PROBABILISTIC ANALYSIS OF BALLISTICS AND BLAST

RESISTANCE WALLS

A Research Proposal

BY

Aminu AHMED(P17EGCV9021)

Supervisory Committee:Dr. J. M. KauraProf. O.S. AbejideDr. A. Ocholi

Submitted to the;

Department of Civil Engineering

Faculty of engineering

Ahmadu Bello University Zaria

IN PARTIAL FULFILMENT OF PHILOSOPHY DEGREE IN

STRUCTURAL ENGINEERING

22nd Oct. 2017

CHAPTER ONE

1.1 Introduction

This paper will deals with the problem of reinforce composite structures resistance under

influence of extreme loads from seismic and blast explosion

The complexity of nonlinear response history analysis (NRHA) has been the motivation

for the development of simplified analyses to determine the nonlinear response of

structures. A good example of these simplified analyses is pushover analysis, in which

the structure undergoes an incremental lateral load up to a target point. Among the

existing pushover procedures, e.g. Poursha et al. (2009), Kreslin and Fajfar (2011),

Brozovicˇ and Dolsˇek (2014) and Kaats and Sucug˘ lu (2014), modal pushover analysis

(MPA) (Chopra and Goel, 2002) is a widely accepted method in the engineering

community.

For the analysis of structures under seismic excitation, the imposed acceleration applies

to the structure’s base; however, in blast wave or wind loads, the imposed external load

directly affects the building with an unexpected load pattern. Based on this, a new

formulation for the modal analysis of structures will be develop in this study. Since the

effect of higher modes in this case can be more significant, we have to utilize an energy-

based approach for pushover analysis, ensuring that higher mode pushover curves are not

subjected by the reversals observed in previous studies (Hernandez- Montes et al., 2004;

Soleimani et al., 2017). To evaluate the presented method, MPA-B, a benchmark two

dimensional shear wall is to be modeled as a case study on which a blast wave will be

imposed.

There is to consider the shock wave propagation in the structural grains and the effect of

the explosion and space wave propagation to structures is to be evaluated. The wave

propagation is simulated on 3D FEM model in software Abaqus/Ansys on the base of the

experimental results Yong Lu and Zhongqi Wang. This work focuses on deterministic

analysis of progressive collapse of a typical RC and other composites civil engineering

structures, induced by a blast event. The probabilistic analysis will be perform by taking

into account the uncertainties in loading such as planar configuration and amplitude of

the blast loading. A standard Monte Carlo simulation method is to be employ to generate

various realizations of the uncertain parameters within the problem. For a given

realization, various component-level dynamic analyses will be preform within a certain

range of magnitude, in order to quantify and locate the damage induced by shock wave

on structural elements

1.2 Statement of Problem

Recent world events such as bombings in London, Madrid, Istanbul and even Nigeria

have highlighted the susceptibility of many civilian structures to terrorist attack.

Explosives directed towards vulnerable structures may cause considerable damage and

loss of life. As a result, there is now a desire to increase the blast resistance of many types

of existing structures. This has led to experimental and finite element (FE) research in

retrofitting concrete and masonry structures with fibre reinforced polymer (FRP)

composites for blast protection.

1.3 Justification

Extreme loading conditions such as man-made malicious actions, fires or natural events

could induce local failure mechanisms (e.g., a loss of a member) which may trigger

progressive collapse. By progressive collapse, it is intended disproportionate

magnification of a minor failure event within the structure. The design or the assessment

of a critical infrastructure needs to address the possibility of such an extreme

circumstance taking place during its effective life-time. It is observed that blast-induced

progressive collapse mechanisms involve non-linear structural behavior similar to that

due to earthquakes.

1.4 Aim and Objectives

1.4.1 Aim

The aim of this study is to evaluates, optimize and provides an effective scheme for the

blast resistance of CFRP and other composite walls

1.4.2 Objectives

The specific Objectives include;

(1) Establish the design basic variables of the existing blast cases as well as the design

variables for the flexural strengthening of the resistance walls using CFRP

(2) To perform a 3D analysis and understand the structural behavior under the influence of

blast pressure using Modeling and Simulation techniques in commercial Finite Element

Analysis (FEA) software; ABAQUS/CAE 6.13 with the view to perform optimum CFRP

strengthening of the wall.

