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ScienceDirect Procedia Engineering 150 (2016) 1891 – 1897

International Conference on Industrial Engineering, ICIE 2016

Research of Total Mechanical Energy of Steel Roof Truss during Structurally Nonlinear Oscillations E. Ufimtsev*, M. Voronina South Ural State University, 76, Lenin Avenue, Chelyabinsk, 454080, The Russian Federation

Abstract A mathematical model for the calculation of the discrete dissipative system total mechanical energy during free oscillations based on the method of time analysis was given. The features of this parameter behavior (jumps, asymptotes) for elastic structurally nonlinear systems were shown. A possible usage of the mathematical model is illustrated by the calculation of flat steel roof truss during oscillations caused by a sudden failure of one of the bearing elements due to corrosion effects. © 2016 2016The TheAuthors. Authors. Published Elsevier © Published by by Elsevier Ltd.Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016. Peer-review under responsibility of the organizing committee of ICIE 2016 Keywords:time analysis method; mathematical model; discrete dissipative system; corrosive; free oscillations; dynamic response; mechanical energy.

1. Introduction The structures of modern buildings exposed to different loads (impulses, explosions, impacts, chemical corrosion, etc.), which can cause a nonlinear behavior of constructions. Such nonlinearity is widely recognized in literature: physical [1-3], structural [4-6] and geometrical [7-9]. Different parameters of a reaction such as kinematic, force, energy are determined for estimation of force resistance. Truss – it is one of widely used types of frame systems in construction. Trusses are used in roofing of buildings, hangars, railway stations, etc. According to materials consumption they are economical and easy to produce. This paper deals with research of energy parameters (potential, kinetic and total mechanical energy) of the dynamic response of a truss which is modeled as a discrete dissipative system (DDS) in case of its structurally nonlinear work when a design scheme is changed during the analysis [10]. At the same time the model makes

* Corresponding author. Tel.: +7-351-267-9000; fax: +7-351-265-4785. E-mail address: [email protected]

1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016

doi:10.1016/j.proeng.2016.07.188

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E. Ufimtsev and M. Voronina / Procedia Engineering 150 (2016) 1891 – 1897

transition into an intermediate state, caused by a failure of one of the bearing elements as a result of strength loss in the context of corrosion attack. For solution of this dynamic problem the authors use a time analysis method (TAM), developed for an elastic DDS and based on the study on characteristic matrix quadratic equations (MQE) [11]. The process of calculating the response is divided into quasi-linear time intervals t [ti, ti+1], where stiffening, damping and inertial parameters of a design dynamic model (DDM) are constant. The moments of turn-off (failure) of elements from operation are accepted as boundary (critical) points ti. The transition from an i to a new (i+1) state is considered as momentary. The solution according to TAM on the quasi-linear interval is developed in the closed form of Duhamel’s integral in a similar way to the elastic analysis. Use of TAM is illustrated by the example of the calculation of statically loaded flat steel truss during free oscillations caused by a sudden failure of one of the bearing elements. 2. Equation of motions of the discrete dissipative system The differential equation of motion (1) of a structurally nonlinear truss with n degrees of freedom in case of static load Q in the process of free oscillations on any quasi-linear interval t [ti, ti+1] together with initial conditions of the problem (2) takes the form: ° M i Y Ci Y t K i Y t ® °Y Y t , Y Y t , 0 0 i ¯

Q,

(1),(2)

where Mi, Ci, Ki ± Mn (R) – matrices of mass, damping and stiffness of the system found in the i state; Y(t) – vector of required nodal displacement of DDS. The integration of homogeneous ODE (1) is connected with construction of a fundamental matrix Ɏ(t) = e Si t , where Si ± Mn (C) – matrix which is a solution (root) of MQE:

M i Si 2 Ci Si Ki

0.

