Blast Resistant Design for Roof System

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BLAST RESISTANT DESIGN FOR ROOF SYSTEMS

A Thesis presented to the Faculty of the Graduate School of the

University of Missouri Columbia

In Partial Fulfillment of the Requirements for the Degree

Master of Science

by

MARK ANDREW MCCLENDON

Dr. Hani Salim, Thesis Supervisor

DECEMBER 2007


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The undersigned, as appointed by the Dean of the Graduate School, have entitled the

thesis entitled

BLAST RESISTANT DESIGN FOR ROOF SYSTEMS

Presented by Mark A McClendon

a candidate for the degree of Master of Science in Civil Engineeringand hereby certify that in their opinion it is worthy of acceptance.

________________________________________________

Dr. Hani Salim

________________________________________________Dr. Sam Kiger

________________________________________________

Dr. Craig Kluever


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ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to my advisor, Dr. Hani Salim,

Assistant Professor of Civil Engineering, University of Missouri-Columbia. His guidance

has been invaluable during the course of this study. His good nature, patience, frankness,

and technical expertise had a profound impact on my academic experience and personal

goals.

I would also like to thank Dr. Sam Kiger, C.W. La Pierre Distinguished Professor

and Director of the Center for Explosion Resistant Design, University of Missouri-

Columbia. His enthusiasm and technical knowledge helped guide this investigation.

A special thanks is given to Dr. Perry Green, Technical Director, Steel Joist

Institute and especially Tim Holtermann from Canam Steel for donating testing materials

and continuous design input.

Gratitude is extended to my fellow students, Aaron Saucier, Rhett Johnson, John

Hoemann, Tyler Oesch, and others whose efforts were in my favor. Without their hard

work and dedication, testing would have been overwhelming. Furthermore, I would like

to thank all the members at the University of Missouri who have provided assistance.

Finally, I extend my sincerest thanks to my friends who have kept me sane

throughout my academic experience, and my family, especially my mother Verna and my

aunt Wilhelmina.


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TABLE OF CONTENTS

ACKNOWLEDGEMENTS..............................................................................................ii

LIST OF ILLUSTRATIONS...........................................................................................vi

LIST OF TABLES...........................................................................................................xii

ABSTRACT....................................................................................................................xiii

1 Introduction

1.1Problem Statement............................................................................1

1.2Thesis Objective.................................................................................2

1.3Thesis Overview.................................................................................3

2 Review of Literature

2.1Introduction.......................................................................................4

2.2Types of Blast Loads.........................................................................4

2.3Effects on Structures.........................................................................8

2.4Blast Resistant Design.......................................................................9

2.4.1 Single-Degree of Freedom Models..................................10

2.4.2 CONWEP..........................................................................14

2.5Roof Systems....................................................................................14

2.5.1 Equivalent Blast Load.....................................................14

2.5.2 Resistance Functions........................................................16


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3 Verification of Equivalent Blast Load Procedure

3.1Introduction.....................................................................................19

3.2Review of ANSYS LS-DYNA.........................................................20

3.3Static Load Simulations..................................................................21

3.3.1 Preprocessing....................................................................22

3.3.2 Loading and Solution.......................................................24

3.3.3 Post Processing.................................................................24

3.3.4 Results...............................................................................24

3.4Dynamic Load Simulations.............................................................31

3.4.1 Preprocessing....................................................................32

3.4.2 Loading and Solution.......................................................34

3.4.3 Post Processing.................................................................35

3.4.4 Results...............................................................................35

3.5Blast Load Simulations...................................................................43

3.5.1 Application of Blast Loads..............................................43

3.5.2 Results (Positive Phase Only) .........................................45

3.5.3 Results (Positive and Negative Phase) ...........................51

3.5.4 Equivalent Blast Loading................................................55

3.6Field Test..........................................................................................60

3.7Summary..........................................................................................72

4 Static Resistance Function of Open Web Steel Joists

4.1Introduction.....................................................................................73

4.2Analytical Resistance Function......................................................74


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4.3Experimental Verification..............................................................78

4.3.1 Testing Samples................................................................78

4.3.2 Test Set-Up.......................................................................80

4.3.3 Testing Apparatus............................................................85

4.3.4 Results...............................................................................86

4.3.4.116K5 Joist Test.......................................................86

4.3.4.226K5 Joist Test.......................................................93

4.3.4.332LH06 Joist Test..................................................99

4.4Summary........................................................................................104

5 Conclusion and Recommendations

5.1Conclusions....................................................................................105

5.2Recommendations..........................................................................107

REFERENCES...............................................................................................................108


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LIST OF ILLUSTRATIONS

Figure 2-1: Generalized Blast Pressure History.............................................................6

Figure 2-2: Blast Loading on Structure...........................................................................8

Figure 2-3: Damped, Single-Degree-of-Freedom System.............................................10

Figure 2-4: Idealized Dynamic Response Curves for Triangular Loading................12

Figure 2-5: Equivalent SDOF System............................................................................13

Figure 2-6: Equivalent Load Factor and Blast Wave Location Ratio........................15

Figure 2-7: DAHS Equivalent Loading Technique......................................................16

Figure 2-8: DAHS Equivalent Load...............................................................................16

Figure 3-1: BEAM 4 ANSYS element.........................................................................23

Figure 3-2: Static Loading #1; Maximum loading = 375 lb/ft.....................................25

Figure 3-3: Static Load #2; Maximum Load = 375 lb/ft..............................................26

Figure 3-4: Static Load #3; Maximum Loading = 375 lbs/ft........................................27

Figure 3-5: Static Load #4...............................................................................................28

Figure 3-6: Static Load #5...............................................................................................29

Figure 3-7: Percentage Error of Maximum Deflection................................................30

Figure 3-8: Percentage of Error of Max Deflection Location......................................31

Figure 3-9: BEAM 161 ANSYS LS-DYNA element..................................................33

Figure 3-10: Dynamic Pulse Load #1.............................................................................37

Figure 3-11: Dynamic Load #2.......................................................................................38

Figure 3-12: Dynamic Load #3.......................................................................................39


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Figure 3-13: Percentage Error in Maximum Response................................................40

Figure 3-14: Percentage Error in Time of Maximum Response.................................41

Figure 3-15: Percentage Error in Response Period......................................................42

Figure 3-16: Locations of CONWEP measurements....................................................44

Figure 3-17: Blast Pressure Distribution.......................................................................44

Figure 3-18: Actual Blast Loading.................................................................................46

Figure 3-19: Loading Scenario #1..................................................................................46

Figure 3-20: Impulse Comparison.................................................................................47



Figure 3-23: Impulse Comparison..................................................................................49

Figure 3-24a: Response Comparison of Loading Scenarios using LS-DYNA...........49

Figure 3-24b: Response Comparison of Loading Scenarios using LS-DYNA (without

Scen. #1).............................................................................................................50

Figure 3-25: Dynamic Response for Loading Scenario #2...........................................51

Figure 3-26: Actual Blast Load Positive and Negative Phase...................................52

Figure 3-27: Loading Scenario #1 Positive and Negative Phase...............................52

Figure 3-28: Loading Scenario #2 Positive and Negative Phase...............................53

Figure 3-29: Negative Impulse Comparison..................................................................53

Figure 3-30: Response of Scenario #2 Positive and Negative Phase.........................54

Figure 3-31: Response Comparison of Positive Only and Pos. & Negative Data for

Loading Scenario #2.........................................................................................55

Figure 3-32: TM-855 Equivalent Blast Load................................................................57


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Figure 3-33: Verification of Equivalent Load Response..............................................58

Figure 3-34: Comparison of Dynamic Response...........................................................59

Figure 3-35: FRP Panel Test...........................................................................................60

