© 2018 Kyle Willemsufdcimages.uflib.ufl.edu/UF/E0/05/27/66/00001/WILLEMS_K.pdf · 4...
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AN ENERGY FLOW BASED APPROACH TO STRUCTURAL RESPONSE ASSESSMENT By KYLE WILLEMS A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2018
Transcript of © 2018 Kyle Willemsufdcimages.uflib.ufl.edu/UF/E0/05/27/66/00001/WILLEMS_K.pdf · 4...
AN ENERGY FLOW BASED APPROACH TO STRUCTURAL RESPONSE ASSESSMENT
By
KYLE WILLEMS
A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
2018
© 2018 Kyle Willems
To my parents, fiancée, family and friends, who have supported me through all of my endeavors
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ACKNOWLEDGMENTS
I would like to thank AFRL for funding this research, Dr. Ohrt for his technical
support, Dr. Krauthammer for his guidance and support throughout this project, and all
of the other students at CIPPS for their assistance in my successful completion of this
research.
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TABLE OF CONTENTS page
ACKNOWLEDGMENTS .................................................................................................. 4
LIST OF TABLES ............................................................................................................ 7
LIST OF FIGURES .......................................................................................................... 8
ABSTRACT ................................................................................................................... 10
CHAPTER
1 INTRODUCTORY REMARKS ................................................................................ 11
1.1 Problem Statement ........................................................................................... 11 1.2 Objective ........................................................................................................... 12
1.3 Research Significance ...................................................................................... 12 1.4 Scope and Organization ................................................................................... 12
2 LITERATURE REVIEW .......................................................................................... 14
2.1 Introduction ....................................................................................................... 14 2.2 Abnormal Loadings ........................................................................................... 14
2.3 Simplified Analysis and Structural Responses .................................................. 17
2.3.1 Single Degree of Freedom Systems ........................................................ 17
2.3.2 Dynamic Structural Model Responses ..................................................... 20 2.3.2.1 Flexural response .......................................................................... 21
2.3.2.2 Direct shear response .................................................................... 21 2.3.3 Failure Modes of Structural Elements Due to Dynamic Loads ................ 23
2.3.3.1 Flexural failure ............................................................................... 24
2.3.3.2 Direct shear failure ......................................................................... 25 2.4 Development of Load-Impulse Diagrams .......................................................... 26
2.4.1 Characteristics of Load-Impulse Diagrams .............................................. 27
2.4.2 Dynamic Loading Regimes ...................................................................... 29 2.4.3 Analytical Solutions for Load-Impulse Diagrams ..................................... 32
2.4.3.1 Closed form solutions ..................................................................... 32
2.4.3.2 Energy balance method ................................................................. 33 2.4.4 Numerical Approaches for Load-Impulse Diagrams ................................ 35
2.4.4.1 Branch tracing technique ............................................................... 36 2.4.4.2 Asymptotic search algorithm .......................................................... 37
2.4.4.3 Constant pressure search algorithm .............................................. 38 2.4.4.4 Pivot point search algorithm ........................................................... 39
2.5 Energy Flow Approach ...................................................................................... 40 2.5.1 Characteristics of Energy Flow ................................................................ 40 2.5.2 Derivation of Energy Components Spectrum .......................................... 43
2.5.2.1 Derivation of energy functions ........................................................ 43
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2.5.2.2 Distinction of loading regimes ........................................................ 45
2.5.2.3 Plot of energy component spectrum............................................... 48
2.5.3 Energy Based Solutions for Load-Impulse Diagrams .............................. 50 2.5.4 Storage Tank Analogy ............................................................................. 57 2.5.5 Energy Tank Analogy .............................................................................. 57
2.6 Development of an Energy Based Equivalent to Load-Impulse Diagrams ........ 61 2.7 Numerical Simulations using Dynamic Structural Analysis Suite ...................... 67
2.8 Summary .......................................................................................................... 68
3 METHODOLOGY ................................................................................................... 70
3.1 Introduction ....................................................................................................... 70 3.2 Validation of P-I to E-R Conversion Equations ................................................. 70
3.3 Relevance of Load Pulse Shape Factor (Beta) ................................................. 72 3.3.1 Influence of Beta Value on P-I and E-R Diagrams Domains ................... 73 3.3.2 Proposed Relationship between Beta Values and Scaled Distance ........ 73
3.4 DSAS P-I Pivot Point Improvements ................................................................. 74 3.5 Energy Flow Analysis ........................................................................................ 76
4 RESULTS AND DISCUSSIONS ............................................................................. 78
4.1 Introduction ....................................................................................................... 78 4.2 Validation of Energy Conversion Equations ...................................................... 78
4.2.1 Analytical E-R vs. DSAS Converted P-I (Rectangular Load) ................... 79 4.2.2 Analytical E-R vs. DSAS Converted P-I (Triangular Load) ...................... 80
4.3 Importance of Beta Value ................................................................................. 82 4.3.1 Influence of Beta Value on Three Domains of a P-I Diagram .................. 82
4.3.1.1 Impulsive domain ........................................................................... 82 4.3.1.2 Quasi-static domain ....................................................................... 83
4.3.1.3 Dynamic domain ............................................................................ 83 4.3.2 Proposed Beta Value Plot on UFC 3-340-02 Blast Charts ...................... 83 4.3.3 Examples of Beta Value Influence on P-I and E-R Diagrams .................. 86
4.4 DSAS Improved P-I Pivot Point Analysis .......................................................... 88 4.5 DSAS Existing P-I vs. Improved P-I Analysis Comparison ............................... 90 4.6 Energy Flow Analysis ........................................................................................ 94
5 CONCLUSIONS AND RECOMMENDATIONS ....................................................... 97
5.1 Conclusions ...................................................................................................... 97
5.2 Recommendations for Future Research ........................................................... 99
LIST OF REFERNECES ............................................................................................. 100
BIOGRAPHICAL SKETCH .......................................................................................... 102
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LIST OF TABLES
Table page 2-1 Analytical expressions for dynamic reaction of beam [11]. ................................. 23
2-2 Load regime summary [3]. .................................................................................. 31
2-3 Structural properties of dynamic analysis [2]. ..................................................... 41
2-4 Results from dynamic analysis [2]. ..................................................................... 41
2-5 Energy functions for undamped SDOF under block loads [2]. ............................ 45
2-6 Max energy functions for undamped SDOF under block loads [2]. ..................... 47
2-7 Energy components for undamped SDOF under block load [2]......................... 49
2-8 Energy solutions for simple SDOF systems [1]. .................................................. 52
2-9 Summary of energy based solutions for P-I diagrams [2]. .................................. 53
4-1 Comparison of original P-I runtime to improved P-I runtime ............................... 92
4-2 Energy deposition amounts from energy releasing events. ................................ 94
4-3 Equivalent triangular load from energy depositions. ........................................... 96
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LIST OF FIGURES
Figure page 2-1 Idealized Loading Profiles a) Rectangular Load b) Triangular Load. .................. 16
2-2 Response spectra for simplified loads [6]. .......................................................... 19
2-3 Equivalent SDOF systems for structural element [5]. ......................................... 20
2-4 Deformed shape for direct shear response [5].................................................... 22
2-5 Flexure shear interactions for RC beams without stirrups [15]. .......................... 25
2-6 Shear stress-slip relationship for direct shear. .................................................... 26
2-7 Typical response spectrum [7]. ........................................................................... 27
2-8 Typical load-impulse diagram [7]. ....................................................................... 28
2-9 Branch-tracing technique. ................................................................................... 36
2-10 Search algorithm (a) Flexure, (b) Direct shear [5]. .............................................. 37
2-11 Search algorithm [4]. .......................................................................................... 38
2-12 Polar coordinate search algorithm [6]. ................................................................ 39
2-13 Load pulses and responses for P-I domains [2].................................................. 40
2-14 Energy transition history for Impulsive Regime [2]. ............................................. 42
2-15 Energy transition history for Dynamic Regime [2]. .............................................. 42
2-16 Energy transition history for Quasi-Static Regime [2]. ........................................ 42
2-17 Typical energy components spectrum [2]. .......................................................... 48
2-18 Typical response spectrum and load-impulse diagram [5]. ................................. 51
2-19 Common Resistance Functions. ......................................................................... 51
2-20 Illustration of energy based solutions for P-I diagrams [2]. ................................. 56
2-21 Comparison of P-I diagrams solutions [2]. .......................................................... 56
2-22 Energy tank analogy for two simplified response modes [2]. .............................. 58
2-23 Flow chart for the energy tank analogy [2]. ......................................................... 60
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2-24 Typical energy based P-I diagrams for multi-failure modes [2]. .......................... 61
2-25 Energy components spectrum for block loads [2]. .............................................. 63
2-26 Energy difference spectrum for rectangular load pulse [2]. ................................ 63
2-27 Energy difference vs energy input rate diagram [2]. ........................................... 64
2-28 Typical energy-based P-I diagrams for idealized load pulse [2]. ........................ 66
4-1 E-R diagram comparison for rectangular load pulse. .......................................... 80
4-2 E-R diagram comparison for rectangular load pulse. .......................................... 80
4-3 E-R diagram comparison for triangular load pulse. ............................................. 81
4-4 E-R diagram comparison for triangular load pulse. ............................................. 81
4-5 Calculated β values. ........................................................................................... 84
4-6 Calculated β values. ........................................................................................... 85
4-7 Simplified load pulses and their corresponding β values. ................................... 86
4-8 P-I diagrams for a range of simplified load pulses. ............................................. 87
4-9 E-R diagrams for a range of simplified load pulses. ........................................... 87
4-10 Impulsive and quasi-static asymptotes with initial pivot point. ............................ 88
4-11 New pivot point that captures portions of asymptotes. ....................................... 89
4-12 New pivot point and P-I curve that results from DSAS. ...................................... 90
4-13 Load pulses for P-I runtime comparison. ............................................................ 91
4-14 Triangular Load P-I Comparison. ........................................................................ 92
4-15 Exponential Load P-I Comparison. ..................................................................... 93
4-16 Bi-Linear Load P-I Comparison. ......................................................................... 93
4-17 E-R diagram for a flexural failure ........................................................................ 94
4-18 E-R diagram with overlaid energy deposition curve. ........................................... 95
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Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science
AN ENERGY FLOW BASED APPROACH TO STRUCTURAL RESPONSE
ASSESSMENT
By
Kyle Willems
August 2018
Chair: Theodor Krauthammer Major: Civil Engineering
A load-impulse (P-I) diagram is a well-accepted tool that is commonly used for
structural strength design as well as for potential damage assessment. This tool has
been used for many years to aid structural engineers in their assessments. Recently, an
energy-based alternative to P-I diagrams, known as Energy vs. Energy Rate (E-R)
diagrams, was proposed. Two simple conversion equations were also proposed to
transform P-I data into E-R data. This diagram serves as another tool for a structural
engineer to use and gives the ability to analyze a structural element from an energy
based perspective.
This research provides an in depth check of the validity of the energy based P-I
diagram by comparing the energy approach to generally accepted forms of structural
analysis. The analytical solutions for rectangular and triangular loads that were
previously proposed will be checked against software, such as DSAS, for validation. In
addition, improvements to the DSAS P-I Analysis module will be presented to increase
the numerical efficiency of the program. The energy flow based approach will be used to
analyze structural elements under severe dynamic loads.
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CHAPTER 1 INTRODUCTORY REMARKS
1.1 Problem Statement
A load-impulse (P-I) diagram and an energy input-energy input rate (E-R)
diagram are examples of a response spectra that allow for a quick evaluation of the
state of a structural element subjected to a dynamic load [1] [2]. The P-I diagram allows
the user to work in the load vs. time domain and determine what levels of load and
impulse are required for a structure to reach a desired response level [1] [3]. The E-R
diagram allows the user to work in the energy vs. time domain to determine what levels
of input energy and input energy rate must be delivered to the system to achieve the
desired response.
Energy is the driving force behind every part of structural analysis. Loads bring
energy into a structural system and that energy is dissipated by the system, resulting in
deformations and displacements. By acquiring a better understanding of how structures
react from an energy standpoint, a protective design engineer will be able to design
safer structures that are more resilient to severe dynamic loads.
The first use of E-R diagrams was performed for a linear elastic single-degree of
freedom (SDOF) system subjected to a simplified load pulse [2]. In order to apply the
use of E-R diagrams to realistic structural systems, the accuracy and implementation of
E-R diagrams for non-linear systems must be studied. This includes looking at how the
diagrams work when using non-simplified loads as well as more complex structures.
The study of energy flow and energy rate as it relates to structural response will be the
focus of this research study.
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1.2 Objective
The objective of this study is to validate the P-I to E-R conversion equations,
which were presented by [2], for specific cases. The study of energy flow, as it relates to
the various domains of P-I and E-R diagrams, will also be explored. The following goals
must be achieved for the objective of this research to be met:
Validate and confirm the P-I to E-R conversion equations for linear-elastic SDOF systems subjected to idealized load pulses using various methods.
Determine the relevance of the load pulse shape factor, 𝛽
Propose methods to generate P-I and E-R diagrams more efficiently
Examine more realistic scenarios using the E-R diagram approach
1.3 Research Significance
This research will present a method that allows for a more efficient use of P-I and
E-R diagrams. The findings from [2] for the behavior of linear-elastic SDOF systems will
be applied to nonlinear systems with varying load pulses. This will expand the domain
that the E-R diagram is currently valid for. By doing this, more realistic cases can be
studied and evaluated. The findings from this research can help further the
understanding of how energy flow influences a structural system, and can be used as a
tool for protective systems assessment and design in the future.
1.4 Scope and Organization
This research will be focused on validating the conversion equations from P-I
diagrams to E-R diagrams for multiple cases. Several cases must be explored to fully
validate the use of E-R diagrams for cases other than linear-elastic SDOF systems
subjected to simplified load pulses. The calculation of P-I diagram asymptotes, P-I
points, and energy flow relationships will be discussed in this proposal. Several
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methodologies are presented in an effort to streamline the process of getting to a P-I or
an E-R diagram.
An in-depth literature review is presented in Chapter 2. This section provides a
summary of what work has been done on the topic of P-I and E-R diagrams. The
research approach is presented in Chapter 3. The results are described in Chapter 4,
and conclusions and future research recommendations will be discussed in Chapter 5.
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CHAPTER 2 LITERATURE REVIEW
2.1 Introduction
Load-impulse (P-I) diagrams are routinely used to preliminarily design protective
structures by defining safe response limits for a given loading scenario. For a typical
dynamic structural analysis, a designer is usually interested in the final states of a
structure rather than a detailed report of the response history [1]. Results from dynamic
analyses may be shown in the form of a response spectrum, such as a load-impulse
diagram.
Numerous studies have been conducted to derive load-impulse diagrams both
theoretically and numerically [1] [4] [5] [6]. It was determined that closed-form solutions
for load-impulse diagrams can be found for specific simplified load pulses on idealized
structures. Several numerical approaches have also been developed to determine P-I
diagrams for any given structure with a defined resistance function [1]. In addition to
using numerical procedures, P-I diagrams are also able to be generated by using
energy balance principles, such as Conservation of Momentum and Conservation of
Mechanical Energy [2] [7].
Damage criterion can be based on experimental data [8], limited deformation [3]
and residual load carrying capacity. The damage criterion is selected by the user
depending on need and definition of the criterion.
2.2 Abnormal Loadings
In typical structural analysis and design, structures are designed for various
types of loads, such as dead, live, wind, and seismic loads. Other loads, which may
have a low probability of occurring, may cause catastrophic failures if they do occur.
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These events are known as abnormal loadings. Most of these types of loads are time-
dependent, such as a blast or impact load. Abnormal loadings can be caused due to
faulty design/construction processes as well as natural, accidental, or intentional events.
