal and vertical asymptotes are equal to 0.5 and 1, respectively. However, the coordinates of any non-dimensional PI curve can be manipulated to make both of its asymptotes equal to unity, thus forming a normalised PI curve. Fig. 1 shows that the coordinates of any normalised PI curve are I/Imin and Pr/Prmin. To convert a normalised PI curve to a PI curve of a given structural member, the axes, and thus the coordinates, should be multiplied by Imin and Prmin. For a given RD curve of a structural member, the equations for Imin and Prmin can be determined using the law of conservation of mechanical energy [2]:

Imin ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffi 2MER

Prmin ¼

(1)

ER ym

(2)

where M ¼ equivalent mass; ym ¼ maximum deflection; and Z ym ER ¼ strain energy, or: ER ¼ RðyÞdy, in which R(y) ¼ RD 0

function. Fig. 1 also displays the three different regions, or loaderesponse relationships, of the normalised PI curve. The regions are the impulse controlled region, the dynamic region and the quasi-static region. Fig. 2 displays examples of the interaction between the load time history and the deflection time history for all three loade response relationships; impulse controlled (a), dynamic (b) and quasi-static (c). For each loaderesponse relationship, the duration of the load, td, and the time to reach maximum deflection, tmax, are also illustrated. The relationship between tmax and td, in Fig. 2, indicates the main notion which distinguishes the three loaderesponse relationships from each other. As shown, for the impulse controlled

Fig. 1. Normalised PI curve for elastic member subjected to triangular pulse load.

J. Dragos, C. Wu / International Journal of Impact Engineering 62 (2013) 1e12

3

Fig. 2. Loaderesponse relationship for three regions of a PI curve; impulse controlled (a), dynamic (b), and quasi-static (c).

region, td is much less than tmax; for the dynamic region, td is comparable to tmax; and for the quasi-static region, td is much larger than tmax. From this observation, a new parameter can be introduced, sr, where:

sr ¼ td =tmax

(3)

sr is an important parameter, because it determines which loaderesponse relationship, and thus, which region of the PI curve a point is located. Therefore, if sr is much less than 1, the response is impulse controlled. If sr is approximately equal to 1, the response is dynamic and if sr is much larger than 1, the response is quasi-static. Furthermore, each point along the normalised PI curve corresponds to a unique sr value, ranging from 0 to infinity. It can be observed in Fig. 2 that the shape of the pulse load, for all cases, is exponential. However, the shape of the pulse load between 0 and tmax, shown as the shaded regions in Fig. 2, is exponential for the dynamic case only. For the quasi-static case, the shape of the pulse load between 0 and tmax is almost rectangular, as the impulse acting on the system after tmax has no effect on response. Also, for the impulse controlled case, the shape of the pulse load between 0 and tmax is quite different as its centroid, along the time axis, is quite close to 0 relative to tmax. From this observation, the concept of effective pulse arises. The difference between a pulse load and an effective pulse load is that the pulse load is the load which acts from 0 to td. However, the effective pulse load is the part of the load acting during the time in which the member is responding to the load, which is from 0 to tmax. Therefore, although a given PI curve is specific to a given pulse load shape, the effective pulse load shape changes within different regions of the PI curve, as shown in Fig. 2. Thus, the effective pulse load becomes much more important than the pulse load itself for determining a point on a PI curve. All of this is assuming that the maximum deflection occurs at the end of its first response cycle, at tmax, as seen in Fig. 2.

(a)

The normalised PI curve, shown in Fig. 1, is specific to an elastic system being subjected to a pressure time history which is of triangular shape. Therefore, the shape of the pulse load is triangular. Although all normalised PI curves have asymptotes at unity, the shape of the normalised PI curve can vary. There are two main factors which affect the shape of the normalised PI curve. The first is the shape of the pulse load and the second is the shape of the RD function of the member itself. Dimensional studies of the influence of RD function shape on normalised PI curves have shown that its effects cannot be ignored [2,20,21]. Fig. 3(a) shows three normalised PI curves of members with differing RD function shapes, being elastic, rigid plastic and of typical bilinear shape, subjected to a triangular pulse. The shapes of the three corresponding RD functions, elastic, rigid plastic and of typical bilinear shape, can be seen in Fig. 3(b). In Fig. 3(b), ym is the maximum displacement and Ru is the ultimate resistance. Also, Fig. 3(b) displays the point of yield of the bilinear RD function in which y/ym ¼ yel/ym and R/Ru ¼ Ry/Ru. This point has a significant effect on the shape of the bilinear RD function and thus has a significant effect on the shape of the normalised PI curve of such a member. However, for any given pulse shape, the elastic and rigid plastic normalised PI curves form the lower and upper boundaries, respectively, of the normalised PI curve for a bilinear RD shape, where the point of yield dictates where, relative to the bounds, this curve lies. 3. Analytical solution The general differential equation which is analytically solved in order to determine a point on a PI curve can be seen in Eq. (4):

€ þ RðyÞ ¼ FðtÞ My

(4)

where R(y) ¼ RD function; F(t) ¼ force time history; and M ¼ equivalent mass. As the response times for structural members

(b)

Fig. 3. Three normalised PI curves (a) corresponding to three RD function shapes (b).

4

J. Dragos, C. Wu / International Journal of Impact Engineering 62 (2013) 1e12

subjected to blasts are very short, damping has very little effect on the response, and is therefore neglected. In the current study, the load function, F(t) as seen in Fig. 4, is chosen to be a bilinear step function in which the first region F1(t) is linear descending, and the second region F2(t) is of constant force. The peak load is represented by Fo and the duration of the pulse load is represented by td. Two parameters, r and s, can also be observed in Fig. 4, which can be manipulated to alter the shape of the pulse load. The reason for choosing a bilinear pulse load is such that, by manipulating r and s, various pulse shapes can be obtained. This aids in determining various points on the normalised PI diagram due to various pulse shapes. To determine a point on a PI curve, the general differential Eq. (4) should be solved such that the following initial and final conditions are satisfied: Initial conditions:

yð0Þ ¼ 0

pulse load. An example of the difference between the effective impulse and the actual impulse can be seen in Fig. 2(c). The shaded region represents the effective impulse which is not equal to the impulse of the actual pulse load. To determine the coordinates of a single point on the PI curve corresponding to a given effective pulse load, the duration and the peak reflected pressure are manipulated such that the initial and final conditions are satisfied. Once the duration, peak reflected pressure, and thus impulse, are determined, the peak reflected pressure and the impulse are divided by the minimum peak reflected pressure (2), Prmin, and the minimum impulse (1), Imin, respectively. This is carried out to determine where this point lies on a normalised PI diagram. This produces the same results as solving the differential Eq. (4) in its dimensional form, such as done by Fallah & Louca [2]. From this, a database of coordinates of points of the normalised PI curve corresponding to various effective pulse shapes is determined for elastic, plastic and various bilinear RD function shapes. 3.1. Analytical solution for an elastic SDOF model

