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1

Analytical study on dynamic response of cantilever flexible retaining wall

Yang Xiuzhu, Liu Xinyuan, Zhao Shuang, Yu Jun

Archives of Civil Engineering

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2023

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Vol. 69, nr 2

435--453

EN

Based on wave mechanics theory, the dynamic response characteristics of cantilever flexible wall in two-dimensional site are analyzed. The partial derivative of the vibration equation of soil layer is obtained, and the general solution of the volume strain is obtained by the separation of variables method. The obtained solution is substituted back to the soil layer vibration equation to obtain the displacement vibration general solution. Combined with the soil-wall boundary condition and the orthogonality of the trigonometric function, the definite solution of the vibration equation is obtained. The correctness of the solution is verified by comparing the obtained solution with the existing simplified solution and the solution of rigid retaining wall, and the applicable conditions of each simplified solution are pointed out. Through parameter analysis, it is shown that when the excitation frequency is low, the earth pressure on the wall is greatly affected by the soil near the wall. When the excitation frequency is high, the influence of the far-field soil on the earth pressure of the wall gradually increases. The relative stiffness of the wall, the excitation frequency and the soil layer damping factor have a significant effect on the dynamic response of the flexible retaining wall.

2

The Generalized Differential Transform Method for solution of a free vibration linear differential equation with fractional derivative damping

Das Deepanjan

Journal of Applied Mathematics and Computational Mechanics

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2019

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Vol. 18, nr 2

19--29

EN

In the present paper, the Generalized Differential Transform Method (GDTM) is used for obtaining the approximate analytic solutions of a free vibration linear differential equation of a single-degree-of-freedom (SDOF) system with fractional derivative damping. The fractional derivatives are described in the Caputo sense.

3

Exact controllability of a string to rest with a moving boundary

Gugat Martin

Control and Cybernetics

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2019

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Vol. 48, No. 1

69--87

EN

We consider the problem of steering a ﬁnite string to the zero state in ﬁnite time from a given initial state by controlling the state at one boundary point while the other boundary point moves. As a possible application we have in mind the optimal control of a mining elevator, where the length of the string changes during the transportation process. During the transportation process, oscillations of the elevator-cable can occur that can be damped in this way. We present an exact controllability result for Dirichlet boundary control at the ﬁxed end of the string that states that there exist exact controls for which the oscillations vanish after ﬁnite time. For the result we assume that the movements are Lipschitz continuous with a Lipschitz constant, whose absolute value is smaller than the wave speed. In the result, we present the minimal time, for which exact controllability holds, this time depending on the movement of the boundary point. Our results are based upon travelling wave solutions. We present a representation of the set of successful controls that steer the system to rest after ﬁnite time as the solution set of two point-wise equalities. This allows for a transformation of the optimal control problem to a form where no partial diﬀerential equation appears. This representation enables interesting insights into the structure of the successful controls. For example, exact bang-bang controls can only exist if the initial state is a simple function and the initial velocity is zero.

4

Zakrzywione pręty cienkościenne o przekroju bisymetrycznym

Bijak R., Chodor L., Kołodziej G.

Czasopismo Inżynierii Lądowej, Środowiska i Architektury / Journal of Civil Engineering, Environment and Architecture

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2017

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z. 64, nr 3/I

583--594

PL

W pracy przedstawiono geometrycznie nieliniową teorię wstępnie skręconych i zakrzywionych w przestrzeni prętów cienkościennych. W przykładach numerycznych analizowano pręty kołowe w zakresie liniowym dla dwóch warunków brzegowych: (1) podparcia widełkowego na obu końcach oraz (2) zamocowania na jednym końcu. Zastosowano 3-węzłowy element izoparametryczny z całkowaniem zredukowanym dla 2 punktów Gaussa. Porównania wyników numerycznych z rozwiązaniami analitycznymi pokazują niewielkie różnice między nimi. Z kolei, jeżeli dodatkowo przyjmiemy hipotezę Bernoulliego dla zginania i Własowa dla skręcania, z prezentowanej teorii można prosto wyprowadzić równania różniczkowe giętno-skrętnej utraty stateczności łuków kołowych. Wzory na uogólnione odkształcenia są tak sformułowane, że możemy również w prosty sposób uwzględniać wpływ imperfekcji geometrycznych pręta.

