Reliability-based design methods for protective structures

Structural safety ELSEVIER

StructuralSafety15 (1994)17-33

Reliability-based design methods for protective structures * L.A. Twisdale *, R.H. Sues, F.M. Lavelle Applied Research Associates, Inc., 6404 Falls of Neuse Road, Suite 200, Raleigh, NC 27615, USA

Abstract This paper summarizes the results of research to develop reliability-based design methods for use with the United States Air Force Protective Construction Design Manual (PCDM). The application of these methods is based on easy-to-use, reliability-based design factors (RBDFs). The research focused on five fundamental areas: (1) airblast and aboveground structure response; (2) groundshock and belowground structure response; (3) fragmentation effects; (4) projectile penetration; and (5) protective structure systems reliability. The scope of the structural response research was limited to reinforced concrete structures; however, the load uncertainty analyses are generally applicable to protective structures of any material. The research also uncovered several areas of significant bias in the analysis methods, which could not be corrected without changes to the fundamental approach in the PCDM. RBDFs were not developed for these cases; however, either specific ways to improve the model prediction accuracy were recommended or improved models were provided and demonstrated. Key words: Reliability-based design; Protective structures; Weapon effects; Structural response

1. Introduction Protective structures are hardened structures that are designed to protect assets for war fighting capabilities. The design of protective structures for non-nuclear weapons involves unique loadings that challenge the designer to develop concepts that are both survivable and economical. The design threats include weapon effects from standoff bursts as well as penetrating weapons and contact detonations. The loadings produced by these threats are high-intensity transients; hence, nonlinear response is fundamental to the protective concept and the design performance criteria. The traditional methods for protective structural design are based primarily on deterministic threats, analysis procedures, and design performance criteria. Uncertainties, while recognized as being important motivators for research and

* Discussion is open until March 1995 (please submit your discussion paper to the Editor, Ross B. Corotis). * Corresponding author. 0167-4730/94/$07.00Q 1994ElsevierScienceB.V. All rights reserved SSDZ0167-4730(94)00008-E

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Safety I5 (1994) 17-33

experimental investigations, have not traditionally been treated in protective design and survivability analyses for non-nuclear weapon effects. Many developments in conventional weapons effects and protective design ha&their origin in the National Defense Research Committee (NDRC) program in the 1940s [ll. For example, development of cube-root scaling procedures for weapons effects, analytic calculations of ’ airblast from bare charges, use of semianalytic penetration equations, and the. pioneering : development of groundshock from buried bursts in different soils are but a few of the studies ~ produced by the NDRC project. The Army manual TM 5-855-1, Fundamentals of Protective Design for Conventional Weapons, published in 1949 and periodically updated [2], was the first attempt to synthesize these data into a single document with results of extensive structural damage produced by World War II. From about 1950 through the early 197Os,much of the research in protective structures was devoted to nuclear weapons. Improvements were made in numerical modeling of weapons effects, dynamic response of aboveground and shallow-buried structures, airblast, blast-induced ground shock, cratering and ejecta, and dynamic response of earth materials. Some of this technology was incorporated into Protection from Non-Nuclear Weapons [3]. However, many of the approaches for nuclear effects used plane wave loading approximations, which produced significant errors when applied to much smaller explosions characterized by 3-D loading waveforms. The publication of the Air Force PCDM in 1989 [4] included several important advances in non-nuclear effects and protective design, including: (1) penetration of weapons into soils, concrete, and rock rubble; (2) groundshock beneath contact explosions on and within burster layers and from weapons that explode in backfill materials; (3) structure-medium interaction; (4) definition of short duration airblast penetration into facilities from external explosions; and (5) shock isolation of internal equipment. In addition, probabilistic design concepts were introduced for the first time in a non-nuclear protective structures design document. However, these concepts were introductory and limited to a few example problems. Following the completion of the PCDM, a research effort [5] was funded to investigate probability-based design methods for protective structures for non-nuclear weapons. The scope was limited to the analysis and design methods in the PCDM, which are based largely on simplified models and single degree of freedom response. The approach adopted was based on probability-based limit state design, which is particularly relevant to protective construction where the accepted design philosophy has long embraced the concept of limit states for both nuclear and non-nuclear weapon effects. Nonlinear response and local damage are acceptable to some degree, provided that structural integrity is maintained and critical assets are protected. Design uncertainties are large due to the extreme load effects and the nonlinear dynamic response behavior of protective structures under these loads. The final products of this effort centered on easy-to-use RBDFs and the basic design equation for acceptable performance $-C>A-E