In protective design, these types of events must be accounted for. To account for
abnormal loadings, it is important to first understand how structures behave under these
loads and what design methodology is required to minimize the possible negative
effects of such loadings. Structural response under abnormal loadings is typically both
dynamic and non-linear, both in material behavior and geometrically.
In many fields of engineering, there is a growing concern about explosions and
other accidents that cause damage. Blast events typically occur from chemical,
physical, or nuclear explosions. Impact events often occur from severe collisions.
Impact and blast events can affect many mechanisms; however, this study will focus
only on the effects of these abnormal loadings of structures and structural elements.
Local damage is important to consider when dealing with an impact load on a
structure. This includes penetration, perforation, scabbing, and punching shear. In
addition to considering local damage, it is also important to consider the global
response of a structure, such as flexural bending.
A blast event is characterized by a sudden release of energy. This sudden
release of energy causes a shockwave to form in front of the explosive gases. As these
explosive gases expand, the pressure from the shockwave decreases. The pressure
decreases until it becomes slightly lower than the atmospheric pressure, which causes
a reversal of pressure flow with the direction now heading back towards the explosion
location. The pressure eventually returns to atmospheric equilibrium [8]. The
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overpressure formed from the blast event is typically the most important load in blast
design, but other pressures could also influence the response of a structure. Blast wave
loading is transient in nature, which can be assumed to be spatially uniform for most
cases, except for cases of close range explosions.
Pressure-time history plots for both impact and blast events can be complex and
difficult to utilize in design. These plots are often idealized to simplify the design
process. Simplified, idealized load pulses, such as rectangular and triangular load
pulses, are commonly used as approximations of these complex loadings. A
conservative estimation of a blast load can be seen in Figure 2-1(b), where Line I
characterizes a linear decay of the blast pressure. Line II in Figure 2-1(b) and the
dashed line Figure 2-1(a) represent simplified loads that preserve the impulse, meaning
the area under the actual and idealized P-T history are equal [9].
Figure 2-1. Idealized Loading Profiles a) Rectangular Load b) Triangular Load.
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2.3 Simplified Analysis and Structural Responses
As mentioned previously, loading functions are commonly simplified to make
analyses easier to perform. This process is also done to structural systems to reduce
the computational efforts needed to perform an analysis. A very common example of
this simplification is a reduction of a structural element into a single degree of freedom
(SDOF) system. Even though a large amount of information that has been gathered
over recent history about dynamic structural analysis, numerous design manuals and
computational tools rely on the ability to use SDOF systems [10].
2.3.1 Single Degree of Freedom Systems
All structures contain more than one degree of freedom (DOF). These structures
can however be represented as a series of SDOF systems for analysis [1] and used to
find various responses of the structure. These include the displacement at any given
time, 𝑥(𝑡), the velocity at any given time, �̇�(𝑡), and the acceleration at any given time,
�̈�(𝑡). When a structure is subjected to a time-dependent load and is displaced from its
original position, the response can be described by the equation of motion, as shown in
the following equation [11].
𝑀�̈�(𝑡) + 𝐶�̇�(𝑡) + 𝐾𝑥(𝑡) = 𝐹(𝑡) (2-1)
where,
M = Concentrated mass of the system 𝐶 = Damping coefficient which defines energy dissipation 𝐾 = Elastic spring stiffness 𝐹(𝑡) = Time-varying forcing function
�̈�(𝑡) = Acceleration at any given time
�̇�(𝑡) = Velocity at any given time 𝑥(𝑡) = Displacement at any given time
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Equation 2-1 is used in dynamic analysis to account for two forces that are not
seen in static analysis. These two forces are the inertial force, 𝑀�̈�(𝑡), and the viscous
damping force, 𝐶�̇�(𝑡). The effect from the viscous damping force is usually small;
however, the effects from the inertial force could be substantial and may control the
response [1]. Due to the very short durations of loading and times to maximum
response of a structure in blast design, the effects of viscous damping are typically
ignored. The dynamic equation of motion for an undamped, elastic SDOF system is
shown in the following equation.
𝑀�̈�(𝑡) + 𝐾𝑥(𝑡) = 𝐹(𝑡) (2-2)
It is important to select a SDOF system that accurately describes the response of
the actual structure. The deflection of the SDOF system is equivalent to the point of
interest on the structure, such as the mid-span of a beam with a uniformly distributed
mass. The time-scale of the loading and response is not altered when using a SDOF
system, so the time of a response on the SDOF system corresponds directly to the time
of the response on the actual structure [11].
The two phases in which a maximum response may occur are the forced-
vibration phase and the free-vibration phase. The forced-vibration phase occurs during
the time of loading, 0 ≤ 𝑡 ≤ 𝑡𝑑. The free vibration phase occurs after the loading has
ended, 𝑡 > 𝑡𝑑. Peak responses, such as maximum displacement or acceleration, can be
plotted in response spectra. A plot of response spectra that summarizes the maximum
response of SDOF systems subjected to various load pulses over a large range of
natural periods (T) can be seen in Figure 2-2.
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Figure 2-2. Response spectra for simplified loads [6].
Although a SDOF system may not sufficiently describe the comprehensive
response of a structure, it does allow for a quick and easy solution that provides useful
information to the designer. Some of this information includes dynamic characteristics
such as frequencies and responses. SDOF systems are often used in preparation for a
more intensive and advanced analysis [10]. When creating a response spectrum, such
as a P-I diagram, a large number of dynamic analyses need to be completed. This
makes using simple SDOF systems more attractive than other intensive methods, such
as Timonshenko beam or Mindlin plate, based on computation time [1].
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2.3.2 Dynamic Structural Model Responses
Krauthammer et al. [12] offered an analytical technique to obtain dynamic
structural responses of reinforced concrete beams and one-way slabs under an
impulsive load. This technique is based on the well-accepted equivalent SDOF system
approach as outlined by Biggs [11]. The method presented by Krauthammer et al. [12]
utilizes variable response-dependent parameters for SDOF systems such as equivalent
mass and equivalent load. The method also considers two loosely coupled equivalent
SDOF systems, such as flexure and direct shear, instead of the traditional SDOF
approach, which normally only takes flexure into account. This loosely-coupled
equivalent SDOF system can be seen in Figure 2-3. The two main responses that will
be discussed in the study are the global flexural response and the local direct shear
response.
Figure 2-3. Equivalent SDOF systems for structural element [5].
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2.3.2.1 Flexural response
The equation of motion can be modified for the flexural response shown in Figure
2-3(b) [12]. This new form of the equation is shown below.
�̈�(𝑡) + 2𝜉𝜔�̇�(𝑡) +𝑅
𝑀𝑒=
𝑃𝑒(𝑡)
𝑀𝑒 (2-3)
where,
𝜉 = Flexural viscous damping ratio 𝜔 = Flexural circular natural frequency 𝑅 = Flexural dynamic resistance function
𝑃𝑒(𝑡) = Equivalent loading function
𝑀𝑒 = Equivalent mass of system The response described above has a nonlinear relationship since the variable
system parameters, such as the flexural resistance, is dependent on the response of
the system. This requires the use of a numerical integration tool, such as the Newmark-
Beta method [13]. The Newmark-Beta method is commonly used because of its ease of
application and stability [14].
2.3.2.2 Direct shear response
The equation of motion can be modified for the direct shear response shown in
Figure 2-3(c). This form of the equation can be written as [12]:
�̈�(𝑡) + 2𝜉𝑠𝜔𝑠�̇�(𝑡) +𝑅𝑠
𝑀𝑠=
𝑉(𝑡)
𝑀𝑠 (2-4)
where,
𝜉𝑠 = Direct shear damping ratio 𝜔𝑠 = Direct shear circular natural frequency 𝑅𝑠 = Direct shear dynamic resistance function
𝑀𝑠 = Equivalent shear mass of system 𝑉(𝑡) = Dynamic shear force (reaction)
�̈�(𝑡) = Direct shear slip �̇�(𝑡) = Direct shear velocity
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The nonlinear equation of motion for direct shear, Equation 2-4, can also be
solved numerically using the Newmark-Beta method [13]. The terms shear mass and
dynamic shear force are briefly described in more detail next.
The equivalent shear mass is calculated from the assumed mode/deformed
shape of a structural element with a direct shear failure [2]. The failure can either occur
at one end prior to the other end of the element, or at both ends at the same time. In the
case where one element end fails before the other, the deformed shape of the beam is
assumed to be triangular, as seen in Figure 2-4. The shear mass in this case is one-half
of the total mass of the beam. In the second case, where both ends fail simultaneously,
the shear mass is the total mass of the beam.
Figure 2-4. Deformed shape for direct shear response [5].
The dynamic shear force, also known as the dynamic reaction, can be found by
considering dynamic equilibrium of the structural element. Biggs [11] listed analytical
expressions for dynamic shear forces for beams with idealized boundary conditions.
These expressions can be seen in Table 2-1. The expressions for dynamic reactions
are described using the dynamic resistance function R(t) and the forcing function P(t). It
is also possible to express the same reactions in terms of inertia force, the product of
mass and acceleration, as well as the forcing function [2].
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Table 2-1. Analytical expressions for dynamic reaction of beam [11].
Type of Loading Strain Range Dynamic Reaction 𝑉(𝑡)
Uniformly distributed Elastic 0.39𝑅(𝑡) + 0.11𝑃(𝑡)
Plastic 0.38𝑅𝑚∗ + 0.12𝑃(𝑡)
Point load at mid-span Elastic 0.78𝑅(𝑡) − 0.28𝑃(𝑡)
Plastic 0.75𝑅𝑚 − 0.25𝑃(𝑡)
Point load at one-third of span Elastic 0.525𝑅(𝑡) − 0.025𝑃(𝑡)
Plastic 0.52𝑅𝑚 − 0.02𝑃(𝑡)
𝑅𝑚∗ is the maximum plastic resistance
2.3.3 Failure Modes of Structural Elements Due to Dynamic Loads
There are multiple mechanisms in which a reinforced concrete (RC) element may
fail under dynamic loads. Failure mechanisms can be divided into two categories. The
first category is local failure, which includes penetration, perforation, and punching
shear. The other category is global failure, such as flexural failure, which can be looked
at with or without the effects of shear. There are two principle shear modes that should
be considered when looking at RC. These shear modes are diagonal shear, often called
diagonal tension/compression, and direct shear, which is commonly referred to as
pure/dynamic shear. The element must exhibit some sort of flexural behavior for a
diagonal shear case; however, no flexural behavior is needed for a direct shear case.
This is because the stresses caused in a direct shear case are localized and due to
some sort of discontinuity, either geometrically or in the loading. It was found that for
diagonal shear, the failure extends over a length roughly equal to the member’s depth
[1].
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2.3.3.1 Flexural failure
There are many ways in which RC beams can fail. One failure mechanism is a
flexural mode which causes plastic hinges to form at locations that reach the ultimate
bending capacity of the section. Members may also fail in a combined flexure-shear
mode. This mechanism happens when tension/flexural cracks form in the shear span of
the beam. In any case, failure is defined as the condition in which the structural element
cannot withstand any increase in external loading without undergoing large irreversible
deformations. Normal moment-curvature relationships are going to be used to describe
the flexural behavior of RC beams. In addition to this, three basic assumptions must be
made [2]:
Plane sections remain plane before and after bending
The strain in the concrete and reinforcement at the same level is the same
Stresses in both the reinforcement and the concrete can be calculated by using the stress-strain curves for each material.
One factor that has been shown to significantly decrease the flexural strength of
a RC beam section is the influence of diagonal shear [1]. There have been many
studies that considered the effects of shear reinforcement, span to depth ratio, and
ultimate moment [12] [15] [16]. The true ultimate moment capacity of beams was found
to be dependent on both the shear reinforcement ratio and the span to depth ratio of the
beams. The results from the study performed by [15] are shown in Figure 2-5. To find
the ratio of the actual moment capacity of the section, 𝑀𝑢, to theoretical moment
capacity of the section, 𝑀𝑓𝑙, two variables must be solved for. These variables are the
shear span to depth ratio, 𝑎 𝑑⁄ , and the reinforcement ratio, 𝜌 = 𝐴𝑠 𝑏𝑤𝑑⁄ . Analytical
models were proposed to account for the decrease in flexural strength due to the effects
25
of diagonal shear [16]. The models presented by [16] have been incorporated into the
structural analysis code DSAS [17].
Figure 2-5. Flexure shear interactions for RC beams without stirrups [15].
2.3.3.2 Direct shear failure
A direct shear failure is characterized by a sudden localized failure in a RC
element. These failures can appear at locations of discontinuity, either a geometric
discontinuity or a loading discontinuity. One common area of geometric discontinuity are
edges of members. Direct shear failure has been studied in limited detail [10] [12] [16].
A model for direct shear was proposed by [18] which is based on the shear
stress to shear slip relationship. This model describes shear transfer in RC members
with well anchored main reinforcement, with the absence of compressive forces in the
static region. The original stress-slip relationship was later enhanced to account for
compression membrane and rate effects that result from dynamic loads [16]. This
26
enhanced relationship was created by applying an increase factor of 1.4 to account for
the effects mentioned above, as shown in Figure 2-6.
Figure 2-6. Shear stress-slip relationship for direct shear.
2.4 Development of Load-Impulse Diagrams
During the Second World War, iso-damage curves were created from
experimental and real-life event results. Some of the data came from bombs that were
released on brick houses in the United Kingdom. The results gathered were used to
determine safe standoff distances in the UK [8]. While load-impulse diagrams are
traditionally used to assess structural elements, they can also be used to evaluate
human response to blast events, since the human body acts as a mechanical system
[3].
27
2.4.1 Characteristics of Load-Impulse Diagrams
A response spectrum is defined as a plot of a maximum response to the ratio of
load duration to natural period, 𝑡𝑑 / T. Response spectra are commonly used to simplify
the design of dynamic systems for a given type of load [19].
Figure 2-7. Typical response spectrum [7].
An example of a response spectrum is shown in Figure 2-7. This figure shows a
plot of maximum displacement to static displacement. Static displacement is calculated
by dividing the peak pressure by the stiffness of the structure. The axes in Figure 2-7
can be transformed into a new set of axes, as shown in Figure 2-8. By using a new set
of axes, the same response spectra can be represented in different ways to show all
possible combinations of load and impulse that result in the same level of structural
response [7]. One example of this is shown in Figure 2-8. The ratio of the maximum
28
displacement to the static displacement can be inverted and the old abscissa can be
multiplied by a new ordinate.
Figure 2-8. Typical load-impulse diagram [7].
Since 𝑃𝑡𝑑 = 𝐼 can be substituted in and the constant, 2𝜋, can be removed, the
new abscissa can be established. This results in a dimensionless load-impulse diagram.
The response of the structure is a function of the loading impulse, maximum dynamic
displacement, mass and structural stiffness of the system [6].
A typical response spectrum and a load-impulse diagram both describe the
relationship between a time dependent characteristic of the structure and the maximum
value of the response limit [5]. The response spectrum shows deformations as a
function of scaled time, while the load-impulse diagram shows combinations of peak
load and impulse that cause a certain level of damage [3].
29
On a load-impulse diagram, the load is typically defined by a peak force or
pressure. The impulse is the area under the load-time history curve. The curve that
represents a load-impulse diagram separates the diagram into two distinct regions.
Combinations of peak load and impulse that fall to the right and above the defined curve
represent damage that is more than the allowable limit defined by the curve. Points that
fall directly on the load-impulse curve will also represent damage higher than the limit
defined. Combinations that fall below or to the left of the curve represents cases that
would not trigger the maximum response that the curve defines. Load-impulse diagrams
can be developed for any limit of deformation or damage level. Utilizing multiple curves
on a single plot allows for a better understanding of what a specific load will do to a
structure. The curves divide the plot into multiple regions, with each region representing
a specific level of damage [2].