Figure 3-36: Static Resistance Function for FRP Panels.............................................61

Figure 3-37: Roof Panel Schematic................................................................................63

Figure 3-38: Pressure-time history at 35 ft....................................................................64

Figure 3-39: Impulse-time history at 35 ft.....................................................................65





Figure 3-44: Deflection of R1 panel at near quarter-point..........................................68

Figure 3-45: Deflection of R1 panel at far quarter-point.............................................69

Figure 3-46: Deflection of R1 panel at midpoint...........................................................70

Figure 3-47: Deflection of R2 Panel at near quarter-point..........................................71

Figure 3-48: Deflection of R2 Panel at far quarter-point.............................................71

Figure 3-49: Deflection of R2 Panel at midpoint...........................................................72

Figure 4-1: Resistance Function for 16K5 Joist............................................................75

Figure 4-2: Resistance Function for 26K5 Joist............................................................75

Figure 4-3: Resistance Function for 32LH06 Joist.......................................................76

Figure 4-4: Maximum deflection of elasto-plastic, one-degree-of-freedom system for

triangular pulse load.........................................................................................77

Figure 4-5: Bearing seat plates for 16K5 and 26K5 Joists...........................................81


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Figure 4-6: Bearing Seat Plates for 32LH06 Joist.........................................................82

Figure 4-7: Steel Joist Institute Specifications for horizontal bridging......................82

Figure 4-8: Lateral bracing welded to strong floor...................................................83

Figure 4-9: Lateral bracing placement.......................................................................84

Figure 4-10: String Potentiometer..................................................................................84

Figure 4-11: 16-Point Loading Tree...............................................................................86

Figure 4-12: 16K5 trusses prior to testing.....................................................................87

Figure 4-13: 16K5 Joist Failure Sequence 1 of 5 Initial bending..........................88

Figure 4-14: 16K5 Joist Failure Sequence 2a of 5 Failure of lateral bracing.......88

Figure 4-15: 16K5 Joist Failure Sequence 2b of 5 Failure of lateral bracing......89

Figure 4-16: 16K5 Joist Failure Sequence 3 of 5 Failure of horizontal

bridging..............................................................................................................89

Figure 4-17: 16K5 Joist Failure Sequence 4 of 5 Continued out-of-plane

bending...............................................................................................................90

Figure 4-18: 16K5 Joist Failure Sequence 5 of 5 Failure of bearing seat

weld.....................................................................................................................90

Figure 4-19: Static Response for 26K5 Joist System....................................................91

Figure 4-20: Midpoint Static Response for an individual 16K5 joist compared to

existing methods................................................................................................92

Figure 4-21: 26K5 Joist System prior to test.................................................................94

Figure 4-22: 26K5 Joist Failure Sequence 1 of 5 Deformation of Tension

Chord.................................................................................................................94


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Figure 4-23: 26K5 Joist Failure Sequence 2a of 5 Failure of End Tension

Member..............................................................................................................95

Figure 4-24: 26K5 Joist Failure Sequence 2b of 5 Failure of End Tension

Member..............................................................................................................95

Figure 4-25: 26K5 Joist Failure Sequence 3 of 5 Failure of Secondary Web

Member..............................................................................................................96

Figure 4-26: 26K5 Joist Failure Sequence 4a of 5 Failure of End Tension

Member..............................................................................................................96

Figure 4-27: 26K5 Joist Failure Sequence 4b of 5 Failure of End Tension

Member..............................................................................................................97

Figure 4-28: 26K5 Joist Failure Sequence 5a of 5 Connection Plate Failure.......97

Figure 4-29: 26K5 Joist Failure Sequence 5b of 5 Connection Plate Failure.......98

Figure 4-30: Static Response for 26K5 Joist System....................................................98

Figure 4-31: Midpoint Static Response for an individual 26K5 joist compared to

existing methods................................................................................................99

Figure 4-32: Initial Deformation of 32LH06 Joist......................................................100

Figure 4-33: 32LH06 Joist System prior to test..........................................................101

Figure 4-34: 32LH06 Joist Failure Sequence 1a of 1 Compression Web Member

Buckling...........................................................................................................101

Figure 4-35: 32LH06 Joist Failure Sequence 1b of 1 Buckling of Secondary Web

Member............................................................................................................102

Figure 4-36: 32LH06 Joist Failure Sequence 1c of 1 Continued Buckling and

Bending of Tension Chord.............................................................................102


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Figure 4-37: Static Response of 32LH06 Joist System...............................................103

Figure 4-38: Midpoint Static Response of individual 32LH06 Joist comparing

existing methods..............................................................................................103


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LIST OF TABLES

Table 3-1: Minimum Time Steps by Element Size.......................................................36

Table 3-2: Aspect Ratio...................................................................................................36

Table 3-3: Peak Pressure vs. Range...............................................................................44

Table 3-4: CONWEP Blast Data....................................................................................56

Table 4-1: Resistance Data from Engineering Calculations........................................77

Table 4-2: Resistance Data from SBEDS Calculations................................................78


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ABSTRACT

The design of structures to resist explosive loads has become more of a concern to the

engineering community. Different structural members, such as walls, have been

thoroughly evaluated under blast loads. This research focuses on the design techniques

for the loading on roof structures and the resistance of open web steel joists, a common

roof component. Blast loads are dynamic, impulsive and non-simultaneous over the

length of a roof. To design against explosions, a procedure has been developed to devise

a uniform dynamic load on a roof that matches the response from blast loads. The

objective of this research is to test this procedure and compare its results to the

deflections from blast loads. This research uses finite element analysis to evaluate the

responses from numerically calculated blast loads and compares them to the equivalent

loading response. The numerical pressures are calculated using the Conventional

Weapons Effects Program (CONWEP) (Hyde, 1992) and the Single-degree-of-freedom

Blast Effects Design Spreadsheet (SBEDS) from the Army Corps of Engineers Protection

Design Center. Also, the response of experimentally measured roof blast pressures is

compared to the equivalent loading response. While the responses from finite element

modeling matched the experimental responses, the equivalent loading procedure did not

adequately predict the initial peak deflection or the maximum deflection.


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The response of several structural members used in roof construction, such as hot-rolled

steel beams and reinforced concrete slabs, are well documented and understood. Open

web steel joists (OWSJ) are other types of common roof components. Their responses

under loading are not clearly defined, and current methods extrapolate techniques used in

the design and analysis of hot-rolled steel beams and reinforced concrete. The resistance

function currently used for these members are linear elastic and perfectly plastic after the

elastic deflection limit. It is believed that the failure mechanisms of OWSJ significantly

are not accurately being taken into account. Three tests consisting of different steel joist

pairs are performed. The resistance function is computed from these results and

compared to current methodologies. The current resistance methods calculate larger

maximum loads than the experimental values and the assumption of a perfect plastic

post-peak response ignores the buckling failure of web members. It is recommended that

additional research is to be done on the prediction of blast pressures on roofs and on the

development of an equivalent uniform dynamic load. It is also recommended that an

analytical resistance function for OWSJ be clearly defined, which includes all failure

limit states.


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Chapter

1Introduction

1.1 Problem Statement

Many research has been performed on the design of structures under explosive

threats. The design and response for a wall system subjected to blast loading and its

components are very well known. However, there is a gap in the knowledge for the roof

system component under explosive loads.

The analysis of any system with dynamic loads, such as blast pressure due to a

bomb, can be generalized with an equation of motion as shown in Eqn. 1.1. This equation

can be solved by various numerical procedures to calculate the dynamic response (Biggs,

1967). The parameters in this equation are the mass, the resistance, and the load.