(1)

where I) = RBDF on nominal capacity C and A = RBDF on nominal load effect E. In some cases, a load factor of unity was appropriate, whereas in others the load and capacity factors were combined into a single factor. In general, a table of t,l~and A values were developed

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Safety IS (1994) 17-33

19

corresponding to different levels of design survivability, P,. A value of P, is selected based on the limit state (functional or ultimate) and the mission of the facility. This paper summarizes the results of this research effort [5], which is currently being published as an addendum to the PCDM [6]. Since this is the first effort aimed at developing probabilistic methods for protective structural design, the focus of the RBD work was at a fairly basic level, i.e., data collection and analysis, uncertainty quantification, and probabilistic analysis methods for individual failure modes. Following additional background on protective structures, we review the basic approach for developing the RBDFs, summarize the specific areas where design factors were developed, and identify gaps and biases in the PCDM procedures.

2. Background Notwithstanding the emphasis on mobility in modern warfare, fixed position hardened structures have an important role in enhancing the survivability of key assets and command and control functions. For example, modern airbases in regions of potential conflict include a variety of protective structures, both aboveground and belowground. In many respects, modern protective structural concepts for non-nuclear weapons differ little from classical military fortifications with massive walls and underground bunkers. The reason for this is that both mass and strength are required to decelerate and defeat heavy projectiles. In-situ materials provide an economical source of hardening, and burying or berming a structure is still an often-used military option. The recent conflict in Iraq illustrates the effectiveness of deeply buried and hardened structures even with a limited air defense system. For threats from standoff explosions, small arms, and light weapons, aboveground structures can be protected, albeit with heavy materials such as reinforced concrete and steel. Lightweight protection provided by advanced composites with high-strength fibers remains too expensive for large-scale use in structures and is practical only for mobile vehicles and personnel body armor, subjected to a much more limited threat spectrum. The protective structural design process is basically identical to the civil structural design process, following the traditional design iteration steps: (1) develop a design concept; (2) analyze for each failure mode; (3) evaluate acceptability of predicted response; and (4) modify design and repeat analysis until all criteria are satisfied. However, s,everal differences exist in protective design. One is that the loads are computed for a specified list of threats, which are determined separately from a complex analysis of enemy capability, doctrine, active defense, etc. Since the threat analysis is decoupled from the protective structural analysis, the predicted structure survivability (reliability) is conditional to the threats explicitly considered. Hence, the reliability of a protective structure may change dramatically if the threat changes during its design lifetime (such as direct hits from precision-guided munitions) or if the original design basis threat was in error. For each threat (often specified by weapon type, impact conditions, or standoff distance), the weapon effects include airblast, fragment impulse and penetration, projectile penetration, groundshock, cratering, and ejecta. A brief summary of these effects, structural elements, failure modes, and protected components is given in Fig. 1.

LA. Thisdale et al. /Structural Safety IS (1994) 17-33

20

Overlay

m ~~~~~~ g&&g&

1,2,3,6 Walls, etc.

*a

1. Arch

4. Doors

10. Rock Rubble ovalay

5. Soil Berm

a

Fig. 1. Interac ion between weapon effects, strl ctural elements, failure modes, and protected systems.

A key objective of protective design is to absorb the weapons effects at the envelope of structure. If the weapon is allowed to penetrate the structure, then the effects are generally lethal to both personnel and equipment. Hence, control against local failures (penetration, spall, breaching, etc.) is central to protective design. Often, a layered system consisting of a burster slab (see Fig. 2), a rock rubble overlay, or a special geometry deflection grid, is used to deflect the trajectory and/or force the bomb to detonate at a given standoff from the structure.

Fig. 2. buried command and control center.