_ yð0Þ ¼ 0 Final conditions:

yðtmax Þ ¼ ym _ max Þ ¼ 0 yðt td ¼ sr tmax To determine all points along a PI curve, the value of sr is varied from 0 to infinity. However, as the aim is to determine points on a normalised PI diagram corresponding to various effective pulse load shapes, sr is set to equal 1. Therefore, the response time, tmax, is set to equal the duration of the pulse load, td. This ensures that the pulse load chosen is also the effective pulse load acting on the structural member. By setting tmax to equal td means that for a given pulse load, and an effective pulse load, a single point along the PI curve can be determined. It should be noted that the choice of a bilinear load function allows effective pulse loads to be chosen which correspond to various loaderesponse relationships. For example, if the parameters are chosen such that, r ¼ 1 & s ¼ 1, a rectangular effective pulse is chosen. This corresponds to a quasistatic loaderesponse relationship similar to that depicted in Fig. 2(c). On the other hand, if the parameters are chosen such that, r ¼ 0 & s ¼ 0.1, an impulse controlled loaderesponse relationship occurs, similar to that depicted in Fig. 2(a). Therefore, manipulation of the shape of the effective pulse can be used to determine various points along a PI curve. The only difference is that the impulse of the effective pulse load is known, but not the impulse of the actual

To solve an elastic SDOF model for the bilinear pulse load in Fig. 4, Eq. (4) can be altered to form two differential equations:

  ð1  rÞt € þ ky ¼ F1 ðtÞ ¼ Fo 1  My std

(5)

€ þ ky ¼ F2 ðtÞ ¼ rFo My

(6)

To solve Eq. (5) at the intersection of F1 and F2 with time t1 ¼ s  td, y(t1) and y0 (t1) should be determined. y(t1) and y0 (t1) then become the initial conditions for part 2, in which Eq. (6) is solved. To be consistent, in part 2 a transformation is undertaken such that t / t  t1. Therefore, to determine y and y0 at time tmax, y(tmax  t1) and y0 (tmax  t1) should be determined, respectively. To satisfy the final conditions, td, Fo and thus Pr are manipulated. For the elastic case, this cannot be achieved algebraically and so was done using an equation solver. As Pr and td are known, the impulse can be determined. Then, I/Imin and Pr/Prmin are determined. For the elastic case, I/Imin is labelled Iel and Pr/Prmin is labelled Prel. Dimensional studies by Li & Meng [16] on PI curves for elastic SDOF systems subjected to various pulse loads showed that only the shape of the pulse load affects the non-dimensional or normalized PI curve. Therefore, as the shape of the bilinear effective pulse load can be altered by changing r and s, a database of Iel and Prel values can be obtained for varying effective pulse load shapes, seen in Tables 1 and 2, respectively. 3.2. Analytical solution for a rigid plastic SDOF model For a rigid plastic SDOF model, Eq. (4) can be altered to form another two differential equations:

Table 1 Database of Iel values corresponding to various bilinear pulse shapes (r & s). Iel

Fig. 4. Bilinear force time history used in SDOF analysis.

s\r

0

0.2

0.4

0.6

0.8

1

0.02 0.2 0.4 0.6 0.8 1

1.000 1.003 1.015 1.039 1.085 1.165

1.564 1.369 1.253 1.209 1.215 1.264

1.569 1.487 1.392 1.339 1.325 1.352

1.570 1.535 1.485 1.440 1.419 1.432

1.570 1.557 1.534 1.511 1.499 1.495

1.579 1.579 1.579 1.579 1.579 1.579

J. Dragos, C. Wu / International Journal of Impact Engineering 62 (2013) 1e12 Table 2 Database of Prel values corresponding to various bilinear pulse shapes (r & s). Prel

s\r

0

0.2

0.4

0.6

0.8

1

0.02 0.2 0.4 0.6 0.8 1

126.5 11.92 5.600 3.533 2.543 2.000

4.975 3.993 3.024 2.386 1.949 1.666

2.497 2.325 2.040 1.781 1.578 1.428

1.666 1.621 1.524 1.418 1.324 1.250

1.250 1.238 1.210 1.175 1.140 1.111

1.000 1.000 1.000 1.000 1.000 1.000

  ð1  rÞt € þ Ru ¼ F1 ðtÞ ¼ Fo 1  My std

(7)

€ þ Ru ¼ F2 ðtÞ ¼ rFo My

(8)

The same process to determine the solutions to Eqs. (5) and (6) is used to determine the solutions to Eqs. (7) and (8) in order to yield I/Imin and Pr/Prmin, which are labelled Ipl and Prpl, respectively. All equations, satisfying the final conditions, can be solved algebraically, allowing Eqs. (9) and (10) to be obtained:

Ipl

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s r r 2 ð1  Þ þ ¼ s ð1  rÞð3  2sÞ 6

(9)

1 Prpl ¼ s rÞ þ r ð1  2

(10)

Eqs. (9) and (10) are functions of the parameters r and s only. This means that for the rigid plastic case, only the shape of the effective pulse load affects the coordinates of a point on the normalised PI curve. 3.3. Analytical solution for a bilinear SDOF model The bilinear RD function, as shown in Fig. 3(b), is a linear step function and can be described by the following:

R1 ðyÞ ¼ ky

if

y < yel

(11)

R2 ðyÞ ¼ k2 ðy  yel Þ þ Ry

if

yel < y < ym

#1: #2: #3: #4:

R(y) R(y) R(y) R(y)

¼ ¼ ¼ ¼

R1(y) R2(y) R1(y) R2(y)

& & & &

F(t) F(t) F(t) F(t)

¼ ¼ ¼ ¼

4. Derivation of equations for normalised PI curves

F1(t) F1(t) F2(t) F2(t)

There are two possible ways in which the model progresses from Case #1 to Case #4. Each progression consists of 3 parts. The chosen progression depends on whether t1 is less than tel, time at which deflection at yield, yel, occurs, or vice versa. Each progression is as follows: Progression #1: Case #1 / Case #2 / Case #4 Progression #2: Case #1 / Case #3 / Case #4

