Analysis Method for Reinforcing Circular Openings in Isotropic Homogeneous Plate-Like Structures Subjected to Blast Loading

Hindawi Advances in Civil Engineering Volume 2018, Article ID 4071732, 21 pages https://doi.org/10.1155/2018/4071732

Research Article Analysis Method for Reinforcing Circular Openings in Isotropic Homogeneous Plate-Like Structures Subjected to Blast Loading

Sebastian Mendes and Liling Cao

ornton Tomasetti, 40 Wall Street, 18th Fl., New York, NY 10005, USA

Correspondence should be addressed to Sebastian Mendes; smendes@thorntontomasetti.com

Received 26 May 2018; Accepted 25 July 2018; Published 5 September 2018

Academic Editor: Chiara Bedon

Copyright © 2018 Sebastian Mendes and Liling Cao. (is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An analysis method is formulated to predict the peak bending stress concentrations around a small circular opening in an idealized isotropic homogeneous, linear elastic-perfectly plastic plate-like structure subjected to uniform blast loading. (e method allows for the determination of corresponding concentrated bending moments adjacent to the opening for the design of reinforcement that can prevent the formation of localized plasticity around the opening during a blast event. (e rapid formation and growth of localized plasticity around the opening can lead to a drastic reduction of the plate-like structure’s local and global stability, which could result in catastrophic failure of the structure and destruction of the entity it is protecting. A set of elemental formulas is derived considering one-way and two-way rectangular plate-like structures containing a single small circular opening located where ﬂexure predominates. (e derived formulas are applicable for elastic global response to blast loading. Abaqus was employed to conduct numerical veriﬁcation of the derived formulas considering various design parameters including material properties, plate dimensions, position of opening, and explosive charge size. (e formulas demonstrate a good correlation with FEA albeit with a conservative inclination. (e derived formulas are intended to be used in tandem with dynamic SDOF analysis of a blast load-structure system for ease of design. Overall, the proposed method has the potential to be applicable for many typical conditions that may be encountered during design.

1. Introduction A rational approach to the blast-resistant design of structures was articulated by Newmark in 1956 [2] and Biggs Plate-like structures are often used in buildings to protect in 1964 [3]. Since that time, the performance of various civil occupants and life-safety systems from high-energy explo- structures such as reinforced concrete frames, reinforced sions [1]. Examples of plate-like structures include hardened masonry walls, and timber panels subjected to blast loading blast walls and strengthened floor slabs for protecting against has been investigated [4–6]. Importantly, the performance of blast infill pressure. Hardened blast walls are strategically the building envelope, essentially glazing and glass facades, positioned within a building to protect means of egress and under air blast has been researched [7, 8]. In direct relation equipment critical for the functioning of life-safety systems to this paper, the dynamic behavior of metallic plate-like after the blast event. (e blast walls are nonload bearing and structures subjected to blast loading has been extensively designed to resist blast infill pressure by one-way flexural investigated through analytical, numerical, and experi- behavior. Strengthened floor slabs are located in building mental studies [9–18]. At present, blast-resistant design is areas where prevention of floor collapse is determined to be standardized in several publications including ASCE 59-11 imperative in thwarting progressive collapse after the blast Blast Protection of Buildings [1] and UFC 3-340-02 Structures event [1]. (e floor slabs are designed to carry conventional to Resist the Effects of Accidental Explosions [19]. (e global gravity and diaphragm loads while also resisting blast infill dynamic response of plate-like structures can remain in the pressure by two-way or one-way flexural behavior. elastic regime, or in extreme cases can enter the plastic 2 Advances in Civil Engineering regime. (e dynamic response is in part dependent upon the structural damping. (e structure continues to oscillate in structural material, support conditions, explosive charge free vibration after the blast pressure has dissipated pro- weight, standoff, and the ratio of the reflection surface to the vided the structure remains intact. For all practical pur- target area [1, 3, 19]. poses, the deformation response corresponds to the Plate-like structures designed for blast loading are often fundamental mode with the mechanical energy balance required to have circular penetrations to accommodate oscillating between kinetic energy and internal strain en- mechanical ducts or electrical conduits. (e size and posi- ergy [3, 19]. tioning of the penetrations depends upon the function of the A common industry standard analysis method for de- entity being protected. For example, a steel hardened blast signing structural elements for blast loading is to transform wall protecting a communication room within a building the load-structure system into an equivalent undamped may require a small circular opening to accommodate single degree-of-freedom (“SDOF”) system possessing electrical conduits. (e presence of a small circular opening a linear elastic-perfectly plastic response mode [1, 19]. (e in a plate-like structure undergoing out-of-plane de- system is loaded by a triangular pressure-time history curve formation due to blast loading induces especially high representing the actual blast pressure-time history, as shown bending stress concentrations around the opening, as in Figures 3(a) and 3(b). (is procedure involves employing demonstrated in Figures 1(a) and 1(b). (ese stress con- factors to transform the load, stiffness, ultimate resistance, centrations can cause localized plasticity even though the and mass into equivalent values for use in an SDOF analysis initial global response of the structure primarily remains in such that the displacement response remains consistent the elastic strain range. (e rapid formation and growth of between the real and equivalent systems [3]. (e trans- localized plasticity around the opening can lead to a drastic formation factors are in part dependent upon the deformed reduction of the plate-like structure’s local and global sta- shape and boundary conditions of the plate-like structure. bility, which could result in catastrophic failure of the (e displacement response of the SDOF system can be structure and destruction of the entity it is protecting. Re- solved for using various numerical methods such as the inforcement can be added to strengthen the region around constant-velocity procedure. Peak stresses occur very early the opening to resist bending stress concentrations that can in the displacement response, and thus, inherent structural cause localized plasticity, as shown in Figure 1(c). damping can be neglected in the analysis. (ere is a considerable amount of literature pertaining to (e structural properties of a plate-like structure can be stress concentrations around flaws subjected to far-field designed to optimize its strength and performance for stress. (e plane solution for stress concentrations around a given blast loading scenario. For reasons of practicality, circular openings was derived by Kirsch in 1898 [20]. Stress a structure can be designed to resist the peak forces expe- concentrations at cracks were later investigated by Griffith, rienced during the free vibration phase while maintaining Westergaard, and Irwin [21–23]. More recent studies have prescribed performance requirements. Performance re- investigated stress concentrations around irregularly shaped quirements generally allow a structure to attain certain levels holes and circular holes in anisotropic materials [24–30]. of ductility and deformation depending upon its structural (e influence of openings on stability has been investigated composition. Maximum response limits for structural [31–33]. components subjected to blast effects are recommended by A high-energy explosion is characterized by a rapidly ASCE 59-11 [1]. expanding hemispherical wave of high-pressure gas called In designing a plate-like structure containing a small a shock wave. (e reflected pressure and impulse from the circular opening for blast effects, it may be desirable to shock wave upon or within a structure is a primary cause of approximately calculate the distribution of peak bending damage from blast effects [19]. (e reflected pressure and stress around the opening such that suitable reinforcement impulse are dependent upon several parameters including can be designed to resist stress concentrations that can the explosive charge weight, range to target, angle of in- cause localized plasticity. (e response of the plate-like cidence, and shielding effects from the building façade or any structure may be adversely influenced if any localized intermediate partition walls [1]. (e reflected pressure can plasticity around the opening is allowed to form and grow be described by a pressure-time history curve, where the over subsequent cycles of free vibration. Specifically, the positive area beneath the curve is the reflected blast impulse. kinetic energy of the plate-like structure may be pro- A blast pressure-time history curve is characterized by a gressively dissipated or stored as plastic strain energy sharp increase in positive pressure followed by a nearly linear around the opening, potentially resulting in large de- decay to atmospheric pressure, as shown in Figure 2(a). A formations, rupture, or dynamic fracture during the initial simplified triangular pressure-time history curve can be cycle or subsequent oscillations, as depicted in Figure 4(b) constructed from the peak reflected pressure and impulse [34–36]. (e use of oversimplified analytical methods for for use in design, as shown in Figure 2(b) [1]. calculating the stress concentrations may result in an overly (e blast impulse imparts inertial energy into a struc- conservative design of the reinforcement. Conversely, the ture, causing it to undergo dynamic deformation [3]. In the use of complex analytical or computational methods can case of plate-like structures, the blast impulse causes the result in a more efficient design; however, these methods structure to rapidly deform out-of-plane wherein the de- may be time-consuming and require a significant allotment formation response of the structure is dependent upon its of computational capacity thus rendering them impractical flexural stiffness, ultimate resistance, mass, and inherent for design. Advances in Civil Engineering 3