(3) Develop performance functions for the flexural modes of failure of the strengthened blast

resistance walls.

(4) Formulate First order reliability analysis and genetic algorithms to develop a computer

programs for the implementation of the genetic algorithms GA-based First Order

Structural Reliability Method FORM using MATLAB.

(5) Perform sensitivity analysis using the developed program in order to investigate the effect

of uncertainties on the safety of the strengthened blast resistance structure

(6) To verify the effectiveness of seismic retrofitting schemes against explosions and the

eventual progressive collapse

1.5 Scope and Limitations

1.5.1 Scope

This study involves simulation and optimization of the strengthening formulation for the

blast resistance wall with CFRP laminates using Finite Element Analysis (FEA) in

SIMULIA ABAQUS software. The study also involves sensitivity analysis of the

strengthened resistance wall in order to assess the effect of uncertainties on the safety of

the wall with MATLAB tool.

1.5.2 Limitations

Geometric data of the wall would be obtained through actual experiments conducted by

other researchers, data on the strength and load actions would be considered with

reference to the existing data in the literature on blast design.

CHAPTER TWOLITRETURE REVIEW

2.1 General

Basically, the solution methods for design optimization problems can be classified into

Optimality Criteria (OC) methods and Mathematical Programming methods. In the

Optimality Criteria method (Zou and Haftka 1995), the optimality conditions for a given

type of problem are derived based on the Karush-Kuhn-Tucker condition or by heuristic

assumptions, and then the optimal design satisfying these condition is to be sought using

different forms of resizing rules. Such methods are recognized to be especially efficient

for problems involving a large number of design variables.

The Mathematical Programming method may be broadly classified into gradient-based

methods (requiring derivatives of the functions) (Johnson 1961, Arora 1990) and non-

gradient or direct methods (requiring no derivatives) (Fox 1971, Rao 1996, Deb 2003).

The use of the gradient-based method for minimization is first presented by Cauchy.

Modern optimization methods are pioneered by Courant’s paper on penalty functions,

Dantzig’s paper on the simplex method for linear programming and Karush, Khun and

Tucker who derived the

“KKT” optimality conditions for constrained problems (Johnson 1961).

Particularly in the 1960s, several numerical methods to solve nonlinear optimization

problems were developed. Mixed integer programming received an impetus from the

branch and bound technique, originally developed by Land and Doig, and the cutting

plane method by Gomory (Fox 1971).

Methods of unconstrained minimization include the Conjugate gradient methods of

Fletcher and Reeves, and the variable metric methods of Davidon-Fletcher-Powell (DFP)

(Siddall 1972).

The Constrained optimization method is pioneered by Rosen’s gradient Projection

method, Zoutendijik’s method of feasible directions, the generalized reduced gradient

method by Abadie, Carpentier and Hensgen and Fiacco and McCormick’s SUMT

techniques (Siddall 1972). The traditional interval search methods, using Fibonacci

numbers or the golden section ratio

are followed by the efficient hybrid polynomial-interval methods of Brent.

Sequential Quadratic Programming (SQP) methods for constrained minimization are then

developed. The Development of interior methods for linear programming started with the

work of Karmarkar in 1984 (Papalambras and Wilde 1988).

In the 1960s, side-by-side with developments in gradient-based methods, there were

developments in non-gradient methods, principally Rosenbrock’s method of orthogonal

directions, the pattern search method of Hooke and Jeeves, Powell’s method of Conjugate

directions, the simplex method of Nedler and Meade and the method of Box (Haug and

Arora 1979).

Most recent among the direct methods are genetic algorithms (Holland 1975, Goldberg

2002) and the simulated annealing algorithm which originated from Metropolis. Special

methods that exploit some particular structure of a problem have also been developed

(Rao 1996). Dynamic programming originated from the work of Bellman, who stated the

principle of optimal policy for system optimization. Geometric programming originated

from the work of Duffin Peterson, Zener. Pareto optimality was developed in the context

of multiobjective optimization.