2.4.2 Dynamic Loading Regimes
The response of a structural element can be divided into three zones in structural
dynamics. These zones are related to the ratio of the natural period of the structure to
the duration of the loading [3]. The three zones, or regimes, are known as the impulsive,
quasi-static, and dynamic regimes.
In the impulsive regime, the load duration is significantly lower than the response
time of the structure. This means that the load is applied and removed well before the
structure sustains significant deformations [8]. The deformation in this zone is directly
related to the amount of impulse imparted to the system during the loading, which is
why it is called the impulsive zone. The mass and stiffness of the structure influence the
deformation in this region. In a P-I diagram, the impulsive regime reaches an asymptotic
value, known as the impulsive asymptote.
30
In the quasi-static regime, the load duration is significantly higher than the
response time of the structure. This means that the load is applied and the structure
reaches its maximum response before the load is removed. In this region, the maximum
dynamic deformation is two times the static deflection. The deformation in this region
depends only on the peak load and the structure’s stiffness. The load-time history and
mass of the structure do not influence the response of the structure. This region of the
P-I diagram also reaches an asymptotic value, known as the quasi-static asymptote.
The final region of the diagram is the dynamic region. This region connects the
impulsive and quasi-static regions. The duration of loading and the response of the
structure are roughly equivalent. The deformations in this region depend completely on
the actual load-time history. The pressure, impulse, mass and structural stiffness all
influence the response of the structure [3].
For an undamped, perfectly elastic system, [3] quantified the three loading
regions for a structure subjected to an exponentially decaying load, as shown in Table
2-2. For rectangular, triangular, and sinusoidal load pulses with 𝑡𝑑/𝑇 ratios of less than
0.25, the structural response can be assumed to be impulsive Biggs [11].
31
Table 2-2. Load regime summary [3].
Illustrations
Loading
range Impulsive Dynamic Quasi-Static
Pressure
Design
range
High Intermediate Low
Design load Impulse Pressure-Time Pressure
Incident
pressure ≫ 100 𝑝𝑠𝑖 < 100 𝑝𝑠𝑖 < 10 𝑝𝑠𝑖
Load
duration Short Intermediate Long
Response
time Long Intermediate Short
𝑡𝑚
𝑡𝑑
𝑡𝑚
𝑡𝑑> 3
𝑡𝑚
𝑡𝑑> 3
𝑡𝑚
𝑡𝑑< 0.1
Approximate
limits
𝑡𝑑
𝑇< 0.0637 0.0637 <
𝑡𝑑
𝑇< 6.37
𝑡𝑑
𝑇> 6.37
32
2.4.3 Analytical Solutions for Load-Impulse Diagrams
Analytical solutions for load-impulse diagrams can be found in closed form as
well as by using principles of energy conservation. Each of the two methods will be
described below.
2.4.3.1 Closed form solutions
Load-impulse diagrams can be solved mathematically for a few special cases.
This type of solution requires an idealized structure that undergoes a simplified load,
such as a rectangular or triangular load, and are known as closed-form solutions.
Exact solutions for P-I diagrams are shown below. For the case of an undamped
elastic SDOF system, with a circular natural frequency, 𝜔, subjected to a rectangular
load pulse with a loading duration, 𝑡𝑑, the displacement functions are [2] [7]:
𝑥(𝑡) =𝑃0
𝐾(1 − 𝑐𝑜𝑠𝜔𝑡) 0 ≤ 𝑡 ≤ 𝑡𝑑 (2-5)
𝑥(𝑡) =𝑃0
𝐾(𝑐𝑜𝑠𝜔(𝑡 − 𝑡𝑑) − 𝑐𝑜𝑠𝜔𝑡𝑑 𝑡 ≥ 𝑡𝑑 (2-6)
The dimension-less force and impulse terms can be written as:
�̅� =𝑃0/𝐾
𝑥𝑚𝑎𝑥 (2-7)
𝐼 ̅ =𝐼
𝑥𝑚𝑎𝑥√𝐾𝑀 (2-8)
Finally, the expressions that represent the horizontal and vertical axes of the P-I
diagram also represent the structural responses in the forced and transient domains.
�̅�𝑠𝑖𝑛𝐼 ̅
2�̅�=
1
2 (2-9)
�̅� =1
2 (2-10)
33
In a similar fashion, the same relationships can be derived for an undamped
elastic SDOF system subjected to a triangular load pulse with zero rise time. While
these expressions are slightly more complex than the previous derivations, they still
yield an exact energy solution for the triangular load pulse. The P-I relationship is
defined by the following expressions [5]:
(2𝐼 ̅
�̅�2)2 = 2 + (
2𝐼 ̅
�̅�)2 −
4𝐼 ̅
𝑃sin (
2𝐼 ̅
�̅�) − 2𝑐𝑜𝑠 (
2𝐼 ̅
�̅�) 1 ≤ 𝐼 ̅ ≤ 1.166 (2-11)
(2𝐼 ̅
�̅�) = 𝑡𝑎𝑛 [(
2𝐼 ̅
�̅�) (1 −
1
2�̅�)] 𝐼 ̅ ≥ 1.166 (2-12)
2.4.3.2 Energy balance method
The energy balance method relies on the principle of conservation of mechanical
energy to calculate the asymptotes for P-I diagrams. These energy formulations
separate the impulsive and quasi-static loading regimes. This is the method that was
originally proposed by Baker [3] for P-I diagrams. Baker’s approximation for calculating
a P-I diagram using energy principles worked very well when finding the impulsive and
quasi-static asymptotes, but failed to give an accurate transition throughout the dynamic
region. A similar approach to can be used to calculate an exact dynamic region for
simplified load pulses [2].
The impulsive asymptote can be obtained by assuming that the total energy
imparted to the system initially is imparted as kinetic energy. The expression is obtained
by equating the initial kinetic energy imparted to the strain energy of the structure at
maximum deformation, strain energy maximum. This yields Equation 2-15, which shows
that the impulsive asymptote is directly related to the input of kinetic energy and the
mass of the system.
34
During quasi-static response, the load is presumed to be constant before the
maximum response is reached, and the main energy mode is total work done. The
expression for the quasi-static asymptote is found by equating the maximum strain
energy to the work done by the load on the structure. The maximum strain energy can
be found by calculating the area under the load vs. deflection curve, which is known as
the resistance curve. The mathematical expressions for these energy formulations is
shown below:
𝐾𝐸 = 𝑆𝐸 𝑖𝑚𝑝𝑢𝑙𝑠𝑖𝑣𝑒 𝑎𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒 (2-13)
𝑊𝐸 = 𝑆𝐸 𝑞𝑢𝑎𝑠𝑖 − 𝑠𝑡𝑎𝑡𝑖𝑐 𝑎𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒 (2-14)
KE is the kinetic energy of the structure at 𝑡𝑑, SE is the strain energy o the
structure at the maximum response, and WE is the work done by the load while
displacing the system from zero to maximum displacement. For a perfectly elastic
system, the following energy expressions hold true:
𝐾𝐸 =𝐼2
2𝑀 (2-15)
𝑊𝐸 = 𝑃0𝑥𝑚𝑎𝑥 (2-16)
𝑆𝐸 =1
2𝐾𝑥𝑚𝑎𝑥
2 (2-17)
where,
𝐾𝐸 = Kinetic energy 𝑊𝐸 = Work energy 𝑆𝐸 = Strain energy
𝐼 = Total impulse 𝑀 = Lumped mass of system 𝑃0 = Peak load 𝐾 = Elastic stiffness
𝑥𝑚𝑎𝑥 = Maximum displacement
35
2.4.4 Numerical Approaches for Load-Impulse Diagrams
The closed-form approaches listed above all require a simplified structure and a
simplified load pulse. More advanced problems require the use of numerical
approaches to solve. These numerical methods generate multiple points that fall directly
on the P-I curve for a specific structural response and allow for curve fitting between
points. Each point that is calculated represents a specific load and impulse that causes
the desired response.
Calculating all possible combinations of load and impulse for a specified
response is computationally very expensive. In order to cut down on the number of
points that must be calculated, a search algorithm must be adopted to solve for a
sufficient number of points. Numerical approaches allow for much more complex
systems, including non-linear resistance functions and intricate loading functions. The
numerical approaches describe the behavior of the P-I curve in the dynamic region
accurately [1].
There are many methods that can be used to generate a P-I diagram. Some
methods employ a branch tracing technique, while others utilize specified search areas
that the P-I diagram should fall within. Still other methods set a specified pressure (or
impulse) and solve for the remaining values in specified increments. One of the most
efficient methods performs a radial search that will always yield a solution, given that
the starting point is set to be in the “fail” zone, which is found above and to the right of
the P-I diagram. A few techniques and algorithms that were previously developed are
detailed below.
36
2.4.4.1 Branch tracing technique
The branch tracing procedure [20] uses an energy balance method to estimate
the location of the impulsive and quasi-static asymptotes and a branch tracing algorithm
to generate the dynamic region. This process is shown in Figure 2-9. By finding the
slope between two previous points, a prediction point is created. Correction steps are
required to bring the point within a specified tolerance of the actual P-I curve. This
method assumes that the P-I curve is smooth and continuous, but at some points, the
P-I curve may violate these assumptions. This causes the method to become unstable,
with data points overlapping zones [5].
Figure 2-9. Branch-tracing technique.
37
2.4.4.2 Asymptotic search algorithm
The asymptotic search method was developed by Soh and Krauthammer [5] to
produce numerically stable P-I diagrams. After calculating the asymptotes, a large
number of dynamic analyses are run within a specified limit of the asymptote. The limits
are decreased until the P-I curve is formed. While this method begins by using
conservation of energy to calculate the asymptotes, it also requires a large number of
dynamic analyses to be performed along the asymptotes, which is computationally
expensive. Figure 2-10 illustrates the technique proposed by Soh [5], and shows rough
limits to where the search intervals would be. Soh [5] analyzed reinforced concrete
beams for flexure, diagonal shear, and direct shear by using two loosely coupled SDOF
systems.
Figure 2-10. Search algorithm (a) Flexure, (b) Direct shear [5].
38
2.4.4.3 Constant pressure search algorithm
The constant pressure search algorithm derives P-I diagrams without calculating
the impulsive and quasi-static asymptotes [4]. The curve is found by holding the
pressure constant and checking whether the P-I combination is in the “safe” or
“damaged” zone. If the point is found to be “safe”, the impulse is increased. If the point
is found to be “damaged”, the impulse is reduced. A threshold point can be found
between the “safe” and “damaged” points, which is what defines the curve, as shown in
Figure 2-11. Once a combination is found, the pressure is decreased and the curve is
generated by iterative calculations using the same procedure.
Figure 2-11. Search algorithm [4].
39
2.4.4.4 Pivot point search algorithm
The numerical procedures mentioned previously produce fairly accurate results,
but have a few disadvantages, mainly that they are computationally expensive and
calculate a large amount of useless data points. Additionally, the methods presented by
Soh and Krauthammer [5] and Ng and Krauthammer [4] were limited to specific cases,
and did not allow for different structural elements or loading.
The pivot point search algorithm [6], uses a polar coordinate system and the
bisection method to obtain P-I diagrams. A pivot point, (𝐼𝑝, 𝑃𝑝), located in the “damaged”
zone, is set as the origin of the polar coordinate search. The specific combination of
pressure and impulse along the radial search is calculated using the bisection method.
The angle and radius are mapped out and points on the P-I curve are calculated. Since
the radial search is predetermined, this process allows for the use of multi-core
processing and a more rapid result. This process can be applied to any structural
system where a resistance function can be found [7].
Figure 2-12. Polar coordinate search algorithm [6].
40
2.5 Energy Flow Approach
The principle of energy conservation states that the energy input during an
impact event should be equal to the output energy under all circumstances. When a
structural element is exposed to an external load, the input energy will transfer to other
forms of energy, such as kinetic or strain energy, and some energy may be dissipated
by damping. Kinetic energy results in a reaction or motion of a structural element. Strain
energy refers to the energy that is absorbed by the material. Energy may be dissipated
through natural damping in the system, or by external, hysteretic damping. It is
important to understand how the input energy is transferred to other forms of energy, all
of which must add up to the amount of input energy.
2.5.1 Characteristics of Energy Flow
The response of a structure under impact loading is related to the duration of
loading and the natural period of the structure [3]. The load-impulse relationship is
separated into three sections; impulsive, dynamic, and quasi-static.
Figure 2-13. Load pulses and responses for P-I domains [2].
41
To see how energy transfers in each domain, three dynamic analyses were
conducted. The same SDOF system was subjected to three rectangular load pulses of
the same peak load and varying load durations. The structural properties used in these
analyses are listed in Table 2-3. The results from these analyses can be seen in Table
2-4, as well as in Figure 2-14, Figure 2-15, and Figure 2-16. The load durations chosen
here were chosen to show the three regimes of the load-impulse diagram [2].
Table 2-3. Structural properties of dynamic analysis [2].
Case
Effective mass, 𝑀𝑒
(𝑘𝑖𝑝 𝑔⁄ )
Effective stiffness, 𝐾𝑒
(𝑘𝑖𝑝 𝑖𝑛⁄ )
Peak load, 𝐹𝑜 (𝑘𝑖𝑝)
Load duration, 𝑡𝑑 (𝑠𝑒𝑐)
𝑡𝑑𝑇⁄
A 0.2 10 5 0.002 0.0023
B 0.2 10 5 0.3 0.338
C 0.2 10 5 5 5.627
Table 2-4. Results from dynamic analysis [2].
Case 𝑢𝑚𝑎𝑥 (in)
𝑡𝑚 (sec)
𝐾𝐸𝑚𝑎𝑥 (10−4𝑘𝑖𝑝 − 𝑖𝑛)
𝑆𝐸𝑚𝑎𝑥 (10−4𝑘𝑖𝑝 − 𝑖𝑛)
𝑊𝐸𝑚𝑎𝑥 (10−4𝑘𝑖𝑝 − 𝑖𝑛)
Regime
A 0.009 0.222 3.906 3.906 3.906 𝑡𝑚 ≫ 𝑡𝑑
B 0.874 0.371 12124 38150 38150 𝑡𝑚 ≅ 𝑡𝑑
C 8.1 0.438 12500 50000 50000 𝑡𝑚 ≪ 𝑡𝑑
*tm – the time when maximum response reached
42
Figure 2-14. Energy transition history for Impulsive Regime [2].
Figure 2-15. Energy transition history for Dynamic Regime [2].
Figure 2-16. Energy transition history for Quasi-Static Regime [2].
43
2.5.2 Derivation of Energy Components Spectrum
In order to derive an energy components spectrum, one must start by defining
the governing energy equations, ensuring that all of the energy is accounted for at each
time step. The three distinct loading regimes, impulsive, dynamic, and quasi-static, also
have unique characteristics of energy flow. The energy components spectrum allows
the user to understand the energy transition at the end of loading for a range of natural
periods (T).
2.5.2.1 Derivation of energy functions
One way to distinguish the three regions of a P-I diagram mentioned previously is
by defining an energy flow relationship for each region. This allows energy terms, such
as kinetic energy, strain energy, and work energy to be displayed as time dependent
functions [2].
Work energy is defined as the product of the force and the displacement in the
direction of the displacement of the undamped SDOF system. The work done by a
force, F, that moves a SDOF system from displacement A to displacement B is shown
in Equation 2-18.
𝑊 = ∫ 𝐹𝐵
𝐴
𝑑𝑢 (2-18)
The terms in the equation of motion are all multiplied by 𝑑𝑢 = (𝑑𝑢
𝑑𝑡𝑑𝑡) and are
then integrated over a time of interest. For a perfectly elastic SDOF system, an energy
balance equation can be derived from the equation of motion, which is shown in the
following equations.