Head Compression region Circular opening support Circular opening Critical tension Reinforcement region

Base support (a) (b) (c)

Figure 1: (a) One-way plate-like structure undergoing out-of-plane ﬂexure due to blast loading; (b) bending stress concentrations around the opening; (c) reinforcement added around the opening.

Positive Positive P P S0 reflected S0 reflected I I impulse, r impulse, r

Negative Pressure, P ( t ) impulse P P 0 0 t t – t d d d Time Time (a) (b)

Figure 2: Blast pressure-time history curves: (a) typical blast pressure-time history curve; (b) simpliﬁed triangular blast pressure-time history.

Head Stiffness, K support Mass, M

F(t)

z(t)

Plate-like structure under blast loading Base support (a) (b)

Figure 3: Load-structure system transformed into an equivalent undamped SDOF system for analysis of blast load response: (a) blast load- structure system; (b) equivalent undamped SDOF system.

2. Analysis Method circular opening in an idealized isotropic homogeneous plate-like structure subjected to uniform blast loading. (e (e objective is to formulate an analysis method to predict predicted stress concentrations are to be used to design the peak bending stress concentrations around a small reinforcement that can prevent the formation of localized 4 Advances in Civil Engineering

Far-field A bending stress B Corresponding σ Localized σ Y σ Y σ stress distribution plasticity in the for an equivalent elastoplastic elastic system Deformation system A-A Time Local domain Circular around opening Elastoplastic system opening Equivalent elastic system

Plastic hinge in an elastoplastic system Extreme σ Stress distribution for Y tension fiber

an equivalent elastic system t σ Y Plate-like Extreme structure compression fiber Section A-A (a) (b) v v B

σY A

ΔUP ΔU′E,peak ΔUE von Mises stress, σ ε von Mises stress, σ ε

Figure 4: Peak stress concentrations and strain energy in the local domain around the opening for the elastoplastic system and equivalent elastic system during initial oscillation: (a) peak unidirectional stress concentrations adjacent to the opening and parallel to far-eld bending stress; (b) deformation response for the elastoplastic system and equivalent elastic system; (c) peak elastoplastic strain energy developed in the local domain around the opening; (d) corresponding elastic strain energy developed in the same domain for the equivalent elastic system.

plasticity. e method is intended to be compatible with 0 U K, dynamic SDOF blast analyses of plate-like structures pos- + ( ) U sessing elastic global response. In accordance with industry where is the internal strainΔ energyΔ in the local domain1 standards, the blast load-structure system is assumed to around the opening and K is the associated kinetic energy possess a linear elastic-perfectly plastic response mode for impartedΔ by the blast impulse. As the plate-like structure the localized stress concentrations around the opening [1]. achieves its initial peak out-of-planeΔ deformation, the kinetic e introduction of an opening in a plate-like structure energy is zero and the total mechanical energy around the inherently modies its exural sti ness. If the opening is small opening is entirely strain energy: compared to the unbraced surface area of the structure free to U U U , deform out-of-plane, then the change in sti ness can be con- peak E + P ( ) sidered negligible (here, the denition of “small” is highly where U is the elastic strain energy and U is the plastic subjective upon the allowable change in dynamic response). E Δ Δ Δ P 2 strain energy. e formation of localized plasticity around Consequently, the global deformation response of a plate-like the opening is indicated by the development of plastic strain structure containing a small opening and subjected to blast Δ Δ energy and accompanying permanent deformation, as loading is e ectively unchanged from the response of an depicted in Figure 4(b). is permanent deformation can identical structure without an opening. is is true unless plastic adversely inuence the plate-like structure’s local and global hinges, rupture, or cracks develop adjacent to the opening such stability and subsequent response. us, the failure criterion that the response is adversely inuenced (Figure 4). is achieved when the plastic strain energy develops, e failure criterion is dened as the formation of expressed as follows: plasticity in the local domain around the circular opening, that is, the development of plastic hinges adjacent to the UP 0. opening due to concentrated bending moments, as depicted ( ) in Figure 4(a). As per the principle of conservation of Reinforcement aroundΔ the> opening should be designed3 mechanical energy (neglecting structural damping), the total to resist concentrated peak bending moments developing mechanical energy around the opening remains constant as plastic hinges adjacent to the opening and corresponding to the plate-like structure oscillates through out-of-plane de- Upeak when (3) is satised. In terms of strain energy, the formation during free vibration [37]: reinforcement should be designed to absorb Upeak in Δ Δ Advances in Civil Engineering 5 a purely elastic form. In accordance with the principle of Formulate the blast Transform an blast conservation of mechanical energy, the peak elastoplastic loading scenario load-structure system strain energy ΔUpeak around the opening during the ﬁrst into an equivalent cycle of vibration is in the most conservative case eﬀectively Perform SDOF undamped SDOF system analysis considering equal to the peak elastic strain energy, ΔU′ , , in the same E peak no circular opening domain derived from an equivalent elastic system, as b depicted in Figures 4(c) and 4(d) [34, 35]. (us, the con- Simple support x centrated peak bending moments can be solved by evalu- U′ Theoretical ating the bending moments corresponding to Δ E,peak: location of the