As both progressions begin with Case #1, during this case, if tel is less than t1 then Progression #1 should be undertaken. However, if t1 is less than tel then Progression #2 should be undertaken. As for the elastic and rigid plastic SDOF models, when progressing from one case to the next, the final conditions of the current case become the initial conditions for the subsequent case. For this to occur, a transformation of the time variable should be made such that the time at which the current case ends (either tel or t1), should be subtracted from the time variable of the subsequent case. To satisfy all initial and final conditions, the manipulation of 3 parameters should be undertaken, tel (time at which yel is reached), Pr and td. As in the elastic case, due to the difficulty in solving the equations algebraically, an equation solver should be used to simultaneously manipulate all three parameters until all conditions are satisfied. Then, from Pr and td, the impulse can be determined. Finally, I/Imin and Pr/Prmin can then be determined. It was found that, for a given effective pulse shape, the two main factors affecting I/Imin and Pr/Prmin were the ratio of the yield resistance to the ultimate resistance (Ry/Ru) and the ratio of the yield deflection to the ultimate deflection (yel/ym). Fallah & Louca [2] used Buckingham’s Pi-theorem to determine two nondimensional parameters which were said to define the shape of the bilinear RD function. The two parameters were the inverse ductility, a, and the hardening/softening index, j. Although yel/ym is equal to the inverse ductility used by Fallah & Louca [2], the hardening/softening index is simply a function of both yel/ym and Ry/Ru. Therefore, as both yel/ym and Ry/Ru are non-dimensional, the current approach yields the similar results as Fallah & Louca [2], but is presented differently. For various effective pulse load shapes (r and s), databases of I/ Imin and Pr/Prmin values for varying yel/ym and Ry/Ru can be determined. For example, for the effective pulse load shape corresponding to r ¼ 0.2 & s ¼ 0.2, the database of I/Imin and Pr/Prmin for various RD function shapes (yel/ym and Ry/Ru) can be seen in Table 3. It should be noted that only elastic plastic hardening RD functions are investigated in the current study. This corresponds to the condition, Ry/Ru  yel/ym. It should also be noted that, if Ry/Ru ¼ 1 and yel/ym ¼ 0, then the RD function is rigid plastic. Finally, if Ry/ Ru ¼ yel/ym, then the RD function is elastic.

(12)

where yel is the deflection at yield; Ry is the resistance at yield; k is the slope of the first region; and k2 is the slope of the second region. Typically, the first region, R1, is the elastic region and the second region, R2, is the strain hardening or hardening region. It should be noted that in the current study, the case in which k2 is negative is neglected, and thus strain softening is neglected. As the pulse load is also a bilinear step function, 4 cases of the differential Eq. (4) should be solved for: Case Case Case Case

5

Using the database of I/Imin and Pr/Prmin values, equations are established to determine a single point on the normalised PI curve, for elastic and rigid plastic SDOF systems for any given effective pulse load shape. Therefore, parameters which can define the shape of the effective pulse load should be determined, which are labelled as effective pulse shape parameters. The equations which are established are therefore a function of these effective pulse shape parameters. 4.1. Effective pulse shape parameters To describe the shape of an effective pulse load, three effective pulse shape parameters are defined. However, first both the unit Table 3 Database of I/Imin and Pr/Prmin for various RD shapes for a given effective pulse shape.

r ¼ 0.2,s ¼ 0.2

Ry/Ru/yel/ym

0.7

0.4

0.2

0.1

0

Pr/Prmin I/Imin Pr/Prmin I/Imin Pr/Prmin I/Imin

0.5

e e 3.975 1.384 3.873 1.446

3.940 1.408 3.812 1.495 3.688 1.575

3.820 1.499 3.683 1.595 3.555 1.685

3.763 1.559 3.631 1.663 3.507 1.768

3.785 1.699 3.680 1.854 3.579 2.003

0.75 1

6

J. Dragos, C. Wu / International Journal of Impact Engineering 62 (2013) 1e12

pulse load and the effective unit pulse load are defined in order to eliminate the influence of magnitude and duration. The unit pulse load, an example of which can be seen in Fig. 4, is a pulse load with duration, td ¼ 1 and maximum force, Fo ¼ 1. If F(t) represents the pulse load with duration td, and with maximum load Fo, then Eq. (13) can be used to convert this pulse load to a unit pulse load.

Funit ¼

Fðtd $tÞ Fo

Z1 h i2 Feff ;unit ðtÞ dt Cy ¼

As can be seen in Fig. 2, the shape of the pulse load acting on the structural member during its response changes based on where along the PI curve the point lies. Therefore, as mentioned previously, the effective unit pulse load is of high importance. If the unit pulse load, duration of 1 and maximum load of 1, is now Funit(t), then the effective unit pulse load can be described by Eq. (14):

Iunit ¼

Ieff;unit ¼

Feff;unit ðtÞdt ¼ 0

Z1 t$Feff ;unit ðtÞdt

0

Ieff;unit

¼

Ieff;unit

(a)

Funit ðtÞdt

(17)

(18)

I ¼ Iunit Pr td

(19)

4.2. Elastic normalised PI curves based on effective pulse shape parameters The aim of this section is to determine formulae to calculate the coordinates of a single point on the elastic normalised PI curve, Iel & Prel, based on the effective pulse shape parameters. To scope out what equations need to be derived, Eq. (3) is re-arranged and substituted into Eq. (19) to yield Eq. (20):

Iel ¼ Iunit Prel sr tmax

(20)

(15)

where Iunit is a constant and can be determined using Eq. (18). Also, this approach requires that sr be chosen and is therefore considered known. As two parameters, Pr and tmax, are unknown, two equations are required to calculate these parameters. For a normalised elastic SDOF system, both Imin and Prmin are equal to 1. From Eqs. (1) and (2), it can then be determined that the angular frequency, u ¼ 2. When the above mentioned elastic SDOF system is subjected to various effective pulse loads, such that failure occurs at time tmax, tmax lies between p/4 and p/2. Knowing this, Eq. (21) can be used to determine tmax:

(16)

tmax ¼

t$Funit ðt=sr Þdt

0

2Ieff;unit

The importance of Eq. (18) is that if the peak reflected pressure and duration of the pulse load are known, its impulse can be determined using Eq. (19):

0

Z1 Cx ¼

Funit ðt=sr Þdt

0

0

where the term sr is described in Eq. (3). As Feff,unit is the effective unit pulse load, Eq. (14) ensures that tmax ¼ 1. This means that it is the unit version of a pulse load which ends at the point in which the structural member reaches its maximum deflection. Fig. 5 illustrates the difference between the unit pulse load and the effective unit pulse load, by displaying the effective pulse load for three different loaderesponse relationships: impulse controlled (Fig. 5(a)); quasi-static (Fig. 5(b)); and dynamic (Fig. 5(c)). These three loaderesponse relationships also correspond to points within the three regions of the normalised PI curve seen in Fig. 1. In Fig. 5(c), the shape of the effective pulse load matches the shape of the actual pulse load, as sr ¼ 1. However, although for all three cases the actual pulse load is triangular, for the cases seen in Fig. 5(a) and (b), the shapes of the effective pulse loads are quite different. Three parameters which define the shape of the effective pulse load will be defined which are functions of the effective unit pulse load. The three parameters are: the effective unit impulse (Ieff,unit); the effective unit centroid in the t-axis (Cx); and the effective unit centroid in the F-axis (Cy), and can be seen in Eqs. (15), (16) and (17), respectively.