(3)

In spectrum of Si there are internal dynamic characteristics: eigenfrequencies, damping coefficients and natural modes [11]. In case of transition of the system to a (i+1) state in the left sides of equations (1) and (3) matrices Mi+1, Ci+1, Ki+1, on the basis of which a new value of Si+1 is found, are recalculated. In this research the authors have agreed that in case of such transition elements of the mass matrix have their initial values. Thus, the described modeling of a structurally nonlinear process enables us to apply the developed algorithm to the problem, when the nonlinear analysis is considered as a sequence of computations of elastic systems. 3. Parameters of dynamic response of the damaged system The system of equations of the response of statically loaded truss in case of sudden failure of connection takes the form [12]: °Y t ® ° Z (t) ¯

2 Re{Z (t)}, Y (t )

2 Re{Si Z (t)}, Y (t )

2 Re{Si 2 Z (t)} M i –1Q,

(4)

–1 Ɏ(t)U i –1 Ɇ i [– Si Y0 Y0 @ > Ɏ(t) – E ] U i Si Q.

where: t = t – ti; Ɏ( t ) = e Si t ; Ui = Mi Si + SiɌ Mi + Ci. This system helps to determine from unified positions the response of quasi-linear DDS in the i state on the interval t [ti, ti+1] when solving the dynamic problem (1), (2).

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The formulas (4) specify nodal kinematic parameters: displacement, velocity and acceleration. On their basis the force characteristics are determined (vectors of restoring, dissipative and inertia forces, respectively): R t

K iY t , F t

Ci Y t , I t

(5)

– Mi Y t .

With the help of nodal displacement Y(t) we can consider parameters of a strain-stress state (SSS) of bars of the structure: deformations İ(t), axial force N(t) and direct stress ı(t), etc. The characteristics (4), (5) enable us to conduct a comprehensive analysis of force resistance of the truss model at the specified time interval in the process of oscillations. Nevertheless, the analysis quality can be improved, having specified energy parameters of the response: total potential, kinetic and total mechanical energy. The total mechanical energy of the system E(t) is determined by a sum of total potential U(t) and kinetic energies K(t) [13]:

E t U t K t .

(6)

The summand U(t) depends on a mutual arrangement of points of the system and is determined by a difference between work of internal forces (strain energy) and potential of external forces: U(t) = Ui(t) – Ue(t) [14]; the summand K(t) is a measure of motion of these points in the process of oscillations [12]. For the discrete system the values of the right side in (6) are determined by sums:

U t

n ¦ ɉk t , K t k 1

n ¦ ɉk t . k 1

For the structure which is under dead load Q, the parameters according to [15] can be written in a vector form depending on the response parameters (4), (5): U t

T

Y T (t ) M i Y (t ) / 2.

¬ª R t / 2 – Q ¼º Y t , K t

(7)

where Ui(t) = R(t)T Y(t) / 2; Ue(t) = QT Y(t). 4. Behavior of response parameters in case of transition of the discrete system to a new state At the moment of transition of the elastic structurally nonlinear system to a new state (when t = ti) some parameters of the dynamic response have breaks which are determined by corresponding differences:

°'Y ti ° ®' R ti ° °' E ti ¯

– Y t , R t – R t , E t – E t , –

Y ti

i

–

i

i

–

i

i

– Y t , ' Y t Y t – Y t , 'F t F t – F t , 'I t I t – I t , 'U t U t – U t , 'K t K t – K t . ' Y ti

Y ti i

i

i

i

–

i

i

–

i

–

i

i

i

i

–

i

–

i

–

i

i

(8)

i

The times ti– and ti+ correspond to target values, calculated at ti before and after the failure of the bearing element, respectively. According to [12, 16] analytical expressions of closure of force and kinematic parameters take the next form:

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E. Ufimtsev and M. Voronina / Procedia Engineering 150 (2016) 1891 – 1897 – M i –1 (' K i Y ti 'Ci Y ti ), °'Y ti 0, ' Y ti 0, ' Y ti ® °' R t ' K Y t , ' F t 'C Y t , ' I t ' R t ' F t , i i i i i i i i i ¯