44
𝑚�̈� + 𝑘𝑢 = 𝐹(𝑡) (2-19)
∫ 𝑚�̈� 𝑑𝑢 + ∫ 𝑘𝑢 𝑑𝑢 = ∫ 𝐹(𝑡)𝑑𝑢 (2-20)
∫ 𝑚𝑑�̇�
𝑑𝑡𝑑𝑢 + ∫ 𝑘𝑢 𝑑𝑢 = ∫ 𝐹(𝑡)𝑑𝑢 (2-21)
1
2𝑚�̇�2 +
1
2𝑘𝑢2 = ∫ 𝐹(𝑡)𝑑𝑢 (2-22)
∫ 𝑚𝑑�̇�
𝑑𝑡
𝑑𝑢
𝑑𝑡𝑑𝑡 + ∫ 𝑘𝑢
𝑑𝑢
𝑑𝑡𝑑𝑡 = ∫ 𝐹(𝑡)
𝑑𝑢
𝑑𝑡𝑑𝑡 (2-23)
∫ 𝑚𝑑�̇�
𝑑𝑡
𝑑𝑢
𝑑𝑡𝑑𝑡 + ∫ 𝑘𝑢
𝑑𝑢
𝑑𝑡𝑑𝑡 = ∫ 𝐹(𝑡)
𝑑𝑢
𝑑𝑡𝑑𝑡 (2-24)
Where the expressions from Equation 2-24 can be represented as:
∫ 𝑚�̈� �̇�𝑑𝑡 = 𝐾𝐸(𝑡) (2-25)
∫ 𝑘𝑢�̇� 𝑑𝑡 = 𝑆𝐸(𝑡) (2-26)
∫ 𝐹(𝑡)�̇�𝑑𝑡 = 𝑊𝐸(𝑡) (2-27)
Therefore, by substituting Equations 2-25 through 2-27 into Equation 2-24, the
relationship between kinetic energy, strain energy, and work energy in the time domain
can be expressed as shown in the following equation.
𝐾𝐸(𝑡) + 𝑆𝐸(𝑡) = 𝑊𝐸(𝑡) (2-28)
Based on the principle of conservation of energy, the total imparted energy must
equal the internal energy in the system. This internal energy includes elastic strain
energy as well as work done by permanent deformations. The total strain energy can be
broken down into elastic and plastic strains.
45
For an undamped elastic SDOF system exposed to a rectangular load pulse, the
expressions of strain and kinetic energy can be found from structural response functions
[2]. Energy functions for the forced and free vibration phases can be seen in Table 2-5.
Table 2-5. Energy functions for undamped SDOF under block loads [2].
Response domain
Term Forced-vibration Free-vibration
Time range 𝑡 0 < 𝑡 ≤ td td < 𝑡
*Response functions
𝑢(𝑡) 𝐹𝑜
𝑘(1 − cos 𝜔𝑡)
𝐹𝑜
𝑘(cos 𝜔(𝑡 − 𝑡𝑑 ) − cos 𝜔𝑡)
Kinetic energy functions
𝐾𝐸(𝑡) 𝐹𝑜
2
2𝑘sin(𝜔𝑡)2
𝐹𝑜2
2𝑘(sin 𝜔(𝑡 − td ) − sin 𝜔𝑡)2
Strain energy functions
𝑆𝐸(𝑡) 2𝐹𝑜
2
𝑘sin(
𝜔𝑡
2)4
𝐹𝑜2
2𝑘(cos 𝜔(𝑡 − td ) − cos 𝜔𝑡)2
*Assuming 𝑢(0) = �̇�(0) = 0
2.5.2.2 Distinction of loading regimes
A static deflection is found by applying a peak load, 𝐹𝑜 , statically to a structure.
The ratio of the maximum dynamic deflection, discussed previously, to the static
deflection, is known as the dynamic load factor, DLF.
𝐷𝐿𝐹 =𝑢𝑚𝑎𝑥
𝑢𝑠𝑡 =
𝑢𝑚𝑎𝑥
𝐹𝑜 𝑘⁄= 𝜑(𝜔𝑡𝑑 ) = 𝜑 (
𝑡𝑑
𝑇) (2-29)
The ratio of load duration to natural period, 𝑡𝑑
𝑇 or 𝜔𝑡𝑑 (
𝑡𝑑
𝑇=
𝜔𝑡𝑑
2𝜋), significantly
influences the response of a structure to blast or impact loading. This ratio has been
chosen to differentiate the loading regimes. After deriving the energy functions for a
load pulse, the energy flow in the three regimes can be analyzed and compared [2].
The maximum value of the specified energies will occur in either the forced or
free vibration phase, similar to maximum structural response, depending on the ratio of
46
𝑡𝑑
𝑇. For example, the maximum strain energy in the forced vibration phase can be found
by equating the derivative of Equation 2-30 to zero [2].
𝑆𝐸𝑓𝑜𝑟𝑐𝑒 =2𝐹𝑜
2
𝑘sin(
𝜔𝑡
2)4 (2-30)
𝑆�̇�𝑓𝑜𝑟𝑐𝑒 = 4𝐹02𝜔 cos
𝜔𝑡𝑚
2sin(
𝜔𝑡𝑚
2)3 𝑘⁄ = 0 (2-31)
where 𝜔𝑡𝑚 = 𝑛𝜋 & 𝑛 = 1,3,5, …
The value of 𝑛 in Equation 2-31 has been taken as an odd number since an even
number would cause a strain energy minimum as well. The largest value of strain
energy will occur in the forced vibration phase if the smallest value of 𝑡𝑚 , found by
taking n = 1, is less than 𝑡𝑑.
𝜋
𝜔≤ 𝑡𝑑
𝑡𝑑
𝑇 ≥ 0.5 (2-32)
𝑆𝐸𝑚𝑎𝑥 =2𝐹𝑜
2
𝑘 (Forced − vibration) (2-33)
If Equation 2-32 does not hold true, the maximum strain energy will occur in the
free-vibration phase. In this case, the maximum value is found by setting the derivative
of Equation 2-34 to zero [2].
𝑆𝐸𝑓𝑟𝑒𝑒 =𝐹𝑜
2
2𝑘(cos 𝜔(𝑡 − 𝑡𝑑) − cos 𝜔𝑡)2 (2-34)
𝑆�̇�𝑓𝑟𝑒𝑒 = −𝐹0
2
𝑘[cos(𝜔𝑡𝑚 − 𝜔𝑡𝑑) − cos(𝜔𝑡𝑚)][𝜔 sin(𝜔𝑡𝑚 − 𝑡𝑑) − 𝜔 sin(𝜔𝑡𝑚)] = 0 (2-35)
where 𝜔𝑡𝑚 = 𝑛𝜋 + 𝜔𝑡𝑑 / 2 & 𝑛 = 1,3,5, …
0 ≤ 𝜔𝑡𝑑 ≤ π 0 ≤𝑡𝑑
𝑇 < 0.5 (2-36)
𝑆𝐸𝑚𝑎𝑥 =2𝐹𝑜
2
𝑘sin(
𝜔𝑡𝑑
2)2 (Free − vibration) (2-37)
47
The same process shown above to find the maximum strain energy can be used
to find the maximum values of kinetic and work energy, 𝐾𝐸𝑚𝑎𝑥and 𝑊𝐸𝑚𝑎𝑥. The results
of these derivations can be seen below in Table 2-6.
Table 2-6. Max energy functions for undamped SDOF under block loads [2].
Response domain Forced-vibration Free-vibration
Term *tm Range Max tm Range **Max
Maximum kinetic energy
𝐾𝐸𝑚𝑎𝑥 𝜋
2𝜔 𝜔td ≥
𝜋
2
𝐹𝑜2
2𝑘
2𝜋 + 𝜔td
2𝜔 0 ≤ 𝜔𝑡𝑑 <
𝜋
2
2𝐹𝑜2
𝑘sin(
𝜔𝑡𝑑
2)2
Maximum strain
energy 𝑆𝐸𝑚𝑎𝑥
𝜋
𝜔 𝜔𝑡𝑑 ≥ 𝜋
2𝐹𝑜2
𝑘
𝜋 + 𝜔td
2𝜔 0 ≤ 𝜔𝑡𝑑 < 𝜋
2𝐹𝑜2
𝑘sin(
𝜔𝑡𝑑
2)2
Maximum external energy
𝑊𝐸𝑚𝑎𝑥 𝜋
𝜔 𝜔𝑡𝑑 ≥ 𝜋
2𝐹𝑜2
𝑘 tm= td 0 ≤ 𝜔𝑡𝑑 < 𝜋
2𝐹𝑜2
𝑘sin(
𝜔𝑡𝑑
2)2
* tm – the time when maximum value reached ** Note that there is no work done by external load after the load has been removed from the system. The maximum value occurs at t = td for a rectangular load pulse.
When the maximum response occurs in the forced-vibration phase, the values of
𝑆𝐸𝑚𝑎𝑥 is equal to 𝑊𝐸𝑚𝑎𝑥. This implies that 𝐾𝐸(𝑡𝑚 ) is equal to zero in this case. If the
maximum response occurs in the free-vibration phase, the values of 𝑆𝐸𝑚𝑎𝑥 is equal to
𝐾𝐸𝑚𝑎𝑥, which indicates that 𝑆𝐸(𝑡𝑑 ) is equal to zero in this case.
Table 2-6 shows that the maximum values of the energies are constant when
they occur in the forced vibration phase. If the maximum values occur in the free-
vibration phase, the maximum values vary depending on the peak load (𝐹𝑜 ), stiffness
(𝑘), load duration (𝑡𝑑 ), and natural frequency (𝜔) [2]. Note that Table 2-6 also lists terms
for range over which each of the cases is valid for both forced and free vibration as well
as the time to maximum response for each case.
48
2.5.2.3 Plot of energy component spectrum
By collecting each of the three energy components at 𝑡 = 𝑡𝑑, (𝐾𝐸𝑡𝑑 , 𝑆𝐸𝑡𝑑
, 𝑊𝐸𝑡𝑑),
an energy component spectrum can be plotted [2]. This plot is a response spectrum
which shows the energy transition at the end of loading durations based on the ratio of
𝑡𝑑
𝑇. Figure 2-17 shows an example of an energy component spectrum. Note that each of
the energy terms, work energy (WE), strain energy (SE), and kinetic energy (KE), are all
normalized by a ratio of the peak load (Fo) squared to the elastic spring stiffness (k).
Figure 2-17. Typical energy components spectrum [2].
49
For a specified loading condition on an elastic SDOF system, the structural
responses, such as displacement and velocity, etc., are reflected by the resultant
energy components. Table 2-7 shows that each of the energy components can be
defined by two equations, one for the free vibration phase and one for the forced
vibration phase.
Table 2-7. Energy components for undamped SDOF under block load [2].
Response domain Forced-vibration Free-vibration
Term Range Value Range Value
Kinetic energy
𝐾𝐸𝑡𝑑
𝐹𝑜2
𝑘⁄
𝑡𝑑
𝑇 ≥ 0.5 0 0 ≤
𝑡𝑑
𝑇 < 0.5
1
2sin(2𝜋
td
T)2
Strain energy
𝑆𝐸𝑡𝑑
𝐹𝑜2
𝑘⁄
𝑡𝑑
𝑇 ≥ 0.5 2 0 ≤
𝑡𝑑
𝑇 < 0.5 2 sin(𝜋
td
T)4
Work done by external
load
𝑊𝐸𝑡𝑑
𝐹𝑜2
𝑘⁄
𝑡𝑑
𝑇 ≥ 0.5
2 0 ≤
𝑡𝑑
𝑇 < 0.5 2 sin(𝜋
td
T)2
The energy balance equation is held remains valid across the entire domain, as
seen in Figure 2-17 and Table 2-7. Tsai [2] notes that the total imparted energy does
not change with respect to time, as shown in Equation 2-38. This is shown in the
following equation, where the work energy at the end of the load duration is equal to the
work energy at the time of maximum response.
𝑊𝐸𝑡𝑑= 𝑊𝐸𝑡𝑚
(2-38)
In the impulsive domain, energy enters the system at a very high rate. The strain
energy at the end of the load duration is zero in this domain. The energy balance
equation can be simplified as shown in Equation 2-39. By substituting terms between
50
Equations 2-38 through 2-39, the impulsive asymptote equation can be solved for, as
shown in Equation 2-40 [2].
𝑊𝐸𝑡𝑑= 𝐾𝐸𝑡𝑑
(2-39)
𝐾𝐸𝑡𝑑= 𝑆𝐸𝑡𝑚
(2-40)
In the quasi-static domain, energy enters the system at a very low rate. Since a
structure reaches its maximum response when velocity reaches zero, the kinetic energy
at maximum displacement is equal to zero. Therefore, the energy balance equation at
the structure’s maximum response is given by Equation 2-41.
𝑊𝐸𝑡𝑚= 𝑆𝐸𝑡𝑚
(2-41)
In the dynamic domain, there are no simplifications that can be made to simplify
the energy balance equation [2].
𝐾𝐸𝑡𝑑+ 𝑆𝐸𝑡𝑑
= 𝑊𝐸𝑡𝑑= 𝑊𝐸𝑡𝑚
= 𝑆𝐸𝑡𝑚 (2-42)
2.5.3 Energy Based Solutions for Load-Impulse Diagrams
As discussed in Section 2.4.2.2, the impulsive and quasi-static asymptotes can
be found by using the following equations [2].
𝐾𝐸 = 𝑆𝐸 (Impulsive asymptote) (2-43)
𝑊𝐸 = 𝑆𝐸 (Quasi − static asymptote) (2-44)
The KE term denotes the work done by a force applied to the structure, while the
SE term represents the work done by the internal restoring forces [2].
Figure 2-18 shows a typical response spectrum and load-impulse diagram.
Derivations of the impulsive and quasi-static asymptotes can be seen in Figure 2-18(b)
and Table 2-8.
51
Figure 2-18. Typical response spectrum and load-impulse diagram [5].
Figure 2-19. Common Resistance Functions.
52
Table 2-8. Energy solutions for simple SDOF systems [1].
SDOF system
Perfect elastic
Rigid plastic
Elastic-plastic
KE 1
2𝑀𝑣2
𝐼2
2𝑀
𝐼2
2𝑀
𝐼2
2𝑀
SE ∫ 𝑅 𝑑𝑥 1
2𝑘𝑥𝑚𝑎𝑥
2 𝑅𝑝𝑥𝑚𝑎𝑥 𝑘𝑥𝑒𝑙2 (
𝑥𝑚𝑎𝑥
𝑥𝑒𝑙
−1
2)
WE ∫ 𝑃 𝑑𝑥 𝑃𝑜𝑥𝑚𝑎𝑥 𝑃𝑜𝑥𝑚𝑎𝑥 𝑃𝑜𝑥𝑚𝑎𝑥
Impulsive
asymptote KE=SE
𝐼
√𝑘𝑀𝑥𝑚𝑎𝑥
= 1 𝐼
√𝑅𝑝𝑀𝑥𝑚𝑎𝑥
= √2 𝐼
√𝑘𝑀𝑥𝑚𝑎𝑥
=√2𝜇 − 1
𝜇
Quasi-Static
asymptote WE=SE
𝑃𝑜𝑘⁄
𝑥𝑚𝑎𝑥
=1
2
𝑃𝑜
𝑅𝑝
= 1 𝑃𝑜
𝑘⁄
𝑥𝑚𝑎𝑥
=2𝜇 − 1
2𝜇2
The asymptotes calculated using the energy balance method can be used to
decrease computational efforts. The energy balance method, however, does not apply
to the transition between the asymptotes, the dynamic region. It was recommended by
Baker [3] to use Equation 2-45 to estimate the dynamic region. This estimation is not a
precise solution to the dynamic region of the P-I diagram [2].