My R F t&& ( )+ = (1.1)

where M = mass

R = resistanceF(t) = applied load


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While the mass of any system can be easily calculated, the resistance and load are

more complex. Explosive loads are highly impulsive, non-simultaneous, and non-

uniform. For walls, the blast pressures are mostly uniform on the entire face of the wall,

and they can be accurately estimated using available codes such as CONWEP (Hyde,

1992). In roof systems, these loading pressures change with respect to time and with

respect to distance. There is currently a loading procedure developed by the Army Corps

of Engineers that equate the response from a blast load to the response from an equivalent

uniform dynamic load that is more suited to design (UFC, 2002). However, this

procedure has not been verified using experimental or numerical data, as far as the author

is aware. Therefore, there is a need to explore and validate the methods and results this

technique uses.

The resistance of many structural systems used in construction of roof slabs is

known. In addition to concrete and hot-rolled steel members, open web steel joists are

also common roof components, and they are in a high demand of use due to their low

weight and relatively high resistance. Current techniques used in design assume a linear

elastic, pure plastic load-deflection curve. It is believed that such truss systems exhibit

different failure modes than those currently used in design. Due to their high use in

military and commercial buildings, current design practices should be researched.

1.2 Thesis Objective

The overall objective of this research is to develop an analysis and design

procedure for open web steel joist roof system under blast loading. To achieve this goal,

the following two specific tasks are realized:


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Analyze the roof loading procedure specified by the United Facilities Criteria

(UFC 3-340-01, 2002) and compare its validity using numerical simulation and

field test data.

Analyze and compare current methods of developing the resistance function

for open web steel joists with experimental data using static tests.

1.3 Thesis Overview

Chapter 2 covers a literature review of the methods and techniques used in the

thesis. This chapter includes an explanation detailing the current knowledge of explosives

and blast waves. It also contains numerical integration techniques for dynamic response

calculation. The equivalent blast load procedure is discussed, as well as methods for

determining the resistance function for open web steel joists.

In Chapter 3 the ANSYS LS-DYNA program is verified for static and dynamic

loads before using the program to develop responses for roof models subjected to blast

loads. These responses are compared to the response from the equivalent blast loading.

Also, the equivalent blast loading procedure is compared to field test data of a roof

subjected to explosive loads.

In Chapter 4, the static testing procedure for open web steel joist samples is

presented. The results from the static tests are compared to current resistance function

techniques used by design engineers. The failure mechanisms present in open web steel

joists are also discussed.

Finally, Chapter 5 summarizes the analysis of the research and presents

recommendations for future work.


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Chapter

2Literature Review

2.1 Introduction

Understanding and solving the problem statement requires basic knowledge of

blast effects and the responses for roof structures. In this chapter current techniques used

to describe blast effects will be discussed. In addition, several design methods for roof

structures will be introduced.

2.2 Blast Waves

Explosions can be caused by physical, nuclear, or chemical events. A common

example of a physical explosion occurs when a pressure vessel fails. Pressure is released,

along with fragments of the vessel. Changing the structure of atomic nuclei produces

nuclear explosions. Vast amounts of energy can be released in short periods by breaking

the bonds between protons and neutrons. Nuclear fission divides the nuclei of heavy

atoms. Nuclear fusion combines nuclei from light atoms. Nuclear explosions generate

kinetic energy, internal energy, and thermal energy.


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The rapid oxidation of fuel elements develops chemical explosions. This reaction

releases heat and produces gas, which expands. Low-end explosives create quasi-static

loads. High explosives (chemical and nuclear) in a surrounding medium, such as air or

water, cause shock waves in the medium. The blast releases high-pressure gases at high

temperatures. These gases naturally expand, and the surrounding medium is consequently

compressed (Smith and Hetherington, 1994).

The compressed medium, or for the specific case of air, forms a shock front. The

shock front travels in a radial direction. As the explosive gases cool and slow their

movement, the amount of overpressure the shock front carries decreases. The gases

release energy to reach equilibrium towards the atmospheric pressure. However, due to

the high pressure and mass of the gases, more expansion is necessary to actually reach

equilibrium. This causes the pressure in the shock wave to drop below the atmospheric

pressure. After sufficient underpressure is expended, the state returns to the

atmospheric pressure. The air behind the shock front also places a load, a drag force, on

objects encountered (Smith and Hetherington, 1994).

The general shape of a pulse shape is shown in Figure 2-1. Important factors

pertinent to burst pressures include the peak pressure, the duration, the air density behind

the shock front, the velocity of the shock front, and the impulse of the blast pressure.

There are several derived equations that calculate the shock front velocity Us, peak

dynamic pressure qs, and air density behind the shock front sbased on the peak

overpressure, ambient air pressure, and the speed of sound in air at the ambient pressure.

Examples of some of these equations are shown in Eqns. 3.1, 3.2, 3.3 (Smith and

Hetherington, 1994).


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Figure 2-1: Generalized Blast Pressure History (TM 5-1300, 1990).

U p p

pas

s= +

6 7

7

0

0

0 (2.1)

ss

s

p p

p p

= +

+

6 7

7

0

0

0 (2.2)

q p

p ps

s

s

=+

5

2 7

2

0( ) (2.3)

where ps= peak static overpressure

p0= ambient air pressure in front of shock wave

0= air density in front of shock wave

a0= speed of sound in air at air pressure


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The peak overpressure is related to a factor called the scaled distance, Z (Eqn.

2.4). This is proportional to the distance from the charge and the cubed root of the charge

mass. Typically the charge mass is measured in terms of TNT, and other types of

explosives are converted to this mass type. As the distance increases, the maximum

pressure of the shock wave decreases. The total duration of the shock burst actually

increases. It should also be noted that at any particular range, the peak overpressure of the

blast wave decays exponentially to the atmospheric pressure (Biggs, 1967).

Z RW

= 1 3/ (2.4)

where R = distance from blast source

W = mass of charge in terms of TNT

When blast waves strike a surface, the overpressure increases. The pressure from

the expanding gases build up since there is no medium to compress and displace.

Therefore, the burst pressures from a surface explosion are larger than an explosion

occurring in the air. The overpressure of a surface burst is approximately twice that of a

free-air burst (Smith and Hetherington, 1994). Next, the discussion of blast loads

continues to how they affect buildings and objects.


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2.3 Effects on Structures

Three main loading conditions are available, as explained by Smith and

Hetherington, 1994. In the first type a relatively large shock wave reaches a structure

relatively small enough that the blast wave encloses the entire structure. The shock wave

effectively acts on the entire structure simultaneously. Additionally, there is a drag force

from the rapidly moving wind behind the blast wave. The structure is, however, massive

enough to resist translation. The second condition also involves a relatively large shock

wave and a target much smaller than the previous case.

The same phenomena happen during this case, but the target is sufficiently small

enough to be moved by the dynamic, drag pressure. In the final case, the shock burst is

too small to surround the structure simultaneously and the structure is too large to be

shifted. Instead of simultaneous loading, each component is affected in succession. For a

typical building, the front face is loaded with a reflected overpressure.

Figure 2-2: Blast Loading on Structure (Forbes, 1999).


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As stated in the previous section, reflected pressure accumulated when the blast

wave meets material denser than the medium in which it is traveling. Air molecules are

stopped by this material and compressed further by the shock front behind them. This

reflected pressure decays over time. Blast pressures simultaneously affect walls with the

same pressure-time history on the entire wall area. This means that while the blast

pressures change with respect to time, these pressures are evenly distributed on every

portion of the wall. Also, due to the perpendicular orientation of the front wall to the

shock wave, the blast pressure is reflected and magnified (Forbes, 1999). The roof and

sides of the building react instead to incident or side-on pressures. Similar to how the

peak pressure decays and load duration grows as the range from the charge increases, the

incident pressure attenuates as it transverses the roof and sides, as shown in Fig 2-2. In

the presence of a sloped roof, the pressure magnifies due to reflection. After the roof and

sides are surpassed, the pressures converge on the back of the building. Again, a

reflecting effect amplifies the overpressure. In addition to this blast wave diffraction, drag

forces load the structure (UFC, 2002).