(9)

where 'Ki = Ki–1 – Ki, 'Ci–1 = Ci – Ci – accordingly, the residual stiffness and damping matrices characterizing the degree of damage of the DDS at its sudden restructuring in the non-linear operation. The values Y(ti) and Y (ti) are calculated at the end of the interval t [ti–1, ti]. They are initial conditions (2) for the interval t [ti, ti+1]. The dependencies (9) show that at the moment of transition of the structurally nonlinear system to a new state some parameters of the response (displacement and velocity) remain as continuous time variables, which is guaranteed by starting conditions (2), other parameters have breaks related to a sudden change of stiffening ('K) and damping properties of the structure. To get equations of energy discrepancy we write necessary parameters of DDS (see (7)) at the time moments ti– and ti+. In terms of ti– stiffness of the structure is determined by the matrix K(ti–) = Ki–1, vectors of nodal displacement and velocity are equal to Y(ti–) = Y(ti), Y (ti–) = Y (ti), respectively, the vector of restoring forces is determined as R(ti–) = Ki–1 Y(ti) (5). After the component failure (in the context of ti+) the same parameters take on values: K(ti+) = Ki, Y(ti+) = Y(ti), Y (ti+) = Y (ti), R(ti+) = Ki Y(ti). Later, having applied the indicated values to (8) we get the required discrepancy of energy parameters (6), (7):

' E ti

'U ti

T

Y ti ' K i Y ti , ' K ti

0.

(10)

The value of an energy change 'E(ti) = 'U(ti) corresponds to the released energy, which is equal to a stored in the bearing element potential energy of strain at the moment of its failure. For the truss bars this value according to [17] is calculated in the following way:

U

0,5 N 2 L / EA ,

(11)

where L, A – bar length and cross section area, respectively; N – axial force; E – elasticity modulus. The breaks in acceleration (see (8)) and energies at the time moment ti indicate a disturbance of equilibrium in the system during its structural transformation. The work of DDS is conducted in the following way. Trying to get an equilibrium position corresponding to new stiffness the structure is involved in the oscillating process in the course of which redistribution of forces occurs, since internal forces should balance external forces, as well as inertia and strain forces. If resistance of the system to structural failure is ensured, further structural transformations don’t happen and oscillations decay. Otherwise, one or several load-bearing elements are disabled (in case of loss of strength or stability), which in turn may lead to a collapse of the whole structure. The assessment of resistance is determined on the basis of values of a rank and a determinant of the stiffness matrix of the system Ki. 5. Numerical implementation of a problem

The usage of the described mathematical model of structurally nonlinear calculation is illustrated on the basis of time analysis of a steel roof truss of the metalwork galvanizing plant situated in the city of Shumikha of Kurgan region (Russia). The truss DDM is given in Fig. 1. This structure is 3 times statically indeterminate and has n = 29 dynamical degrees of freedom connected with nodal mass mj. The truss material is steel ɋ245 (ȿ = 206000 MPa). The truss elements have stiffness [18]: top chord – 2Ŀ125u8 mm; bottom chord – 2Ŀ100u7 mm; diagonal web elements – 2Ŀ110u8 mm, 2Ŀ100u7 mm, 2Ŀ63u6 mm; vertical web members – 2Ŀ56u5 mm.


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Fig. 1. Calculated dynamic model of the roof truss of the galvanizing plant