𝑆𝐸 = 𝑊𝐸𝑡𝑎𝑛ℎ2√𝐾𝐸
𝑊𝐸 (2-45)
Tsai [2] proposed a general energy based solution to define the entire domain of
a P-I diagram. This solution works for a simplified system with a simplified load profile.
Table 2-9 lists the energy solutions for a perfectly elastic SDOF system.
53
Table 2-9. Summary of energy based solutions for P-I diagrams [2].
Load-response
illustration
Loading regime Impulsive Dynamic Quasi-Static
Load duration Short Medium Long
Response time Long Medium Short
𝑡 = 𝑡𝑑 𝑊𝐸𝑡𝑑= 𝐾𝐸𝑡𝑑
𝑊𝐸𝑡𝑑= 𝐾𝐸𝑡𝑑
+ 𝑆𝐸𝑡𝑑 𝑊𝐸𝑡𝑑
= 𝐾𝐸𝑡𝑑+ 𝑆𝐸𝑡𝑑
𝑡 = 𝑡𝑚 𝑊𝐸𝑡𝑚= 𝑆𝐸𝑡𝑚
𝑊𝐸𝑡𝑚= 𝑆𝐸𝑡𝑚
𝑊𝐸𝑡𝑚= 𝑆𝐸𝑡𝑚
Absorbed energy 𝐾𝐸𝑡𝑑=
𝐼2
2𝑀 𝑊𝐸𝑡𝑑
=1
2𝑀𝑣𝑡𝑑
2+1
2𝑘𝑢𝑡𝑑
2 𝑊𝐸𝑡𝑚= 𝐹𝑜𝑢𝑚𝑎𝑥
Asymptotes 𝐾𝐸𝑡𝑑= 𝑆𝐸𝑡𝑚
𝐾𝐸𝑡𝑑+ 𝑆𝐸𝑡𝑑
= 𝑆𝐸𝑡𝑚 𝑊𝐸𝑡𝑚
= 𝑆𝐸𝑡𝑚
*𝑊𝐸𝑡𝑑= 𝑊𝐸𝑡𝑚
The maximum strain energy for a given response, such as 𝑢𝑚𝑎𝑥, is
𝑆𝐸𝑡𝑚= ∫ 𝑘𝑢𝑑𝑢
𝑢𝑚𝑎𝑥
0
(2-46)
where
𝑢𝑚𝑎𝑥 = Maximum displacement 𝑢 = Displacement 𝑘 = Structural stiffness 𝑆𝐸𝑡𝑚
= Maximum strain energy
As stated previously, the equation for the quasi-static asymptote can be derived
from setting the maximum strain energy equal to the work done [2]. By rearranging
Equation 2-41, the minimum load required to active the desired response is:
𝐹𝑚𝑖𝑛 =𝑆𝐸𝑡𝑚
𝑢𝑚𝑎𝑥 (2-47)
54
The equation for the impulsive asymptote can be derived by setting the kinetic
energy at the end of loading equal to the strain energy at the time of maxim response
[2]. The minimum impulse needed to activate the desired response is given by Equation
2-48, where M is the lumped mass of the system.
𝐼𝑚𝑖𝑛 = √2 𝑀 𝑆𝐸𝑡𝑚 (2-48)
The load and impulse combinations that deliver an adequate amount of energy to
the system to reach a desired response must satisfy the Law of Conservation of
Mechanical Energy [2].
∫ 𝐹(𝑡)𝑑𝑢 = ∫ 𝑀𝑑�̇�
𝑑𝑡𝑑𝑢 + ∫ 𝑘𝑢 𝑑𝑢 (2-49)
Tsai [2] suggested to use an incremental load as a variable to calculate the
corresponding displacement and velocity. The equations for this process are shown
below:
�̅�𝑖 = 𝐹𝑚𝑖𝑛 + 𝑖 ℎ (2-50)
�̅�𝑖 =𝑆𝐸𝑡𝑚
�̅�𝑖
(2-51)
�̅�𝑖 = √2�̅�𝑖�̅�𝑖 − 𝑘�̅�𝑖
2
𝑀 (2-52)
where,
𝑖 = Given sequence ℎ = Given increment 𝑘 = Spring stiffness
�̅�𝑖 = Incremental load
𝐹𝑚𝑖𝑛 = Minimum force required
�̅�𝑖 = Displacement corresponded to �̅�𝑖
�̅�𝑖 = Velocity corresponded to �̅�𝑖 and �̅�𝑖
55
The imparted energy, or work done, is represented as �̅�𝑖�̅�𝑖. The load duration for
each incremental load step can be found using Equation 2-53, a theoretical value
assuming a rectangular (block) load pulse [11].
𝑡�̅�𝑖=
acos (1 −𝑘�̅�𝑖
�̅�𝑖)
𝜔
(2-53)
The impulse delivered from each incremental load can be calculated with
Equation 2-54.
𝐼�̅� = 𝛽�̅�𝑖𝑡�̅�𝑖 (2-54)
where 𝛽 is a load pulse shape factor [2]. Values for the load pulse shape factor for
rectangular, triangular, and sinusoidal load pulses are 𝛽= 1, 1
2, and
2
𝜋 , respectively. Tsai
does not go into great detail about load pulse shape factors other than the three
previously mentioned. The idea of looking at other values of 𝛽 will be explored in
Chapter 3.
A set of load and impulse combinations are plotted in Figure 2-20 to show the
transition in the dynamic domain. Figure 2-21 shows the energy solution (discussed
above), closed form solution, and Baker’s approximation for a rectangular load pulse.
The energy solution matches very well to the closed form solution. Baker’s
approximation does not match the closed form solution throughout the dynamic regime
[2].
56
Figure 2-20. Illustration of energy based solutions for P-I diagrams [2].
Figure 2-21. Comparison of P-I diagrams solutions [2].
57
2.5.4 Storage Tank Analogy
A basic understanding of how energy is absorbed and dissipated in a structure
was developed by Clough [14]. A storage tank represents the structure. The elastic
strength of the structure is represented by the height of the overflow nozzle. The
structures capacity to dissipate energy in the form of plastic deformations is shown by
the smaller tank on the left of the large tank. Structural damping is depicted as the small
nozzle on the bottom right of the larger tank. The inlet pipe, which brings energy into the
system, has a valve that restricts the energy flow based on the type of structure. The
valve will be wide open for linear-elastic structures. The valve will be partially closed for
nonlinear-inelastic structures. The size of the overflow tank represents the amount of
plastic deformation the structure can undergo. A very small overflow tank represents a
brittle structure that may fail very soon after plastic deformations begin.
2.5.5 Energy Tank Analogy
Clough [14] proposed an energy tank analogy to describe the energy transition
from one response to another. This analogy has been modified to more clearly define
the energy flow through a structure subjected to impact and blast loads, while
considering two response modes, local and global responses [2].
In the modified energy tank analogy, only kinetic energy and strain energy are
present. The energy flow from one tank to another does not include energy losses.
Figure 2-22 shows how the tanks are separated into two modes, local and global. The
faucet on the upper right of the figure signifies the amount of energy that is delivered to
the system from the impact or blast load. The volume of each tanks represents each
responses ability to absorb energy [2]. Energy may transition horizontally from kinetic
energy to elastic strain energy without causes damage to the structure. Damage is
58
caused when the elastic strain energy limit is reached, such as when Tank A overflows
into Tank B, and the horizontal transition is between the elastic strain energy tank and
the plastic strain energy tank. Vertical flow in the diagram represents energy transition
from a local response to a global response, such as from a direct shear response to a
flexural response [2].
Figure 2-22. Energy tank analogy for two simplified response modes [2].
59
The energy tank analogy presented by Tsai [2] looks at the energy transition from
𝑡 = 0 to 𝑡 = 𝑡𝑚. The sum of the energy in all the tanks must still be equal to the amount
of energy that was imparted to the system.
In Figure 2-22, a structure’s local response, such as direct shear, is shown by
Tanks A and B. As energy transitions from the KE, kinetic energy, side to the SE, elastic
strain energy side, elastic deformations begin. If the SE side of Tank A overflows into
Tank B, permanent damage will occur. If the capacity of Tank B is reached, the
structure will fail in the local response.
If the energy imparted to the system does not cause a local failure, the energy
will transition vertically and will begin to fill the global response tanks. The same
horizontal transitions will occur in the global response tanks that were detailed for the
local response tanks [2].
The energy balance equation for the analogy detailed in Figure 2-22 can be
written as,
∑ ∆ 𝐸(𝑡) = 0 → 𝐸𝑖𝑛𝑝𝑢𝑡(𝑡) = 𝐸𝑖𝑚𝑝𝑎𝑟𝑡𝑒𝑑(𝑡) + 𝐸𝑜𝑢𝑡𝑝𝑢𝑡(𝑡) (2-55)
𝐸𝑖𝑚𝑝𝑎𝑟𝑡𝑒𝑑(𝑡) = 𝐸𝑡𝑎𝑛𝑘_𝐴(𝑡) + 𝐸𝑡𝑎𝑛𝑘_𝐵(𝑡) + 𝐸𝑡𝑎𝑛𝑘_𝐶(𝑡) + 𝐸𝑡𝑎𝑛𝑘_𝐷(𝑡) (2-56)
where,
𝐸𝑖𝑛𝑝𝑢𝑡(𝑡) = Total input energy at a given time
𝐸𝑖𝑚𝑝𝑎𝑟𝑡𝑒𝑑(𝑡) = Imparted energy or work done by external load at a given time
𝐸𝑜𝑢𝑡𝑝𝑢𝑡(𝑡) = Difference between input energy and the work done at a given time
𝐸𝑡𝑎𝑛𝑘_𝐴(𝑡) = Energy stored in the tank A at a given time
𝐸𝑡𝑎𝑛𝑘_𝐵(𝑡) = Energy stored in the tank B at a given time
𝐸𝑡𝑎𝑛𝑘_𝐶(𝑡) = Energy stored in the tank C at a given time
𝐸𝑡𝑎𝑛𝑘_𝐷(𝑡) = Energy stored in the tank D at a given time
60
In addition to the energy tank analogy, Tsai [2] created a flow chart to show the
results of energy flow and the corresponding damage or failures. If a complete local
failure mode is reached, such as penetration or direct shear, no energy can flow to the
global state. However, if the local failure mode is less severe, such as spalling or
scabbing, energy may still transition to the global response mode [2].
Figure 2-23. Flow chart for the energy tank analogy [2].
This approach presented by Tsai [2] allows for an energy based load-impulse
diagram to be created. This plot is created by defining the input energy and the rate at
which the input energy is delivered to the system. An example of an energy based P-I
diagram can be seen in Figure 2-24. The four curves shown on this figure represent
local damage, local failure, global damage, and global failure limits. By determining
where a specific combination of energy input and energy input rate falls on the diagram,
61
the expected structural response can be identified. The actual development of these
energy based P-I diagrams is discussed in the next section.
Figure 2-24. Typical energy based P-I diagrams for multi-failure modes [2].
2.6 Development of an Energy Based Equivalent to Load-Impulse Diagrams
In traditional structural design, the criteria that defines structural damage is
usually a maximum deflection or support rotation angle. Characteristics of energy flow
can be pulled out from a given loading scenario. Tsai [2] suggested using energy terms
to define a response limit for specific loading cases rather than traditional force terms.
With this new index, a structure is capable of displacing to a stable deformed position if
the structure can absorb or dissipate the input energy before the failure state is
reached.
For an energy-based P-I diagram, the energy input and the input rate must be
able to be defined accurately. For a structure subjected to a given load pulse with a
duration (𝑡𝑑), the input energy, input energy rate, and the energy imparted to the system
can be defined by the following equations.
62
𝐸𝑖𝑛𝑝𝑢𝑡 =(𝐼𝑖𝑚𝑝𝑢𝑙𝑠𝑒)
2
2 ∙ 𝑀𝑚𝑎𝑠𝑠=
(𝛽 ∙ 𝐹𝑜 ∙ 𝑡𝑑)2
2 ∙ 𝑀𝑚𝑎𝑠𝑠 (2-57)
𝐸𝑖𝑛𝑝𝑢𝑡_𝑟𝑎𝑡𝑒 =𝐸𝑖𝑛𝑝𝑢𝑡
𝛽 ∙ 𝑡𝑑 (2-58)
𝐸𝑖𝑚𝑝𝑎𝑟𝑡𝑒𝑑 = ∫ 𝑘𝑢𝑑𝑢𝑢𝑚𝑎𝑥
0
(2-59)
where,
𝐸𝑖𝑛𝑝𝑢𝑡 = Input energy
𝐸𝑖𝑛𝑝𝑢𝑡_𝑟𝑎𝑡𝑒 = Input energy rate
𝐸𝑖𝑚𝑝𝑎𝑟𝑡𝑒𝑑 = Energy imparted to the system (absorbed energy)
𝐼𝑖𝑚𝑝𝑢𝑙𝑠𝑒 = Impulse delivered to system (area under load-time history)
𝑀𝑚𝑎𝑠𝑠 = Mass of the system 𝛽 = Load pulse shape factor 𝐹𝑜 = Peak load
𝑡𝑑 = Load duration 𝑢𝑚𝑎𝑥 = Given response limit
The idea of an energy component spectrum was discussed previously. In the
same way that the energy component spectrum was developed, an input energy
spectrum can be created, with each point on the curve representing the same response
limit [2]. The difference between the energy input to the system and the energy imparted
to the system can be found by using the following equation.
𝐸𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 𝐸𝑖𝑛𝑝𝑢𝑡−𝐸𝑖𝑚𝑝𝑎𝑟𝑡𝑒𝑑 (2-60)
Figure 2-26 shows the energy difference spectrum, where load pulses with
shorter durations result in lower energy differences. As the energy input rate increases,
less energy is needed to reach a desired response limit [2]. This can be seen for the
case where a fixed amount of energy is applied to a system and the input energy rate is
changed, as seen in Figure 2-27.
63
Figure 2-25. Energy components spectrum for block loads [2].
Figure 2-26. Energy difference spectrum for rectangular load pulse [2].
64
Figure 2-27. Energy difference vs energy input rate diagram [2].
A typical energy based load-impulse diagram has the horizontal axis defined as
the input energy and the vertical axis as the input energy rate. These diagrams work the
same way that traditional P-I diagrams do, with combinations that fall to the right and
above the curve representing the limit state has been reached [2]. E-R diagrams allow a
designer to look at the response of the structure from an energy standpoint.
The dimensionless terms for the energy input and energy input rate, the axes for
the E-R diagram for an undamped elastic SDOF system, are given by:
�̅� =𝐸𝑖𝑛𝑝𝑢𝑡
𝐸𝑖𝑚𝑝𝑎𝑟𝑡𝑒𝑑=
𝐼2
2 𝑚12 𝑘 𝑢𝑚𝑎𝑥
2=
𝐼2
𝑘 𝑚 𝑢𝑚𝑎𝑥2 (2-61)
�̅� =𝐸𝑖𝑛𝑝𝑢𝑡_𝑟𝑎𝑡𝑒
𝐸𝑖𝑚𝑝𝑎𝑟𝑡𝑒𝑑 𝜔=
(𝛽𝐹𝑜 𝑡𝑑)2
2 𝑚1
𝛽𝑡𝑑
12 𝑘 𝑢𝑚𝑎𝑥
2 𝜔=
𝛽 𝐹𝑜 2 𝑡𝑑
𝑘32 𝑚
12 𝑢𝑚𝑎𝑥
2 (2-62)
�̅�
�̅�= 𝛽𝜔𝑡𝑑 (2-63)
65
Currently, closed-form (analytical) solutions can be found for ideal structures
subjected to a simplified load pulse.