EN

In the paper it was presented nonlinear theory of initially twisted and bended thin walled rods in three dimensional space. Theory was basis for implementation of computer program, which employs 3-node isoparametric finite element with reduced numerical integration (Gaussian integration at 2 points). In numerical examples, circular rods were analyzed in a linear range for two boundary conditions: 1) circular arch with fork-alike supports at the ends 2) fully restrained cantilever. Result were compared then with analytic solutions and good consistency was observed. On the other hand, additional assumption of Bernouli hypothesis for bending and Wlasov theory for twisting allows for easy derivation of differential equations of lateral-torsional buckling for circular arches. Expressions for generalized strains are formulated in a manner which allows taking into consideration influence of geometric imperfections of the rods in the analysis.

5

Problemy utrzymania ruchu podczas modernizacji budowli kolejowych na przykładzie posterunku odgałęźnego Wrocław Grabiszyn

Mazurkiewicz R., Dziekoń D.

Materiały Budowlane

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2016

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nr 6

122--124

PL

Modernizacja infrastruktury kolejowej połączona jest najczęściej z zamykaniem dla ruchu torów szlakowych i z dużymi problemami w utrzymaniu ruchu kolejowego na przebudowywanych odcinkach linii. W przypadku punktów węzłowych sieci o złożonej geometrii układu torowego, zapewnienie przejezdności staje się poważnym problemem organizacyjnym. W artykule podano przykład organizacji ruchu kolejowego w czasie modernizacji posterunku odgałęźnego Wrocław Grabiszyn, realizowanej w latach 2014 – 2015. Przedstawiono zakres prac wykonanych w poszczególnych fazach przebudowy oraz przyjętą w każdej z nich organizację ruchu kolejowego.

EN

The modernization of the railway infrastructure is connected with the closing main tracks for traffic and with big problems of keeping the railway traffic on the reconstructed sections of railway lines. For nodes of the network of the complex geometry of the track system, ensurance the operability becomes a serious and complex problem of organization. The paper is an example of the organization of rail traffic during the modernization of the Grabiszyn branch-off point, realized in 2014-2015. The work done in different phases of reconstruction and the organization of railway traffic adopted in each of them were presented.

6

Analytic bending solutions of rectangular cantilever thin plates

Tian B., Zhong Y., Li R.

Archives of Civil and Mechanical Engineering

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2011

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Vol. 11, no 4

1043-1052

EN

Analytic bending solutions of rectangular cantilever thin plates subjected to arbitrary loads are derived using the double finite integral transform method. Since only the basic governing equations of the plates are used and there are no predetermined functions, the present method overcomes the deficiency of the conventional semi-inverse methods thus serves as a completely rational model in solving plate bending problems. The method can be extended to more boundary value problems of plates such as buckling and vibration.

PL

W pracy sformułowano analityczne rozwiązania dotyczące zginania prostokątnych, wspornikowych płyt cienkich poddanych dowolnym obciążeniom przy zastosowaniu metody podwójnej, skończonej transformaty całkowej. Ponieważ zastosowano tylko podstawowe równania płyt, zaprezentowana metoda przezwycięża niedobór konwencjonalnych metod semi-odwrotnych i może służyć jako całkowicie wymierny model przy rozwiązywaniu problemów zginania płyt. Zaprezentowana metoda może być rozszerzona na dalszą klasę problemów brzegowych płyt w zakresie ich wyboczenia oraz drgań.

7

Analytic and experimental study for light alloy aluminium panels under compression

Fayza A., Zghal A., Bayraktar E.