LA. Twisdale et al. /Structural Safety 15 (1994) 17-33

21

The structure is then designed to survive the groundshock loading from a detonation equal to the standoff distance from the last burster slab. Multiple hits producing cumulative damage in the same local area are generally not considered in design. For high water tables and/or pile foundations, liquefaction can also be an important failure mode. Support systems must be designed to resist severe instructure shock environments. Large facilities are often compartmentalized so that if a weapon does penetrate the structure, the propagation of the explosive effects is limited. For the most part, contemporary protective designs consist of reinforced concrete elements with simple geometries, such as boxes, shells, slabs, and arches. Due to the large and inherent uncertainties in the threat specification and the basic role of passive protection in wartime, the design reliability goal of protective structures is much less than most civil structures, which typically have reliabilities greater than about 1 - 10w4. Based on the conservatisms of past design practices and our current level of understanding of the basic phenomenology, a practical and achievable design reliability goal for most protective structures is on the order of 0.9 to 0.99. In some cases, initial cost considerations may require nominal designs with lower reliabilities. These design reliabilities represent conditional probabilities of survival for the specified threat. A final comment is in order regarding a major effort that is underway to develop a joint military service handbook, accompanied by an electronic version that can also launch supporting analysis codes [7]. Reliability-based design methods are being incorporated into portions of this handbook.

3. Approach for developing RBD factors For each design method that we analyzed from the PCDM, the approach for developing RBD procedures followed four basic steps. These steps are summarized in Fig. 3 as: (1) identify failure modes and limit states; (2) characterize parameter uncertainties; (3) perform the probabilistic response analyses; and (4) perform the reliability assessment. The nominal parameters and models are the prediction equation in the PCDM for each load/response failure mode considered. The experimental data provided the linkage between the nominal procedures and the RBD development process. We used the prediction error analysis method [8] extensively, forming nondimensional statistics that we analyzed in detail to characterize bias and randomness. Fig. 3 also illustrates that, in addition to RBDFs, we developed rank orders of the dominant uncertainties where possible. This systematic, step-by-step process led to one of several results in terms of how well the nominal PCDM design procedures performed. If the systematic errors (biases) were acceptably small and the data were sufficient, RBDFs were developed for use with the nominal PCDM procedures. If we found marginally acceptable errors and the data were sufficient to update the model parameters, we produced RBDFs for the PCDM procedures, using updated parameters. In most of these cases, we qualified the updates to the range of relevant data available. For casesin which the predictions did not agree with the experimental results, we determined if the data and our understanding of the issues and research resources were sufficient to update the PCDM model. If so, we provided an updated or new model, stated necessary limitations, and

22

LA. lbisdale et al. /Stnrctwal

PCDM Nominal Parameters

l

l l

l l

Flexure Spall Shear Penetration Components . .

LoadInputs Capacity Inputs * Materials Systematic Errors Random Uncertainties . .

l

l

_--e-v-

_----

PCDM Nominal Predictions I 7.

PCDM Nominal

+

I

l

l

Safety IS (1994) 17-33

l

. FORM/SORM/FpI Monte Carlo Sensitivity Analysis Dichotomous Regression . . . l l l

_---

-

4, Per$orm Reliability Assessment ___I______--__----__----Model Robustness Parameter Updates Error Analysis . l

7 l

l

__---A

Fig. 3. The four steps of the RBD development process.

gave factors for use with it. If not, and if the PCDM methods were conservative, we provided qualitative discussion and recommended that no additional factors be used.

4. Summary of RBDFs A brief summary of the developed RBD methods vis a vis the PCDM method is given in Table 1 for each conventional weapon effects phenomena considered. Due to space limitations, the resulting RBDFs are given only for 50% and 90% reliability levels. The following subsections highlight the key results. The RBD development processes for two of the effects in Table 1 (free-field airblast and spa11of aboveground structures) are also briefly described. 4.1. Airblast effects

The RBD analyses for airblast effects considered both incident (free-field) and reflected airblast. 4.1.1. Incident airblast

The prediction methodology in the PCDM for incident airblast pressure and impulse are based on the airblast parameter polynomials for uncased TNT hemispherical surface bursts of

LA. lIvisdalc et al. /Structural Safety IS (1994) 17-33

Table 1 Summary of RBD methods for the PCDM PCDM method Phenomena

23

Comments

RBDFs P, = 0.50 P, = 0.90

1. Incident airblust (i) Pressure-surf. tang. (ii) Impulse-surf. tang. (iii) Pressure-half-buried (iv) Impulse-half-buried

- Median centered Slightly unconservative Slightly unconservative Slightly unconservative

0.96 1.12 1.10 1.16

1.40 1.53 1.60 1.60

Based on CHEBS data; Range independent; separate factors for surface target and half-buried bombs.

2. Reflected airblast

1.00 (i) Normal pressure and Assumed unbiased 1.47 impulse (ii) Non-normal pressure Significantly overpredicts Not developed and impulse 3. Aboveground wallflexurul response

Significantly over-predicts Not developed

Factor of four overprediction when used with the PCDM airblast. Closed form expression developed for design reliability.

