Modified Quasi-Static, Elastic-Plastic Analysis for Blast Walls with Partially Fixed Support

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1 Article Modified Quasi-Static, Elastic-Plastic Analysis for Blast Walls with Partially Fixed Suort Pattamad Panedojaman Deartment of Civil Engineering, Faculty of Engineering, Prince of Songkla University, Songkhla 9, Thailand Abstract. Blast walls with artially fixed suort have been used to limit imacts of ressure due to an exlosion. As design criteria, the dynamic resonse of blast wall systems in terms of their ressure load-dislacement relationshi is considered. In addition to a comlicatedly dynamic analysis, the quasi-static, elastic-lastic analysis is recently an alternative method. However, the ressure load-dislacement resonse of the system with the short connection obtained from the quasi-static, elastic-lastic analysis tends to be over conservative comaring with the resonse of the revious exeriments. Therefore, the quasi-static, elastic-lastic model is modified to imrove the rediction accuracy. The model simly consists of two flexural elements connected through an angular sring element. The beam end is suorted by the vertical and horizontal sring. The horizontal sring is simulated based on a cantilever beam mechanism and an axial restraint of the connection member. The uncouled dislacement mode shaes are assumed to reresent the overall elastic-lastic behavior of the systems. The quasi-static equilibrium equations are determined from the minimum otential energy of the system. The redicted resonse of the modified analysis is found to better agree with the revious exerimental resonse of the systems with the short connection. Keywords: Blast walls, artially fixed suort, quasi-static elastic-lastic analysis, ressure load-dislacement resonse. ENGINEERING JOURNAL Volume 6 Issue 5 Received March Acceted May Published October Online at htt://

2 . Introduction Blast wall systems as shown in Fig. have been used to limit imacts of ressure due to an exlosion of hydrocarbon gas on temorary refuges, oeration rooms of offshore oil and gas latforms and chemical laboratories. The function of blast walls is to absorb exlosive energy. As a result, an efficient energy absorbing systems must be design for blast walls. Generally, a non-symmetrical traezoidal anel is used due to its ability to absorb the energy. Guidance issued by the Fire and Blast Information Grou (FABIG) and the Steel Construction Institute (SCI), known as Technical Note 5 [], is develoed to design blast walls. Note that Technical Note 5 [] has its basis in Eurocode []. The blast wall designs, such as moment resistance, shear resistance, local loading effect, buckling resistance, weld detail and structural dynamic resonse, are considered. The resonse of blast walls is comuted in terms of their ressure loaddislacement relationshi. The ressure load is designed u to the dislacement limitation. Main frame Connection Section A-A Blast wall anel A Blast wall system A Partially fixed connection Side view D view Front view Fig.. Detail of blast wall systems. The suort or connection of blast walls as shown in Fig. are normally considered as artially fixed behavior. However, to consider the ressure load-dislacement relationshi, the current designs assume a single lastic hinge formation at the ends of blast walls which is based on a simle bending model. The restrain effect of the suort is ignored in the design. Therefore, the resonse of the wall generally tends to be conservative. Langdon and Schleyer [] found that their exerimental results of blast wall systems with the short connection rovided their ressure load caacity above the standard design caacity (maintaining the dislacement limitation). The results remind the actual caacity of the wall with the suort is larger than that of the simle design. Many researchers have examined methods for assessment of the resonse of blast walls under blast load. The finite element analysis is widely acceted for its accuracy to redict the resonse of blast walls [- 6]. However, the model is required an exensive finite element rogram as well as an exertise user. Aroximate methods have been roosed to be an alternative design method. Biggs [7] develoed the simle design chart to aroximate the structural resonse. The chart to redict the resonse is formulated based on the equivalent single-degree-of-freedom (SDOF) systems, also known as the sring mass system. The method is well known and still used. Kaliszky [8] roosed an imulsively loaded structure consisted of an ideally rigid-lastic material and had a kinematically admissible velocity field. The imulsively loaded structure is used to be substituted with an equivalent SDOF system and be solved statically. The analysis with rigid-lastic material rovides a good agreement with the elasticlastic solution when the ratio of initial kinetic energy to elastic strain energy is larger than, and the load duration is sufficiently short with resect to the natural eriod of the structure [9]. Schubak et al. [] roosed a new rocedure, a beam model, for redicting the dynamic resonse in consideration of axial restrain behavior and flexural behavior. The beam model is subjected to ulse loads. The elastic resonse of blast walls is not considered in the model. The beam model consists of two flexural elements connected through an angular sring element having rigid-lastic characteristics. To investigate 6 ENGINEERING JOURNAL Volume 6 Issue 5, ISSN 5-88 (htt://