For an undamped elastic SDOF system, Biggs [11] described the shock
spectrum when subjected to a rectangular load pulse as
𝑢𝑚𝑎𝑥
𝐹𝑜𝑘⁄
= 2 sin(𝜔𝑡𝑑
2) 0 < 𝜔𝑡𝑑 ≤ 𝜋 (2-64)
𝑢𝑚𝑎𝑥
𝐹𝑜𝑘⁄
= 2 𝜔𝑡𝑑 > 𝜋 (2-65)
For an undamped elastic SDOF system, Soh [5] described the shock spectrum
when subjected to a triangular load pulse as
𝑢𝑚𝑎𝑥
𝐹𝑜𝑘⁄
=√2 + (𝜔𝑡𝑑)2 − 2𝜔𝑡𝑑 sin 𝜔𝑡𝑑 − 2 cos 𝜔𝑡𝑑
𝜔𝑡𝑑 0 < 𝜔𝑡𝑑 ≤ 2.331 (2-66)
𝑢𝑚𝑎𝑥
𝐹𝑜𝑘⁄
= 2 (1 −tan−1 𝜔𝑡𝑑
𝜔𝑡𝑑) 𝜔𝑡𝑑 > 2.331 (2-67)
Equations 2-61 through 2-63 can be substituted into the shock spectrums defined
above, resulting in the expressions for the axes of the normalized E-R diagrams.
For a rectangular load pulse, the forced and free vibration phases can be defined
using the energy expressions shown below [2].
�̅�
√�̅�𝑠𝑖𝑛 (
�̅�
2�̅�) =
1
2 1 ≤ �̅� ≤ 2.467 (2-68)
�̅�
√�̅�=
1
2 �̅� > 2.467 (2-69)
66
For a triangular load pulse, the forced and free vibration phases can be defined
using the energy expressions shown below [2].
1
�̅�(
�̅�
�̅�)
4
= 𝑠𝑖𝑛 (�̅�
�̅�)
2
+ (�̅�
�̅�)
2
− (�̅�
�̅�) 𝑠𝑖𝑛 (
2�̅�
�̅�) 1 ≤ �̅� ≤ 1.36 (2-70)
2�̅�
�̅�= tan [
2�̅�
�̅�−
�̅�1.5
�̅�2] �̅� > 1.36 (2-71)
Figure 2-28. Typical energy-based P-I diagrams for idealized load pulse [2].
67
The impulsive asymptote shown in Figure 2-28 implies that as energy is
delivered to the system at a very high rate, all the energy will be imparted to the system.
The minimum energy required to cause a desired response is the is equal to the
minimum energy input, the energy input value at the impulsive asymptote [2].
For lower input energy rates, more energy is required to be delivered to the
system since the energy difference is larger at lower rates. The quasi-static asymptote
is no longer a straight line, but rather a quadratic.
�̅� =√�̅�
2 (2-72)
2.7 Numerical Simulations using Dynamic Structural Analysis Suite
Dynamic Structural Analysis Suite, commonly referred to as DSAS, is a software
package that has been developed and maintained throughout the years by
Krauthammer since the late 1970s. DSAS analyzes a variety of structures and structural
elements for both static and dynamic load cases [17]. DSAS utilizes SDOF methods to
analyze structural elements, which allows for a quick analysis that usually takes less
than five seconds. The program has built-in features that allow for quick analysis of
columns, beam, slabs, and user defined shapes. The shapes may use steel or concrete
with longitudinal and transverse reinforcement for analysis purposes. DSAS also has
the option to use Ultra High Performance Concrete (UHPC) based on the results from
testing of Ductal. The software package allows for multiple support conditions, dynamic
increase factors (DIFs), as well as the inclusion of diagonal shear and damping. While
DSAS does not provide the in depth analysis that finite element analysis programs
provide, it does allow for a very quick and accurate analysis with low computational
costs [2].
68
2.8 Summary
Impact and blast loading events are considered abnormal loadings for structural
elements. A simplified analysis method, known as the Single Degree of Freedom
method, allows the user to rapidly obtain a structural response for these given abnormal
loads. The two main response / failure modes that will be looked at in this report are
flexure, a global response, and direct shear, a local response.
A common tool that is used to assess structural damage from given dynamic
loads are P-I diagrams. These diagrams can be broken down into three domains, the
impulsive, dynamic, and quasi-static domains. For simplified load pulses on idealized
structures, P-I diagrams can be created using closed-form solutions. For more complex
loading scenarios, one must use an algorithm, such as the ones described in Section
2.4.3, to determine the P-I diagram. The radial search algorithm developed by Blasko et
al. [6] is the most efficient algorithm used to calculate P-I diagrams. One improvement
to this method, determining a more efficient pivot point for the radial search, will be
proposed in Chapter 3.
The driving force in all of the methods mentioned in this section is energy. The
amount of energy that is delivered from the dynamic loads determines how the structure
reacts. The rate at which this energy enters the system is also a critical factor in
determining structural response. An E-R diagram allows the user to look at how much
energy is entering the system and at what rate the energy is entering to determine
structural responses due to energy flow. These diagrams have only been proposed for
simplified load pulses on SDOF systems, so there is a need to determine if the process
is the same for non-simplified load pulses on realistic structures, which will be discussed
in Chapter 3.
69
Various terms are used when describing the Input Energy vs. Input Energy Rate
Diagrams (E-R Diagrams).
The input energy, sometimes referred to as the energy input, is described as the
total amount of energy that is delivered to the structural system. Some of this input
energy will do work on the system, some of it may not. The rate at which the input
energy is delivered to the system is called the input energy rate. The input energy rate
can also be referred to as the power generated by the input energy. The amount of
input energy that does work on the system is called the imparted energy. This type of
energy is also commonly called the absorbed energy. Finally, the amount of input
energy that does not do work on the system is called the energy difference. This value
can be calculated by finding the difference between the input energy and the imparted
energy.
70
CHAPTER 3 METHODOLOGY
3.1 Introduction
As described in Chapter 2, various terms can be used when describing E-R
Diagrams. Understanding what each term represents plays a key role in understanding
how E-R diagrams are created. The input energy is the total amount of energy that is
delivered to the structural system. The input energy rate is the rate at which the input
energy is delivered to the system. This can also be referred to as the power generated
by the input energy. The imparted energy is the amount of input energy that does work
on the system. This type of energy is also commonly called the absorbed energy. The
energy difference is the amount of input energy that does not do work on the system,
which is the difference between the input energy and the imparted energy.
3.2 Validation of P-I to E-R Conversion Equations
Equations 3-1 and 3-2 were proposed by Tsai [2] to convert numerical data from
a P-I to an E-R diagram. This allows for an easy transition into the energy domain if the
mass of the system, peak load, load duration, load shape, and impulse are known.
𝐸𝑖𝑛𝑝𝑢𝑡 =(𝐼𝑖𝑚𝑝𝑢𝑙𝑠𝑒)
2
2 ∙ 𝑀𝑚𝑎𝑠𝑠=
(𝛽 ∙ 𝐹𝑜 ∙ 𝑡𝑑)2
2 ∙ 𝑀𝑚𝑎𝑠𝑠 (3-1)
𝐸𝑖𝑛𝑝𝑢𝑡 =(𝐼𝑖𝑚𝑝𝑢𝑙𝑠𝑒)
2
2 ∙ 𝑀𝑚𝑎𝑠𝑠=
(𝛽 ∙ 𝐹𝑜 ∙ 𝑡𝑑)2
2 ∙ 𝑀𝑚𝑎𝑠𝑠 (3-2)
where,
𝐸𝑖𝑛𝑝𝑢𝑡 = Input energy
𝐸𝑖𝑛𝑝𝑢𝑡_𝑟𝑎𝑡𝑒 = Input energy rate
𝐼𝑖𝑚𝑝𝑢𝑙𝑠𝑒 = Impulse delivered to system (area under load-time history)
𝑀𝑚𝑎𝑠𝑠 = Mass of the system 𝛽 = Load pulse shape factor
𝐹𝑜 = Peak load 𝑡𝑑 = Load duration
71
The conversion equations were proposed for linear-elastic SDOF systems
subjected to idealized load pulses, such as rectangular and triangular loads. In order to
validate the accuracy of the method, various tools will be used to create E-R diagrams
that will be compared to determine if the method is valid.
The first method that will be used was described in Section 2.6. This method
allows the user to create E-R diagrams for rectangular and triangular load pulses
without using any conversion equations. The diagrams created using this method are
normalized and can be used on any linear-elastic SDOF system.
The equations for an E-R diagram subjected to a rectangular load pulse are
given by:
�̅�
√�̅�𝑠𝑖𝑛 (
�̅�
2�̅�) =
1
2 1 ≤ �̅� ≤ 2.467 (3-3)
�̅�
√�̅�=
1
2 �̅� > 2.467 (3-4)
The equations for an E-R diagram subjected to a triangular load pulse are given
by:
1
�̅�(
�̅�
�̅�)
4
= 𝑠𝑖𝑛 (�̅�
�̅�)
2
+ (�̅�
�̅�)
2
− (�̅�
�̅�) 𝑠𝑖𝑛 (
2�̅�
�̅�) 1 ≤ �̅� ≤ 1.36 (3-5)
2�̅�
�̅�= tan [
2�̅�
�̅�−
�̅�1.5
�̅�2] �̅� > 1.36 (3-6)
The second method that will be used to create E-R diagrams began by
determining a shock spectrum for the given load pulse. Numerical equations can be
used to generate a P-I diagram for a specified displacement limit using the given shock
72
spectrum. Finally, the P-I diagrams will be converted using Equations 3-1 and 3-2 to
create E-R diagrams.
The last method that is necessary to validate the conversion equations is to use
the P-I Analysis function in DSAS. A linear elastic beam can be created in DSAS and a
resistance function can be obtained. This resistance function will be the same as the
one used in the shock spectrum analysis method. Using the built-in P-I Analysis function
in DSAS and specifying the same displacement limit as before, a P-I diagram for the
beam can be easily generated. Equations 3-1 and 3-2 will be used to create an
equivalent E-R diagram for the given beam. The results of this study can be seen in
Chapter 4, where comparison charts will show the accuracy of the conversion
equations.
3.3 Relevance of Load Pulse Shape Factor (Beta)
The relationship between the peak load, duration, and impulse can be described
as the load pulse shape factor, 𝛽. For a simplified load pulse, the area under the
normalized load vs. time plot is the load pulse shape factor. The normalized load vs.
time plot can be obtained by dividing all the time values by the load duration, 𝑡𝑑, and all
the load values by the peak load, 𝐹𝑜. For a rectangular load pulse, the load pulse shape
factor is 𝛽 = 1. For a triangular load pulse, the load pulse shape factor is =1
2 . For an
exponentially decaying load pulse, such as a simplified blast load pulse, there are a
range of load pulse shape factors that are all under =1
2 . The influence of the Beta
value on P-I and E-R diagrams will be studied and preliminary results of this study can
be seen in Chapter 4. Based on the results shown by Tsai [2] in Chapter 2, it is
73
assumed that slight changes will be seen in P-I and E-R diagrams that are constructed
using exponentially decaying load functions.
3.3.1 Influence of Beta Value on P-I and E-R Diagrams Domains
As shown in Chapter 2, the shape of the load vs. time plot can play a significant
role in determining what the P-I or E-R diagram looks like, as shown by the difference in
plots for a rectangular load pulse and a triangular load pulse. The Beta value for each of
these load pulses is different, resulting in a slightly different P-I diagram.
A P-I, or E-R, diagram can be broken down into 3 domains. These sections are
called the impulsive, quasi-static, and dynamic domains. Section 4.3 will describe what
effect, if any, the Beta value has on each domain, based on the preliminary results of
the investigation.
3.3.2 Proposed Relationship between Beta Values and Scaled Distance
The blast charts that are included in UFC 3-340-02 [21] that pertain to blast loads
are Figure 2-7 (Positive phase shock wave parameters for a spherical TNT explosion in
free air at sea level) and Figure 2-15 (Positive phase shock wave parameters for a
hemispherical TNT explosion on the surface at sea level). Both charts classify blast
loads based on the scaled distance. The scaled distance for a blast load is calculated
by dividing the standoff distance by the cubed root of the TNT charge weight.
𝑍 =𝑅
𝑊1/3 (3-7)
For a given scaled distance, values such as peak pressure, scaled impulse, and
scaled duration can be read from a single chart. While it is not currently listed as a value
on the UFC charts, one could calculate a new set of data points to overlay onto the UFC
charts that represent a 𝛽 value for each scaled distance. These data points can be
74
calculated by dividing the scaled impulse by the peak pressure and the scaled duration,
as shown in Equation 3-4. Since the charts have values for both incident and reflected
pressures, 𝛽 values can also be calculated for both incident and reflected blasts. The
results of the 𝛽 calculations can be seen in the Section 4.3.
𝛽 =(
𝐼𝑊1/3)
𝑃𝑜 ∗ (𝑡𝑑
𝑊1/3) (3-8)
3.4 DSAS P-I Pivot Point Improvements
When using the structural analysis software, DSAS, to generate P-I diagrams,
DSAS employs a radial search from a specified pivot point to find the combinations of
peak load and impulse that define the curve. This process, which was implemented by
Blasko et al. [6], can be seen in Section 2.4.3.4. In some cases, the pivot point that is
auto-calculated in DSAS works well and results in a solution in a short amount of time.
In other cases, such as exponentially decaying load pulses, the auto-calculated point is
very high, which yields a poorly-defined dynamic region and numerous points on the
asymptotes. To get a well-defined dynamic region using the auto-calculated pivot point,
one would have to decrease the search angle, which results in many more points that
are to be generated.
One way to increase the numerical efficiency of the analysis is to choose a pivot
point that is closer to the P-I curve. The ideal pivot point only calculates a small number
of points on the asymptotes, since the value of the asymptote can be found by using
conservation of mechanical energy. One proposed method of calculating a more
efficient pivot point is detailed below.
75
The first step in calculating a more efficient pivot point is to calculate the
impulsive and quasi-static asymptotes. This can be done by using Equations 3-9 and 3-
10, which are shown below. The asymptotes that are calculated can be looked at as
minimum values that must be reached for a failure to be possible.
𝐼𝑚𝑖𝑛 = √2 𝑀 𝑆𝐸𝑡𝑚 (3-9)
𝑃𝑚𝑖𝑛 =𝑆𝐸𝑡𝑚
𝑢𝑚𝑎𝑥 (3-10)
where,
𝑀 = Lumped mass of the system 𝑆𝐸𝑡𝑚
= Maximum strain energy
𝑢𝑚𝑎𝑥 = Maximum displacement
𝐼𝑚𝑖𝑛 = Impulsive Asymptote 𝑃𝑚𝑖𝑛 = Quasi-Static Asymptote
Next, determine where the dynamic region roughly begins. Calculate an initial
pivot point that corresponds to a 1% offset from the beginning of the impulsive and
quasi-static asymptotes. The values of load and impulse that correspond to a 1% offset
are shown in Equations 3-11 and 3-12.
𝑃1%_𝑜𝑓𝑓𝑠𝑒𝑡 = 1.01 ∗ 𝑃𝑚𝑖𝑛 (3-11)
𝑃1%_𝑜𝑓𝑓𝑠𝑒𝑡 = 1.01 ∗ 𝑃𝑚𝑖𝑛𝑃𝑚𝑖𝑛 =𝑆𝐸𝑡𝑚
𝑢𝑚𝑎𝑥 (3-12)
Each of the values calculated in Equations 3-11 and 3-12 represent a
characteristic of a single P-I point. The matching value of the two specific P-I points
must now be calculated. The two specific P-I points are (𝑃1%_𝑜𝑓𝑓𝑠𝑒𝑡 , 𝐼𝑝𝑖𝑣𝑜𝑡_𝑖𝑛𝑖𝑡𝑖𝑎𝑙) and
(𝑃𝑝𝑖𝑣𝑜𝑡_𝑖𝑛𝑖𝑡𝑖𝑎𝑙 , 𝐼1%_𝑜𝑓𝑓𝑠𝑒𝑡). The initial pivot point is then given by (𝑃𝑝𝑖𝑣𝑜𝑡_𝑖𝑛𝑖𝑡𝑖𝑎𝑙 , 𝐼𝑝𝑖𝑣𝑜𝑡_𝑖𝑛𝑖𝑡𝑖𝑎𝑙).