4. Breaching of aboueground Median centered wall

1.00

5. Spa11of aboveground wall Conservative

New model developed 1.00 1.91

The new model reduces both systematic and random uncertainties.

A: B: A: B:

Did not attempt to correct bias. Site-specific RBDFs can be developed. Two generic RBDFs given for sites A and B, where A = controlled backfill and B = only limited site information available.

6. Gr&nd

1.21

Preliminary; based on assumption that PCDM is unbiased; combined ideal and nonideal effects assuming independence.

shock

(i) Free-field velocity

Conservative for most soil types

(ii) Free-field stress

7. Belowground wallflexural response

1.00 1.00 1.00 1.00

1.87 2.00 1.97 2.55

Significantly overpredicts Not developed

8. Breaching of belowground Slightly conservative wall

0.95

-

Much better results were obtained for 2DOF model with improved resistance function. PCDM model valid only for scaled range < 1.3 ft/lbl/3. RBDFs developed only for P, = 0.25, 0.50, and 0.75.

9. Fragmentation effects

(i) Fragment penetration Conservative or unconservative (ii) Fragment impulse Slightly unconservative

10. Penetration effects (i) Depth of penetration (ii) Spa11 (iii) Perforation (iv) Residual velocity

- Median centered Median centered Median centered Very conservative

RBD procedure developed 1.07 1.22

An RBD method was developed that takes into account weapon standoff and target geometry. Procedure is valid only for structures for which the PCDM equivalent uniform load method is valid.

1.03 1.00 1.00

RBDFs were also developed for striking velocity uncertainties for penetration, spall, perforation, and residual velocity.

0.78

1.23 1.42 1.38 1.00

24

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Safety 15 (1994) 17-33

Kingery and Bulmash [9]. The CHEBS test series data [lo, 111was used to evaluate the PCDM equations. The effective TNT weights used to predict the CHEBS observations were a ftmction of the type of explosive and bomb depth-of-burial. The CHEBS data groups have been formed by pooling together observations from tests having identical bomb orientations (e.g., horizontal surface tangent, vertical half-buried, etc.). Following screening and editing of the database, a total of 325 peak pressures and 320 positive impulses at scaled ranges varying from 0.5 to 2.6 m/kg1j3 (1.25 to 6.5 ft/lb1i3) were retained. Denoting the jth observation of pressure and impulse in each data group by pi and ij and the respective PCDM predictions by Pi and ii, the prediction errors sp and & for the jth observation is simply sp, = pi/pi

and

5ij = ii/ii.

(2)

The statistics of t,,j and tij allow us to quantify the systematic and random components of the PCDM airblast parameter prediction errors. The mean of 5 provides a measure of the systematic component of the prediction error and the higher moments (variance, etc.) indicate the random uncertainty. Plots of the prediction error ratios for each data group are shown in Fig. 4. Also shown on these plots are linear regression fits of the data in log-log space. Statistics of the observed prediction error ratios, as well as tests for systematic bias and lognormality, were analyzed and it was concluded that the prediction methodology is biased towards underpredicting incident impulse. Statistics for the log-log regression fits shown in Fig. 4 indicate that, in most cases, there is a significant range dependence; however, a relatively small percentage of the prediction error variability is “explained” by each regression line. Hence, a range independent model is used in the development of free-field airblast load factors. In order to have the largest data base possible and to simplify the development and implementation of reliability-based design procedures, it is advantageous to pool together the Groups I through IV data into combined data sets-one for peak pressure and one for positive impulse. However, lumping the surface tangent and half-buried data groups together may not be justified. Therefore, separate statistics for the surface tangent (Groups I and III) and half-buried (Groups II and IV) must also be considered. The process of pooling the individual data sets can be justified if it is determined that the difference between the groups is not statistically significant. An analysis of the group to group variability reveals that Group IV is significantly different from the other peak pressure data groups and Group II is significantly different from the other incident impulse data groups. These differences can most likely be attributed to the fact that Groups II and IV consist of data from half-buried bomb tests. The use of an effective TNT weight for half-buried bombs as one-half the weight of a similar surface-tangent bomb appears to be slightly unconservative. Having noted the modeling of half-buried explosives as a potential area for improvement, the pooled data Groups I through IV are accepted as being representative prediction error ratios. Analysis of Variance (ANOVA) was used to determine the final groups for the load factor development. The final ANOVA results given in Table 2 concern the pooling of different bomb orientation groups into a single, combined data set, labeled Groups I-IV. When all of the data sets are pooled together, ANOVA reveals that the group number is a significant classification variable. Further investigation revealed that Group IV is highly different from Groups I, II, and III for peak pressure, while Group II differs from the other positive impulse date groups. The