Journal of Achievements in Materials and Manufacturing Engineering

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2007

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Vol. 25, nr 1

19-22

EN

Purpose: Aluminium alloys have been indispensable for the progress of many technologies during the last decades. In particular, most stiffeners in aerospace structures are composed of aluminium panels often solicited with elastic and plastic bucking under particular boundary and loading conditions. The purpose of this paper is to enhance the difficulties encountered to predict the incipient elastic-plastic buckling for thin aluminium plates under combined compressive loads. Design/methodology/approach: The used methodology was an analytic non linear approach, validated further with an experimental investigation. In fact, the instability of thin elastic-plastic rectangular panels made of 2024 T45 alloys is analyzed. General concept of the Von Kaman's equation with a set of trigonometric and harmonic functions was used in the analytic model. The computation of buckling loads concerns both elastic and plastic instability solutions. Developments in the plastic range were concerned with the (j2d) deformation and the (J2f) flow constitutive laws. Findings: A methodology to develop this kind of analytic resolution is pointed out and has been illustrated for a set of variables. Several 2d and 3d plots, which can be used to predict incipient buckling strengths for plates with flat initial configurations, have been presented for the various load conditions. Research limitations/implications: In the future it will be possible to apply the investigated analytic procedure to other particular cases. Practical implications: Plots obtained with analytic solution can be used to predict incipient buckling strengths for plates with flat initial configurations are presented for the various tests. The interest of three dimensional representations is to indicate when plastic buckling occurs for a square plate under biaxial loading. Originality/value: This paper presents a stable and low cost analytic solution to deal with instability phenomenon in elastic and plastic range for the design of light alloy aluminium plates. This approach is applied to assess the conditions for which plastic buckling can happen when particularly thin aluminium panels are used. This latter, can be implemented in finite element standard codes.

8

Analytic solutions of a first order iterative differential equation x'(z)=x(p(z) + bx(z))

Qiu F., Liu H.

Demonstratio Mathematica

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2006

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Vol. 39, nr 1

81-90

EN

This paper is concerned with a iterative differential equation x'{z) = x{p(z) + bx(z)). By constructing a convergent power series solution y(z) of a companion equation of the form betay'(3betaz) = y'(z)[y(beta2z) - p(y(betaz)]], analytic solutions of the form [y(betay- l(z)) - p(z)]/b for the original differential equation are obtained.

9

Analytic solutions of a nonlinear functional differential equation with proportional delays

Si J.

Demonstratio Mathematica

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2000

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Vol. 33, nr 4

747-752

10

Analytic solutions of a polynomial-like iterative functional equation

Si J., Wang X.

Demonstratio Mathematica

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1999

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Vol. 32, nr 1

95-103

EN

This paper is concerned with a polynomial-like iterative functional equation lambda1f(z) + lambda2f(2)(z) + .. . . + lambdamf[m](z) = F(z), where f(z) is an unknown function, F(z) is a given function, f[i]l denotes the i-th iterate of f, and lambda1, lambda2, . . .,lambdam are complex constants. By constructing a convergent power series solution phi(z) of an auxiliary equation of the form lambda1alpha(alphaz)+lambda2phi(alpha2z)+.....+lambdamphi(alphamz)=F(phi(z)), analytic solutions of the form phi(alphaphi-1(z)) for the original iterative functional equation are obtained.

11

Note on an iterative functional differential equation

Si J., Cheng S.

Demonstratio Mathematica

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1998

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Vol. 31, nr 3

609-614

EN

This note is concerned with an iterative functional differential equation xI(z)=c1x(z) + . . . + Cmx(m) (z), where x(k) (z)=x(x(. . . x(z))) is the k-th iterate of the function x (z) . We apply the techniques developed in a previous paper [1] which deals with the simpler equation xI (z)=x (m) (z) and obtain analytic solutions for the more general equation.