+ h U + U U′ . circular opening Δ E Δ P ≈ Δ E,peak (4) x y ( o, o) In general, the following formulation is proposed: in the first step a blast load-structure system is derived for a plate- Simple support y like structure assumed to not contain a small circular Extract peak opening. (is is achieved by first constructing a triangular Express deformed displacement Δ Peak shape w(x, y) in terms pressure-time history based on a given blast loading sce- during the free of peak displacement nario. Next, the characteristic shape function and flexural vibration phase stiffness are extracted from the assumed deformed shape of the plate-like structure corresponding to the fundamental mode. Mass and load transformation factors are then de- Obtain bending stress Substitute w(x, y) field on the tension surface into Kirchhoff–Love rived from the shape function, mass, and load. (e blast of the plate-like structure plate equations load-structure system is transformed into an equivalent undamped SDOF system and dynamically analyzed in the elastic strain range. (e peak displacement during the free Determine bending stress Consider acquired stress to vibration phase is extracted from the analysis. magnitude at theoretical be loading opening as location of opening (x , y ) In the intermediate step, the deformed shape is expressed o o far-field plane tensile stress in terms of the characteristic shape function and the peak σ Infinite far-field displacement. (e complete peak bending stress field at the Use the Kirsch equations to plane tension surface of the plate-like structure may then be de- formulate expressions for termined by substituting the deformed shape into the Circular elastic bending stress Kirchhoff–Love plate differential equations for expressing opening concentrations around the opening in terms of peak displacement σ bending stresses in terms of curvatures. far-field In the final step, a small circular opening is assumed to lie within an infinite plane and loaded by uniaxial or biaxial FEA Formulate expressions far-field tensile stress. (e tensile stress is determined from verification for concentrated peak the bending stress field obtained in the previous step at the bending moments theoretical location of the opening. (e Kirsch equations are Figure 5: Proposed formulation of the analysis method. used to determine the elastic stress concentrations around the circular opening when loaded by the far-field tensile stress (assuming no yielding). As a result, the peak elastic opposite edges, and a rectangular slab simply supported at all bending stress concentrations around a circular opening four edges. (ese two particular cases encompass a signifi- may in part be expressed in terms of the peak displacement cant share of blast-resistant plate-like structures in- obtained from a dynamic SDOF blast analysis in the elastic corporated into buildings. Numerical verification of the strain range. As per the principle of conservation of me- analysis method is performed using the finite element analysis chanical energy, the concentrated peak bending moments (“FEA”) software Abaqus. (e formulation of the analysis adjacent to the opening producing plastic hinges may be method is outlined in Figure 5. determined from the unidirectional elastic stress concentrations adjacent to the opening (and parallel to the far-field 2.1. Case: Rectangular Wall Simply Supported at Two Opposite tensile stress) exceeding the actual yield point [37, 38]. (is Edges. It is assumed that the deformed shape of a rectan- is under the propositions that the unidirectional elastic stress gular plate-like structure during elastic response to blast predominates the state of stress and is thus used in lieu of the loading conforms to the deflection surface obtained from von Mises stress, and that the elastoplastic transition range a statically applied uniform surface load. Any deflection through the thickness is neglected. (e resulting concen- surface, w(x, y), for an isotropic homogeneous plate-like trated peak bending moments may then be used to design structure must satisfy the boundary conditions and the appropriate reinforcement for preventing localized plasticity governing plate equation [39]: around the opening. z4w z4w z4w q Two common cases of rectangular plate-like structures ∇4w(x, y) � + 2 + � , (5) are considered; a rectangular wall simply supported at two zx4 zx2zy2 zy4 D 6 Advances in Civil Engineering

b z Simple support x ϕ(y)

o –1 Circular y opening with radius, a Flexure h predominates

Simple support y

(a) (b)

Figure 6: (a) One-way rectangular wall with a small circular opening under blast loading; (b) characteristic shape function.

Table 1: Parameters for an equivalent SDOF system of a one-way extracted for deformations remaining within the elastic ﬂexural element simply supported at the two opposite edges [3]. strain range. In general, the peak deformed shape can be expressed in Strain range Real stiﬀness, K Mass factor, KM Load factor, KL terms of the characteristic shape function and peak dis- Elastic (384EI)/(5h3) 0.50 0.64 placement as follows:

wpeak(x, y) � Δpeakϕ(x, y). (8) where q is the uniform surface load and D is the plate Et3 [ ( 2)] flexural rigidity defined by / 12 1 − μ . (e Kirchhoff–Love plate differential equations for A rectangular wall simply supported at the two opposite expressing bending stresses at the surface of a plate in terms edges and uniformly loaded by blast pressure effectively of the peak deformed shape are given as follows [39]: behaves as a one-way flexural element, where bending occurs in the y-axis direction between the supported edges. −tE z2w z2w σ (x, y) � ⎛⎝ peak + μ peak⎞⎠, (9a) Parameters described with respect to the transverse x-axis x 2 1 − μ2 � zx2 zy2 direction remain constant and therefore vanish from the bending equations for the y-axis direction. A small circular opening is assumed located within the wall where flexure −tE z2w z2w σ (x, y) � ⎛⎝μ peak + peak⎞⎠, (9b) predominates. (e one-way rectangular wall and its de- y 2 1 − μ2 � zx2 zy2 formed shape under blast loading are shown in Figures 6(a) and 6(b). where t is the wall thickness and μ is Poisson’s ratio. (e deformed shape, characteristic shape function, Equations (9a) and (9b) can be used to approximate the far- stiffness, and transformation factors for one-way flexural field bending stress inducing stress concentrations around elements are readily available in the literature [3]. (e de- the small circular opening in the wall due to out-of-plane formation in the elastic strain range is described with respect deformation. Substituting (7) into (8) and substituting the to the y-axis in the direction of one-way bending and is given result into (9b) gives an expression for the bending stress σy as follows: at the tension surface of the wall, in the direction of one-way q 3 3 4 flexure, and in terms of the peak displacement: w(y) � �h y − 2hy + y �, (6) 24EI 96tEΔ y2 − hy � σ y � � − peak o o , (10) where h is the wall height, E is Young’s modulus, and I is the y o 5h41 − μ2 � second moment of area about the neutral axis. (e corresponding characteristic shape function is expressed as where yo is the location of the opening along the height of follows: the wall. (e transverse bending stress σx is neglected for one-way flexure since its contribution to the stress con- 16 3 3 4 ϕ(y) � �h y − 2hy + y �. (7) centrations around the opening is minute compared to the 5h4 contribution by σy [40]. (e well-known real stiffness and mass and load Equation (10) can be used to approximate the uniaxial transformation factors for one-way flexural elements are far-field bending stress at the tension surface of the wall and listed in Table 1. loading the circular opening, as shown in Figure 7. (e plane (e parameters in Table 1 can be used to perform an elastic stress field around a circular opening loaded by equivalent SDOF analysis for a given blast loading scenario uniaxial far-field tensile stress is expressed by the Kirsch from which the peak displacement of the wall, Δpeak, can be equations in polar coordinates as follows [20]: Advances in Civil Engineering 7

b σ y y ( o) Simple support x

r xp Reinforcement ry θ σ =0° σ Y Y A-A h σθ(rx) σθ(rx)

rx r A-A B-B θ xp = 90° Simple support

Circular opening y with radius, a Steel reinforcement plates σ (y ) y o 2a

Extreme r r xp xp Steel σθ(rx) tension fiber σ plate Y Steel plate condition t Section A-A σ Figure Y 8: Reinforcement arrangement around the circular opening in the one-way rectangular wall. Plate-like Extreme structure compression fiber ° Section A-A where rx is the radial distance when θ � 90 . (e peak elastic r Plate-like Extreme bending stress distribution along x is depicted in Figure 7. structure tension fiber Localized plasticity around the opening occurs when σ θ(rx) the peak bending stress distribution at the tension surface t exceeds the material yield strength, σY. For reasons of simplicity, the elastoplastic transition range through the wall thickness is neglected and the presence of yielding at Extreme the wall surface is assumed to signify the development of compression fiber a through-thickness plastic hinge, as shown in Figure 7. Section B-B Reinforcement can be positioned around the opening in Figure 7: Far-field bending stress and resulting peak bending stress order to strengthen the regions of high bending stress distribution along rx directly adjacent to the circular opening. concentrations where plastic hinges are expected to occur. For the case of one-way flexure, reinforcement should be positioned on each side of the circular opening and oriented 2 4 in the direction of one-way flexure such that the concentrated σ∞ a σ∞ 3a σ (r, θ) � �1 + � − �1 + � cos(2θ), (11) bending moments directly adjacent to the opening are ef- θ 2 r2 2 r4 fectively resisted, as shown in Figure 8. Each region of re- where a is the radius of the opening, r is the radial distance inforcement should be designed to transfer the concentrated with respect to the center of the opening, θ is the angle with peak bending moment, Mxp, on each side of the opening such that yielding does not occur at the wall surface. In accordance respect to the axis of far-field stress, and σ∞ is the uniaxial far-field tensile stress. In the case of one-way flexure, the with the principle of conservation of mechanical energy given by (4), the concentrated peak bending moment producing peak elastic bending stress distribution, σθ(rx), along the rx radial distance is principally induced by the far-field a plastic hinge can be determined by integrating the peak bending stress in the y-axis direction. Substituting (10) elastic bending stress distribution (assuming no yielding) over ° the radial distance over which yielding does occur and into (11) with σ∞ � σy and θ � 90 results in an expression for the distribution of peak elastic bending stress at the multiplying with the unit elastic section modulus: 2 r +a tension surface of the wall and directly adjacent to the t xp ( ) Mxp � � σθrx � drx, 13 opening: 6 a 2 4 2 2 4 48tEΔpeakhyo − yo � 3a + a rx + 2rx� r � � , (12) where rx + a is the radial distance at which σ � σ . In σθ x − 2 4 4 p θ Y 5μ − 1 �h rx summation form, 8 Advances in Civil Engineering