The stiffness matrix of the truss model K is formed with the help of the finite element method. The mass matrix Ɇ includes the masses of the truss and roof elements. The damping matrix C is developed according to the model of non-proportional damping [11] when the value of logarithmic decrement is į = 0,085 [19]. At the initial moment of time (t0 = 0) the structure rests under gravity Q. The stiffness of the system is determined by the matrix K0, a basic level of static deflections is equal to Yst,0 = K0–1Q. When t0+ = 0, there is a sudden failure of the element 1-15 (shown as a dashed line in Fig. 1) as a consequence of long-term chemical corrosion attack related to the presence of sulfuric acid (H2SO4) vapor in the plant facilities. As a result of this, the truss is changed to the mode of free decaying oscillations. A new stiffness matrix of the system is K1, and the vector of static deflections Yst,1 = K1–1Q. The time analysis of the structure response is conducted with a step of integration ǻt = 0,0002 sec. The initial conditions (2) are the following: Y0 = Yst,0, Y0 = 0. The problem is solved by the elastic approach. Fig. 2 illustrates oscillograms of some response parameters used in formulas (7): kinematic (displacement (a) and velocity (b)) and force (restoring (c) and axial (d) forces). The oscillograms of velocities at the failure moment t0+ have breaks (Fig. 2, b). On the diagrams of restoring forces (Fig. 2, c) one can observe breaks connected with a sudden change of stiffness and damping properties of the system. The oscillograms of axial forces (Fig. 2, d) show an increase in values of forces (up to 100%) for some elements in comparison with static values. In articles [20, 21] similar truss calculations are performed. Besides, the features of characteristics behavior (4), (5) are given and described. The behavior of energy of the truss response (6), (7) is shown in Fig. 3. Oscillograms of total potential (a) and kinetic energy (b) have a decreasing nature. The fragments of graphs show that the change of these characteristics occurs in opposite phases. In the initial state the truss doesn’t have kinetic energy (K(t0) = 0), as it rests; the value of total potential energy U(t0) = –303,89 J corresponds to the level of static deflections Yst,0. According to (6), the value of total mechanical energy is E(t0) = U(t0). At the moment of failure of the element 1-15 the characteristic E(t0) according to (10) is jumped for E(t0+) = [ (Yst,0)Ɍ'K1 Yst,0 ] = –26,99 J to a new level E(t0+) = –330,88 J. In this case 'K1 = K1 – K0. The value |'E(t0+)| corresponds to the stored potential strain energy (11) in the disabled element 1-15 at the moment of time t0+:

U1–15

86,62 u 583,6 / 2 u 20600 u 2 u19,7

26,99 J .

In the process of free decaying oscillations (in case of t > 8 sec) the energy values are approaching to: K(t) = 0, E(t) = U(t) = –511,89 J, corresponding to a new position of static equilibrium of the truss. Meanwhile, as Fig. 3, a shows the parameter E(t) as opposed to U(t) and K(t) is fluently changed from the level E(t0+) to the level E(t). In the period t [0; 0,3] sec, after disabling the diagonal brace the truss is in the transient mode. This is evidenced by high-order harmonics (see fragments in Fig. 3). This phenomenon is connected with a substantial modification of the frequency spectrum of the structure in case of a sudden decrease of its stiffness characteristics. After t § 0,3 sec the system is changed to the mode of specified (harmonic) damped oscillations.

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The realization of the mathematical model and solution of the problem is carried out with the help of MATLAB [22].

Fig. 2. Oscillograms of the dynamic response of the truss model: (a) Displacements; (b) Velocities; (c) Restoring forces; (d) Axial forces in truss rods

Fig. 3. Oscillograms of energy parameters of the truss response: (a) Total potential and total mechanical energy; (b) Kinetic energy

6. Conclusion

1. The formulas of kinetic, force and energy parameters of dynamic response of the discrete system model in case of its elastic structurally nonlinear oscillations are given in the closed form.