LA

Twisdale et al. /Structural

Safety IS (1994) 17-33

25

Unear Regresston

0.1

0.1 0.1

1.0 S@ed Range (m/kg h l/S) GROUP

+

+

+

I

x

x

x

Ii

*

10.0

l

.

iI1

0

0

0

Iv

(a) Peak Pressure 10.0 Unear RegressIon

1.0

x

0.1 1.0 Scaled Range (m/kg “l/Cl) GROUP

+

+

+

I

x

x

Y

Ii

.

.

100 *

111

0

(b) Positive Impulse Fig. 4. CHEBS prediction error ratio scattergram.

0

q

Iv

26

LA. nvisdale et al. /Structural

Table 2 Analvsis of variance results Group variable I (Vertical, nose tangent) II (Vertical, half-buried) IV (Horizontal, half-buried) I-IV (All orientations)

Safety 15 (1994) 17-33

Classification

Probability of a larger F-statistic Impulse Pressure

Test number Test number Test number Gage angle (67.-Y, 90”,112.5”) Group number Group IV vs. I, II, and III Group II vs. I, III, and IV Bomb type (Mk82 vs. Mk83)

0.8697 0.1291 0.0255 u 0.0006 b 0.0001b 0.0001b 0.4406

0.7182 0.4102 0.4051 0.0013 b 0.0625 0.0297 n 0.1039

* Significant variable (5 percent confidence level). b Highly significant variable (1 percent confidence level).

differences can most likely be attributed to the fact that Groups II and IV consist of data from the half-buried tests. The PCDM does not contain a procedure for scaling the explosive weight of partially buried bombs and one-half of the equivalent TNT weight was used to predict the airblast produced in the half-buried CHEBS tests. Since the mean prediction errors for the half-buried data are significantly larger than the prediction errors for the surface tangent data, we developed separate RBDFs for half-buried and surface tangent explosions. A lognormal model of the prediction errors was used with the parameters given in Table 3. The incident airblast load factors are computed by 5(R) = ew[h, + @-‘(R)q,] (3) where R is the desired reliability level, Cpis the standard unit normal function, and pin and a,, are the logarithmic mean and standard deviation, respectively. Airblast load factors computed using Eq. (3) are given in Table 4. The load factors are applied directly to the PCDM predictions for a given range and explosive weight. For example, multiplying the PCDM prediction for positive pressure by 1.55 produces a factored incident pressure that is exceeded by approximately 5 percent of the surface tangent CHEBS observations. 4.1.2. Reflected airblast

Comparisons of the reflected normal pressure to an independent data set indicate acceptable agreement, although the preliminary factors tend to understate the data scatter. No RBDFs Table 3 Incident airblast uncertainty model Model parameters Mean, p cov, 6 = u/p Logarithmic mean, pin Logarithmic standard deviation, u,”

Surface tangent Pressure, tp, 1.00 0.30 - 0.0431 0.2936

Impulse, 5i,, 1.15 0.25 0.1094 0.2462

Half-buried Pressure, tp, 1.15 0.30

Impulse, &, 1.20 0.25

0.0967 0.2936

0.1520 0.2462

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LA lhkdale et al. /Structural Safety IS (1994) X7-33

Table 4 Incident airblast reliability-based load factors Load factor Reliability Surface tangent 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.99

Half-buried

Pressure

Impulse

Pressure

0.59 0.66 0.79 0.96 1.17 1.40 1.55 1.90

0.74 0.81 0.94 1.12 1.32 1.53 1.67 1.98

0.68 0.76 0.90 1.10 1.34 1.60 1.79 2.18

Impulse 0.78 0.85 0.99 1.16 1.37 1.60 1.75 2.06

were developed for non-normally reflected airblast since the PCDM methodology overpredicts pressures and impulse by factors greater than three. See references [5] and [12] for the details. 4.2. Aboveground structure response

Three aboveground wall response phenomena are summarized in Table 1: flexural response, breaching, and spall. 4.2.1. Flexural response