b z x –1 Simple support x Φ(b/2, y) z Φ(x, h/2)

Circular o –1 y opening with radius, a h Flexure x

Simple support o predominates Simple support

Simple support y

(a) (b) (c)

Figure 9: (a) Two-way rectangular slab with a small circular opening under blast loading; (b) shape function along the y-axis; (c) shape function along the x-axis.

r +a t2 xp deformed shape under blast loading are shown in Mxp � � σθrx �iΔrx. (14) Figures 9(a)–9(c). 6 i�a Navier’s method gives the exact solution for the deformed shape of a rectangular plate-like structure simply (e radial distance r over which yielding occurs can be xp supported at all four edges and uniformly loaded by q; the solved by substituting the known constants into (12), setting solution is described by a rapidly converging double trig- the result equal to σ , solving for r , and subtracting a. Y x onometric series [39]. For simplicity, the first term in the Equation (12) may then be substituted into (14) to obtain series is employed to describe the deformed shape: the concentrated peak bending moment. (e summation term in (14) may be readily solved in spreadsheet form by 16q sin((πx)/b) sin((πy)/b) r w(x, y) � , (17) summing the values of σθ at each incremental length Δ x Dπ6(()1/b2 +()1/h2 )2 along the radial distance rxp. M (e nominal elastic bending limit, nr, of a reinforced where b is the slab width and h is the slab length. (e region directly adjacent to the opening should be equal to or characteristic shape function is determined by multiplying M greater than xp, explicitly expressed by the fundamental (17) by its inverse whilst setting x � b/2 and y � h/2: design requirement: πx πy M M . ( ) ∅(x, y) � sin� � sin� �. (18) nr ≥ xp 15 b h (e width of each reinforced region should nominally be (e bending stiffness corresponding to the out-of-plane equal to rxp and extend longitudinally at least beyond the displacement at the center of the slab is determined by extents of the opening, as shown in Figure 8. (e elastic substituting q � 1, x � b/2, and y � h/2 into the inverse of bending limit of the reinforced region is dependent upon the (17) and multiplying with the slab surface area: material and geometric properties of the reinforced section 2 and may be readily determined from design formulas. In Dπ6b2 + h2 � K � . (19) 3 3 general, Mnr is equal to an internal moment at which the 16b h extreme outer fibers of the reinforced section begin to yield, explicitly expressed by the product of the yield strength and Finally, the mass and load transformation factors are the elastic section modulus, S, of the reinforced section [40]: determined by substituting (18) into the known formulas for transforming a multidegree system into an SDOF system [3]: M � σ S. (16) nr Y h b � � m∅2(x, y) dx dy 1 Reinforcement is not required if the peak elastic bending K � 0 0 � , (20a) M mbh 4 stress given by (12) does not exceed σY. h b � � q∅(x, y) dx dy 4 K � 0 0 � , (20b) 2.2. Case: Rectangular Slab Simply Supported at All Four L qbh π2 Edges. A rectangular slab simply supported at all four edges and uniformly loaded by blast pressure behaves as a two-way where m is the mass density. In a manner similar to the one- plate-like element. As in the one-way case, a small circular way case, (19), (20a), and (20b) may be used to transform opening is assumed located within the slab where flexure the load-structure system into an equivalent SDOF system predominates. (e two-way rectangular slab and its and analyzed for a given blast loading scenario. (e peak Advances in Civil Engineering 9

σ x y r y( o, o) y θ σx(x , y ) =90° σ x y o o σθ(ry) x( o, o) σ ry Y σ θ=0° σ

Y Y y p σθ(rx) σθ(rx) r

rx rx θ=0° rx rx θ=90° p p y p r σ Y

σ x y σ r y( o, o) θ( y)

(a) (b) Figure 10: Far-ﬁeld bending stress and resulting peak bending stress distribution along (a) rx and (b) ry directly adjacent to the circular opening.

2 displacement of the slab Δpeak can be extracted for de- 9Dπ Δ πx πy r � � peak �o � �o � formations remaining within the elastic strain range. σθ x − 4 2 2 2 sin sin rxt b h b h Substituting (18) into (8) and substituting the result into (22a) the Kirchhoff–Love plate differential equations given by (9a) a2r2 2r4 and (9b) results in the bending stress field at the surface of · �a4 + x + x��h2μ + b2 �, the slab in tension: 3 3

2 x , y � 9Dπ Δpeak πx πy σx o o �r � � �o � �o � σθ y 4 2 2 2 sin sin ryt b h b h 2 2 2 (22b) 6DΔpeakπ sinπxo �/b � sinπyo �/h �b μ + h � 2 2 4 � , a ry 2ry t2b2h2 · �a4 + + ��h2 + b2μ �. 3 3 (21a)

(e peak elastic bending stress distributions along rx and x , y � σy o o ry directly adjacent to the opening are depicted in Figure 10. 6DΔ π2 sinπx �/b � sinπy �/h �b2 + h2μ � For the case of two-way flexure, reinforcement should be � peak o o , t2b2h2 positioned on each side of the circular opening as required (Figure 11). Each region of reinforcement should be ( ) 21b designed to transfer the concentrated peak bending moment, M or M , on each side of the opening such that where xo is the location of the opening along the width of the xp yp yielding does not occur at the wall surface. In accordance slab and yo is its location along the length of the slab. In the case of two-way flexure, the distribution of peak with the principle of conservation of mechanical energy bending stress at the surface of the slab and directly adjacent given by (4), the concentrated peak bending moment pro- to the opening is dependent upon the far-field bending ducing a plastic hinge is derived by integrating the peak stress in both the x-axis and y-axis directions, as shown in elastic bending stress distribution (assuming no yielding) Figure 10 [38]. (e peak elastic bending stress distribution, over the radial distance over which yielding does occur and multiplying with the unit elastic section modulus: σθ(rx), along the rx radial distance is principally induced by 2 r +a the far-field bending stress in the y-axis direction. Con- t xp ( ) versely, σ (r ) along the r radial distance is principally Mxp � � σθrx � drx, 23a θ y y 6 a induced by the far-field bending stress in the x-axis di- r +a rection. (e peak elastic bending stress distributions along t2 yp M � r dr , (23b) rx and ry are obtained from (11) using the combinations of yp � σθ�y � y 6 a σ∞ and θ tabulated in Table 2. Substituting the combinations of σ∞ and θ as tabulated where rxp + a and ryp + a are the radial distances at which in Table 2 into (11) results in the following equations: σθ � σY. In summation form, 10 Advances in Civil Engineering

Table 2: Combinations of σ∞ and θ for use in (11) for obtaining the peak bending stress distributions along the rx and ry radial distances.