76
To make sure that a small portion of the asymptotes will be calculated, ensuring
that the entire dynamic region is being calculated, it is suggested to divide the initial
pivot point by the 𝛽 value of the load pulse that was used to generate the P-I
asymptotes and initial points. A set of visual steps of this process can be seen in the
Section 4.4.
3.5 Energy Flow Analysis
It is well known that a majority of the energy that is released from an event does
not reach a desired target. In the case of a hemispherical detonation, the energy waves
propagate away from the detonation site in the form of growing hemispheres. It would
be useful to designers to be able to determine how much energy will reach the structure
and at what rate that energy will do so. If these values could be calculated, a designer
could generate an E-R diagram for a desired response of a structure and determine
whether the amount of energy released from a event will cause the desired response.
One proposed way to determine how much energy will reach the structure is to
calculate how much energy is released from the detonation and subtract forms of
energy losses. Using thermodynamics principles, one could calculate how much energy
is lost in the form of heat. Another type of energy loss that happens is energy lost to
ground shock, which has been shown to be roughly 14% of the total available energy
[1]. The largest form of energy loss comes from the fact that the area of the structure
that will feel the energy wave is so small compared to the overall area of the energy
hemisphere. By studying the forms in which energy will be “lost”, one could theoretically
determine how much energy will reach a specific structure and at what rate the energy
will reach the structure.
77
This approach allows the user to analyze a structural element in the energy
domain, by using the E-R diagram to determine if the load from the event would fail the
structure. The user would not have to determine the pressure and load duration to run a
dynamic analysis, but rather could just plot the point corresponding to the energy
released from the event over the E-R diagram to check for failure. To show how this
process works, an E-R diagram for a given structural response, such as a flexural
failure, must be created. Once this is created, one could plot data points representing
different levels of energy deposition on the same plot as the E-R diagram. As mentioned
previously, the data points that fall above and to the right of the E-R diagram indicate
that the response limit has been reached, which in this case would indicate a flexural
failure has occurred. This process will be shown in Chapter 4.
78
CHAPTER 4 RESULTS AND DISCUSSIONS
4.1 Introduction
In the following sections, the results from the methods that were described in
Chapter 3 will be presented. The validation of the P-I to E-R data conversion equations
will be presented, as well as the importance of the load pulse shape factor, Beta. An
example of the improved pivot point for DSAS P-I Analysis will be presented, proving
the increase in numerical efficiency. The E-R approach will be presented using an E-R
diagram and energy deposition points to predict the failure of a beam.
4.2 Validation of Energy Conversion Equations
It was previously proposed that for a simplified load pulse, the following
equations could be used to transform a given peak pressure and total impulse to the
corresponding input energy and input energy rate. The variables that need to be known
for this are the load pulse shape factor, peak load, load duration, and the lumped mass
of the system.
𝐸𝑖𝑛𝑝𝑢𝑡 =(𝐼𝑖𝑚𝑝𝑢𝑙𝑠𝑒)
2
2 ∙ 𝑀𝑚𝑎𝑠𝑠=
(𝛽 ∙ 𝐹𝑜 ∙ 𝑡𝑑)2
2 ∙ 𝑀𝑚𝑎𝑠𝑠 (4-1)
𝐸𝑖𝑛𝑝𝑢𝑡_𝑟𝑎𝑡𝑒 =𝐸𝑖𝑛𝑝𝑢𝑡
𝛽 ∙ 𝑡𝑑 (4-2)
For a simplified load pulse, the impulse can be calculated by the Equation 4-3.
By rearranging Equation 4-3, the ratio of the minimum impulse, the impulsive
asymptote, to the peak load can be calculated.
79
𝐼𝑖𝑚𝑝𝑢𝑙𝑠𝑒 = 𝛽 ∙ 𝐹𝑜 ∙ 𝑡𝑑 (4-3)
𝛽 ∙ 𝑡𝑑 = 𝐼𝑖𝑚𝑝𝑢𝑙𝑠𝑒
𝐹𝑜 (4-4)
Equation 4-4 can now be substituted into Equation 4-2, which allows one to solve
for the energy input and energy input rate without needing to solve for the load duration,
𝑡𝑑, or the load pulse shape factor, 𝛽, if the mass of the system, peak load, and impulse
are known.
𝐸𝑖𝑛𝑝𝑢𝑡 =(𝐼𝑖𝑚𝑝𝑢𝑙𝑠𝑒)
2
2 ∙ 𝑀𝑚𝑎𝑠𝑠 (4-5)
𝐸𝑖𝑛𝑝𝑢𝑡_𝑟𝑎𝑡𝑒 =𝐸𝑖𝑛𝑝𝑢𝑡 ∗ 𝐹𝑜
𝐼𝑖𝑚𝑝𝑢𝑙𝑠𝑒 (4-6)
The conversion equations listed above were validated by deriving an analytical
based E-R diagram for both rectangular and triangular load pulses. The results from the
analytical based E-R diagram, a traditional SDOF system approach, and a DSAS P-I
analysis were compared for validation of the energy conversion equations.
4.2.1 Analytical E-R vs. DSAS Converted P-I (Rectangular Load)
Figure 4-1 shows the comparison between a traditional SDOF approach to
develop a P-I diagram, which was then converted using Equations 4-5 and 4-6, and the
analytical based E-R diagram. This plot shows normalized E-R diagrams for any given
beam subjected to a rectangular load pulse.
80
Figure 4-1. E-R diagram comparison for rectangular load pulse.
Figure 4-2 shows the results from a DSAS P-I analysis, which was also then
converted using Equations 4-5 and 4-6, and the analytical based E-R diagram. This plot
shows the actual E-R diagrams for a given beam. The energy input is measured in
Joules, while the energy input rate is measured in Watts.
Figure 4-2. E-R diagram comparison for rectangular load pulse.
4.2.2 Analytical E-R vs. DSAS Converted P-I (Triangular Load)
Figure 4-3 shows the comparison between a traditional SDOF approach to
develop a P-I diagram, which was then converted Equations 4-5 and 4-6, and the
analytical based E-R diagram. This plot shows normalized E-R diagrams for any given
beam subjected to a triangular load pulse.
Energy Input
En
erg
y I
np
ut
Ra
te
0.5 1 10 1000.5
1
10
SDOF P-I (Converted)Analytical E-R
Energy Input (J)
En
erg
y In
pu
t R
ate
(W
)
200 1000 1000050000
100000
1000000
DSAS P-I (Converted)Analytical E-R
81
Figure 4-3. E-R diagram comparison for triangular load pulse.
Figure 4-4 shows the results from a DSAS P-I analysis, which was also then
converted using Equations 4-5 and 4-6, and the analytical based E-R diagram. This plot
shows the actual E-R diagrams for a give beam. The energy input is measured in
Joules, while the energy input rate is measured in Watts.
Figure 4-4. E-R diagram comparison for triangular load pulse.
Figure 4-1 through Figure 4-4 show results that match up very well to each other.
These results allowed for the energy conversion equations to be validated and shows
the reliability of the equations over various load pulses. The results shown above also
validate the equations by comparing normalized plots as well as actual E-R diagram
plots.
Energy Input
En
erg
y I
np
ut
Ra
te
0.5 1 100.5
1
10
SDOF P-I (Converted)Analytical E-R
Energy Input (J)
En
erg
y In
pu
t R
ate
(W
)
200 1000 1000050000
100000
1000000
5000000
DSAS P-I (Converted)Analytical E-R
82
4.3 Importance of Beta Value
The load pulse shape factor is a unit-less value. Therefore, a single 𝛽 value can
be used to describe many peak loads and load durations. For example, a 𝛽 value of
0.200 can be scaled up to describe a peak load of 100 kips and a load duration of 15
milliseconds, or a load pulse with a peak load of 50 kips and 20 milliseconds. Since the
𝛽 value is the area under the normalized load vs. time plot, the impulse delivered by the
given load is given by the following equation.
𝐼𝑖𝑚𝑝𝑢𝑙𝑠𝑒 = 𝛽 ∙ 𝐹𝑜 ∙ 𝑡𝑑 = (0.200) ∗ (100𝑘𝑖𝑝) ∗ (15𝑚𝑠𝑒𝑐) = 300 𝑘𝑖𝑝 ∗ 𝑚𝑠𝑒𝑐 (4-7)
𝐼𝑖𝑚𝑝𝑢𝑙𝑠𝑒 = 𝛽 ∙ 𝐹𝑜 ∙ 𝑡𝑑 = (0.200) ∗ (50𝑘𝑖𝑝) ∗ (20𝑚𝑠𝑒𝑐) = 200 𝑘𝑖𝑝 ∗ 𝑚𝑠𝑒𝑐 (4-8)
Both of these examples of different exponential load functions that have the
same Beta value will yield the exact same P-I and E-R diagrams.
4.3.1 Influence of Beta Value on Three Domains of a P-I Diagram
A P-I, or E-R, diagram can be broken down into 3 distinct domains. These
sections are known as the impulsive, quasi-static, and dynamic domains. The following
sections will describe what effect, if any, the Beta value has on the three domains of a
P-I Diagram.
4.3.1.1 Impulsive domain
The combinations of peak load and impulse that fall within the impulsive domain
land directly on the impulsive asymptote. As shown in Chapter 2, the calculation of the
impulsive asymptote does not depend on the Beta value. Therefore, the shape of the
load vs. time plot does not influence values in this range.
83
4.3.1.2 Quasi-static domain
The combinations of peak load and impulse that fall within the quasi-static
domain land directly on the quasi-static asymptote. The calculation of the quasi-static
asymptote also does not rely on the Beta value. The data points in this region are not
affected by the Beta value.
4.3.1.3 Dynamic domain
The combinations of peak load and impulse that fall within the dynamic domain
do not land on either the impulsive or quasi-static asymptote. These values must be
calculated using a numerical procedure in which the shape of the load vs. time plot is
significant. Recall that a rectangular load pulse has a 𝛽 = 1 and a triangular load pulse
has a 𝛽 =1
2. Since each of these values yielded a different curve in Chapter 2, it is
logical to assume that a load pulse with a Beta value that is neither 1
2 nor 1 will also
produce a slightly different curve in the dynamic region.
The dynamic domain does not have a set limit on the boundaries. While it is easy
to determine the boundaries of the impulsive and quasi-static domains, by determining
where the asymptotes begin, it is not as easy to set limits on the where the dynamic
domain begins and ends. It is easier to define the dynamic domain as the data points
that fall between the other two domains.
4.3.2 Proposed Beta Value Plot on UFC 3-340-02 Blast Charts
The proposed method of calculating a Beta value for each specific scaled
distance, as described in Section 3.3.2, can be seen in the following two figures. Figure
4-5 shows the proposed curves for Beta values for hemispherical TNT detonations.
84
Since there are differences in values for incident and reflected pressures, there are also
different Beta values associated with each of the types of loads.
Figure 4-5. Calculated β values.
Scaled Distance Z = R/W1/3
Figure 2-7. Positive phase shock wave parameters for aspherical TNT explosion in free air at sea level
0.1 1 10 1000.005
0.01
0.1
1
10
100
1000
10000
100000
Pr, psiPso, psiir/W
1/3, psi-ms/lb1/3
is/W1/3, psi-ms/lb1/3
ta/W1/3, ms/lb1/3
to/W1/3, ms/lb1/3
U, ft/msLw/W1/3, ft/lb1/3
Reflected
Incident
85
Figure 4-6 shows the proposed curves for Beta values for spherical TNT
detonations. Since there are differences in values for incident and reflected pressures,
there are also different Beta values associated with each of the types of loads.
Figure 4-6. Calculated β values.
Scaled Distance Z = R/W1/3
Figure 2-15. Positive phase shock wave parameters for ahemispherical TNT explosion on the surface at sea level
0.1 1 10 1000.005
0.01
0.1
1
10
100
1000
10000
100000
200000200000
Pr, psiPso, psiir/W
1/3, psi-ms/lb1/3
is/W1/3, psi-ms/lb1/3
tA/W1/3, ms/lb1/3
to/W1/3, ms/lb1/3
U, ft/msLw/W1/3, ft/lb1/3
Reflected
Incident
86
4.3.3 Examples of Beta Value Influence on P-I and E-R Diagrams
The shape of the load pulse, 𝛽, alters the shape of P-I and E-R diagrams. While
the asymptotes for the diagrams are calculated independent of the load function, the
dynamic region between the impulsive and quasi-static asymptotes can be significantly
affected by the shape of the load pulse. Figure 4-7 shows a range of normalized load
vs. time plots, each of which has a corresponding 𝛽 value, as shown in the legend.
Each normalized load vs time curve in Figure 4-7 was pulled from Figure 4-5 to show a
range of valid Beta values that correspond to a specific charge of TNT. Each of the load
pulses shown in Figure 4-7 were run through the DSAS P-I Generator to develop P-I
diagrams for each of the load pulses, as can be seen in Figure 4-8.
Figure 4-7. Simplified load pulses and their corresponding β values.
Normalized Time
No
rma
lize
d L
oa
d
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Exponential Load 1 ( = 0.150)Exponential Load 2 ( = 0.203)Exponential Load 3 ( = 0.249)Exponential Load 4 ( = 0.298)Exponential Load 5 ( = 0.351)Exponential Load 6 ( = 0.400)Triangular Load ( = 0.500)
87
Figure 4-8. P-I diagrams for a range of simplified load pulses.
After using the P-I to E-R conversion equations proposed by Tsai [2], as shown
in Equations 4-1 and 4-2, Figure 4-9 shows the range of E-R diagrams for the load
pulses described in Figure 4-7.
Figure 4-9. E-R diagrams for a range of simplified load pulses.
Impulse (psi-sec)
Pre
ss
ure
(p
si)
2 10 100 20050
100
1000
5000
Exponential Load 1Exponential Load 2Exponential Load 3Exponential Load 4Exponential Load 5Exponential Load 6Triangular Load
Input Energy (Joules)
Inp
ut
En
erg
y R
ate
(W
att
s)
50000 100000 1000000 1E+75000000
1E+7
1E+8
Exponential Load 1Exponential Load 2Exponential Load 3Exponential Load 4Exponential Load 5Exponential Load 6Triangular Load
88
4.4 DSAS Improved P-I Pivot Point Analysis
As described in Section 3.4, a more efficient P-I pivot point is being proposed to
increase the numerical efficiency of the DSAS P-I analysis. This pivot point allows for
less points to be calculated on the impulsive and quasi-static asymptotes and more to
be calculated in the dynamic transition region, yielding a more efficient and smoother
curve as the output. A visual representation of the steps described in Chapter 3 can be
seen below.
The first step in calculating the new P-I pivot point is to calculate the asymptotes.
Once the asymptotes are calculated, determine where the dynamic region roughly
begins. This can be done by calculating an initial pivot point that corresponds to a 1%
offset from the beginning of the impulsive and quasi-static asymptotes. The two blue
points shown in Figure 4-10 indicate the rough beginning and end of the dynamic
region.
Figure 4-10. Impulsive and quasi-static asymptotes with initial pivot point.
Impulse (psi-sec)
Pre
ss
ure
(p
si)
2 10 100 20050
100
1000
10000
Impulsive AsymptoteQuasi-Static AsymptoteInitial Pivot Point
89
Figure 4-11 shows the improved new pivot point and the range of radial search
for DSAS. This point is found by dividing the initial pivot point by the Beta value,
increasing the value of the pivot point to include a small portion of the impulsive and
quasi-static asymptotes.
