Inelastic Stress Analysis Of Curved Beams With Bending And -Books Download

Inelastic Stress Analysis of Curved Beams with Bending and
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classical theory of elasticity the proposed computational model adopts the generalized Hook s law in plane stress with. consideration of the plastic strains defined by the total deformation theory The swift type nonlinear hardening law is. considered for the inelastic constitutive relations of the material with its behaviour governed by von Mises yield criterion. With the introduction of a certain form of Airy stress function that satisfies the compatibility equation in polar coordinates. the strain compatibility equation with the corresponding boundary conditions can be simplified as a second order ordinary. differential equation of a properly defined generalized stress function dependent on radius of the wide curved beam. Solution of this ODE equation becomes a boundary valued problem that can be solved numerically in both elastic and. inelastic stages To validate the proposed model inspired by the work of Eraslan and Arslan 8 9 a preliminary numerical. study of the stress analysis in the elastic stage has been conducted Encouragingly the numerical solution matches. perfectly with its analytical counterpart given by classical theory of elasticity 10. 2 Computational Model, An in plane flexural damper under a load 2P at a distance l from the ends of the circular arch is illustrated in Fig. 1 a The ends of the arch are subjected to a bending moment M Pl and a shear P simultaneously as indicated in Fig 1 b. Taking advantage of symmetry with respect to the vertical axis the task can be reduced as solving a problem of a circular. cantilever subjected to coupling of shear and moment at its end as indicated in Fig 1 c. a in Plane Flexural Damper b Free body of the Circular Arch c Circular Cantilever. Fig 1 The Damper and the Portion Considered in Analysis. a P M Coupling b End Shear P only c Pure Bending M. Fig 2 Resolving Diagram of the Loads on the Circular Cantilever. 2 1 Fundamental of Solid Mechanics, The curved beam with inner radius a and outer radius b is considered as shown in Fig 2 It is most convenient to. start with the well developed fundamentals of elasticity in polar coordinates 10 For plane stress problems the. equilibrium equations of the curved beam with a constant thickness t z can be expressed as. ICSENM 107 2, Where r are respectively the normal stress in the radial and tangential directions and r is the shear stress. These stress components can be further written in terms of an Airy stress function r as. Which satisfy the equilibrium equation 1 The strains are related to the displacement fields u v respectively for. the radial and tangential displacements as, where r are respectively the normal strain in the radial and tangential directions and r is the shear strain. The strain compatibility equation according to 11 is expressed as. r r2 r r 0 4, And the compatibility can further be expressed in terms of the Airy stress function r as.
2 1 1 2 2 1 1 2,r r r r 2 2 r 2 r r r 2 2, The von Mises yielding criterion adopted in this study is defined as. vM r 2 r 2 3 r y 6, And the material becomes partially plastic wherever the inequality holds The generalized Hooke s law is written in. accordance with the total deformation theory 12 as. ICSENM 107 3,where the plastic strains are defined by. In which the equivalent plastic strain of a swift type nonlinear hardening law is considered as. EQ for vM y 9,0 for vM y, Where parameters m and H characterize the post yielding behaviour of the material. 2 2 Pure Bending, Under the condition of pure bending as indicated in Fig 2 c the shear stress vanishes and the equilibrium equation.
reduces to, If a generalized stress function Y1 r is defined according to Eraslan and Arslan 9 as. Y1 r r r 11,and as a result, Using the strain displacement relations in eq 3 and taking into account the fact of shear strain being zero the. strain compatibility eq 4 can be simplified as,ICSENM 107 4. where M is an integration constant Substitution of the generalized Hooke s law 7 for the strains and expressing. the stresses in terms of the generalized stress function Y1 Eq 13 can be written in the form of a second order ordinary. differential equation of Y1 as,r 2Y1 rY1 Y1 r M rp p r 14. or in brief,Y1 F r Y1 Y1 15,With the following boundary conditions.
Y1 a Y1 b 0 16a,1 z Pl t z 16b, The 4th order Runge Kutter algorithm is adopted for solving the ODE problem iteratively along with Newton s. method in the shooting process to accelerate convergence of the boundary valued problem as suggested by Eraslan and. Arslan 9 Once Y1 is solved the stress components can be found by eqs 11 and 12 immediately. 2 3 End Shear, With a shear force acting at the end of the curved beam as indicated in Fig 2 b the condition becomes more. complicated with the presence of shear stress and all the stress components are now dependent on both r and However. if we loan of the Airy stress function r in the form of. r f r sin 17, From the classical theory of elasticity 10 for the very same problem to start with then the stress components can be. written as,r f f sin 18a,r f 2 f cos 18c, Let s define another generalized stress function Y2 r as. ICSENM 107 5, Then the stress components in eq 18 can be written in terms of Y2 r to be.
Y2 2 sin 20b,r 2 cos 20c, As a result the generalized Hooke s law 7 can be modified as. r rP Y2 2 sin 21a,P Y2 2 2 sin 21b,r rp cos 21c, Upon substitution of eq 21 for the strains into the strain compatibility equation 4 and integrating it with. respect to the radius r leads to,r Y2 rY2 4Y2 p,r r r rp 22. or in brief,Y2 F r Y2 Y2 23,With the following boundary conditions. Y2 a Y2 b 0 24a,a r dr t z, The same procedure as described in Section 2 2 can be adopted to solve the boundary valued problem of eqs 23.