Peak bending stress Plane stress distribution Uniaxial far-ﬁeld tensile stress, σ∞, Angle, θ, with respect to the axis of distribution obtained from (11) obtained from (21) far-ﬁeld stress ° σθ(rx) σθ(rx, θy) σy 90 ° σθ(ry) σθ(ry, θx) σx 90

r +a t2 xp b Mx � � σ rx � Δrx, (24a) p 6 θ i Simple support i�a x

r +a t2 yp M � � r r . ( ) Reinforcement yp σθ�y �iΔ y 24b 6 i�a r yp h (e radial distances rxp and ryp over which yielding occurs can be solved for by substituting the known constants Simple support r Simple support into (22a) and (22b), setting the results equal to σY, solving xp for rx and ry, and subtracting a. Equations (22a) and (22b) may then be substituted into (24a) and (24b) to obtain the concentrated peak bending moments. (e summation terms Simple support in (24a) and (24b) may be readily solved in spreadsheet form y by summing the values of σθ at each incremental length Δrx or Δry along the radial distance rxp or ryp, respectively. Figure 11: Reinforcement arrangement around a circular opening As in the one-way case, the nominal elastic bending limit in the two-way rectangular slab. Mnr of each reinforced region directly adjacent to the opening should be equal to or greater than the corresponding concentrated peak bending moment demand, Mxp 2.3.FEAVerificationofAnalyticalSolution. Abaqus was used or Myp, as expressed by the fundamental design requirement to perform numerical verification of the peak elastic bending given by (15). (e widths of the reinforced regions resisting stress distribution formulas given by (12), (22a), and (22b). A Mxp and Myp should nominally be equal to rxp and ryp, series of plate-like structures each containing a small circular respectively, and extend longitudinally to at least beyond the opening were modeled using general purpose S4R shell extents of the opening, as shown in Figure 11. Re- elements to verify the formulas. (e global mesh size was set inforcement is not required if the peak elastic bending stress to 10 cm, with a much finer mesh size of 0.20 cm defined in given by (22a) and (22b) does not exceed σY. the immediate region around the circular opening. (e (e derived formulas (12), (14), (22a), (22b), (24a), and typical test model configuration is shown Figure 12. Eight (24b) employ several assumptions for the purpose of test groups each containing three test models were analyzed functionality. (e formulas assume isotropic homogeneous, considering various design parameters including blast linear elastic-perfectly plastic material properties and neglect loading conditions, model dimensions, material properties, the elastoplastic transition phase through the plate-like positions of opening, and support conditions. (e design structure thickness. (e former assumption is accurate for parameters are summarized in Tables 3 and 4. Idealized most metallic structural materials such as alloy steel but less isotropic homogeneous, linear elastic material properties applicable for anisotropic nonhomogeneous materials such were assigned to the shell sections as outlined in Table 5. as reinforced concrete [38]. (e latter assumption is suffi- (e blast effects software ConWep was used to calculate cient for relatively thin plate-like structures having a thick- the peak reflected pressure and impulse upon each model ness-to-length ratio ranging between 0.1 and 0.01 [41]. considering various explosive charge weights and range to Additionally, the approach of using the plane stress obtained the target surface as listed in Table 6. (e reflecting surface from the Kirchhoff–Love plate equations for loading a cir- was set equal to the target surface to account for clearing cular opening in an infinite plane neglects the influence of effects [42]. Triangular blast pressure-time histories were nearby edge supports. However, this is compensated in that developed based on the calculated pressure and impulse and reinforcement around openings near edge supports should used as a basis for determining the design parameters. (e be designed primarily for shear rather than flexure; openings design parameters were optimized such that the blast load in regions where flexure predominates are more accom- response remained within the elastic strain range. (e modating to the assumption of an infinite plane. Finally, the typical blast loading scenarios are depicted in Figure 13. formulas apply only for blast load global responses (e primary objective of the test groups was to dem- remaining primarily within the elastic strain range. Con- onstrate a correlation between the derived peak elastic sequently, the method is inherently elemental; however, the bending stress distribution formulas and the FEA results. method is acquiescent to further analytical or empirical (is is also indirect verification of the concentrated peak refinement. bending moments given by (14), (24a), and (24b), and the Advances in Civil Engineering 11

b x ry o y x r o x h

2a Circular opening

(a) (b)

Figure 12: Typical test model conﬁguration: (a) test model mesh; (b) ﬁne mesh around opening.

Table 3: Test group properties. Test Support Width, b Height, h Circular hole diameter, Circular hole location, Circular hole location, Material1 group conditions (m) (m) 2a (cm) xo (m) yo (m) Carbon TM-1 One-way 8 4 15 4 2 steel Carbon TM-2 One-way 8 4 15 4 1 steel TM-3 One-way Titanium 8 4 25 4 2 TM-4 One-way Titanium 8 4 25 4 1 Carbon TM-5 Two-way 8 4 15 4 2 steel Carbon TM-6 Two-way 8 4 15 6 1 steel TM-7 Two-way Titanium 10 8 30 5 4 TM-8 Two-way Titanium 10 8 30 8 2 1Refer to Table 5 for material properties.

Table 4: Test model thicknesses. corresponding peak elastic strain energy developed around the circular opening. Test groups TM-1 through TM-4 in- t Test model (ickness, (cm) vestigated the stress concentrations around the circular TM-1a 4 opening under one-way flexural behavior and were intended TM-1b 6 to simulate an idealized one-way hardened blast wall. Test TM-1c 8 groups TM-5 through TM-8 investigated the stress con- TM-2a 4 TM-2b 6 centrations under two-way flexural behavior and were TM-2c 8 intended to simulate an idealized two-way strengthened slab. TM-3a 2 (e peak elastic bending stress distribution formulas spe- TM-3b 3 cifically assume isotropic homogeneous, linear elastic ma- TM-3c 4 terial properties, and consequently, the test groups were TM-4a 2 modeled with such material properties for the purpose of TM-4b 3 impartial comparison with the formulas. (e intention was TM-4c 4 therefore not to scrupulously simulate the real-life behavior TM-5a 4 of the considered test models but rather to investigate the TM-5b 6 correlation of the derived formulas with FEA whilst con- TM-5c 8 TM-6a 4 sidering the aforementioned design parameters. TM-6b 6 (ree steps were defined in Abaqus for the analysis of TM-6c 8 each test model. In the first step, the support conditions were TM-7a 3 defined. In the second step, a dynamic implicit step was TM-7b 4 created with a duration equal to the blast load duration. (e TM-7c 5 appropriate triangular pressure-time history was built into TM-8a 3 the load definition and uniformly applied to the model TM-8b 4 surface representing the target area. History output requests TM-8c 5 for the out-of-plane displacement and stress field around the 12 Advances in Civil Engineering

Table 5: Test model shell element material properties [40, 43]. 3 Material Yield strength, σY (MPa) Young’s modulus, E (GPa) Poisson’s ratio, ] Density, ρ (kg/m ) Carbon steel 345 200 0.30 7860 Titanium 880 114 0.34 4430

Table 6: Blast pressure and impulse imposed upon test groups. Test group Equivalent TNT charge mass (kg) Range to target (m) Uniform peak pressure1 (kPa) Uniform impulse1 (kPa·msec) TM-1 500 50 63.61 592.54 TM-2 500 50 63.61 592.54 TM-3 350 40 76.19 614.05 TM-4 350 40 76.19 614.05 TM-5 450 65 39.59 409.96 TM-6 450 65 39.59 409.96 TM-7 275 60 34.82 382.45 TM-8 275 60 34.82 382.45 1Reﬂecting surface set equal to the target surface to account for clearing eﬀects.