Z1

¼

Z1

(14)

Z1

2Ieff ;unit

½Funit ðt=sr Þ2 dt

For a pure quasi-static load in which the effective pulse load is rectangular and sr ¼ infinity, similar to that in Fig. 5(b), Ieff,unit ¼ 1, Cx ¼ 0.5 & Cy ¼ 0.5. Conversely, for a pure impulse controlled load in which sr ¼ 0, like that in Fig. 5(a), Ieff,unit ¼ 0 & Cx ¼ 0. Just as these extreme loaderesponse relationships form the bounds of the PI curve, they also form the bounds in which the effective pulse shape parameters can take: 0 < Ieff,unit  1; 0 < Cx  0.5; and 0 < Cy  0.5. Finally, the unit impulse can also be defined. This is the impulse of the unit pulse load, and is defined in Eq. (18):

(13)

Feff;unit ¼ Funit ðt=sr Þ

0

Z1

(b)

ael p

(21)

4

(c)

Fig. 5. Three effective unit pulse loads corresponding to the regions; impulse controlled (a), quasi-static (b), and dynamic (c).

J. Dragos, C. Wu / International Journal of Impact Engineering 62 (2013) 1e12

where ael is a parameter which ranges from 1 to 2, depending on the effective pulse load being applied. Therefore, this parameter is solely a function of the shape of the effective pulse load. When ael ¼ 1, tmax corresponds to its lower bound occurring due to a pure impulse controlled load (sr ¼ 0) which causes free body vibration and causes failure at time tmax. When ael ¼ 2, tmax corresponds to its upper bound which occurs when a purely quasi-static (rectangular, sr ¼ infinity) load acts on the member and causes failure at time tmax. The database of Iel and Prel values determined in Section 3.1, corresponding to various effective pulse load shapes, was then used to determine a database of values for ael. Each ael value within the database corresponded to a unique effective pulse load shape, and thus unique effective pulse shape parameters. After an investigation of the influence of the effective pulse shape parameters on ael, it was observed that the effective pulse shape parameter which had the most influence over ael was Cx. A function was then empirically determined which correlated the most with the database of ael values and can be seen in Eq. (22):





1

ael ¼ 1:5 þ sin1 8Cx2  1 p

Prel ¼ 

Ieff ;unit

determined, tmax and Prpl. However, in contrast to the elastic case, for the rigid plastic case a purely mathematical approach can be used to determine both Prpl and tmax. The differential equation which needs to be solved can be seen in Eq. (4), such that R(y) ¼ Ru. This can then be re-arranged and integrated to obtain My0 at time t*.

Zt*   * ¼ My_ t FðtÞdt  Ru t *

For a point to lie on the PI curve, y0 (t* ¼ tmax) ¼ 0. Therefore, this leads to Eq. (27).

Ztmax Ru tmax ¼

FðtÞdt

It is known that for the normalised case, Ru ¼ 1. Furthermore, as Eq. (27) is simply the effective impulse, the following can be deduced:

tmax ¼ Ieff ¼ Prpl tmax Ieff;unit

(23)

bel

/Prpl ¼

i

bel ¼ 1:083  1:044Ieff;unit 1  Feff;unit ð1Þ Cy2

1 Ieff;unit

My ¼

1 FðtÞdtdt  Ru t *2 2

(30)

Once again, to lie on the PI curve, y(t* ¼ tmax) ¼ ym:

Ztmax Z t Mym ¼

1 FðtÞdtdt  Ru tmax2 2

(31)

0

0

Also, Eq. (27) can be substituted into Eq. (31), and re-arranged, to obtain the following:

2 Mym ¼ 4tmax

Ztmax

Ztmax Z t FðtÞdt 

0

3 1 FðtÞdtdt 5 þ tmax 2

0

0

Ztmax FðtÞdt 0

(32) Eq. (32) is arranged in such a way because the use of integration by parts shows that:

Ztmax

Ztmax t$FðtÞdt ¼ tmax

0

4.3. Rigid plastic normalised PI curves based on effective pulse shape parameters

(29)

0

(24)

where Feff,unit(1) represents the value of the effective unit pulse load at t/tmax ¼ 1. Although Eqs. (23) and (24) do not predict Prel with full accuracy, in comparison to the database the errors involved are typically within 2.5%. As functions for Prel and ael have been derived, the coordinates of any point, due to any effective pulse load, on the elastic normalised PI diagram can be determined.

(28)

As expected, Eq. (29) agrees with the database of Prpl values determined from Eq. (10). Eq. (26) can be integrated again to determine My at time t*:

0

According to Eq. (23), it can be seen that for a purely impulse controlled load in which Ieff,unit approaches 0, Prel approaches infinity. Also, for a purely quasi-static load in which Ieff,unit ¼ 1, Prel ¼ 1. A deeper investigation was then undertaken to empirically determine a function for bel. The investigation was done by minimising the absolute errors between Prel determined via Eq. (23) and that determined from the database, which yielded Eq. (24).

Ztmax Z t FðtÞdt 

0

FðtÞdtdt 0

(33)

0

Therefore, by substituting Eq. (33) into Eq. (32), the following can be determined:

Like the elastic case, the aim is to determine a method to calculate the coordinates of a point on the rigid plastic normalised PI curve, Ipl & Prpl, based on the effective pulse shape parameters. Eq. (20) also applies to the rigid plastic case, which yields Eq. (25):

Ipl ¼ Iunit Prpl sr tmax

(27)

0

Zt * Z t

h

(26)

0

(22)

From Eq. (22), it can be seen that for a purely impulse controlled load in which Cx ¼ 0, ael ¼ 1. Also, for a purely quasi-static load in which Cx ¼ 0.5, ael ¼ 2. Although the function does not predict ael with full accuracy, in comparison to the database the errors involved are typically within 2%. A comparison of the database of Prel values against the effective pulse shape parameters showed that Prel was approximately inversely proportional to Ieff,unit, as can be seen in Eq. (23):

1

7

(25)

As for the elastic case, from Eq. (25) it can be observed that two separate equations for two parameters need to be

Ztmax Mym ¼  0

1 t$FðtÞdt þ tmax 2

Ztmax FðtÞdt

(34)

0

It is known that for the normalised case, in which Imin ¼ Prmin ¼ 1, Mym ¼ 0.5. It can be seen that the first integral term in Eq. (34) is the centroid along the t-axis of the effective pulse load multiplied by the effective impulse. Also, that the

8

J. Dragos, C. Wu / International Journal of Impact Engineering 62 (2013) 1e12

second integral term is the effective impulse. After substitution of the aforementioned terms and some re-arranging, Eq. (35) can be derived.