3 elastic resonse of beams, the beam model was develoed by using energy methods to rovide equilibrium equations and incororating the elastic-lastic resonse of the model with different structural geometries of blast wall systems []. Later, variable suort restraints of blast walls were included in the beam model []. The model analysis [] considers membrane action, strain hardening effects and flexible boundary conditions. Due to an effect of the artially fixed connection on the overall resonse [], the beam model with artially fixed connection was develoed and dynamically analyzed []. The beam model consists of beam elements and sring elements connected to form a continuous system, as shown in Fig.. The sring constants used in the analysis must be re-determined based on revious test data. The re-determined sring constants cause limitation of using the beam model with the artially fixed connection. (a) Fig.. Blast wall with artially fixed connection: (a) blast wall system and (b) beam model. (b) In addition to the comlicatedly dynamic analysis, the quasi-static, elastic-lastic model [] was recently roosed to avoid the re-determined sring constants. The analysis is simlified by using static analysis and considered the elastic resonse of blast walls. Characteristics of the srings are determined based on the dislacements of the beam and geometry of the connection. However, comaring with the exerimental resonse, the redicted resonse of the system with the short connection is significantly conservative []. Therefore, this study develos the modified quasi-static, elastic-lastic analysis to imrove its accuracy to redict the resonse of blast walls with the artially fixed suort. Normally, the resonse is considered at the mid san which rovides the maximum dislacement [,, 5]. The ressure-dislacement at the mid san obtained from the roosed model and the revious models are investigated. The roosed model can simly be used as a art of blast wall design.. Modified Quasi-Static, Elastic-Plastic Model The half-san blast wall system with the artially fixed suort as shown in Fig. (a) is investigated. By using the original quasi-static, elastic-lastic model [], the connection members are fundamentally simulated as a vertical inned member on the both sides. However, in the modified quasi-static, elasticlastic model, the connection members fixed at the suort are fundamentally simulated as a restraint angular sring based on a cantilever beam behavior. The other end is modeled as a in at the beam end described. The fundamental model of the system described in Fig (b). Note that, for the full-san blast wall system, the model consists of consists of two flexural elements connected through an angular sring element at the mid-san. Subsequently, the connection member is simly modeled by the vertical and horizontal sring described in Fig (c). k x and k y are the horizontal and vertical sring constant, resectively. The modified quasi-static, elastic-lastic analysis to investigate the structural resonse is develoed based on the original quasi-static, elastic-lastic equations []. ENGINEERING JOURNAL Volume 6 Issue 5, ISSN 5-88 (htt:// 7

4 Connection member Connection member (a) (b) k x k y (c) Fig.. Structural model of the modified quasi-static, elastic-lastic analysis: (a) blast wall system; (b) fundamental model; and (c) simlified model. Secified to redict the overall elastic-lastic behavior, the dislacement rofile of the beam model under ressure loading consists of the elastic dislacement mode of the connection w (x), the elastic dislacement mode of the beam w (x) and the lastic large dislacement mode w (x). w ( x) C () w ( x) C cos x / L () w ( x) C x / L () in which C, C and C are the maximum dislacement of each mode as shown in Fig. ; and L is the halfsan length of the system. The uncouled mode shaes allow to searately determining the elastic and lastic effects in the quasi-static analysis. The mode shaes are used to formulate the total energy due to flexural and membrane behavior of the structural elements. The structural resonse is analyzed based on a consideration of the minimum otential energy of the beam system. As a result, a comlex mechanism can be simlified in the analysis. w C L C w C cos x / L w C x / L C C Fig.. Dislacement mode shaes of the beam model.. Elastic Stage Under the elastic stage of the beam, the dislacement rofile combines the fundamental flexural mode shae of a simly suorted beam and the vertical suort dislacement. The axially restraint (i.e., 8 ENGINEERING JOURNAL Volume 6 Issue 5, ISSN 5-88 (htt://