One-way hardened z Two-way blast wall strengthened slab z SS Circular opening

Target surface SS SS x Circular y x opening SS SS Target surface y Equivalent SS Range TNT charge Range Equivalent TNT charge SS = simple support SS = simple support (a) (b)

Figure 13: Typical blast loading scenarios: (a) test groups TM-1 through TM-4; (b) test groups TM-5 through TM-8. opening were defined. In the final step, a second dynamic concentrated peak bending moments were calculated using implicit step was created with a duration of 0.50 seconds and (14), (24a), or (24b) for use in designing any required re- with a time step of 0.001 seconds to simulate the free vi- inforcement around the circular openings. (e overall nu- bration phase. (e blast load was set to inactive, and the merical verification procedure is outlined in Figure 14. history output requests for the displacement and stress field around the opening were propagated from the second step. 3. Results and Discussion For each analysis, the history output requests for the displacement were used to determine the time step at which the (e peak out-of-plane displacements from the equivalent peak displacement occurred. (e peak unidirectional undamped SDOF analyses and FEA are summarized in bending stress distribution at the tension surface and along Table 7. (e displacements were obtained at the center of rx or ry at that specific time step was then extracted from the each test model at x � b/2 and y � h/2. (e typical analysis. displacement-time history is characterized by undamped Concurrently, a series of dynamic SDOF blast analyses periodic motion in the z-axis direction, as exemplified in were performed on the test groups to replicate the analysis Figure 15(a) for the test model TM-1c. (e peak displace- procedure that would be followed in the proposed method. ment is achieved at the first peak during the free vibration For each test model, an SDOF analysis was conducted in phase. For the one-way test models, the SDOF analyses order to obtain the peak out-of-plane displacement. (e slightly underestimate the peak displacements compared to peak displacement was then substituted into (12), (22a), or FEA. (e percentage difference between SDOF and FEA (22b) and the resulting peak bending stress distribution tends to reduce as the thickness of the plate-like structure along rx or ry was compared to the FEA results. Finally, the increases. Conversely, the SDOF analyses overestimate the Advances in Civil Engineering 13

Formulate blast Construct a triangular loading scenario pressure-time history

FEA verification Equivalent undamped with Abaqus SDOF analysis

Develop Rectangular Transform an load-structure Perform SDOF test models wall or slab system into an equivalent analysis considering undamped SDOF system no circular opening Material assignments and section thicknesses Extract peak Opening location displacement during the Input into (12), free vibration phase (22a), or (22b) to obtain S4R shell peak bending stress Analysis elements distribution at tension surface along rx or ry

Step 1 Define support conditions Input into (14), (24a), or (24b) to Dynamic implicit step: History output requests for obtain concentrated peak Step 2 define blast load stress field and out-of-plane bending moments magnitude and amplitude displacement around the opening

Dynamic implicit step: History output Step 3 free vibration requests propagated

Determine the time Extract peak step of peak displacement Comparison displacement

Extract bending stress distribution at tension surface along rx or ry

Figure 14: Numerical veriﬁcation procedure.

peak displacements for the two-way test models compared to TM-1c. (e peak bending stress distributions along rx as FEA, and the percentage difference tends to increase as the obtained from (12) and FEA are shown in Figure 16 for thickness increases. (ese minor differences may be derived the one-way test groups TM-1 through TM-4. (e stress from the slight differences in global stiffness between the distributions along rx and ry as obtained from (22a) and SDOF models and the FEA models. (e stiffness in the FEA (22b), respectively, and FEA are shown in Figures 17 and models is in part dependent upon the fineness of mesh and 18 for the two-way test groups TM-5 through TM-8. accounts for the presence of the circular opening. On the Equation (12) demonstrates a good correlation with the contrary, the stiffness in the SDOF models is in part de- FEA results. Equations (22a) and (22b) tend to slightly pendent upon the assumed deformed shape and neglects the overestimate the stress concentrations compared to FEA. presence of the opening. It is noted that, as per the proposed (e correlation is strongest when the opening is located analysis method and verification procedure, both the SDOF in regions where flexural stress predominates (Figures 6 and FEA models were analyzed without considering re- and 9). As may be expected, the correlation decreases as inforcement in order to predict the unaltered (i.e., unre- the opening approaches any edge supports by virtue of inforced) performance of the plate-like structures. the supports having an increasing influence upon the far- Equations (12), (22a), and (22b) predicting the peak field bending stress. In all cases, the derived formulas bending stress distributions around the opening are directly predict slightly greater stress concentrations directly dependent upon the peak out-of-plane displacement. In this adjacent to the opening perimeter compared to FEA. way, these formulas can be directly coupled to the peak Overall, the formulas exhibit closer correlation to the displacement response of a preceding dynamic SDOF blast FEA results for the one-way test models than for the two- analysis to readily design any required reinforcement way models. (ere is no apparent correlation between the around a circular opening. Typical bending stress contours test model thickness and the correlation of stress are shown in Figures 15(b) and 15(c) for the test model concentrations. 14 Advances in Civil Engineering

Table 7: Peak displacement from SDOF and FEA.

Test model SDOF peak (cm) FEA peak (cm) % di erence One-way test models TM-1a 10.3Δ Δ12.3 17.7 TM-1b 4.51 5.36 17.2 TM-1c 2.48 2.89 15.3 TM-2a 10.3 12.3 17.7 TM-2b 4.51 5.33 16.7 TM-2c 2.48 2.87 14.6 TM-3a 76.2 92.8 19.6 TM-3b 33.8 40.2 17.3 TM-3c 18.9 22.5 17.4 TM-4a 76.2 90.9 17.6 TM-4b 33.8 39.7 16.1 TM-4c 18.9 22.2 16.1 Two-way test models TM-5a 10.8 9.34 14.5 TM-5b 4.64 3.89 17.6 TM-5c 2.54 1.96 25.8 TM-6a 10.8 9.35 14.4 TM-6b 4.64 3.89 17.6 TM-6c 2.49 1.96 23.8 TM-7a 97.3 78.1 21.9 TM-7b 54.6 44.1 21.3 TM-7c 34.9 28.1 21.6 TM-8a 97.3 76.8 23.5 TM-8b 54.6 43.1 23.5 TM-8c 34.9 27.6 23.4

TM-1c: blast load-time history TM-1c: displacement-time history 2250 4 2000 3 1750 2 1500 1 1250 0 1000 –10.00 0.10 0.20 0.30 0.40 0.50 750 –2

Load, F ( t ) (kN) 500 –3 250 Displacement, z ( t ) (cm) –4 0 0.00 0.10 0.20 0.30 0.40 0.50 Time, t (s) t Time, (s) SDOF FEA (a) b x ry o y x o rx h

Circular opening 2a

(b) (c)

Figure 15: Test model TM-1c SDOF and FEA analysis results: (a) blast load time history and corresponding displacement-time history from SDOF and FEA; (b) bending stress contours in the y-axis direction on the tension surface of TM-1c; (c) stress contours around the opening. Advances in Civil Engineering 15