2.4 Blast Resistant Design

Engineers employ several different methods for the structural design of structures

resisting blast loads. This can range from simplified hand calculations to utilizing

computer programs programmed to deal with complex loading and resistances. Any

method used takes into account the knowledge of blast loads described in the previous

section and the component resistances that make up the structure (Morison, 2006).


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2.4.1 Single-Degree-of-Freedom Model

The first explosive design methods are based on the single-degree-of-freedom

(SDOF) model. This model idealizes an entire structure or structural component as one

point in the structure. The resistance at this point is also taken as the resistance for the

entire structure. The equation of motion for a linear elastic, damped single-degree-of-

freedom system follows (Biggs, 1967).

My ky cy F f t&& & [ ( )]+ + = 1 (2.5)

where M = mass of structure

k = structural stiffness

c = damping coefficient

F1= constant force value

f(t) = nondimensional time value

Figure 2-3: Damped, Single-Degree-of-Freedom System.

y

F1

M


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The dot superscripts represent partial derivatives with respect to time. On dot

signifies velocity; two dots mean acceleration. For simplicity, the damping coefficient is

assumed to be zero. The equation simulates the response of lumped mass-spring system

(Biggs, 1967). Solution of this equation (without damping) leads to Duhammads, or

Convolution, Integral (Eqn. 2.6). This gives the deflection-time history.

y y t y

t y f t d stt

= + + 0 0cos &

sin ( ) sin ( )

(2.6)

where y0= initial displacement

&y = initial velocity

= circular frequency k M/

yst= static deflection F1/k

f() = non-dimensional load-time function

The military has used two main SDOF methods to deal with explosive threats.

The first technique, the Modal SDOF method, turns up in a 1946 manual Fundamentals

of Protective Design (Non-Nuclear) EM 1110-345-405, reissued in 1965 as TM 5-855-1

and not superseded until 1986. In this method, the response period is taken as the natural

period of the first mode under free vibration. Normalized curves were created to aid in

calculation of maximum deflection for various dynamic loads and linear elastic, pure

plastic resistance curves. The most notable dynamic load used was a triangular load with

zero rise time.

An example of this non-dimensional chart is shown. This method only analyzes

simplified systems and was not ideal for more general loading histories and resistance

functions. It also undercalculates the deflection response and reaction response (Morison,

2006).


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Figure 2-4: Idealized Dynamic Response Curves for Triangular Loading (Morison, 2006).

The equivalent SDOF method was widely published in 1957 in parts of the

USACE manual Design of Structures to Resist the Effects of Atomic Weapons, EM

1110-345-415 Principles of Dynamic Analysis and Design, and EM 1110-345-416

Structural Elements Subjected to Dynamic Loads. While the Modal SDOF method uses

the actual mass and loading in the system, this alternative method determines equivalent

values of mass, resistance, and loading for the lumped mass-spring system based on the

distribution of the structures mass and the systems loading.

Using the conservation of kinetic energy, internal strain energy and external work,

transformation factors can be found to change the systems mass, resistance and loading

into their lumped mass counterparts. These transformation factors are also based on the

deflected shape function of the system as a whole from a particular reference point,

usually the middle point of a structure (Biggs, 1967).


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k

Accurate shape functions result in accurate solutions. When the shape function is

not as well known, the solution becomes more approximate and more errors are entered.

The static deflection shape under the same loading distribution as the dynamic load is

relatively easy to calculate and results in very accurate answers. The equation of motion

for this equivalent SDOF system is given in Eqns. 2.6a and 2.6b. This can be solved in

the same manner as the Modal SDOF models. Analysis of the equation allows for the use

of only one transformation factor KLM, the ratio of the mass factor to the load factor. It

should also be noted that the transformation factors for load and resistance are the same

(Biggs, 1967).

M y k y F te e e

&& & ( )+ = (2.6a)

or

K My K ky K F t M L L&& & ( )+ = (2.6b)

where KM= mass transformation factor

KL= load/resistance transformation factor

M = total mass

k = stiffness

F = total load

y

p

Figure 2-5: Equivalent SDOF System.


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2.4.2 CONWEP

The Conventional Weapons Effect Program (CONWEP) (Hyde, 1992) uses blast

test data and empirical equations to calculate load data for different types of explosives

with varying loading sequences. This program outputs parameters of the pressure-time

history including side-on and reflected pressure, incident and reflected impulse, time of

arrival, positive phase duration, shock front velocity, and peak dynamic pressure. This

program only computes data corresponding to the positive pulse of a blast load.

2.5 Roof Systems

This section discusses design techniques for roof systems subjected to blast loads.

Looking at the general equation of motion, the mass of a structure is a well-known value.

The loading function and, more importantly, the resistance are the less realized

components. Different methods for determining these quantities are presented.

2.5.1 Equivalent Blast Load

The Unified Facilities Criteria 3-340-01 (which supersedes Army Manual TM 5-

855-1/Airforce AFPAM 32 1147/NAVYFAC P-1080/DSWA DAHSCWEMAN-97)

formulates an equivalent uniform blast loading for the roof and sidewalls of a structure.

The shape of this load is a triangular pulse with an unequal rise and decay times. The

equation is shown in Eqn. 2.7. The load factor is a function of the Lwb/L, which is the

ratio of the length of the blast wave at the back of the roof to the length of the roof. The

drag coefficient is a function of the dynamic pressure. This equivalent load is calculated

using values from the CONWEP program.


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P C P C qor E sob d ob= + (2.7)

where Por= peak pressure

Psob= peak incident pressure at back of roof

CE= equivalent load factor

qob= dynamic pressure at back of roof

Cd= side-on element dynamic drag coefficient

CONWEP data is taken at a range equal to the radial distance from the blast

source to the back of the roof. The length of the shock wave is calculated from the shock

front velocity and the positive phase duration of the shock front. Using Fig. 2-6, the load

factor and the ration of D/L can be computed, where D is the point on the roof where the

peak pressure is said to occur. The shape of the loading function, as shown in Fig. 2-8,

demonstrates that the maximum pressure on the roof takes time to build. As the shock

front transverses the roof, the changing pressure-time history accumulates to a peak

value. In Fig. 2-8, tfis the time the shock wave hits the roof. The time to reach the peak

pressure point is defined as td. The loading ends after the blast wave reaches the end of

the roof and the duration of the loading expires, tband tofb,respectively.

Figure 2-6: Equivalent Load Factor and Blast Wave Location Ratio (DAHS, 2002).


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Figure 2-7: DAHS Equivalent Loading Technique.

Figure 2-8: DAHS Equivalent Load (TM 5-1300, 1990).

2.5.2 Resistance Function

There are several different types of roof structures. Some types include concrete

slab and I-beam composites, open web steel joists (OWSJ) and metal decking, steel joists,

purlins and decking.


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For these systems, the I-beam and the steel truss are the significant components

with respect to design. The behavior of I-beams is well-known and documented.

However, the response of open web steel joists to failure is not well documented.