Figure 4-11. New pivot point that captures portions of asymptotes.
The results of a DSAS P-I analysis using the improved pivot point can be seen in
Figure 4-12. The pivot point that was chosen allows the radial search to capture a small
portion of the two asymptotes as well as the entire dynamic region. By using a pivot
point that is closer to the actual P-I curve, the user may select less data points to be
calculated and still achieve a high-resolution result.
Impulse (psi-sec)
Pre
ss
ure
(p
si)
2 10 100 20050
100
1000
10000
Impulsive AsymptoteQuasi-Static AsymptoteFinal Pivot Point
90
Figure 4-12. New pivot point and P-I curve that results from DSAS.
By employing this method for calculating P-I diagrams in DSAS, not only are the
results clearer but the time it takes to run the analysis is significantly reduced. By not
allowing the program to calculate hundreds of points that fall on either of the
asymptotes, the number of data points needed to generate a smooth curve may be
reduced, resulting in a shorter runtime. A comparison of the increase in efficiency can
be seen in the following section.
4.5 DSAS Existing P-I vs. Improved P-I Analysis Comparison
Three types of load functions were analyzed for this comparison. The first was a
triangular load pulse, the second was an exponential decay load pulse, and the third
was a bi-linear load pulse. Each of the three load pulses can be seen in Figure 4-13
below.
Impulse (psi-sec)
Pre
ss
ure
(p
si)
2 10 100 20050
100
1000
10000
DSAS (Triangular Load)Impulsive AsymptoteQuasi-Static Asymptote
91
Figure 4-13. Load pulses for P-I runtime comparison.
When generating a P-I diagram, the magnitudes of the pressure and time from
the loading function do not influence the curve. The shape of the load function,
previously mentioned as the Beta value, is the factor that changes the P-I diagram. The
three loads were chosen with the same peak pressure and load duration to illustrate the
difference in load pulse shape.
Each of the three load pulses were run through the DSAS P-I generator using the
default pivot point to determine the default P-I diagram. Next, an improved pivot point
was calculated for each of the three load pulses using the process previously described.
The three load pulses were then run using the improved pivot point. The actual runtime
was recorded for each of the 6 analyses. The runtime results and improvements can be
seen in Table 4-1 below. The improved P-I pivot point allowed for runtimes to be cut
down to less than 20% for all cases that were examined, with some cases showing a
runtime of almost 10% of the original.
Time (sec)
Pre
ss
ure
(p
si)
0 0.003 0.006 0.009 0.0120
3
6
9
12
Triangular LoadExponential LoadBi-Linear Load
92
Table 4-1. Comparison of original P-I runtime to improved P-I runtime
Type of Load Triangular Exponential Bi-Linear
Load Pulse Shape Factor Beta = 0.5 Beta = 0.4 Beta = 0.25
DSAS Default Runtime 3:15 3:00 3:35
DSAS Improved Runtime 0:21 0:53 0:23
Percent of Original Runtime 11% 16% 11%
Although the runtimes were significantly lower for the improved pivot point
analyses, the results still needed to be checked to ensure the accuracy of the solution
that was found. The following three figures show the results of the P-I analyses with the
improved analysis overlaid on top of the original, or default, solution.
Figure 4-14. Triangular Load P-I Comparison.
Impulse (psi-sec)
Pre
ss
ure
(p
si)
2 10 100 50050
100
1000
5000
Original P-I DiagramImproved P-I Diagram
93
Figure 4-15. Exponential Load P-I Comparison.
Figure 4-16. Bi-Linear Load P-I Comparison.
From the results of the three cases shown above, it can be easily seen that the
improved pivot point still captures the beginning of the asymptotes while allowing for a
very smooth transition through the dynamic zone in the middle. By allowing the user to
generate these diagrams in a fraction of the time it currently takes while still ensuring
the accuracy of the solution, the method for the improved P-I pivot point increases the
numerical efficiency of the solution when running the DSAS P-I Analysis.
Impulse (psi-sec)
Pre
ss
ure
(p
si)
2 10 100 50050
100
1000
5000
Original P-I DiagramImproved P-I Diagram
Impulse (psi-sec)
Pre
ss
ure
(p
si)
2 10 100 50050
100
1000
5000
Original P-I DiagramImproved P-I Diagram
94
4.6 Energy Flow Analysis
The E-R diagram approach can be very useful to an engineer when working in
the energy domain. Take an example of a beam that is subjected to various energy
depositions. The E-R diagram for a flexural failure was generated using the process that
was previously described, as shown in Figure 4-17.
Figure 4-17. E-R diagram for a flexural failure
As described in Chapter 3, one could theoretically calculate the amount of energy
and the rate at which that energy is flowing at certain specified locations. Assuming this
process was previously performed, Table 4-2 shows the energy deposition that would
be seen from four different energy releases at a specific location.
Table 4-2. Energy deposition amounts from energy releasing events.
Energy Source Deposition 1 Deposition 2 Deposition 3 Deposition 4
Input Energy (kJ) 59.57 73.94 89.62 115.58
Input Energy Rate (kW) 262771.39 314608.49 368977.16 455100.76
Input Energy (kJ)
Inp
ut
En
erg
y R
ate
(k
W)
50 100 1000 100005000
10000
100000
1000000
Flexural Failure
95
These energy deposition points can now be overlaid on top of the flexural failure
E-R diagram to illustrate which depositions would cause the desired response, which is
a flexural failure in this case. This can be seen in Figure 4-18. The energy deposition
points from Table 4-2 increase in input energy from Deposition 1 to Deposition 4. The
curve for Energy Deposition Amounts in Figure 4-18 shows Depositions 1-4 in order
from left to right.
Figure 4-18. E-R diagram with overlaid energy deposition curve.
By plotting both of these curves on the same plot, one can visually tell which
points would cause the flexural failure. From Figure 4-18, energy depositions 1 and 2
would not cause the failure to occur, while depositions 3 and 4 would cause the failure.
Once the E-R diagram is created, dynamic analyses are not needed to determine the
level of structural response. Only the amount of input energy and input energy rate that
reach the structural element from an energy releasing event need to be known to
analyze the structure.
Although dynamic analyses are not needed to analyze the structural element
once the E-R diagram was created, each of the energy deposition cases were analyzed
individually to show the accuracy of the approach. The equivalent mass of the element
Input Energy (kJ)
Inp
ut
En
erg
y R
ate
(k
W)
50 100 1000 100005000
10000
100000
1000000
Flexural FailureEnergy Deposition Amounts
96
that was used was 62.57 kg and the surface area that the energy deposition was
applied to was 460.808 in2. From these characteristics of the structural element, the
conversion equations shown in Equations 4-5 and 4-6 can be rearranged to solve for
the equivalent pressure and impulse for each deposition.
These resulting triangular loads were used in DSAS to analyze the element using
traditional dynamic analysis methods to show the accuracy of the E-R approach. Table
4-3 shows the equivalent load characteristics and the results from the DSAS analysis.
The results shown below match the expected results mentioned previously, showing the
energy depositions 1 and 2 would not fail the beam, while 3 and 4 would cause a
flexural failure.
Table 4-3. Equivalent triangular load from energy depositions.
Deposition Number
Energy Input (kJ)
Energy Input Rate (kW)
Impulse (psi*sec)
Pressure (psi)
Duration (msec)
DSAS Result
1 59.6 262771 1.332 5876 0.453 Failure NOT Detected
2 73.9 314608 1.484 6314 0.470 Failure NOT Detected
3 89.6 368977 1.634 6726 0.486 Failure DETECTED
4 115.6 455101 1.855 7306 0.508 Failure DETECTED
97
CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS
5.1 Conclusions
This study was aimed at validating the conversion equations from P-I data to E-R
data and applying the methodology to more realistic scenarios. By comparing multiple
cases using converted P-I data from DSAS and the analytical based energy solutions
that were described in Chapter 2, the conversion equations were successfully validated.
The validation was done on linear-elastic SDOF systems using simplified load pulses.
By using the conversion equations, one is simply changing the axes on a P-I diagram to
represent the same response in terms of energy input and energy input rate.
Another focus in this research was to determine the role that the load pulse
shape factor played in the calculation of P-I and E-R diagrams. The load pulse shape
factor, Beta, was originally calculated for simplified load pulses only, such as
rectangular and triangular loads. It was noticed that there were differences in the
dynamic region of the diagrams when the load used to calculate the diagram was either
rectangular or triangular. Recall that the Beta value is the area under the normalized
load vs. time curve and that a rectangular load has a Beta value of 1.0, while a
triangular load has a Beta value of 0.5. By noticing that there are many potential other
Beta values in addition to these two, the role of Beta was investigated. It was found that
every different Beta value yielded a slightly different P-I and E-R diagram throughout the
dynamic region. However, the asymptotes for all of the cases still converged to the
same value, regardless of the Beta value, showing that the Beta value only plays a
significant role throughout the dynamic transition region between the impulsive and
quasi-static asymptotes. This is an interesting discovery which will allow the user to
98
potentially use a simplified load pulse to determine a P-I or E-R diagram if the loading
that is to be expected is mostly impulsive in nature. The simplified load pulses allow for
a more efficient process in creating the diagrams, and their use can be justified if the
asymptotes are the more critical sections of the diagram due to the given loadings that
could occur.
After validating the conversion equations from P-I to E-R diagrams and
determining the role the Beta value played, the numerical efficiency in calculating these
diagrams was studied. It was found that the current P-I Analysis module in DSAS auto-
calculates a pivot point for the radial search that encompasses a large amount of points
on the P-I diagram. The proposed method to increase numerical efficiency requires the
user to calculate the impulsive and quasi-static asymptotes before selecting a pivot
point. Using the methodology described in Section 3.4, the more efficient pivot point can
be found. By using this method, analysis runtimes drop to less than 20% of the current
runtime, and some even drop to less than 10%. This is a great increase in numerical
efficiency, which can potentially save time and money for the user.
The final aspect of this research was to examine more realistic scenarios using
the E-R diagram approach. This was done by creating an E-R diagram for a beam that
represented a flexural failure. A series of a “load points”, or energy depositions, were
plotted on the same graph as the E-R diagram. From this, it was clear that two of the
four energy depositions would fail the beam in flexure. The four energy depositions
were transformed into pressure vs. time plots using the conversions equations
mentioned previously. These loads were checked in DSAS to ensure the expected
results from the E-R diagram. Each of the four points caused the beam to react exactly
99
as would be expected. Two of the loads did not cause a failure, while the final two loads
caused a flexural failure. This confirmed that the approach is valid and can be used to
analyze a structural element based on expected amounts of energy that will be
deposited to the element.
5.2 Recommendations for Future Research
From the progress that was made with this research, there are a few
recommendations for future research to take this process a step farther. All of the
analysis runs that were examined assumed that the structural element was uniformly
loaded with pressure only. However, this is not always the most accurate way to
describe a load. Analyzing an element under a single or double point load using the E-R
approach could prove to be a very useful tool to designers. Also, considering the effects
of fragment loading would also help improve the accuracy of the approach, bringing
results potentially closer to real world examples.
The improvements that were recommended for DSAS have not been integrated
into the software program at this time. The following items would be beneficial to
incorporate into the software package DSAS. The first would be to integrate the P-I to
E-R conversion equations into DSAS. This would allow the program to output the E-R
diagrams as well as the P-I diagrams once the analysis is completed. The second would
be to incorporate the improved pivot point into DSAS so that the auto-calculated point
uses the methodology described in Chapter 3. This would increase the numerical
efficiency of DSAS.
100
LIST OF REFERNECES
[1] T. Krauthammer, Modern Protective Structures, CRC Press, Taylor and Francis Group, 2008.
[2] Y. K. Tsai and T. Krauthammer, "Energy Based Load-Impulse Diagrams for Structural Elements," University of Florida, 2015.
[3] W. E. Baker, P. A. Cox, P. S. Westine, J. J. Kulesz and R. A. Strehlow, Explosion hazards and evaluation, Amsterdam, NY: Elsevier Scientific Pub Co, 1983.
[4] P. H. Ng, "Pressure-Impulse Diagrams for Reinforced Concrete Slabs," The Pennsylvania State University, 2004.
[5] T. B. Soh, "Load-Impulse Diagrams of Reinforced Concrete Beams Subjected to Concentrated Transient Loading," The Pennsylvania State University, 2004.
[6] J. R. Blasko, T. Krauthammer and S. Astarlioglu, "Pressure-Impulse Diagrams of Structural Elements Subjected to Dynamic Load," U.S. Army Engineer Research and Development Center (ERDC), Vicksburg, MS, 2007.
[7] T. Krauthammer, S. Astarlioglu, J. Blasko, T. B. Soh and P. H. Ng, "Pressure-Impulse Diagrams for the Behavior Assessment of Structural Components," International Journal of Impact Engineering, pp. 35:771-83, 2008.
[8] P. D. Smith and J. G. Hetherington, Blast and Ballistic Loading of Structures, London: Butterworth-Heinemann Ltd, 1994.
[9] N. Jones, Structural Impact, Cambridge: Cambridge University Press, 1989.
[10] T. Krauthammer, "Blast Mitigation Technologies: Development and numerical Considerations for Behavior Assessment and Design," in International Conference on Structures under Shock and Impact, 1998.
[11] J. M. Biggs, Introduction to Structural Dynamics, New York, NY: McGraw-Hill Book Company, 1964.
[12] T. Krauthammer, S. Shahriar and H. M. Shanaa, "Response of RC Elements to Severe Impulsive Loads," Journal of Structural Engineering, pp. 116(4):1061-79, 1990.
[13] N. M. Newmark and E. Rosenblueth, Fundamentals of Earthquake Engineering, Englewood Cliffs, NJ: Pearson Prentice Hall, 1971.
101
[14] R. W. Clough and J. Penzien, Dynamics of Structures, Second ed., New York, NY: McGraw-Hill Book Company, 1993.
[15] G. N. J. Kani, "Basic Facts Concerning Shear Failure," ACI Journal, pp. 63(6):675-92, 1966.
[16] T. Krauthammer, S. Shahriar and H. M. Shanaa, "Analysis of Concrete Beams Subjected to Severe Concentrated Loads," ACI Structural Journal, pp. 84(6):473-80, 1987.
[17] S. Astarlioglu and T. Krauthammer, "Dynamic Structural Analysis Suite (DSAS) User Manual," CIPPS, 2012.
[18] A. H. Mattock and N. M. Hawkins, "Shear Transfer in Reinforced Concrete - Recent Research," Journal of the Pre-stressed Concrete Institute, p. 17(2):55–75, 1972.
[19] J. W. Tedesco, W. G. McDougal and C. A. Ross, Structural Dynamics: Theory and Applications, Menlo Park, CA: Addison-Wesley Longman Inc, 1999.
[20] M. P. M. Rhijnsburger, J. R. Van Deursen and J. C. A. M. Van Doormaal, "Development of a Toolbox Suitable for Dynamic Response Analysis of Simplified Structures," in DoD Explosive Safety Seminar, 2002.
[21] U.S. Army Corps of Engineers, "Unified Facilities Criteria (UFC) 3-340-02," U.S. Department of Defense, 2014.
[22] T. Krauthammer, M. ASCE, N. Bazeos and T. J. Holmquist, "Modified SDOF Analysis of RC Box-Type Structures," Journal of Structural Engineering, vol. 112, no. 4, 1986.
102
BIOGRAPHICAL SKETCH
Kyle began his undergraduate studies at the University of Florida in the fall of
2012. He received a Bachelor of Science in Civil Engineering from the University of
Florida in December 2016 and subsequently began to pursue his master’s degree in
January 2017. Kyle received a Master of Science in Civil Engineering with a
concentration in structures in August 2018.