1000 600 800 500 400 600 300 ) (MPa) ) (MPa) 400 x x 200 ( r ( r 200 θ θ 100 σ σ 0 0 03691215 18 21 24 27 30 0 3 6 9 12 15 18 21 24 27 30 Radial distance, rx (cm) Radial distance, rx (cm) Equation (12) Equation (12) FEA FEA (a) (b) 500 750 400 600 300 450 ) (MPa) 200 ) (MPa) 300 x x ( r 100 ( r 150 θ θ σ 0 σ 0 03691215 18 21 24 27 30 0 3 6 9 12 15 18 21 24 27 30 Radial distance, rx (cm) Radial distance, rx (cm) Equation (12) Equation (12) FEA FEA (c) (d) 500 400 400 300 300 200 ) (MPa) ) (MPa) 200 x x 100 ( r ( r 100 θ θ σ σ 0 0 0 3691215 18 21 24 27 30 0 3 6 9 12 15 18 21 24 27 30 Radial distance, rx (cm) Radial distance, rx (cm) Equation (12) Equation (12) FEA FEA (e) (f) 2000 1250 1500 1000 1000 750 ) (MPa) ) (MPa) 500 x x

( r 500 ( r 250 θ θ σ 0 σ 0 03691215 18 21 24 27 30 33 03691215 18 21 24 27 30 33 Radial distance, rx (cm) Radial distance, rx (cm) Equation (12) Equation (12) FEA FEA (g) (h) 1000 1500 1250 750 1000 500 750 ) (MPa) ) (MPa) x x 250 500 ( r ( r θ θ 250 σ σ 0 0 03691215 18 21 24 27 30 33 03691215 18 21 24 27 3330 Radial distance, rx (cm) Radial distance, rx (cm) Equation (12) Equation (12) FEA FEA (i) (j) 1000 800 750 600 500 400 ) (MPa) ) (MPa) x x

( r 250 200 ( r θ θ σ 0 σ 0 03691215 18 21 24 27 3330 03691215 18 21 24 27 3330 Radial distance, rx (cm) Radial distance, rx (cm) Equation (12) Equation (12) FEA FEA (k) (l)

Figure 16: Comparison of (12) and FEA derived peak bending stress distributions along rx for the test groups TM-1 through TM-4 (one- way test models): (a) TM-1a; (b) TM-1b; (c) TM-1c; (d) TM-2a; (e) TM-2b; (f) TM-2c; (g) TM-3a; (h) TM-3b; (i) TM-3c; (j) TM-4a; (k) TM- 4b; (l) TM-4c. 16 Advances in Civil Engineering

1000 800 800 600 600 400 ) (MPa) 400 ) (MPa) x x 200 ( r 200 ( r θ θ σ 0 σ 0 0 3 6 9 12 15 18 21 24 27 30 0 3 6 9 12 15 18 21 24 27 30 Radial distance, rx (cm) Radial distance, rx (cm) Equation (22a) Equation (22a) FEA FEA (a) (b) 600 600 450 450 300 300 ) (MPa) ) (MPa) x x 150 150 ( r ( r θ θ σ σ 0 0 0 3 6 9 12 15 18 21 24 27 30 0 3 6 9 12 15 18 21 24 27 30 Radial distance, rx (cm) Radial distance, rx (cm) Equation (22a) Equation (22a) FEA FEA (c) (d) 400 300 300 225 200 150 ) (MPa) ) (MPa) x x 100 75 ( r ( r θ θ σ σ 0 5 0 3 6 9 12 15 18 21 24 27 30 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Radial distance, rx (cm) Radial distance, rx (cm) Equation (22a) Equation (22a) FEA FEA (e) (f) 1250 1000 1000 750 750 500 ) (MPa) ) (MPa) 500 x x 250 ( r ( r 250 θ θ σ σ 0 0 0 4 8 12 16 20 24 28 32 36 40 0 4 8 12 16 20 24 28 32 36 40 Radial distance, rx (cm) Radial distance, rx (cm)

Equation (22a) Equation (22a) FEA FEA (g) (h) 800 600 600 450 400 300 ) (MPa) ) (MPa) x x 200 150 ( r ( r θ θ σ σ 0 0 0 4 8 12 16 20 24 28 32 36 40 0 4 8 12 16 20 24 28 32 36 40 Radial distance, rx (cm) Radial distance, rx (cm)

Equation (22a) Equation (22a) FEA FEA (i) (j) 400 300 300 225 200 150 ) (MPa) ) (MPa) x x 100 75 ( r ( r θ θ σ σ 0 0 0 4 8 12 16 20 24 28 32 36 40 0 4 8 12 16 20 24 28 32 36 40 Radial distance, rx (cm) Radial distance, rx (cm)

Equation (22a) Equation (22a) FEA FEA (k) (l)

Figure 17: Comparison of (22a) and FEA derived peak bending stress distributions along rx for the test groups TM-5 through TM-8 (two- way test models): (a) TM-5a; (b) TM-5b; (c) TM-5c; (d) TM-6a; (e) TM-6b; (f) TM-6c; (g) TM-7a; (h) TM-7b; (i) TM-7c; (j) TM-8a; (k) TM- 8b; (l) TM-8c. Advances in Civil Engineering 17

600 400 450 300 300 200 ) (MPa) ) (MPa) y y 150 100 ( r ( r θ θ σ σ 0 0 0 3 6 9 12 15 18 21 24 27 30 093 6 15 1812 21 24 27 30 Radial distance, ry (cm) Radial distance, ry (cm)

Equation (22b) Equation (22b) FEA FEA (a) (b) 250 300 200 225 150 150 ) (MPa) ) (MPa) 100 y y 75 ( r ( r 50 θ θ σ σ 0 0 0 3 6 9 12 15 18 21 24 27 30 093 6 15 1812 21 24 27 30 Radial distance, ry (cm) Radial distance, ry (cm) Equation (22b) Equation (22b) FEA FEA (c) (d) 200 125 150 100 100 75 ) (MPa) ) (MPa) 50 y 50 y ( r ( r 25 θ θ σ 0 σ 0 306 9 12 15 18 21 24 27 30 02 468 10 12 14 16 18 20 22 24 26 28 30 Radial distance, ry (cm) Radial distance, ry (cm)

Equation (22b) Equation (22b) FEA FEA (e) (f) 1000 800 750 600 500 400 ) (MPa) ) (MPa) y y 250 200 ( r ( r θ θ σ σ 0 0 0 448 12 16 20 24 28 32 36 0 0 4 8 12 16 20 24 28 32 36 40 Radial distance, ry (cm) Radial distance, ry (cm)

Equation (22b) Equation (22b) FEA FEA (g) (h) 600 400 450 300 300 200 ) (MPa) ) (MPa) y y 150 100 ( r ( r θ θ σ σ 0 0 0 4212 1 68 20 248 32 36 40 024 8 12 16 20 248 363240 Radial distance, ry (cm) Radial distance, ry (cm)

Equation (22b) Equation (22b) FEA FEA (i) (j) 300 250 225 200 150 150 ) (MPa) ) (MPa) 100 y 75 y ( r ( r 50 θ θ σ 0 σ 0 0 4 8 12 16 20 28 322436 40 0 428212 16 20 438 32 6 40 Radial distance, ry (cm) Radial distance, ry (cm)

Equation (22b) Equation (22b) FEA FEA (k) (l)

Figure 18: Comparison of (22b) and FEA derived peak bending stress distributions along ry for the test groups TM-5 through TM-8 (two- way test models): (a) TM-5a; (b) TM-5b; (c) TM-5c; (d) TM-6a; (e) TM-6b; (f) TM-6c; (g) TM-7a; (h) TM-7b; (i) TM-7c; (j) TM-8a; (k) TM- 8b; (l) TM-8c. 18 Advances in Civil Engineering