Conventional engineering designs assume a simple bilinear linearly elastic, perfect plastic

resistance function. The Steel Joist Institute (SJI) has developed charts to ascertain the

maximum allowable load per length for every truss as a function of joist type and length

(SJI, 2005). The allowable deflection is L/360, where L is the effective joist length. In

addition, the Steel Joist Institute provides an equation for the approximate joist moment

of inertia, based on the maximum live load and the effective length of the truss. This

equation is described in Eqn. 2.8. The derivation for this moment of inertia originates

from the stiffness of a simply supported beam under a uniform load and includes a 15%

increase in deflection due to elongation in the truss web. Along with the equivalent

SDOF method, current engineering practices define a resistance function shape for use in

design.

I w Lj LL span= 26 767 103 6. ( )( ) ( ) (2.8)

where wLL= maximum allowable live load of joist (lb/ft)

Lspan= effective length (ft)

A current engineering method of deriving the resistance formula takes into

account the properties of the joist members and the Steel Joist Institute load

specifications. The truss can be designed as a simply supported beam with a uniformly

distributed load. Using the design tensile strength of the web members as a limit, the

elastic section modulus of the joist cross-section can be computed.


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Using this value and the joist material characteristics, an effective moment

capacity can be computed. The maximum moment for a simply supported beam under

uniform load can be determined from the load and the length of the beam. Included into

this maximum moment calculation are strength increase factors for the joist materials.

Using this relationship, the ultimate resistance can be computed. Similarly, using the

deflection equation for the same system, the stiffness and elastic deflection limit can be

calculated. The maximum response is calculated using numerical techniques and/or

idealized charts based on the ratios of load duration to the joist period and resistance

magnitude to the forced load magnitude.

SBEDS (2004) uses a slightly different approach. For one, the moment of inertia

is calculated from the actual cross-sections of the joist, including the top chord, the

bottom chord, and some factor for the web. These calculations prove to result in higher

moments of inertia for a given steel truss compared to the Steel Joist Institutes

approximate calculation. The program defines a value Lshear, which is the maximum joist

length designed according to SJI with the maximum total allowable load. For K-Series

joists this maximum total load is 550 lb/ft. For the deeper joists LH and DLH, this

maximum shear load is based on individual truss and is detailed in the SJI Load Tables

document. The effective moment capacity is calculated from the maximum shear load

and the Lshearquantity, assuming a simply-supported beam with a distributed load.

SBEDS program back-calculates a resistance value from the joist length and effective

moment capacity. This resistance value is also factored using strength increase factors.

The elastic deflection limit and stiffness are calculated using the simply supported beam

assumption mentioned earlier.


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Chapter

3Verification of

Equivalent BlastLoad Procedure

3.1 Introduction

Blast loads are dynamic, impulsive, and non-uniform. An experimental blast load

has been formulated to emulate the response of a blast load acting on a roof system. This

loading resembles a triangular pulse with a rise time and decay time. This loading shape

has been in use for over fifty years. With the advent of more advanced computer

technology and finite element analysis programs, current techniques for expressing blast

loads on the roofs of structures can be evaluated. This chapter explains the programs and

procedures used in blast and dynamic loading estimation.


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3.2 Review of ANSYS Program

Finite element analysis is a numerical method of deconstructing a structure or

system into smaller, simpler discrete regions. These regions, also called finite elements,

are linked and controlled by several governing equations, including equations of

equilibrium, compatibility equations and constitutive relations. The computer solves

these equations simultaneously using different numerical schemes to express the behavior

of the structure as a whole. In general the number, size, and shape of the elements control

the program results.

ANSYS was created in 1970 by Dr. John Swanson (Swanson Analysis Systems,

Inc.). The program ANSYS ED Release 9.0 is used in this research. This is a student

level version meant for academic use. It supports many of the capabilities present in the

full version of ANSYS Multiphysics with some limitations on types and size of elements,

material types, and degrees of freedom. ANSYS is able to construct static and dynamic

structural analysis (both linear and non-linear), heat transfer problems, fluid flow

problems, acoustics, and electromagnetic analyses.

The product ANSYS LS-DYNA is a result of the partnership between ANSYS,

Inc. and Livermore Software Technology Corporation and was first introduced in 1996. It

is an explicit non-linear structural simulation used for dynamic analyses. This is

applicable for structures experiencing large deformations and short time durations,

including impact test simulations, drop test problems, explosive simulations, and metal

forming.


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For ANSYS use, three main steps must be performed preprocessing, completing

a solution, and post processing. First, a two- or three-dimensional model of the structure

or system has to be formed. The entire structure can be modeled or parts of the structure

can be modeled. Symmetry can also be used to simplify the construction of the model.

Next, the different material properties that make up the model have to be described and

applied to the models geometry. The system can now be discretized into elements. The

number, size and type of element is based on the analysis type and the users judgment.

Once the system is idealized into finite elements with their own properties, physical

constraints and different loading types can be positioned on the model. The user directs

the program to solve the problem and output results. These results can be illustrated using

tables, graphs, or contour plots. The ANSYS program includes a graphical user interface

(GUI) to aid in executing these steps. To gain confidence in ANSYS, static and dynamic

analyses were performed using simple models. These analyses are also beneficial in

validating the use of ANSYS in analytical experimentation. The next sections depict the

methods used for static and dynamic loads.

3.3 Static Load Simulations

The ANSYS program was used to compare deflection curves derived from

kinematic boundary conditions with the results obtained from ANSYS finite element

modeling. A linearly elastic, isotropic BEAM-type element was used in the modeling. Its

dimensions were 4 x 18 x 20. The beam was modeled as being simply supported. The

material of the beam is modeled as steel.


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3.3.1 Preprocessing

The element type used in this analysis is called BEAM4. It is described as a

uniaxial element with tension, compression, torsion, and bending capabilities. The

element contains two nodes at either end. Each node (shown as I and J in Fig. 3-1) has six

degrees of freedom from translations in the local x, y, and z directions and rotations in the

local x-, y-, and z-axes. This element is used in static analyses to simulate the reactions of

an elastic, untapered section in 3D. Figure 3-1 shows the rotational aspects of the beam

element. The numbers depict the six different surfaces on the element. This technique is

commonly used during the loading stage to determine on which face to apply the load.

For different elements, the ANSYS program supplies different ways of detailing

each section. The GUI offers an automated section development that allows users to

choose the shape of the element cross-section and its dimensions. For BEAM4 elements,

a set of real constants is inputted. These constants include the cross-sectional area, the

weak and strong moments of inertia, the cross-sectional dimensions, and the mass density

per unit length.

Different material models for several structural materials are available in the

ANSYS program. For this model, a linear elastic, isotropic material model was used. The

modulus of elasticity (and also the mass density) is chosen for this particular material

type.


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Figure 3-1: BEAM 4 ANSYS element (ANSYS, 2004).

For the creation of a simple beam, keypoints are chosen as the locations of the

beams ends. A straight line is drawn between these keypoints. A third keypoint is also

drawn to define the orientation of the element. In the Figure 3-1 this point is denoted as

node K. The next step is to mesh the line. This will apply elements along the lines

geometry. The user defines the number and size of the finite elements. The ANSYS ED

version limits the number of possible nodes, lines, and elements. The user can apply the

real constants and material attributes to the element mesh and use some node K to orient

the elements. The line is then meshed into the number of prescribed elements.


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3.3.2 Loading and Solution

The structure under consideration is a simply supported beam. To that effect, the

node at one end of the beam is constrained from movement in the x, y, and z directions;

the node at the other end of the beam is constrained in the x and y direction. Next, loads

are placed on one surface of the beam elements to simulate bending about the weak axis

of the beams cross-section. This model is then solved.

3.3.3 Post-Processing

ANSYS provides several post processing options. A list of the nodes and

positions in the global directions can be provided, or a graphical representation of the

original and deflected position of the model can be drafted. A contour of the stresses in

the beam model can be generated.