1 tmax ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  2Cx

(35)

As expected, when Eq. (35) and Eq. (29) are input into Eq. (25), the values obtained all completely agree with the database of Ipl values determined using Eq. (9). Similar to the elastic case, these equations can be used to determine the coordinates of any point, due to any effective pulse load, on the rigid plastic normalised PI diagram. 5. Normalised PI curves for bilinear SDOF systems Equations to determine a single point on the normalised PI curve for elastic and rigid plastic members have been determined. The focus now shifts to determine such a point for members with a bilinear RD function. The aim of this section is to determine I/Imin and Pr/Prmin for members with a bilinear RD function subjected to an arbitrary effective pulse load using the extensive database of values obtained in Section 3.3, such as in Table 3. An example illustrating the approach that is employed to achieve this goal can be seen in Fig. 6. It assumes that to determine a given point on the normalised PI curve, sr has been chosen and is thus known. From this assumption, the effective pulse load and its corresponding effective pulse shape parameters can be determined. Therefore, its corresponding point on the elastic and rigid plastic normalised PI curves, illustrated as coordinates in Fig. 6, can be determined. From Fig. 6, it can also be seen that, for the same sr value and therefore the same effective pulse load, I/Imin for a member with a bilinear RD function always lies between the coordinates of the elastic and rigid plastic cases. This occurs as long as the second region of its RD function has a positive slope. Therefore, as long as the condition: k  k2  0 holds. The main factors which determine where the coordinates, I/ Imin & Pr/Prmin, lie are: Ry/Ru, yel/ym and the effective pulse shape parameters. As the elastic case can be thought of as the lower bound of the bilinear RD function, when Ry/Ru ¼ yel/ym, then the point (I/Imin, Pr/Prmin) lies on the point (Iel, Prel). Conversely, as the rigid plastic case can be regarded as the upper bound of the bilinear RD function, when Ry/Ru ¼ 1 and yel/ ym ¼ 0, then the point (I/Imin, Pr/Prmin) lies on the point (Ipl, Prpl). For all intermediate values of Ry/Ru and yel/ym, the point (I/Imin, Pr/Prmin) lies somewhere between the lower and upper bounds. Therefore, the aim is to derive equations which determine where the point lies, using the extensive database obtained in Section 3.3.

Fig. 6. Normalised PI curves for elastic, rigid plastic and bilinear RD functions.

5.1. I/Imin for bilinear SDOF systems To begin with, an investigation into the effects of the RD function shape (yel/ym & Ry/Ru) on I/Imin for a triangular effective pulse load is undertaken. For this case, Cx ¼ Cy ¼ 1/3 and Ieff,unit ¼ 1/2. From the database, I/Imin values for a triangular effective pulse load can be plotted, as seen in Fig. 7. It shows I/Imin plot against yel/ ym for various Ry/Ru values. From Fig. 7, it can be seen that all points lie between Iel and Ipl, the lower and upper bounds. Fig. 7 also illustrates another term which needs to be introduced, IN. This is the value which I/Imin takes when yel/ym ¼ 0, which is the condition in which no elastic region exists or yel ¼ 0. As shown in Fig. 7, it is important to note that IN varies based on a given Ry/Ru. For a triangular effective pulse load, it was observed that when Ry/Ru ¼ 0, IN ¼ Iel and when Ry/Ru ¼ 1, IN ¼ Ipl, which correspond to the elastic and rigid plastic cases, respectively. More importantly, it was observed that IN varied linearly between Iel and Ipl, thus leading to Eq. (36):

  Ry  Ipl  Iel Ru

 IN ¼ Iel þ

(36)

Fig. 7 illustrates that I/Imin varies non-linearly from the coordinates:

ðyel =ym ; I=Imin Þ ¼ ð0; IN Þ

(37)

To the coordinates:



ðyel =ym ; I=Imin Þ ¼ Ry =Ru ; Iel

(38)

Therefore, a new term is introduced, I, as seen in Eq. (39):

I ¼

IN  I=Imin IN  Iel

(39)

This term, I, can be plotted against yel/ym to produce Fig. 8: It can be seen from Fig. 8 that the term, I, is only a function of yel/ ym and Ry/Ru. Therefore, a function, V, which correlated the most with the plots of I in Fig. 8 was obtained and can be seen in Eq. (40):

V ¼

tan1

sin1 ðyel =ym Þ hrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii

(40)

1 ðRu =Ry þ1Þðym =yel 1Þ

As this function correlates well with the plots in Fig. 8, it can be said that V ¼ I. Therefore, Eqs. (36), (39) and (40) can all be combined and re-arranged to provide Eq. (41):

 I=Imin ¼ Iel þ

  Ry  Ipl  Iel ð1  VÞ Ru

Fig. 7. Plot of I/Imin against yel/ym and Ry/Ru.

(41)

J. Dragos, C. Wu / International Journal of Impact Engineering 62 (2013) 1e12

9

The term, kep, ranges from 0.89, for a rectangular effective pulse load, to 1.0, for a triangular effective pulse load. With the two new functions introduced, Si and Zi, the modified equation for determining I/Imin is as follows:

 I=Imin ¼ Iel þ

Fig. 8. Plot of I against yel/ym and Ry/Ru.

It should be noted that Eq. (41) only applies to the case in which a member is subjected to a triangular effective pulse load. Therefore, the database of I/Imin values for other various effective pulse loads was used to compare against that obtained from Eq. (41). It was observed that, although Iel and Ipl were known for each effective pulse load, Eq. (41) was not accurate enough at determining I/Imin between these two extremes for different effective pulse loads. Therefore, some modifications to this equation had to be made to rectify this difference. It was found that, for some effective pulse loads with large values of Ieff,unit, linear interpolation was not accurate enough for determining IN. Therefore, to accommodate this scenario, a modification to Eq. (36) was made by introducing a new term, Si. This new term, Si seen in Eq. (42), was empirically determined using the database and is a function of only the effective pulse shape parameters.