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2. The relation of parameters of the dynamic response with a change of stiffness and damping properties of the structure in terms of a sudden failure of one of the bearing elements is established. 3. The time analysis of elastic response of the discrete mode of the roof truss under dead load in case of a sudden failure of one of the bearing elements as a result of corrosion attack is conducted. 4. The features of energy parameters of the truss response (jumps, levels, etc.) in the process of its elastic structurally nonlinear oscillations with regard to static loading are detected. References [1] R. Klaf, D.Zh. Penzien, Dynamics of Structures, Stroyizdat, Moscow, 1979. [2] A.N. Potapov, Time Analysis of Elastic-plastic Finite Dissipative Systems under Nonstationary Effects, Vestnik YuUrGU, Seriya Stroitel'stvo i arkhitektura. 13 (2005) 57–62. [3] M.I. Erkhov, The Theory of Ideal Plastic Bodies and Structures, Nauka, Moscow, 1978. [4] V.V. Bezdelev, Numerical Simulation of the Dynamic Stress-strain State of Buildings under Seismic Actions in Order to Optimize Parameters of Damping Devices, Journal for Computational Civil and Structural Engineering. 4 (2008) 24–25. [5] N.V. Klyueva, O.V. Azhzeurov, K.A. Shuvalov, On the Issue of Development of Experimental Studies of Deformation and Fracture of Spatial Design Systems under Beyond Design Impacts, Yugo-Zapadnogo gosudarstvennogo universiteta. 46 (2013) 111–116. [6] P.G. Eremeev, Prevention Avalanche (Progressive) Collapse of Supporting Structures of Unique Large-span Structures during Emergency Loads, Stroitel'naya mekhanika i raschet sooruzheniy. 2 (2006) 65–72. [7] S.V. Shlychkov, S.P. Ivanov, S.G. Kuzovkov, Yu.V. Loskutov, Calculation of Geometrically Nonlinear Structures Using Finite Element Method, Izvestiya vysshikh uchebnykh zavedeniy, Tekhnicheskie nauki, Mashinostroenie i mashinovedenie. 4 (2008) 145–152. [8] E.V. Simon, Calculation of Geometrically Nonlinear Rod Systems in Mixed Form, Internet-vestnik VolgGASU, Seriya Politematicheskaya. 3 (2012). URL: http://vestnik.vgasu.ru/attachments/Simon-2012_3(23).pdf. [9] A.A. Sventikov, Geometrically Nonlinear Calculation of the Hanging Rod Designs, Part 2: The Matrix Calculation of Suspension Systems, Stroitel'stvo i arkhitektura. 1 (1970) 18–27. [10] A.V. Perelmutrer, Conversations about the Structural Mechanics, SCAD Soft Publ., BIA Publ., Moscow, 2014. [11] A.N. Potapov, Dynamical Analysis of Discreet Dissipative Systems under Nonstationary Loadings, SUSU Publ., Chelyabinsk, 2003. [12] A.N. Potapov, Analysis of Vibrations of Structures with Breaked Ties, Morskie intellektual'nye tehnologii. 3 (2011) 45–48. [13] Elementary Physics, Volume 1: Mechanics, Heat, Molecular Physics, Fizmatlit Publ., Moscow, 2000. [14] N.A. Alfutov, Fundamentals of Calculation on the Stability of Elastic Systems, Mashinostroenie Publ., Moscow, 1978. [15] G.V. Vasilkov, Evolutionary Theory of the Life Cycle of Mechanical Systems: Theory of Structures, URSS Publ., Moscow, 2013. [16] A.N. Potapov, E.M. Ufimtsev, Time Analysis of the Structures with the Breaking Ties under the Pulse Load, in: Proceedings of the XXIV International Conference Integration, Partnership and Innovation in Construction Science and Education, Moscow. (2011) 229–233. [17] V.A. Ikrin, Strength of Materials with Elements of the Theory of Elasticity and Plasticity: A Textbook for High Schools in the Direction of Building, BIA Publ., Moscow, 2005. [18] GOST 8509-93, Hot-Rolled Angle Equilateral, Assortment, 1993. [19] M.F. Bernshteyn, V.A. Il'ichev, B.G. Korenev, Dynamic calculation of buildings and structures, Stroiyizdat Publ., Moscow, 1984. [20] A.N. Potapov, E.M. Ufimtsev, Oscillations of Systems with a Breaking Ties, in: Proceedings of the international-term scientific and practical conference Theory and practice of calculation of buildings, structures and structural elements, Analytical and numerical methods. (2011) 292– 301. [21] E.M. Ufimtsev, Time Analysis of Physically and Structurally Nonlinear Oscillations of Truss Structures in the Pulse Loading, Part 2: An Example of the Calculation of the Elastic Structurally Nonlinear Oscillations of the Roof Truss, in: Proceedings of the 1st International Scientific-Practical Conference Building, Ecology: Theory, Practice, Innovation, Chelyabinsk. (2015) 106–108. [22] V.P. D'yakonov, MATLAB 7.*/R2006/R2007: Self-teacher, DMK Press Publ., Moscow, 2008.