5 horizontal restraint) at the beam end is assumed to be neglected due to its small dislacement. The shae function w(x) is aroximated as w( x) C C cos x / L () For the half-san blast wall system, the flexural strain energy of the beam element U b, the otential energy loss PE due to the loading and the total otential energy V of the system are given as follows: LM ( x) EI Ld w (5) Ub dx dx EI dx L PE w( x) dx (6) b (7) V U PE U PE in which M(x) is the bending moment; E is the Young s modulus of elastic; and I is the second moment of area of the blast wall anel. The quasi-static equilibrium equations are determined from the minimum otential energy of the system which is V C i (8) Eq. (8) rovides the formulation of C and C as follows: where L C (9) k C k y 6 L EI () Acnt E L () 5 y cnt A cnt and L cnt are the area and the length of the connection member. Limited by the lastic moment caacity M of the beam at the mid-san, the maximum elastic dislacement of the beam C max is given in Eq. (). The corresonding load at the elastic limiting can be comuted in Eq. ()... Plastic Stage M max L C () EI 5 max EI C () 6L The lastic deformation of the beam after lastic hinge formation at the mid-san is aroximated by the shae function w (x). Due to the large dislacement of the lastic hinge, the connection member is assumed to stretch and rotate as shown in Fig. 5(a). The beam ends are considered to be restrained both axially and vertically due to the resistance of the connection in terms of the horizontal force F x and the vertical force F y. Movement of the corresonding connection sring is described in Fig. 5(b). The movement is assumed to less affect the overall large dislacement of the beam. ENGINEERING JOURNAL Volume 6 Issue 5, ISSN 5-88 (htt:// 9

6 F x F x u F y (a) k x y k y k x u (b) k C Fig. 5. Large dislacement geometry and forces of connection: (a) fundamental model, and (b) simlified model. From the large dislacement geometry of the blast wall systems in Fig. 5(b), the horizontal dislacement u corresonding to w (x), the axial stretching of the beam during bending, the horizontal movement of the connection δ and the angle of the connection rotation α are formulated as follows: u L L C () FL x AE (5) u (6) u sin sin Lcnt Lcnt (7) where A is the area of the blast wall anel. Since the value of is normally very small comaring with the value of u, the value of δ can be aroximated to be same as the value of u in Eq. (7). Based on the vertical force equilibrium, the resistance force of the connection in y direction F y is evaluated as Fy L (8) whereas F x is deended on the combination of the axial restraint F x in Eq. (9) and the restraint of the angular sring F x based on the cantilever beam mechanism in Eq. (). Due to blast wall systems are generally designed to failure at its blast wall anel such as in the buckling mode [], F x is assumed to be only in the elastic stage. Therefore, F x and the corresonding k x is given in Eq. () and Eq. (), resectively. Fx Fy tan L tan (9) EI Fx L () cnt F F F () x x x 5 ENGINEERING JOURNAL Volume 6 Issue 5, ISSN 5-88 (htt://

7 k x F x () F x in Eq. () is evaluated based on comatibility of Eq. (5) and Eq. (6). A trial and error rocess is required to solve F x. To consider the total otential energy, the strain energy of the beam element U m,b and the horizontal sring element U m,sx are comuted in Eq. () and Eq. (). The angular sring element can be assumed as a rigid-lastic characteristic in the model [], as shown in Fig. 6. The strain energy of the angular sring at the mid-san U θ is comuted in Eq. (5). AE Umb, L () kx Um, sx () k U M (5) in which k θ is the angular sring stiffness of hardening moment due to hardening strain; and θ is the angle of the beam rotation, as described in Fig. 5, and can be numerically aroximated as C /L. The total otential energy V of the system at the lastic stage is m, b m, sx (6) V U PE U U U PE M M Fig. 6. Rigid-lastic characteristics. The minimum otential energy of V/ C = rovides the relationshi between and C as C C M k C (7) L L L L AE where AE kl x kx kl x AE (8) (9) ENGINEERING JOURNAL Volume 6 Issue 5, ISSN 5-88 (htt:// 5