24 Once Y 2 is solved the stress components can be found by eqs 20 immediately for a given cross section at. 2 4 Bending and Shear Coupling, If the curved beam is subjected to the coupling of bending M and shear P at the end simultaneously the stresses. can be obtained by summing the results from pure bending and end shear derived independently as. r r 1 r 2 25a,ICSENM 107 6, where the subscript 1 denotes those obtained from pure bending and subscript 2 from end shear Note that under the. coupling loading condition the resultant stresses from eqs 25 should be used in checking the yielding condition via the. von Mises criterion 6,3 Numerical Verification, As an effort to validate the proposed model a preliminary numerical study of the stress analysis has been conducted. in the elastic stage where analytical solutions are available for comparison An in plane flexural steel damper in a U shape. considered for component test earlier by the authors is adopted herein The inner radius a of the circular arch of the. damper is 45 mm and the outer radius b is 220 mm The steel plate is 25 mm in thickness and the effective length l of. the straight arm reads 235 mm Parameters of the material are summarized in the following Young s modulus E 200 GPa. Poisson s ratio 0 3 and the yield stress y 250 MPa A compression force P is acting inwards from each end of the. arms through pin connection resulting in a moment of Pl and an end shear of P simultaneously on the bottom of the arch. as indicated in Fig 2 a The yield strength Py 39 75kN determined by setting the normal stress a 2 y where. initial yielding of the damper takes place is considered as the applied load Stress components normalized with respect to. the yield stress y are presented graphically for 2 and 4 in Figs 4 a and 4 b respectively The bar. over the stress symbols in the figures indicates the normalized stress The numerical solution agrees perfectly with its. analytical counterpart given by classical theory of elasticity 10 The normalized von Mises stress vM vM y 1. occurs at the cross section of 2 on the inner surface r a under yield load Py. Fig 3 Design Details of the in Plane Flexural Damper Considered in the Numerical Example. ICSENM 107 7, Fig 4 Comparison of Numerical Solution in the Elastic Stage with the Analytical Counterpart P Py. 4 Conclusion, In this paper the analytical model for inelastic stress analysis of symmetrical curved beams with bending and shear.
coupling represented in a second order ordinary differential equation of a generalized stress function is derived As an. effort to validate the proposed model a preliminary numerical study of the stress analysis in the elastic stage has been. conducted The numerical solution agrees perfectly with its analytical counterpart given by classical theory of elasticity. The model will be further used for the inelastic stress analysis of the in plane flexural damper. Acknowledgements, This research work is granted by the Ministry of Science and Technology of Republic of China under contract. MOST 104 2221 E 009 197,References, 1 A S Whittaker V V Bertero C L Thompson and L J Alonso Seismic Testing of Steel Plate Energy Dissipation. Devices Earthquake Spectra vol 7 no 4 pp 563 604 1991. 2 K C Tsai H W Chen C P Hong and Y F Su Design of Steel Triangular Plate Energy Absorbers for Seismic. Resistant Construction Earthquake Spectra vol 9 no 3 pp 505 528 1993. 3 Y P Wang and C S Chang Chien A Study on Using Pre Bent Steel Strips as Seismic Energy Dissipative Devices. Earthquake Engineering Structural Dynamics vol 38 pp 1009 1026 2009. 4 W K Chan and F Albermani Experimental Study of Steel Slit Damper for Passive Energy Dissipation. Engineering Structures vol 30 pp 1058 1066 2008, 5 J H Park and K H Lee Cyclic Loading Tests of Steel Dampers Utilizing Flexure Analogy Deformation in. Proceedings of the 15 TH World Conference on Earthquake Engineering Lisbon Portugal September 24 28 paper no. 6 Z Guan J Li and Y Xu Performance Test of Energy Dissipation Bearing and Its Application in Seismic Control of. a Long Span Bridge Journal of Bridge Engineering ASCE vol 15 pp 622 630 2010. 7 Y P Wang D H Chen and C L Lee An Experimental Study on in Plane Arch shaped Flexural Damper in. Implementing Innovative Ideas in Structural Engineering and Project Management Proceedings of ISEC 8 Nov 23. 28 Parramatta Australia pp 293 298 2015, 8 A N Eraslan and E Arslan A Concise Analytical Treatment of Elastic Plastic Bending of a Strain Hardening. Curved Beam ZAMM vol 88 no 8 pp 600 616 2008, 9 A N Eraslan and E Arslan A Computational Study on the Nonlinear Hardening Curved Beam Problem.
International J Pure and Applied Mathematics vol 43 no 1 pp 129 143 2008. ICSENM 107 8, 10 S P Timoshenko and J N Goodier Theory of Elasticity Third Ed New York McGraw Hill 1970. 11 A P Boresi K P Chong and J D Lee Elasticity in Engineering Mechanics Third Ed Hoboken New Jersey John. Wiley and Sons Inc 2011, 12 R M Jones Deformation Theory of Elasticity Blacksburg Virginia Bull Ridge Publishing 2009.


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