In addition to these conventional methods, some innovative approaches using analogies

of physics and biology, such as Simulated Annealing, Genetic Algorithm and

volutionary lgorithms (Papadrakakis et al 1998, Deb 2001), are also employed for the

solution of global optimization problems. These approaches are characterized by

gradient-free methods and utilize only function values. Generally, these algorithms

require a large number of function evaluations to achieve convergence, and thus have

limited use in applications involving complicated designs.

Different methods of design optimization are widely used in the design of engineering

structures for the purpose of improving the performance and reducing their costs. The use

of design optimization techniques has rapidly increased, mainly due to the development

of sophisticated computing techniques and the extensive applications of the finite element

method (Deb 2003). Moreover, recently, it is widely recognized that design optimization

methodologies should account for the stochastic nature of engineering systems, and that

concepts and methods of life-cycle engineering should be used to obtain a cost-effective

design during a specified time horizon. To ensure high reliability and safety, uncertainties

inherent to or encountered by the product during the entire life cycle must be considered

and treated in the design process. The various types of uncertainty, the mathematical

models of uncertainty reported in literature, and the optimization methodologies which

include these uncertainties are described below.

2.2 The Structure of Optimization

After we have discussed what optima are and have seen a crude classification of global

optimization algorithms, let us now take a look on the general structure common to all

optimization processes. This structure consists of a number of well-defined spaces and

sets as well as the mappings between them. Based on this structure of optimization, we

will introduce the abstractions fitness landscapes, problem landscape, and optimization

problem which will lead us to a more thorough definition of what optimization is.

2.2.1 Spaces, Sets, and Elements

In this section, we elaborate on the relation between the (possibly different)

representations of solution candidates for search and for evaluation. We will show how

these representations are connected and introduce fitness as a relative utility measures

defined on sets of solution candidates. You will find that the general model introduced

here applies to all the global optimization methods mentioned in this book, often in a

simplified manner. One example for this structure of optimization processes is given in

Figure 2 by using a genetic algorithm which encodes the coordinates of points in a plane

into bit strings as an illustration.

The Problem Space and the Solutions therein Whenever we tackle an optimization

problem, we first have to define the type of the pos- sible solutions. For deriving a

controller for the Artificial Ant problem, we could choose programs or artificial neural

networks as solution representation. If we are to find the root of a mathematical function,

we would go for real numbers R as solution candidates and when configuring or

customizing a car for a sales offer, all possible solutions are elements of the power set of

all optional features. With this initial restriction to a certain type of results, we have

specified the problem space X. Definition 1.18 (Problem Space). The problem space X

(phenome) of an optimization problem is the set containing all elements x which could be

its solution.

Usually, more than one problem space can be defined for a given optimization problem.

A few lines before, we said that as problem space for finding the root of a mathematical

function, the real number R would be fine. On the other hand, we could as well restrict

ourselves to the natural numbers N or widen the search to the whole complex plane C.

This choice has major impact: On one hand, it determines which solutions we can

possible find.

On the other hand, it also has subtle influence on the search operations. Between each

two different points in R, for instance, there are infinitely many other numbers, while in

N, there are not.

In dependence on genetic algorithms, we often refer to the problem space synonymously

phenome. The problem space X is often restricted by 1. logical constraints that rule out

elements which cannot be solutions, like programs of zero length when trying to solve the

Artificial Ant problem and 2. practical constraints that prevent us, for instance, from

taking all real numbers into consideration in the minimization process of a real function.

On our off-the-shelf CPUs or with the Java programming language, we can only use 64

bit floating point numbers.

With these 64 bit, it is only possible to express numbers up to a certain precision and we

cannot have more than 15 or so decimals.

Fig. 2 Genetic Algorithm

Amy coffield and Hojjat adeli (2014) [13] investigated different framing systems for three

seismically designed steel frame structures subjected to blast loading. The blast loads are

assumed to be unconfined, free air burst detonated 15 ft. (4.572 m) from one of the center

columns. The structures are modeled and analyzed using the Applied Element Method, which

allows the structure to be evaluated during and through failure. Failure modes are investigated

through a plastic hinge analysis and member failure comparison. The main conclusion of this

research is that braced frames provide a higher level of resistance to the blast loading.