  Ieff;unit Cx þ 0:97 Si ¼ 0:039 0:5  Cx

(42)

From Eq. (42), it can be seen that for effective pulse loads corresponding to the impulse controlled region of the PI curve, Si is slightly less than 1. Conversely, for effective pulse loads approaching the quasi-static region of the PI curve, Si is slightly greater than 1. For a triangular effective pulse load, Si ¼ 1. The term, Si, is introduced into Eq. (36) to produce Eq. (43):



IN

Ry ¼ Iel þ Ru

S i 

Ipl  Iel



(43)

The function V, seen in Eq. (40), correlates quite well with the plots of I against yel/ym, seen in Fig. 8, corresponding to that of a triangular pulse load only. However, for other effective pulse load shapes, the function V alone is not sufficient for predicting I. Therefore, an extra function, Zi, was introduced. This term is to be multiplied by V to alter the shape it takes between the extremes (yel/ym ¼ 0 and yel/ym ¼ Ry/Ru) but has no effect on its value at the extremes and is described by Eq. (44):



  kep þ 1  kep yymel Zi ¼

R  kep þ 1  kep Ruy

Ry Ru

S i 

 Ipl  Iel ð1  VZi Þ

(46)

In Eq. (46), Iel and Ipl, are determined using the equations provided in Sections 4.2 and 4.3, respectively. Then, all the other terms describe the influence of the shape of the RD function and effective pulse load on I/Imin. Eq. (46) is not completely accurate, but the errors between I/Imin obtained using the equation and that from the database are typically less than 3%. It should be noted that Ipl, Iel and I/Imin, within Eq. (46), can represent either the effective impulse or the actual impulse, although not a combination of both, as they all correspond to the same effective pulse load. However, Si and kep are functions of the effective pulse shape parameters. 5.2. Pr/Prmin for bilinear SDOF systems As was done with I/Imin, an investigation into the effects of the RD function shape on Pr/Prmin for a triangular effective pulse load is undertaken first. For this case, Cx ¼ Cy ¼ 1/3 and Ieff,unit ¼ 1/2. As was done previously, the database is used to plot Pr/Prmin against yel/ym for various Ry/Ru values, as seen in Fig. 9. Firstly, PrN, which corresponds to the Pr/Prmin value of a member with no elastic region (yel/ym ¼ 0), should be introduced. It can be seen in Fig. 9, that for the case corresponding to a triangular effective pulse load, Prpl ¼ Prel ¼ PrN ¼ 2. It can also be seen that, between the extremes, Pr/Prmin is always less than Prel and PrN by a small amount which is typically less than 7% of Prel and PrN. Therefore, what can now be plotted is Prel  Pr/Prmin against yel/ym for various Ry/Ru, as seen in Fig. 10. It can be seen that the plot in Fig. 10 is only a function of yel/ym and Ry/Ru. Therefore, a function, YPr, was then empirically determined which correlates quite well with the plot in Fig. 10, and can be seen in Eq. (47):

YPr ¼

V



Ry Ru

 yymel

DPr

 (47)

where V can be calculated using Eq. (40) and DPr is approximately 3.7 for the case of a triangular effective pulse load. For other effective pulse load shapes, it was observed that this value changes. From Eq. (47), it can be seen that for the elastic case in which yel/ ym ¼ Ry/Ru, Pr/Prmin ¼ Prel. Also, for the rigid plastic case in which yel/ ym ¼ 0 and Ry/Ru ¼ 0, Pr/Prmin ¼ Prel ¼ Prpl.

(44)

where the term, kep, is a function of only the effective pulse shape parameters and is described by Eq. (45):

kep ¼

2 Ieff;unit 9 Cx Cy

(45) Fig. 9. Plot of Pr/Prmin against yel/ym and Ry/Ru for triangular effective pulse load.

10

J. Dragos, C. Wu / International Journal of Impact Engineering 62 (2013) 1e12

Fig. 12. Plot of Prel  Pr/Prmin against yel/ym and Ry/Ru for bilinear effective pulse load. Fig. 10. Plot of Prel  Pr/Prmin against yel/ym and Ry/Ru for triangular effective pulse load.

As Eq. (47) was derived based on a member subjected to a triangular pulse load, in which Prel ¼ Prpl ¼ PrN ¼ 2, the equation is not extensive enough to be applied to all effective pulse loads. For other effective pulse loads, it was observed that Prel was not necessarily equal to Prpl. Therefore, Eq. (47) has to be extended to incorporate this and to also incorporate varying PrN values. To accomplish this, an investigation into the effects of RD function shape on Pr/Prmin for a bilinear effective pulse load corresponding to r ¼ 0.2 & s ¼ 0.2 is undertaken. For this effective pulse load, Pr/Prmin can be plotted against yel/ym, for various Ry/Ru, as seen in Fig. 11. Fig. 11 illustrates that Prel and Prpl are not always equal. It also illustrates that, although Prel forms the upper bound, Prpl does not form the lower bound for Pr/Prmin. Fig. 11 also illustrates PrN, corresponding to yel/ym ¼ 0, for various Ry/Ru values. It was observed that for this effective pulse load, and for all others, linear interpolation can be used to determine PrN, as seen in Eq. (48):

  y =ym Lðyel =ym Þ ¼ ðPrel  PrN Þ 1  el Ry =Ru

(49)

The purpose of this linear function is, for each Ry/Ru value, to plot the difference between the plot in Fig. 12 and its corresponding linear function, L(yel/ym). This is being done in an attempt to obtain similar plots to that in Fig. 10. This new plot is the function, WPr, and can be described by Eq. (50):

WPr ¼ ½Prel  Pr =Prmin   Lðyel =ym Þ

(50)