8 Note that β and β are numerically solved by using Binomial series of the first two terms. The relationshi between and C is solved based on a trial and error rocess. In the lastic stage, the overall dislacement at the mid san can be considered as w(x=) = C + C max + C. Following the rocedure in the elastic and lastic stage, the relationshi between and w(x) can also be comletely lotted.. Quasi-Static Loading Regime To adot the quasi-static analysis, the structural resonse must be classified in the quasi-static loading regime. Biggs [7] investigated the structural resonse of blast wall systems under a ulse load having a triangular shaed ressure ulse with a duration t d. The single-degree-of-freedom (SDOF), elastic sringmass system is used in the investigation. The study describes that dynamic effects significantly influence on the resonse of blast wall systems when t d /T < in which T is the fundamental elastic eriod of vibration of blast wall systems. At the same eak load, the static dislacement calculated agrees with the dynamic dislacement within aroximately 7% when t d /T >.75. Inertia effects are significant when the loading is alied almost instantaneously as in the case of shock loading. The structural resonse of blast walls under a blast load is influenced by the ratio t d /T or ωt d (i.e. ωt d = πt d /T). Three loading regimes are classified as follows []: quasi-static loading regime for ωt d >, dynamic loading regime for. < ωt d, and imulsive loading regime for ωt d.. The fundamental elastic eriod T of vibration of a blast wall is given as where T M / K () M k m () M K k k () L 8EI k () 5 L M and K are the equivalent mass and the equivalent stiffness of blast wall systems for the SDOF analysis, resectively. m and k are the mass and the elastic sring constant of blast wall systems, resectively. The formulation of k in Eq. () is evaluated based on blast wall systems considered as a simle suort with no end fixity and a uniformly distributed loading. k M is the mass transformation factor which is.5 for the elastic stage and. for the lastic stage [7]. k L is the load transformation factor which is.6 for the elastic stage and.5 for the lastic stage [7]. Through the above equations, the loading regime can be classified.. Validation of the Modified Analysis To validate the roosed model, the blast wall system with the connection lengths L cnt is investigated as shown in Fig. 7. The connection lengths L cnt of mm, 8 mm and 7 mm are secified in the analysis and denoted as the short, medium and long connection through the criteria of L cnt /L.,. < L cnt /L <.6 and L cnt /L.6, resectively []. The structural resonse of the system is described in terms of the ressure load-dislacement curve at the mid san. The curves are comared with a series of the exerimental investigation [, 5], the finite element analysis [] and the original quasi-static, elastic-lastic analysis. The blast wall anel roerties are illustrated in Table. The blast wall is made of 6L stainless steel. Note that the study aroximates the hardening moment of resistance in Table through the ressuredislacement curve of the FE analysis []. The value of the hardening moment of resistance can also be evaluated from exeriments. However, the revious exeriments are conducted by using the ¼ scale blast walls. Therefore, the value in this study is aroximated through the ressure-dislacement curve of FE analysis. Without the hardening moment of resistance, the ressure-dislacement curve based on Eq. (9) is over conservative. The revious investigations were conducted through the blast wall systems under ulse ressure load in [, ] as shown in Fig. 8 and under static ressure load in [5]. Due to limitation of the full scale test, the exerimental investigations [, 5] were conducted through the ¼ scale blast walls. For comaring the 5 ENGINEERING JOURNAL Volume 6 Issue 5, ISSN 5-88 (htt://