The single degree-of-freedom (SDOF) response model in the PCDM for aboveground wall flexural response was found to be conservatively biased when compared to half-scale tests. The mean overpredictions in peak wall deflections range from a factor of two when the observed airblast loads are used to a factor of four when the one-way PCDM airblast loads are used. Sensitivity analyses identified the two largest sources of prediction error bias as reflected impulse (particularly for small standoffs) and maximum flexural capacity (particularly if significant membrance action is likely). 4.2.2. Breaching and spa11

An RBD model was developed for breaching of aboveground walls, based on dichotomous regression analysis of the PCDM equations, using data from McVay [13]. The RBD results indicate that the PCDM model was median centered and that for 90% reliability, the wall thickness had to be factored up by a factor of 1.21, as indicated in Table 1. For spall, the PCDM formula is overly conservative and a new, empirically derived spa11 model was developed for cased bombs. The new RBD spa11model is derived from the logistic form

ewP(x)l Pd(X)= 1 + ew[G(x)l

2s

LA. Twisdale et al. /Structural

Safety 15 (I 994) 17-33

where Pd is the probability of damage; x is the set of bomb and target variables; and G(x) is a linear combination of x with coefficients determined from the regression analysis. A Generalized G function was formulated based on the PCDM model as

G=Ao+A,ln(~)+~21n(~)~31~(~)

(5)

where t = wall thickness (feet), W= explosive weight (pounds TNT), C = casing weight, and R = burst standoff (ft). The new RBD spa11model was designed by setting G = 0 and solving for t,: &

= 0.20( $)-O.‘O(

A)-“’

where t, = design wall thickness (ft) required to defeat spall. The logistics fit yields

From Eqs. (4) and (7), the reliability design equation is

t = et,,

where $ = exp[-0.2933ln(+-l.)]

and R = probability of survival. Tabulated RBDFs, which are applied directly to the computed t, value from Eq. (6), are given in Table 5. For example, for 125 lbs effective charge weight, a standoff of 1.5 ft/lb1i3, the nominal wall thickness to prevent spall, is computed from Eq. (6) as 24 inches. For a reliability of 90% against spall, the wall thickness can be computed from Eq. (8) (or using Table 5) as 46 inches. With the new RBD developed spa11model, both the systematic and random uncertainties are reduced when compared to the PCDM spa11model. 4.3. Groundshock effects

The uncertainty in predicting ground shock is both site-specific and range dependent. A reliability-based model is developed using the PCDM prediction procedure coupled with Table 5 Reliability of wall thickness to prevent spa11from a standoff cased bomb Reliability P&) G(x) 0.05 0.10 0.25 0.50 0.75 0.90 0.95

0.95 0.90 0.75 0.50 0.25 0.10 0.05

2.94 2.20 1.10 0.00 - 1.10 - 2.20 - 2.94

RBDF t/t, 0.42 0.52 0.72 1.00 1.38 1.91 2.37

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Safety 15 (1994) 17-33

29

analysis of free-field velocity and stress measurements from the ground shock database. Generic RBDFs are also developed for scaled ranges of 1.0 to 3.0 ft/lb’13 for two cases that represent two different degrees of uncertainty in the site geologic properties: a controlled backfill and a site where only limited soil property information is available. The details of the groundshock RBD methodology are discussed in references [5] and [14]. 4.4. Belowground structure response

The structural model for the flexural response given in the PCDM treats the wall as an equivalent lumped mass SDOF spring system and accounts for effects of structure-medium interaction. Comparisons of the model to experimental results indicates that the PCDM procedure significantly overpredicts wall response (support rotation). Additional comparisons were made with improved resistance functions and a 2DOF model that accounts for motion of the reaction structure. The 2DOF model with improved resistance function had significantly reduced prediction error and a reduced coefficient of variation for the random error. No recommended RBDFs were developed due to the systematic errors in the PCDM model. However, the improved model was recommended for RBD, and we believe the 2DOF prediction error statistics are more indicative of full scale structures (even when the SDOF model is used). For breaching failures, the empirical PCDM model was found to be applicable only for scaled ranges less than 1.3 ft/lb ‘I3 . Due to the limited data, RBD factors were developed for only three reliability levels.: 0.25, 0.50, 0.75. 4.5. Fragmentation effects