Kulkarni (2014) [14] examined the dynamic response of a High Rise Structure subjected to blast

load. The lateral stability of a high rise building modeled using SAP2000. The model building

was subjected to two different charge weights of 800 lbs and 1600 lbs TNT at a two different

standoff distances of 5 m and 10 m. The blast loads are calculated using the methods outlined in

section 5 of TM5-1300 and a nonlinear modal analysis is used for the analysis of the dynamic

load of the blast. The primary performance parameters that will be used to evaluate the behavior

of the building from a global perspective are the total drift and the inter-storey drift. the results

shows that the first storey columns subjected to high pressure they could cause big deformation

and exceed the support reaction so the columns which are close to explosion are damaged which

leads to sudden loss of critical load bearing columns is lost.

Aditya Kumar et. al. (2014) [15] reviewed various loading which can occur during a blast i.e.

the dynamic impact loading, varying rate concentrated loading &transverse blast loading and the

methods applied to analyze those loading phenomena i.e. Single Degree of Freedom (SDOF)

model, Finite Element Model (FEM) & non-linear dynamic analysis. The analysis shows that the

lack of relevant code is the major concern behind the ignorance of this phenomenon while

designing the structure.

M. Amini et. al. (2015) [16] developed the method to nonlinear dynamic analysis of single-

degree-of-freedom (SDOF) systems under exploding loads. Newton-Raphson iterative method

used to develop new formulation for solving nonlinear dynamic problems. A simple step-by-step

algorithm is implemented and presented to calculate dynamic response of SDOF systems. The

validity and effectiveness of the proposed method is demonstrated with two examples. Quartic

B-spline time integration method gains second order of acceleration at each time-step so it

benefits from high order accuracy. The numerical evaluation shows that the proposed method is

a fast and simple procedure with trivial computational effort.

Amy Coffield and Hojjat Adeli (2015) [17] considered six seismically designed steel framed

structures moment resisting frames (MRF), concentrically braced frames (CBF) and eccentrically

braced frames (EBF) each with geometric irregularity in the plan and with a geometric

irregularity in the elevation. The blast loads are assumed to be unconfined, free air burst

detonated 15 ft from one of the center columns. The structures are modeled and analyzed using

the Applied Element Method. Comparative analysis observing roof deflection and acceleration to

determine the effect of geometric irregularity under extreme blast loading conditions. Two

different blast locations are examined. Result shows that for all structural types a vertical or

horizontal irregularity results in a smaller roof deflection in the order of 12–17 %. The

conclusion is concentrically braced frame provides higher level of resistance to blast loading for

irregular structures and geometric irregularity has an impact on the response of a structure

subjected to blast loading.

Sarita Singla et. al. (2015) [18] studied the blast pressure for different TNT and standoff

distance. Blast pressures for different cases are computed using correlation between blast

pressure and blast scaled distance based on charts given in U.S manual. Time history loading is

also obtained with parameters of reflected total over pressure and duration of positive phase of

blast. The result shows that as the distance increases from the building, blast pressure reduces.

CHAPTER THREEMETHODOLOGY

3.1 GeneralFinite Element Analysis (FEA) is to be conducted using Simulia ABAQUS, a FE’s

software. Also a MATLAB (2007) program will be developed for the Structural

Reliability Analysis (SRA) of the bridge deck after repairs, using Genetic Algorithm,

GA-based First Order Reliability Method (FORM).

The proposed blast resistance wall is to be modeled for performance in blast loads.

3.2 DETERMINISTIC AND PROBABILISTIC METHODS

Most problems concerning the reliability of building structures are defined today as a

comparison of two stochastic values, loading effects E and the resistance R, depending on

the variable material and geometric characteristics of the structural element. The

variability of those parameters is characterized by the corresponding functions of the

probability density f R (r )andf E (e). In the case of a deterministic approach to a design, the

deterministic (nominal) attributes of those parameters Rdand Edare compared.