As was done for the triangular effective pulse load case, a plot of Prel  Pr/Prmin against yel/ym for various Ry/Ru values can be constructed, as seen in Fig. 12. It can be seen that, although both Figs. 10 and 12 are plotting Prel ¼ Pr/Prmin, they do not correlate at all. This is because in Fig. 10, when yel/ym ¼ 0, Prel  Pr/Prmin ¼ 0. However, in Fig. 12 when yel/ ym ¼ 0, Prel  Pr/Prmin s 0. Therefore, Eq. (47) cannot be applied to the plots in Fig. 12 and more manipulation of these plots is required. For the case in which Ry/Ru ¼ 0.5, Fig. 12 illustrates the point in which Pr/Prmin ¼ PrN. It also illustrates a linear function, L(yel/ym), which ranges from the coordinates (0, Prel  PrN) to (Ry/Ru, 0). Therefore, for any Ry/Ru value the linear function, L(yel/ym), can be described by Eq. (49):

where the first bracketed function in Eq. (50) represents the plots in Fig. 13, and the second term is the linear function, L(yel/ym). Now that the function, WPr, has been defined, it can be plotted against yel/ym, as seen in Fig. 13. The correlation between Figs. 10 and 13 is quite noticeable. This is because, for both plots, the functions are equal to 0 at the extremes (yel/ym ¼ 0 and yel/ym ¼ Ry/Ru). It should be noted that the function, WPr seen in Eq. (50), is also being plotted in Fig. 10. However, this plot is for the special case in which L(yel/ym) ¼ 0. It can be seen that the difference between the shapes of the plots in Figs. 10 and 13 is quite large between the extremes. This is because the plots of WPr are influenced by the effective pulse load being considered. This is why the function YPr, seen in Eq. (47), cannot be used to predict WPr for all effective pulse loads, only the case of the triangular effective pulse load. Therefore, to use Eq. (47) to predict WPr, it should be extended to account for the influence of different effective pulse loads. This was achieved by introducing two additional terms which are functions of the effective pulse shape parameters. Eq. (47) contains a term, DPr, which for the case of a triangular effective pulse load, is equal to 3.7. After testing of this value for other effective pulse loads, using the database, an expression for DPr was empirically determined and can be seen in Eq. (51):

Fig. 11. Plot of Pr/Prmin against yel/ym and Ry/Ru for bilinear effective pulse load.

Fig. 13. Plot of WPr against yel/ym and Ry/Ru for bilinear effective pulse load.

 PrN ¼ Prel þ

  Ry  Prpl  Prel Ru

(48)

J. Dragos, C. Wu / International Journal of Impact Engineering 62 (2013) 1e12

DPr

  Ieff;unit Cx þ 1:95 ¼ 1:5 0:5  Cx

(51)

Also, a new function was introduced, ZPr, which is to be multiplied by the term, DPr. For a triangular effective pulse load, ZPr ¼ 1, and thus does not exist in Eq. (51). After further testing against the database, it was deemed necessary to add this extra term to modify the shape of WPr, between the extremes, based on the effective pulse load. Therefore, an expression for ZPr was also empirically determined and can be seen in Eq. (52):

ZPr

   k8ep þ 1  k8ep yymel   ¼ k8ep þ 1  k8ep ð0:1Þ

(52)

where kep is purely a function of the effective pulse shape parameters and can be calculated using Eq. (45). Expressions for DPr and ZPr can then be inserted into Eq. (47) to determine the function YPr, seen in Eq. (53):

YPr ¼

V



Ry Ru

 yymel

 (53)

DPr $ZPr

As the function YPr can be used to predict the plots of WPr for all effective pulse loads, such as in Figs. 10 and 13, YPr and WPr can be considered equal. Therefore, Eqs. (50) and (53) can be combined, together with the expression for PrN in Eq. (48), to determine the overall expression for Pr/Prmin, seen in Eq. (54):

Pr =Prmin

 ¼ Prel þ Prpl  Prel 

V DPr $ZPr



Ry yel  Ru ym



(54)

In Eq. (54), Prel and Prpl, are determined using the equations provided in Sections 4.2 and 4.3, respectively. Then, all the other terms describe the influence of the shape of the RD function and effective pulse load on Pr/Prmin. Eq. (54) is not completely accurate, but after testing against the database, it was found that the errors that lie between Pr/Prmin determined using the equation and that from the database are typically within 2%. 6. Technique for constructing a normalised PI curve For a given sr value, the expressions derived in Sections 4 and 5 can be used to determine the coordinates of a single point on the normalised PI curve for a member with a bilinear RD function subjected to an arbitrary pulse load. The steps showing the parameters which need to be determined are outlined below: 1. Establish sr value 2. Determine Ieff,unit, Cx and Cy (effective pulse shape parameters, Section 4.1) 3. Determine Iel & Prel (point on elastic normalised PI curve, Section 4.2) a. Determine ael / tmax b. Determine Prel c. Determine td / Iel 4. Determine Ipl & Prpl (point on rigid plastic normalised PI curve, Section 4.3) a. Determine tmax b. Determine Prpl c. Determine td / Ipl 5. Determine I/Imin & Pr/Prmin (point on bilinear RD normalised PI curve, Section 5) a. Determine terms: kep, Zi, Si, DPr, ZPr & V b. Determine I/Imin c. Determine Pr/Prmin

11

Although, the steps provided only describe the process for determining a single point on the normalised PI curve, the process can be repeated for many chosen sr values, ranging from infinity down to 0. This produces many points, effectively forming the normalised PI curve. Therefore, it is more appropriate to use a spreadsheet application to calculate entire normalised PI curves. The main difficulty in this, is that for every iteration some form of integration needs to be used to determine the effective pulse shape parameters, Ieff,unit, Cx and Cy. This is because for every sr value, the effective pulse shape changes. If the function of the pulse load is defined, and easily integrated, it may be easy to setup equations by hand to determine Ieff,unit, Cx and Cy for every sr value. However, if this is not the case, some other method should be used. As this process is most suitable to a spreadsheet application, a numerical technique for determining Ieff,unit, Cx & Cy, for each sr value has been developed. It can be easily implemented into a spreadsheet application and the method only requires that the unit pulse load, Funit(t), can be typed into the spreadsheet application as a function. To implement this numerical technique, first sr,i needs to be calculated from 0 to m, where m represents the number of points along the normalised PI curve to be calculated. sr,0 should be set to equal infinity, sr,1 should be set to equal some large number, say 200. Then, the following equation can be used to calculate all subsequent sr values:

sr;i ¼ Asr;i1

(55)

where the term, A, represents the resolution of the points along the normalised PI curve. Setting A ¼ 0.97 is an appropriate value for it to take. Then, for each sr,i value, Eq. (13) should be used to calculate Funit(1/sr,i), which is labelled Funit,i for simplicity. According to Eqs. (15)e(17), integration needs to be used to calculate Ieff,unit, Cx and Cy, respectively, for each sr value. However, a numerical technique has been constructed to determine Ieff,unit,n, Cx,n & Cy,n for sr,n based on Ieff,unit,n1, Cx,n1 & Cy,n1, respectively. Applying the trapezoidal rule to Eq. (15), an equation for Ieff,unit,n corresponding to sr,n can be determined:

Ieff;unit;n

" !# n X

1 1 1 ¼ sr;n þ Funit;i1  F sr;i sr;i1 2 unit;i i¼1

(56)

Then, Eq. (56) can be expanded such that the nth slice is separated from all other slices to provide the following:

(

Ieff;unit;n

n1 X

"