Detonation of a cased explosive produces thousands of high-velocity primary fragments. Secondary fragments with lower velocities are also produced from the breakup of elements near the detonation. Analyses performed in Chapter 2 of the PCDM demonstrated that the fragment penetration survival probability depends significantly on weapon standoff and target geometry and that the confidence level selection procedure in the PCDM design fragment penetration methodology has no direct relationship with the actual structure design survival probability. A preliminary RBD framework for selecting a design primary fragment is demonstrated in [5]. A lethality extreme value distribution is derived and the hypergeometric distribution is used to compute the probability that no fragments with lethality greater than the design fragment will impact the structure. Additional research is underway to finalize the design methods for incorporation into the DAHS Handbook [7]. For fragment impulse effects, a table of velocity load factors was also developed. RBD procedures have not been developed for secondary fragments. 4.6. Penetration effects

The PCDM uses the NDRC [I] penetration formula for projectile penetration into concrete. We collected penetration data from several databases and performed prediction error analyses for each effect. The data included a lot of small caliber projectiles, impact velocities less than about 3000 ft/s, and concrete strengths ranging from 4000 to 6500 psi. We found the

30

L.A. Twtkdale et al. /.Stnrctural Safety IS (1994) 17-33

penetration depth equations to be approximately median centered when compared to 534 penetration tests into semi-infinite concrete. For 90% reliability, the wall thickness needs+ be factored up by 1.23 (see Table 1). Dichotomous regression formulations of the spa11and perforation models indicate that the PCDM methods are also essentially median centered, Factors of about 1.4 are required for 90% reliability. Analytic RBD models as well as RBDF tables were developed for both effects. The PCDM residual velocity model was compared to 45 test records and found to significantly overpredict actual residual velocities. In addition to these RBDFs, velocity load factors that account for uncertainty in striking velocity were developed for projectile penetration, perforation, and spall. These factors were developed using Monte Carlo simulations and tabulated as a function of design reliability and coefficients of variation of the projectile impact velocity. 5. Protective structure systems analysis Protective structures are designed to defeat multiple weapon effects from a wide range oi threats. In addition, because the loading, structural response analysis, and failure analysis will be subject to uncertainty, it is often not possible to identify a single critical failure mode. For example, it may not be possible to establish, with certainty, whether wall flexural failure from a standoff burst, breaching, or fragment perforation will dominate. The reliability levels for the RBDFs must be selected so that when the individual failure mode reliabilities are combined, the system reliability meets the structure or facility design criteria. Research has not been performed to enable a prescription of protective system design reliabilities for various structure/facility types and/or missions. Therefore, systems reliability analysis must be performed on a case-by-case basis by combining the results of the individual failure mode analyses. A simplified procedure for performing RBD for a protective system or facility is given in Fig. 5. The system reliability objective RF is established based on the mission objectives, considering facility functional performance. Failure modes are identified using fault trees. (See [5] for a complete set of fault trees for aboveground and belowground structures.) The system reliability model takes into account the failure logic for each performance level, generally reduced to a Boolean expression of AND and OR operators. As indicated in Fig. 5, reliability is allocated to each failure mode and RBD is performed for the controlling failure modes using the developed RBD factors. The final step involves a design check to r_ecomputethe system reliability, using the model developed in Step 3. Iteration is required if R
LA. Twisdale et al. /Structural Safety 15 (1994) 17-33 r-----I I

31

-----------------------, .,~ .:. ‘..

.:.\\

:-i.

>i:..:

i>>..:

. ...\.,

..,.:.

. . ,.

.... .. . .

:

2. Failure Mode Analysis _-_____-__-----_-___----------------------Structure: k Components: m Failure Mode: j

3. System Rdlabllity Model -_---_-____--_-----_---------------------------

ffk

4. Alto&e

No

Reliability to Failure Mode

6. Reliability DesIgnCheck _--____---_-__--------------

Finalize Design Details

Fig. 5. SystemsRBD procedure to achieve protective facility reliability objective RF.

gaps found in the PCDM models examined in this research. It is clear that the procedures in the current Air Force PCDM are not risk-consistent and that designs with various levels of survivability are produced using the nominal methods. The RBD process provides a focused procedure to identify, characterize, and reduce prediction errors in the analysis and design of protective structures. The resulting RBDFs are easy to use and provide the designer with rationally developed safety factors to produce risk consistent designs. The next step is to

32

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nvisdale et al. /Structural

Safety 15 (1994) 17-33

Table 6 Biases and methodology gaps in PCDM methodology Conservative Unconservative Airblast and l Non-normal reflection factors l Free-field impulse aboveground l Structural element resistance 0 Casing factor structure function response l One-way response model l Centerstrip analysis method l