3.3.4 Results

Five different static load configurations were tested. Each configuration contained

only distributed loads. For each static loading, three test beams were modeled. Each beam

was divided into a different number of segments. The tests were designed to determine

how the finite element model deflection compared to expected results and how

discretization of the model affects results. Each static load was tested on a beam model

consisting of 1 segments, 6 segments, or 1 segments, or conversely 20 beam elements,

40 beam elements, or 240 beam elements, respectively.


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The first static load is a uniform static load. The loading amount is designed to

result in a discernable deflection curve, while keeping the stress of the beam in the elastic

range. As can be seen in Figure 3-2, the expected deflection curve as well as the different

curves from the three test beams are fairly accurate. The displacement of the beam

reaches maximum in the center of the beam, as expected. This loading also results in the

greatest amount of deflection from all five curves. The shape of the results is consistent

with known kinematics. From the graph, there appears to be little difference between the

calculated values and each ANSYS test.

0

1

2

3

4

5

6

0 5 10 15 20

Distance from Support (ft)

Deflection(in)

Defl. Eqn.

1' Beam Elements

6" Beam Elements

1" Beam Elements

0

50

100

150

200

250300

350

400

0 5 10 15 20


Load(plf)

Figure 3-2: Static Loading #1; Maximum loading = 375 lb/ft.


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The next static load is a triangular loading that decays over the length of the

beam. The displacements for the three ANSYS tests also appear fairly close to expected

answers. Due to the centroid of the loading, the maximum deflection is off the center of

the beams length. The total load on the beam is half that of the uniform loading, and the

range of the displacement reflects this.

0

0.5

1

1.5

2

2.5

3

0 5 10 15 20


Deflection(in)

Defl.Eqn

1' Beam Element6" Beam Element

1" Beam Element

0

100

200

300

400

0 5 10 15 20


Load(plf)

Figure 3-3: Static Load #2; Maximum Load = 375 lb/ft.

The third static load is also triangular; however it rises to a peak halfway across

the beams length and descends the rest of the length to zero. The total load of this curve

is the same as that of Static Load #2. However, the centroid of the loading is at the center

of the beams length and it is balanced about the center of the beam. Little difference can

be discerned from the different test beams in this graph. The maximum loading is located

in the center of the beam. The final two static loads are neither symmetrical nor are they

continuous along the beam.


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0

0.5

1

1.5

2

2.5

3

3.5

4

0 5 10 15 20


Deflection(in)

Defl. Eqn.

1' Beam Element

6" Beam Element

1" Beam Element

0

100

200

300

400

0 5 10 15 20


Load(plf)

Figure 3-4: Static Load #3; Maximum Loading = 375 lbs/ft.

Static Load #4 consists of three varying uniform loads acting on three different

equal length sections of the beam. While the deflection curves of the previous three static

loads can be easily found in a book or manual (AISC, 2001), the derivation of the

deflections for Static Loads #4 and #5 involves calculating the internal moment in the

beam and integrating twice. Also, the kinematic boundary conditions of the beam state

that the slope and deflection must be continuous along the beam and at the changes in

loading.

The general shape of the displacement curves is customary to the expected results.

The loads center of gravity is about 9 from a support, and the maximum beam

dislocation is located about an inch from the center of the beams length. The total load is

little more than half of Static Load #1, and the deflection is in the right scale.


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The curves for the 1 length beam element and for the 6 length beam element are

discernable Figure 3-5 around the center of the beam. The test with the smallest element

length (conversely the largest number of elements) appears to coincide with that of the

deflection equation.

0

0.5

1

1.5

2

2.5

3

0 5 10 15 20


Deflect

ion(in)

Defl. Eqn.

1' Beam Element

6" Beam Element

1" Beam Element

0

100

200

300

0 5 10 15 20


Load(plf)

Figure 3-5: Static Load #4.

As can be seen by the deflection curves of Static Load #4, the 1 beam segment

overestimates and the 6 curve underestimates the expected result. It can also be seen that

the difference in expected and calculated values decreases with increased discretization.

Static Load #5 has similar results. It is made up of three different uniform loads acting on

equal length sections of the beam, along with one extra section that remains unloaded.

The curves all underestimate the deflection, but they arrive closer to the expected value

as the element size increases. Again, due to the loading schematic, the location of

maximum deflection is offset from the center, and the curves reflect this.


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0

0.25

0.5

0.75

1

1.25

1.5

0 5 10 15 20


Deflection(in)

Defl. Eqn.

1' Beam Element

6" Beam Element

1" Beam Element

0100200300

0 5 10 15 20


Load(plf)

Figure 3-6: Static Load #5.

The ANSYS results appear to resemble the expected values, for the most part. To

further examine this, the maximum deflection and the locations of maximum deflection

are compared to the derived deflection equation. As can be seen from Figures 3-2 through

3-6, the continuous loads have less error than discontinuous loads. In addition, the

symmetrical loads (Static Loads #1 and #3) have very small deviations from the expected

values. And, as expected, the percentage of error decreases as the element size decreases.


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Finally, the location along the beam model of the maximum displacement is

compared. The number of elements in the model determines the number of nodes in the

beam. Each node is spaced the length of an element, and so accuracy of the displacement

and location of displacement is limited by the length of an element. For each loading, the

location of the maximum deflection is no more than half a foot from the center of the

beam. The closest the 1 beam element models can measure is 1, so there is a relatively

larger error for every loading. The smaller the element, the more accurate the

measurement is. For Static Loads #1 and #3, the maximum deflection is located at the

center of the beam, so there is no error for any beam model.

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

4.50%

5.00%

1' 6" 1"

Element Size

Error

Static Load 1

Static Load 2

Static Load 3

Static Load 4

Static Load 5

Figure 3-7: Percentage Error of Maximum Deflection.


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0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

4.50%

5.00%

1' 6" 1"Element Size

Error

Static Load 1

Static Load 2

Static Load 3

Static Load 4Static Load 5

Figure 3-8: Percentage of Error of Max Deflection Location.

From these tests, it can be concluded that ANSYS provides good values for

displacement, with less than 5% error. If available, the smaller element results in close to

exact answers, but larger element sizes can give reasonable answers. The following

section details the process of evaluating the dynamic analysis capabilities of ANSYS.

3.4 Dynamic Load Simulations

The ANSYS program can be also be used to model structures subjected to

dynamic loadings. However, a sub-program called LS-DYNA exists in ANSYS that is

more suited towards explicit solutions and dynamic loads that are applied over short

durations. LS-DYNA computes solutions faster than ANSYS and can supply more

information (i.e. data points). On the other hand, ANSYS has more graphical support

than LS-DYNA.


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The LS-DYNA program is used to verify the maximum deflection of a simply

supported beam under different pulse loadings. An equivalent single degree of freedom

(SDOF) model was used to determine both the maximum deflection of the beam under a

uniformly loaded dynamic load and the shape of the time-history curve of this deflection.

This maximum value and the shape of the graph as a whole were compared to the LS-

DYNA results. The same beam dimensions from the static tests were used. Also, the

magnitude and duration of the dynamic load were constant for each pulse loading.

3.4.1 Preprocessing

The beam element used in this analysis, BEAM 161, consists of a node at either

end of the element (I and J) and an orientation node (node K) on a different line than that

of the beam direction. This element is used for explicit dynamic analyses only, and can

be used to model isotropic, linear elastic materials. It can model beams of different cross-

section shapes. Each node has six physical degrees of freedom translation and rotation

in three directions and also three nodal velocity degrees of freedom and three nodal

acceleration degrees of freedom. These latter are not technically degrees of freedom, but

are computed and stored as such. The BEAM 161 element has two main methods of

calculation. A Hughes-Liu beam element assumes a constant moment along its length and

detects stresses in the center. A Belytschko-Schwer element generates a linearly changing

moment along the element and measures stresses at either end.