Table 8: Concentrated peak bending moments for the one-way test reinforcement by virtue of the peak bending stress con- models. centrations not exceeding the material yield strength. (e concentrated bending moments tend to decrease when the M · Test model xp per (14) (kN m) opening is located in regions near edge supports where TM-1a 8.34 flexural stresses begin to decrease. TM-1b 4.88 (e design of steel reinforcement for the test model TM- TM-1c 1.97 2a is outlined as a typical example. For reinforced regions TM-2a 3.37 TM-2b 1.34 with rectangular cross sections and composed of isotropic TM-2c NR homogeneous, linear elastic-perfectly plastic material, the TM-3a 5.24 elastic bending limit Mnr is given as follows: TM-3b 2.89 σ r d2 Y p (25) TM-3c NR Mnr � , TM-4a 2.07 6 TM-4b 0.13 where rp is the width of the reinforced section equal to rxp or TM-4c NR ryp and d is the total thickness of the reinforced section. NR � no reinforcement required. From Table 8, the magnitude of Mxp for the test model TM- 2a is 3.37 kN·m. Also, from Figure 16, the value of rxp is approximately 2.7 cm. As per the fundamental design re- Table 9: Concentrated peak bending moments for the two-way test models. quirement given by (15), the required thickness of the rectangular reinforced section can be readily determined by Test model Mxp per (24a) (kN·m) Myp per (24b) (kN·m) setting (25) equal to Mxp � 3.37 kN·m, substituting in the TM-5a 15.8 5.70 known material yield strength of σY � 345 MPa, setting TM-5b 7.53 NR rp � 2.7 cm, and solving for d. (e required thickness of the TM-5c 4.19 NR reinforced section is solved to be d � 4.7 cm. By contrast, the TM-6a 1.39 NR nonreinforced thickness of the test model TM-2a is t � 4 cm. TM-6b NR NR (e final idealized reinforced sections on each side of the TM-6c NR NR opening are depicted in Figure 19. TM-7a 2.03 NR TM-7b NR NR TM-7c NR NR 4. Conclusions TM-8a NR NR An analysis method is formulated for predicting the peak TM-8b NR NR TM-8c NR NR bending stress concentrations around a small circular opening in a plate-like structure subjected to uniform blast NR � no reinforcement required. loading. (e method is applicable to plate-like structures possessing isotropic homogeneous material properties and (e concentrated peak bending moments given by (14), assumes a linear elastic-perfectly plastic response mode. (24a), and (24b) for the one-way and two-way test groups are Specifically, the method allows for the determination of listed in Tables 8 and 9, respectively. (ese formulas are concentrated peak bending moments adjacent to the directly dependent upon (12), (22a), and (22b). As per the opening for the design of reinforcement that can prevent the principle of conservation of mechanical energy given by (4), formation of localized plasticity. In terms of mechanical if the peak elastic bending stress concentrations in the local energy, the reinforced regions are designed to absorb the domain around a circular opening are accurate, then the peak strain energy around the opening in a purely elastic concentrated peak bending moments derived from the form. (is is under the supposition that the development of distributions can allow for the design of reinforcement to localized plasticity around the opening is the failure crite- absorb the peak strain energy ΔUpeak around the opening rion. Specifically, the kinetic energy of the plate-like in a purely elastic form. (e summation term in the for- structure may progressively dissipate or build up as plas- mulas was readily solved in spreadsheet form by summing tic strain energy around the opening, potentially resulting in the stress magnitude at each incremental length Δr along the large deformations, rupture, or dynamic fracture during the radial distance rxp or ryp. Considering that the elastoplastic initial cycle or subsequent oscillations [34–36]. transition range through the plate-like structure thickness is A set of elemental formulas is derived to achieve this neglected in the derivation of (14), (24a), and (24b), the objective, focusing upon rectangular one-way and two-way formulas may slightly overestimate the values of Mxp and plate-like structures containing a single small circular Myp. In accordance with the design requirement expressed opening located where flexure predominates. (e formulas by (15), this may result in a conservative design of the are those expressed by (12), (14), (22a), (22b), (24a), and reinforcement. (24b) and are directly dependent upon the predicted peak An inclination towards conservatism in design is gen- displacement response of a blast load-structure system. In erally favorable, especially with respect to blast design since conformance with the proposed method, the formulas are the nominal, unreduced capacities are used [1]. Nonetheless, intended to be implemented in tandem with a dynamic the results show that many of the test models do not require SDOF analysis of the blast load-structure system. Advances in Civil Engineering 19

b=8 m Simple support x Steel y =1 m A-A o 2a=15 cm reinforcement plates Reinforcement Reinforcement d=4.7 cm t=4 cm =4 m 2a=15 cm h r r Steel x =4 m xp=2.7 cm xp=2.7 cm o plate Section A-A Simple support

Figure 19: Idealized reinforced sections on each side of the opening for the test model TM-2a.

Formulate blast Construct triangular total of 24 test models subdivided into eight test groups were loading scenario pressure-time history investigated considering various design parameters. Overall, the formulas demonstrate a good correlation with FEA albeit Perform SDOF Transform a load-structure with a conservative inclination; the formulas tend to slightly analysis considering system into an equivalent overestimate the stress concentrations directly adjacent to the no circular opening undamped SDOF system opening perimeter compared to FEA. Additionally, the formulas neglect the elastoplastic transition phase through the Design a plate-like structure plate-like structure thickness, which serves to add to the Design to satisfy strength and parameters: aforementioned conservatism. However, this conservatism is performance requirements E, μ, b, h, t, and a deemed favorable especially with regard to blast design. (elastic strain range only) e method is limited insomuch by the blast load- structure conguration that can be analyzed by way of an Extract peak displacement equivalent SDOF system. is is because the method is Δpeak during free vibration phase directly dependent upon the peak displacement response, deformed shape, characteristic shape function, and sti ness Substitute into Obtain theoretical inherent to a dynamic SDOF analysis. e method is, (12), (22a), or (22b) location of however, acquiescent to further analytical or empirical re- to obtain σθ(rx) or σθ(ry) circular opening nement by future investigations or experiments to extend x , y ( o o) its applicability. Future studies could investigate the presence of multiple openings in a plate-like structure and No σ Yes Exceeds Y? openings with irregular shapes. e method could also be rened to accommodate plate-like structures composed of r r Reinforcement Solve for xp or yp anisotropic nonhomogeneous materials. around opening e overall premise of the analysis method is to provide not required Substitute into a more e¡cient and viable alternative than overgeneralized (14), (24a), or (24b) to analytical methods or high delity FEA for analyzing and M M obtain xp or yp designing reinforcement around a circular penetration located in a plate-like structure subjected to uniform blast Design reinforcement loading. e proposed method is applicable for many typical to satisfy conditions that may be encountered during design. With M M nr ≥ xp further modications, the method has the potential to be M ≥ M nr yp applicable for an even wider range of structural congu- Figure 20: Proposed analysis and design method. rations, with the remaining atypical conditions reserved for analysis by FEA.

e deformed shape, characteristic shape function, sti ness, Data Availability and transformation factors for a rectangular two-way exural element are formulated. ese same parameters are e data used to support the ndings of this study are readily available in the literature for rectangular one-way available from the corresponding author upon request. exural elements [3]. e proposed method is outlined in Figure 20 and can be readily incorporated into standard Conflicts of Interest SDOF blast design spreadsheets. Abaqus was employed to conduct FEA verication of the e authors declare that there are no conicts of interest derived peak elastic bending stress distribution formulas. A regarding the publication of this paper. 20 Advances in Civil Engineering

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