The deterministic definition of the reliability condition has the form

Rd ≥ Ed (3.1)

and in the case of the probabilistic approach, it has the form

RF = R − E ≥ 0 (3.2)

Where RF is the reliability function, who can be generally expressed as a function of the

stochastic parameters X1, X2to X n used in the calculation of R and E.

RF = g(X1,X2,…..X n) (3.3)

The failure function g(X) represents the condition (reserve) of the reliability, which can

either be an explicit or implicit function of the stochastic parameters and can be single

(defined on one cross-section) or complex (defined on several cross-sections, e.g., on a

complex finite element model).

The most general form of the probabilistic reliability condition is given as follows:

Pf P(R-E 0) P (RF0) Pd (3.4)

WherePdis the so-called design (“allowed “or “acceptable“) value of the probability of

failure. From the analytic formulation of the probability density by the functions f R(r)

and

f E (e) and the corresponding distribution functions ΦR(x) and ΦE(x), the probability of

failure can be defined in the general form:

Pf ∫−¿¿

❑d P f∫−¿¿

❑ f E(x)ΦR(x)dx ∫−¿¿

❑Φ E(x)f R(x)dx

(3.5)

This integral can be solved analytically only for simple cases; in a general case it should

be solved using numerical integration methods after discretization. Together with Pf the

index of reliability β is often used for analyzing the degree of reliability. It is defined

using the linearization condition of the failure function g(X). In the case of the normal

distribution of this function it is valid:

β ❑µRF

σ RF

(3.6)

WhereµRFand σ RF are the mean values and the standard deviations of the reliability

function, respectively.

3.3 RELIABILITY ANALYSIS METHODS

From the point of view of one’s approach to the values considered, structural reliability analyses can be classified in two categories, i.e., deterministic analyses and stochastic analyses. In the case of the stochastic approach, various forms of analyses (statistical analysis, sensitivity analysis, probabilistic analysis) can be performed. Considering the probabilistic procedures, Eurocode 1 recommends a 3-level reliability analysis. Most of these methods are based on the integration of Monte Carlo (MC) simulations.

Straight Monte Carlo methods are based on a simulation of the input stochastic

parameters according to the expected probability distribution. The accuracy of this

method depends upon the number of simulations and is expressed by the variation index:

V Pf = 1

√N Pf (3.7)

Where N is the number of simulations. If the required probability of failure is Pf =10−4,

then by the number of simulations N = 106, the variation index is equal to 10%, which is

an acceptable degree of accuracy.

3.4 LOADING AND LOAD COMBINATIONExternal explosions (accident, terrorist attack) belong to unidentifiable cases of accidental situations. The loads and load combination were considered for deterministic and probabilistic analysis particularly.The combination of actions can be expressed for deterministic analysis as:

(3.8)

WhereEdis a design value of load combination, Gk is characteristic value of permanent load, AEx, kis a design value of accidental load, Qk is a main changing load. The values of Coefficient Ψ are defined in STN EN 1990 tab. A1.1In the case of probabilistic analysis the load combination was taken as

(3.9)

Where gvar, qvar, avar are variable parameters in the form of histograms calibrated to

load effects in accordance of Eurocode ENV 1990 [2002].

3.5 VARIABILITY OF THE INPUT PARAMETERS

The uncertainty of loading, the stiffness and the resistance of the structure can be

effectively analyzed by a sensitivity analysis in the case of a probabilistic approach.

The variability of the subgrade’s stiffness is expressed through the variable values of the

global movesK z−var, of the building as a Winkler’s model. The uncertainties of the

calculation model were taken into account by the various parameters of the model’s

uncertainties ❑Rand loading effects ❑E(μ = 1,σ = 0.1 ) according to a Gaussian normal

distribution.

Table 1: Probabilistic model of the principal values

3.6 RESISTANCE OF STRUCTURE

The reliability of the panel building was checked on the inter storey drift and maximum shear resistance for dynamic and quasi static explosion expression in accordance of Eurocode 1998 (2000) requirements.Reinforced concrete wall structures must be designed to the tension or compression in the combination with bending and shear stresses. The failure function g(V) for the shear is defined in the form

(3.10)

Where VE is the shear force from the action and VR is the sheer force of resistance by 1m of element longitude in accordance of standard EN 1992-1-1.

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