The cross-sectional dimensions are defined as real constants. Next, the material

properties of the beam are defined. The modulus of elasticity and Poissons ratio can be

inputted.


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Figure 3-9: BEAM 161 ANSYS LS-DYNA element.

For this model, the material was assumed to be steel, with a Youngs modulus of

30 x 106psi and a Poissons ratio of 0.3. Also, because this is an explicit solution, the

weight of the structure is considered a necessary value; therefore, another material

property is the mass density. The beam can be modeled very simply with a line drawn

between two keypoints. A third keypoint that is not collinear with either endpoint is also

necessary for element orientation. The line that makes up the beam can be divided into a

certain number of elements or a particular element size. This is necessary from a practical

point to have a node in the center of the beam to measure the maximum deflections. So,

at the very least the beam model has to be divided into 2 elements. After construction of

the line keypoints and line, the beam can be meshed into the desired number of elements

with the set material and geometrical properties.


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3.4.2 Loading and Solution

Three different pulse loadings were used to test the dynamic load capabilities of

LS-DYNA. A rectangular pulse loading with instantaneous acceleration was tested

because it is a very simple, uniform load. The two other loadings tested are triangular

pulse loads. The first triangular pulse load is an instantaneous loading with a linear decay

to zero. The second pulse loading has an equal time to rise from zero and time to decay to

zero. The first triangular pulse loading allows for comparison to results with the

rectangular loading and is similar to the pulse loading from a blast load. The second

triangular load is also used for comparison and is similar to the equivalent loading of a

roof structure subjected to a blast load. All three dynamic loads have equal magnitudes

and durations.

These pulse loads were inputted into the model as array parameters. The loads

were all linear, and very few points need to be inputted. The program interpolates the

points between those given. Two arrays are defined for each load. One array gives the

time values and the other the load values. The elements that make up the beam can be

defined as components where the loads will be acting. The pulse loads are uniformly

distributed across the length of the beam, and are set to act in the center of the beams

cross-section. The test beam is designed as a simply supported, and so translational

constraints are put on the endpoints.

The duration of the analysis can then be specified and the program can be directed

to output ANSYS solutions or LS-DYNA solutions. ANSYS solutions have a limit of

100 points and are usually used to examine the deflection of an entire structure at a

specific time.


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The LS-DYNA solutions have a limit of 1000 points, and because of their large

size are usually limited to observe the time history of important points or nodes on the

beam. The purposes of these tests do not necessitate a very large number of output points.

The length of time under investigation should be the length of at least half a response

period.

3.4.3 Post-Processing

ANSYS has the capabilities to display the deflection of the entire beam model at a

particular time, or graph the displacement time history of a particular node on the beam.

For these tests the time history of the center node of the beam is graphed.

3.4.4 Results

For each dynamic load the model divides the beam into sets of 2 elements,

6 elements, 10 elements, 20 elements, 40 elements, and 240 elements; or element lengths

of 10, 3 4, 1, 6, and 1 respectively. The LS-DYNA program calculates the time

steps used in the numerical integration based on the dimensions and properties of the

beam and the element length (ANSYS, 2004).

t L

c

=0 9. (3.1)

where L = length of element

c = wave propagation velocity


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c E

=

(3.2)

where E = modulus of elasticity

= mass density

The wave propagation velocity (Eqn. 3.2) is calculated from the Youngs

Modulus and density of the beam, and the smallest time step (Eqn. 3.1) used in the

program is the ratio of the smallest element length in the model and the propagation

velocity. For all models the element length is uniform. Six models were used with

different lengths of beam element. The following table shows the number of beam

elements, respective lengths, and used and corresponding time steps.

Table 3-1: Minimum Time Steps. Table 3-2: Aspect Ratio.

Number

of

Elements

Element

Length

(ft)

Smallest

Time Step

(sec)

2 120 0.0005342

6 40 0.0001781

10 24 0.0001068

20 12 5.342E-05

40 6 2.671E-05

240 1 4.451E-06

Number

of

Elements

Length:

Depth

Ratio

2 30:1

6 10:1

10 6:1

20 3:1

40 3:2

240 1:4

As can be seen, the more elements in a beam (i.e. the shorter the element length),

the smaller the time steps are. It would be assumed that the smaller the time steps used in

numerical integration, the more accurate the solution would be. It must be noted that too

small of a time step could lead to a numerical instability in which the solution is not

bounded and the resulting deflection approaches infinity. This was not encountered in

this investigation.


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The ANSYS test data was compared to a time history calculated using a single

degree of freedom (SDOF) model and load and mass factors to correct for the shape of

the load and the shape of the beam (refer to Sec. 2.4.1 for details).

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60 70 80 90

Time (msec)

Deflection(in)

2 elem (LS-DYNA)

6 elem (LS-DYNA)

20 elem (LS-DYNA)

240 elem (LS-DYNA)

Duhammad's Integral

0

200

400

0 2 4 6Time (msec)

Load(p

lf)

Figure 3-10: Dynamic Pulse Load #1.

Dynamic Load #1 is a rectangular pulse load with an instantaneous rise and

decay. Several deflection time history curves are shown in Figure 3-10. The beam model

with two elements and only one node between the endpoints drastically underestimates

the response. Using six elements results in a displacement within 20% of the theoretical

response. Coincidentally, this is the result without including the load and mass factors.

Subsequent decreases in element length arrive to a more accurate response. The period

response also becomes more and more exact. The beginning curvature of the graphs

demonstrates zero initial displacement and zero initial velocity. However, the more

elements modeled, the more warped the graphs become.


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Dynamic Load #2 is a triangular pulse load with an instantaneous rise and a linear

decay to zero. The time history curves resemble those for Dynamic Load #1. This

displays the precision of the ANSYS program. Again, the graphs show a zero initial

deflection and a zero initial velocity (Fig. 3-11). The accuracy of the answer increases

with the decrease in element length. The maximum displacement is less than that of the

first dynamic loading, which is the expected result. The response period is the same,

which is consistent with the period of Dynamic Load #1 and with the natural period of

the beam model. Unfortunately, the curves also become more and more wavy with the

addition of elements. In addition to the shape, the waviness affects the time of maximum

deflection. This causes a difference between the calculated and expected time of

maximum response.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30 40 50 60 70 80 90

Time (msec)

Deflection(in)

2 elem (LS-DYNA)

6 elem (LS-DYNA)

20 elem (LS-DYNA)240 elem (LS-DYNA)

Duhammel's Integral

0100200300400

0 2 4 6

Time (msec)

Load(plf)

Figure 3-11: Dynamic Load #2.


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Dynamic Load #3 is also a triangular pulse load with an equal time to rise and

time to decay. The time history curves are very similar to that of Dynamic Load #2 in

terms of shape and magnitude. The similar shapes are due to the accuracy of the analysis.

The close responses result from the equal impulse of the pulse loads. Due to the beam

models properties, it is an impulsive sensitive system. The resistance of the beam is less

related to the peak load than to the pulses impulse.

The shape of the curves relates to the aspect ratio of the beam elements. The depth

of the beam, and thus the depth of each element, measures 4 inches. The aspect ratio of

the largest beam element (as shown in Table 3-2) is 30:1, and the aspect ratio of the

smallest beam element is 1:4. The elements with aspect ratios less than 5:1 are specified

as deep beams, and have different kinematic and equilibrium conditions than those of

regular beams. The ANSYS code chosen was designed only for regular beams.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30 40 50 60 70 80 90

Time (msec)

Deflection(in)

2 elem (LS-DYNA)

6 elem (LS-DYNA)

20 elem (LS-DYNA)

240 elem (LS-DYNA)

Duhammel's Integral

0

200

400

0 2 4 6Time (msec)

Load(plf)

Figure 3-12: Dynamic Load #3.