9 resonse of the full scale blast wall with that of the ¼ scale blast wall [], the dislacement of the ¼ scale blast wall is scaled u which is multilied by whereas the ressure values are not changed. The detail of scaling u the ¼ scale blast walls is described in []. All ulse ressure loads in [, ] are considered as the quasi static load based on the classification of the structural resonse. Dimension in mm Main frame L flex 6 8 Blast wall anel Detail of connection Detail of blast wall Fig. 7. Detail of the investigated blast wall system. Table. Blast wall anel roerties []. Proerties Magnitude Cross-section area of rofile mm Profile material density 797. kg/m Mass of beam section/unit length 68.5 kg/m Second moment of area 68. cm Young s modulus. N/mm Initial lastic moment 5.5 knm Hardening moment of resistance 55. knm/rad Base yield stress 65. N/mm Design dynamic yield stress. N/mm By using the quasi-static, elastic-lastic analysis, the analyzed ressure load corresonding to the initial lastic moment of the blast wall is.75 bar. Such ressure load is found to differ from the exeriments which are about.9 bar. The reason may due to the design dynamic yield stress of. N/mm does not reresent the real value of the exeriment. The dynamic yield stress normally deends on details of its dynamic test. Based on the exeriment of [6], the dynamic yield stress of 6 L stainless steel varies from N/mm to N/mm. Therefore, to lessen the effect of the various dynamic yield stresses, the normalized ressure load / is used in the comarison. ENGINEERING JOURNAL Volume 6 Issue 5, ISSN 5-88 (htt:// 5

10 Pressure t d Time Fig. 8. Triangular ulse load. Comarisons of the normalized ressure-dislacement curve at the mid san between the roosed model and the original model are shown in Fig. 9. For the blast wall system with the medium or long connection, it is found that the curves obtained from the modified analysis slightly differ from the original method. In other words, the horizontal resistance due to the cantilever beam model less affects the ressure resistance of the system with the medium or long connection. Furthermore, the modified analysis and the original method agree well with the revious exeriments both static and imulse load. However, for the blast wall system with the short connection, the cantilever beam model of the connection is significantly imroved the ressure resistance of the systems. Comaring with the revious investigations, the modified analysis rovides more accuracy to redict the structural resonse of the blast wall systems. / 5 5 / 5 5 cnt 6 8 / 5 Dislacement (mm) cnt 6 8 Dislacement (mm) ช ดข อม ล ช ดข อม ล ช ดข อม ล ช ดข อม ล ช ดข อม ล5 ช ดข อม ล6 L of mm L of 8 mm cnt L of 7 mm 6 8 ช ดข อม ล ช ดข อม ล ช ดข อม ล ช ดข อม ล ช ดข อม ล5 ช ดข อม ล6 Dislacement (mm) ช ดข อม ล ช ดข Proosed อม ลanalysis ช ดข Original อม ล quasi static analysis [] ช ดข Static อม ล ressure test [5] ช ดข Pulse อม ล5 ressure tests [] ช ดข FE Analysis อม ล6 [] ช ดข อม ล ช ดข อม ล ช ดข อม ล ช ดข อม ล ช ดข อม ล5 ช ดข อม ล6 Fig. 9. Comarisons of the normalized ressure -dislacement curve. 5. Conclusion To redict the ressure load-dislacement resonse of blast wall systems, the study rooses the modified quasi-static, elastic-lastic analysis. The fundamental model of the quasi-static, elastic-lastic analysis consists of two flexural elements connected through an angular sring element. The angular sring element at the mid-san is simulated based on the rigid-lastic behavior. The beam suort is simulated as the vertical and horizontal sring. The horizontal sring resistance is formulated through a cantilever beam 5 ENGINEERING JOURNAL Volume 6 Issue 5, ISSN 5-88 (htt://