1 1 1 ¼ sr;n  Funit;i þ Funit;i1 s s 2 r;i r;i1 i¼1 ) 

1 1 1 þ Funit;n þ Funit;n1  sr;n sr;n1 2

!#

(57)

Then, as: n1 X

"

i¼1

1 1 1 þ Funit;i1  F sr;i sr;i1 2 unit;i

!# ¼

Ieff;unit;n1

sr;n1

(58)

Therefore, by substituting Eq. (58) into Eq. (57) leads to the following:

Ieff;unit;n ¼ sr;n

 Ieff;unit;n1

sr;n1

þ

 

1 1 1  Funit;n þ Funit;n1 sr;n sr;n1 2 (59)

The same technique, of separating out the nth slice from all previous slices as used above, can be applied to Eqs. (16) and (17) to determine Cx,n and Cy,n corresponding to sr,n:

12

Cx;n ¼

J. Dragos, C. Wu / International Journal of Impact Engineering 62 (2013) 1e12

sr;n2

( Cx;n1 Ieff;unit;n1

Ieff;unit;n  

s2r;n1 1

sr;n



1

  1 Funit;n Funit;n1 þ sr;n sr;n1 2 )

References

þ

sr;n1 (60)

Cy;n ¼

sr;n

 Cy;n1 Ieff;unit;n1

Ieff;unit;n

 

sr;n1 1

sr;n



1

 1 Funit;n2 þ Funit;n12 4  þ

sr;n1 (61)

Eqs. (59)e(61) are all expressed in such a way for convenience so that only these equations need to be input into the cells of a spreadsheet to calculate Ieff,unit, Cx and Cy for each sr value. In terms of the overall outline provided earlier, Eq. (55) is used in step 1 and Eqs. (59)e(61) are used in step 2. Finally, Sections 4.2, 4.3 and 5 are used for steps 3, 4 and 5, respectively. These steps are repeated from sr,0 to sr,m to determine m points along the normalised PI curve. 7. Conclusion A new concept, the effective pulse load, was introduced. Then, using the SDOF model, an entire database of points on the normalised PI diagram for elastic, rigid plastic and elastic plastic hardening structural members corresponding to various effective pulse load shapes was determined. Then, three parameters which define the shape of the effective pulse load were derived. Using the database, the influence of effective pulse load shape on the coordinates of a point on the normalised PI curve for both elastic and rigid plastic members was studied. Furthermore, the database was then used to determine the influence of both the shape of the effective pulse load and the shape of the bilinear RD curve on the coordinates of a point on the normalised PI curve. The general procedure to determine a given point on the normalised PI curve for an arbitrary effective pulse shape and RD function shape was outlined. A method for iterating through this procedure to determine many points, forming an entire normalised PI curve, was then provided in order for it to be implemented into a spreadsheet application. Although the relationships were developed based on various bilinear pulse loads, as the equations are a function of the effective pulse shape parameters, they can be used to determine normalised PI curves for any pulse load shape.

[1] Abou-Zeid BM, El-Dakhakhni WW, Razaqpur AG, Foo S. Modeling the nonlinear behavior of concrete masonry walls retrofitted with steel studs under blast loading. Journal of Performance of Constructed Facilities 2011;25(5):411e21. [2] Fallah AS, Louca LA. Pressure-impulse diagrams for elastic-plastic-hardening and softening single-degree-of-freedom models subjects to blast loading. International Journal of Impact Engineering 2006;34(4):823e42. [3] Jones J, Wu C, Oehlers DJ, Whittaker AS, Marks S, Coppola R. Finite difference analysis of RC panels for blast effects. Engineering Structures 2009;31(12): 2825e32. [4] Crocker MJ, Hudson RR. Structural response to sonic booms. Journal of Sound and Vibration 1969;9(3):454e68. [5] Florek JR, Benaroya H. Pulse-pressure loading effects on aviation and general aircraft structures-review. Journal of Sound and Vibration 2004;384(1e2): 421e53. [6] Design of blast resistant buildings in petrochemical facilities. Reston, VA: American Society of Civil Engineers (ASCE); 1997. [7] Ballot version 2Blast protection of buildings. Reston, VA: American Society of Civil Engineers (ASCE); 2008. [8] UFC-3-340-02. Structures to resist the effect of accidental explosions. Washington, DC: US Department of the Army, Navy and the Air Force; 2008. [9] Mays GC, Smith PD. Blast effects on buildings e design of buildings to optimize resistance to blast loading. London: Thomas Telford; 1995. [10] El-Dakhakhni WW, Mekky WF, Rezaei SHC. Validity of SDOF models for analyzing two-way reinforced concrete panels under blast loading. Journal of Performance of Constructed Facilities 2010;24(4):311e25. [11] Fischer K, Haring I. SDOF response model parameters from dynamic blast loading experiments. Engineering Structures 2009;31(8):1677e86. [12] Ma GW, Shi HJ, Shu DW. PeI diagram method for combined failure modes of rigid-plastic beams. International Journal of Impact Engineering 2007;34(5): 1081e94. [13] Abrahamson GR, Lindberg HE. Peak load e impulse characterization of critical pulse loads in structural dynamics. Nuclear Engineering and Design 1976;34(1):35e46. [14] Li QM, Jones N. Foundation of correlation parameters for eliminating pulse shape effects on dynamics plastic response of structures. Journal of Applied Mechanics 2005;72(2):172e6. [15] Krauthammer T, Astarlioglu S, Blasko J, Soh TB, Ng PH. Pressure-impulse diagrams for the behavior assessment of structural components. International Journal of Impact Engineering 2007;35(8):771e83. [16] Li QM, Meng H. Pressure-impulse diagram for blast loads based on dimensional analysis and single-degree-of-freedom model. Journal of Engineering Mechanics 2002a;128(1):87e92. [17] Campidelli M, Viola E. An analytical-numerical method to analyze single degree of freedom models under airblast loading. Journal of Sound and Vibration 2007;302(1e2):260e86. [18] Dragos J, Wu C, Oehlers DJ. Simplification of fully confined blasts for structural response analysis. Engineering Structures 2013;56(2013):312e26. [19] Youngdahl CK. Correlation parameters for eliminating the effect of pulse shape on dynamic plastic deformation. ASME, Journal of Applied Mechanics 1970;37(3):744e52. [20] Li QM, Meng H. Pulse loading shape effects on pressure-impulse diagram of an elastic-plastic, single-degree-of-freedom structural model. Journal of Engineering Mechanics 2002b;44(9):1985e98. [21] Huang X, Ma GW, Li JC. Damage assessment of reinforced concrete structural elements subjected to blast loads. International Journal of Protective Structures 2010;1(1):21e43.