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To further explore the ANSYS results, the maximum deflection, time of

maximum response, and period are compared to the expected values. They are compared

according to the number of finite elements and the type of loading. Figure 3-13 shows a

clear decrease in maximum deflection error with an increase in element number. This

trend is common for every pulse load.

Figure 3-14 presents the error in calculating the time of maximum deflection. The

beam model results in a value within 3% of the SDOF models prediction. Unfortunately,

the accumulation of elements past that level also increases the amount of error. The error

in the magnitude of maximum displacement for the 20-element beam is less than 2% for

all loads.

0%

10%

20%

30%

40%

50%

60%

2 6 10 20 40 240

Number of Elements

Error

Dynamic Load #1

Dynamic Load #2

Dynamic Load #3

Figure 3-13: Percentage Error in Maximum Response.


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0%

5%

10%

15%

20%

25%

30%

35%

2 6 10 20 40 240

Number of Elements

Error

Dynamic Load #1

Dynamic Load #2Dynamic Load #3

Figure 3-14: Percentage Error in Time of Maximum Response.

Finally, the period of response is examined for the test beams. Figure 3-15 shows

a marked decrease in error with the increase in element number. The error for the 20-

element beam is about 3%, while the error for the 240-element beam is close to zero.


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0%

4%

8%

12%

16%

20%

2 6 10 20 40 240

Number of Elements

Error

Dynamic Load #1

Dynamic Load #2

Dynamic Load #3

Figure 3-15: Percentage Error in Response Period.

In conclusion, the ANSYS/LS-DYNA models result in answers close to the

expected values. None are the answers are perfect, and additional programming with the

LS-DYNA and other finite element analysis software needs to be performed to determine

if these are currently the most accurate outcomes. For the current purposes, the error is

small enough to be applicable. In the next section, the response of beams under blast

loads is explored. For these analyses, an aspect ratio between 3:1 and 3:2 appears to

result in the best solutions in terms of maximum deflection, time of maximum deflection,

and response period. An element length of 1 is used for the next section.


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3.5 Analysis of Blast Load Response Using LS-DYNA

The same steps as before are used to generate a model of a simply-supported

beam. Twenty beam elements make up the beam, corresponding to an element length of

1 each. The responses of the blast loads are compared to the response of the equivalent

dynamic blast load using ANSYS. The response of the equivalent loading is also derived

using Duhammads integral.

3.5.1 Application of Blast Loads

The pressures on a structure from a blast are non-uniform and highly impulsive.

The load pulses on a roof contain a positive downward phase and a negative section

phase. The peak pressures decay and the load duration increases as the blast wave

traverses the roof length. Using ANSYS, different sections of the beam require different

loading characteristics at different times. The loading response is analyzed in stages.

First, the positive pressure phase is idealized using a triangular pulse with an

instantaneous rise time and a time to decay. For each subsequent section of the beam, the

peak load decreases and the time history of the loading changes to represent the

accentuation of the blast pulse. Next, the same process is repeated using both positive and

negative phase data. Finally data from an actual blast test is used in analysis.

To reflect the variable loads, the beam model is divided into five differing loading

sections. The loads on each section have a different initial load time, peak load, and final

time. The loads are inputted as time and load array parameters similar to the pulse

loadings in Sec. 3.3. Instead of one set of time and load arrays, there are five sets of time

and load arrays. The test beam is divided into five separate components, where each set

of time and load data is applied to each component.


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Figure 3-16: Locations of CONWEP measurements.

Figure 3-17: Blast Pressure Distribution.

The application CONWEP (Hyde, 1992) generates time of arrival, incident peak

pressure, positive phase duration, and incident impulse output used in the arrays. In

CONWEP, a hemispherical surface explosive of 1000 lbs of TNT is analyzed at a range

of 30 feet from the simply-supported beam. Pressure information is recorded at ranges of

30 feet (the front end of the beam), 35 feet, 40 feet, 45 feet, and 50 feet (corresponding to

the back end of the beam). These ranges are shown in Figure 3-16.

Table 3-3: Peak Pressure vs. Range.

30 35 40 45 50

Range of

Data

Origin

(ft)

Peak

Pressure

(psi)

Range of

Data

Distribution

30.0 134.2 30-32'

35.0 94.92 32-37'

40.0 69.83 37-43'

45.0 53.23 43-48'

50.0 41.75 48-50'


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These five loading patterns are distributed along the beam to replicate actual blast

characteristics. The pressure from the 30 feet range is distributed (as shown in Fig. 3-17)

over the first two feet of the beam, from 30 to 32. The pressure from the 35 feet range is

allocated from 32 to 37. The 40, 45, and 50 feet range data is applied over the next 6, 5,

and 2 feet, respectively. Table 3-3 displays the peak pressure and the range to which it

relates. The beam model supports line loads instead of surface loads. The pressures are

multiplied by the width of the cross-section to calculate the associated distributed load.

3.5.2 Results (Positive Phase Only)

The actual blast loads can be seen in Fig. 3-18. In the actual blast pressures, the

peak decays in a nonlinear manner to zero. The first trial positive phase pressures decay

linearly over the same length of time (see Fig 3-19). An important check in this analysis

compares the impulses from the actual data to the approximate data. It can be shown that

the first trial impulses are from 2 to 7 times that of the actual pulses (see Fig. 3-20). To

correct this, the pulses were modified to a bilinear decay. From the simple dynamic

models performed earlier, it can be ascertained that the beam model is impulse sensitive.

Therefore, the actual shape should not matter, and the arrangement of the bilinear decay

should be irrelevant.

This decay model is performed with two separate examples (Figs. 3-21 and 3-22).

One shape follows the pattern of decreasing primary and secondary peaks, along with

increasing durations. The third scenario keeps the bilinear decay form without following

a discernable configuration.


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0

20

40

60

80

100

120

140

0 5 10 15 20 25 30Time (msec)

IncidentPressure(psi)

Beam Segment 1

Beam Segment 2

Beam Segment 3

Beam Segment 4Beam Segment 5

Figure 3-18: Actual Blast Loading.

0

500

1000

1500

2000

2500

0 0.005 0.01 0.015 0.02 0.025 0.03

Time (sec)

Load(lbs/in)

Beam Segment 1

Beam Segment 2

Beam Segment 3

Beam Segment 4

Beam Segment 5

Figure 3-19: Loading Scenario #1.


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0

200

400

600

800

1000

1200

1400

1600

30.0 35.0 40.0 45.0 50.0

Distance (ft)

IncidentImpulse(psi-ms

ec) Real Data

Trial #1 Data

Figure 3-20: Impulse Comparison.

0

500

1000

1500

2000

2500

0 0.005 0.01 0.015 0.02 0.025 0.

Time (sec)

Load(lb/in)

Beam Segment 1

Beam Segment 2

Beam Segment 3

Beam Segment 4

Beam Segment 5



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0

500

1000

1500

2000

2500

0 0.005 0.01 0.015 0.02 0.025 0.03

Time (sec)

Load(lbs/in

)

Beam Segment 1

Beam Segment 2

Beam Segment 3

Beam Segment 4

Beam Segment 5


To further continue validation of the input data, the beam is divided into four

equal-sized components instead of five different sized components. Four different pulses

load these sections. For Scenario #4, loads from 30, 35, 40, and 45 feet range act on the

four segments. This is repeated in Scenario #5 using loads from the 35, 40, 45, and 50

feet range.

The dynamic responses from these five loading combinations are plotted together

(Fig. 3-23) and the im

Documents

Blast Resistant Design for Roof System