11 mechanism and an axial restraint of the connection member. The dislacement mode shaes of the elastic dislacement mode of the connection, the elastic dislacement mode of the beam and the lastic large dislacement mode are used to analyze the overall elastic-lastic behavior. The uncouled mode shaes allow to searately determining the elastic and lastic effects in the quasi-static analysis. The total energy of flexural and membrane behavior is derived by using the dislacement of the model corresonding to the mode shaes. The structural resonse is analyzed based on a consideration of the minimum otential energy of the beam system. The ressure load-dislacement resonse is solved through a trial and error rocess. To validate the modified analysis, the normalized ressure-dislacement curve at the mid san is used to lessen the effect of the various dynamic yield stresses. The redicted ressure resistance at a given dislacement of the blast wall system with the short connection is found to be significantly imroved comaring with that of the original model. The redicted resonse also agrees with the exerimental results. However, the modified analysis less affects the resonse of the systems with the medium or long connection. The comarison describes the modified analysis is required to better redict the resonse of the blast wall systems with the short connection. The modified quasi-static, elastic-lastic analysis can be considered as a fundamental tool of blast wall design. Acknowledgement We would like to sincerely thank all the journal reviewers who dedicated their time to rovide valuable comments. References [] Fire and Blast Information Grou (FABIG), Design guide for stainless steel blast walls, Technical Note 5, SCI, June, 999. [] CEN, Design of steel structures, Part, sulementary rules for cold-formed thin gauge members and sheeting, ENV 99--, 996. [] G. S. Langdon, and G. K. Schleyer, Inelastic deformation and failure of rofiled stainless steel blast walls. Part I: exerimental investigations, Int. J. Imact Eng., vol.,. -69, 5. [] L. A. Louca, J. W. Boh, and Y. S. Choo, Design and analysis of stainless steel rofiled blast barriers, J. Constr. Steel Res., vol. 6, ,. [5] W. Jun,L. Jingbo,Y. Qiushi, Effect of shock wave on fabricated anti-blast wall and distribution law around the wall under near surface exlosion, Trans. Tianjin Univ., vol , 8. [6] M. A. Faruqi, J. Grisel, A. Salem, and J. Sai, A arametric study for the efficient design of corrugated blast wall anels used in etrochemical facilities, ARPN J. Eng. Al. Sci., vol. 5, no., ,. [7] J. M. Biggs, Introduction to Structural Dynamics, NewYork, McGraw-Hill, 96. [8] S. Kaliszky, Large deformations of rigid-lastic structures under imulsive and ressure loading, J. Struct. Mech., vol., no.,. 95-7, 97. [9] N. Jones, Structural Imact, Cambridge, Cambridge University Press, 989. [] R. B. Schubak, D. L. Anderson, and M. D. Olson. Simlified dynamic analysis of rigid-lastic beams, Int. J. Imact Eng., vol. 8, no.,. 7-, 989. [] G. K. Schleyer, and M. Mihsein, Develoment of mathematical models for dynamic analysis of structures, Conference Proceedings on Structural Design against Accidental Loading-As Part of the Offshore Safety Case, London,...-, 99. [] G. K. Schleyer, and S. S. Hsu, A modelling scheme for redicting the resonse of elastic lastic structures to ulse ressure loading, Int. J. Imact Eng., vol., ,. [] G. S. Langdon, and G. K. Schleyer, Inelastic deformation and failure of ¼ scale rofiled stainless steel blast wall anels. Part II: analytical modelling considerations, Int. J. Imact Eng., vol., no.,. 7-99, 5. [] G. K. Schleyer, and G. S. Langdon, Pulse ressure testing of ¼ scale blast wall anels with connections Phase II, HSE Research Reort, 6. ENGINEERING JOURNAL Volume 6 Issue 5, ISSN 5-88 (htt:// 55

12 [5] G. S. Langdon, and G. K. Schleyer, Resonse of Quasi-statically Loaded Corrugated Panels with Partially Restrained Boundaries, Exer. Mech., vol. 7,. 5-6, 7. [6] G. S. Langdon, and G. K. Schleyer, Unusual strain rate sensitive behaviour of AISI 6L austenitic stainless steel, IMechE J. of Strain Analysis, vol. 9, no.,. 7-86,. 56 ENGINEERING JOURNAL Volume 6 Issue 5, ISSN 5-88 (htt://