Ground shock and belowground structure response

Fragmentation

Free-field velocity for strong and/or stiff materials close in 0 Structural element resistance function

l

l

Projectile penetration

Cylindrical model for fragment impulse

l

l

l

Gap l Partially buried bombs 0 Spa11equation l Scale model vs. full-scale behavior for breach and spa11

Fragment penetration formulas Use of standard fragment shape, normal collinear impact, zero rotational velocity Cylindrical spray pattern Penetration for modem highspeed projectiles (e.g., 30 mm A/C cannon)

l

l

l

Free-field velocity for weak and/or compressible materials close in Velocity waveform decay. for weak and/or compressible materials

Mean fragment velocity

l l

l

l

l

l

l

l

l

Dynamic shear Breaching model very limited Use of same stress and velocity waveform Bomb detonatlon between burster slab and structure Design fragment selection grossly incorrect Weight distribution for large fragments Fragment loading Residual velocity model very limited Rebar size/spacing

perform the additional RBD and systems reliability analyses needed for implementation throughout future editions of the Joint Services DAHS Handbook. Acknowledgment The research was funded by the Air Force Civil Engineering Support Agency (now Wright Laboratory, Airbase Survivability Branch) at Tyndall Air Force Base. References [l] M.P. White (Ed.), Terminal Ballistics, Part III, Effects of Impact and Explosion, Summary Technical Report of Division 2, National Defense Research Committee, Washington, DC, 1946. [2] Department of the Army, Fundamentals of Protective Design for Conventional Weapons, TJv58551, Waterways Experiment Station, Vicksburg, MS, 1984. [3] R.E., Crawford, Protection from non-nuclear weapons, AFWL-TR-70-127, Air Force Systems Command, Kirtland AFB, NM, 1971.

LA.

Twisdaleet al. /Structural Safety15 (1994)17-33

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[4] J.L. Drake et al., Protective construction design manual, ESL-TR-87-57, Air Force Engineering Services Laboratory, Tyndall Air Force Base, FL, 1989. [5] L.A., Twisdale, R.H. Sues and F.M. Lavelle, Reliability-based analysis and design methods for reinforced concrete protective structures, Final Report F08635-89-C-0121, Air Force Civil Engineering Support Agency, Tyndall Air Force Base, FL, 1992. [6] R.H., Sues, L.A. Twisdale and F.M. Lavelle, Protective construction design manual, (Change 6) “Reliabilitybased design for reinforced concrete protective structures” (Draft), Air Force Civil Engineering Support Agency, Tyndall Air Force Base, FL, 1992. [7] J.E., Welter, Joint services program for the design and analysis of hardened structures to conventional weapons effects, 6th Int. Symp. on the Interaction of Non-Nuclear Munitions with Structures, Panama City, FL, May 1993. [8] A.H-S., Ang, Structural risk analysis and reliability-based design, J. Struct. Diu., AXE, 99(9) (1973) 1891-1910. [9] C.N. Kingery, and G., Bulmash, Airblast parameters from TNT spherical air burst and hemispherical surface burst, ARBRL-TR-02555, Ballistic Research Laboratory, 1984. [lo] J.M., Carson, D. Morrison, and R.J., Hampson, Conventional high-explosive blast and shock (CHEBS) test series: Mark-83 general-purpose bomb tests, Air Force Weapons Laboratory, AFWL-TR-84-27, 1984. [ll] J.M. Carson, and D., Morrison, Conventional high-explosive blast and shock (CHEBS) test series: Mark-83 general-purpose bomb tests, Air Force Weapons Laboratory, AFWL-TR-86-53, Parts l-3, 1987. [12] L.A., Twisdale, R.H. Sues, F.M. Lavelle and D. Miller, Research to develop reliability-based design methodology for protective structures, 5th Znt. Symp. on the Interaction of Conventional Munitions with Protective Structures, Germany, April 1991. [13] M.K, McVay, Spa11 damage of concrete structures, Technical Report, SL-88-22, U.S. Army Waterways Engineering Stations, Vicksburg, MS, 1988. [14] R.H., Sues, J.L. Drake and L.A. Twisdale, Reliability-based safety factors for ground shock loads in protective construction, 6th Int. Symp. on the Interaction of Non-Nuclear Munitions with Structures, Panama City, FL, May 1993.