Snap-through of pin-ended shallow arches of variable thickness with torsional spring supports

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1 IABSE-JSCE Joint Conference on Advances in Bridge Engineering-II, August 8-,, Dhaka, Bangladesh. ISBN: Amin, Okui, Bhuiyan (eds.) Snap-through of pin-ended shallow arches of variable thickness with torsional spring supports Ali Asghar Atai Department of Mechanical Engineering, Islamic Azad University, Karaj Branch, Iran Mahnaz Panahiazar Department of Mechanical Engineering, Islamic Azad University, Karaj Branch, Iran ABSTRACT: Shallow arches are interesting structures that can be used as skeleton of large structural systems or in toggle mechanical or thermal switches. An important and interesting, and sometimes undesired, phenomenon in these structures is snap-through which is considered as a kind of buckling resulted from buildup of internal compressive forces that make the structure unstable and force it to undergo a large displacement toward a new stable configuration. It is this displacement that is interesting in applications such as toggle switches and undesired in stationary structures such as truss domes. This phenomenon can be analyzed from a potential energy point of view, although solving the governing equation of the deformation could also give the point of snap-through. Researches on this subject are mostly focused on arches with constant cross section and/or simple pin supports. In this study, that work is extended to a variable thickness with spring supports. The governing nonlinear equation is solved by a simple but accurate difference scheme, and by varying the choice of thickness function and the support spring constants, interesting results are presented in which the critical limit point load is amplified or diminished. It is shown that by proper choice of the spring constants, the effect of thickness variation can be nullified. Two cases of pin supports and fixed supports are investigated and verified as limiting cases of spring constants. Also, the simple case of constant thickness with spring supports is presented for validation. Several cases of thickness variations which narrow or thicken towards the arch center point are considered. The results of the numerical difference scheme are also duplicated by a finite element analysis. INTRODUCTION Shallow structures such as arches and caps are used extensively as supporting members or covering outer structures and bodies of vehicles, and also as the key part in mechanical and thermal switches. Their instability could result in the disaster of collapse of the structure or could cause an electrical circuit to be closed or opened. In this instability, the structure is under the action of a compressive load normal to the one- or twodimensional continuum. The constraints on these structures are such that the external loading causes a compressive internal force to be formed along or in the plane of the structure. As the external loading is increased and tries to flatten the structure, this compression builds up and reaches a point that the structure cannot withstand in an stable equilibrium configuration and therefore, the structure passes (jumps) through a series of unstable configurations until it reaches another stable equilibrium configuration. Because of applications and hazards this jump brings with itself, acquiring suitable knowledge on the phenomenon is of great importance. Several researches and investigations have been carried out on the subject for the simple case of constant or variable thickness without spring supports [-6]. Because some support conditions are better modeled with a spring type, such as a clamped arcg or a quasi-clamped one in which the arch is welded to the base plate, some investigations on the subject have considered spring-type supports [7-]. In all these works, a constant thickness assumed for the arch. This work investigates the effect of thickness and support stiffness variation on the limit load of pin-ended shallow arches. Next section discusses the problem formulation mainly based on the notation used in [5]. The numerical method of solution is briefly described at the end of that section. Some examples are discussed in section 4. Some verifying cases based on previous works and also the effects of different parameters on critical snap loading are presented.

2 PROBEM FORMUATION AND SOUTION PROCEDURE Figure shows a variable thickness pin-ended shallow arch with torsional spring supports. The equation of centerline of the loaded arch is represented by w( and its undeformed state is denoted by w (, in which x shows the horizontal coordinate of the arch. The span of the arch is denoted by and the maximum height of the centerline is denoted by e. For shallow arches, the ratio e/ will be considered to be lower than.5 []. The initial shape of the centerline is considered to be a parabola with the equation w ( = 4ex( / () and it has been investigated by the authors that for constant thickness shallow arch, the shape of the centerline has very little effect on the critical snap load [] (in a numerical example, centerlines in the shape of parabola, circular, and elliptical arches yielded the same snap load within less than one percent of difference). The ends are pinned with torsional spring supports which for the time being, are considered to be of unequal stiffnesses k and k at the ends and respectively. This kind of support causes the development of an axial loading throughout the arch and its presence and buildup as the external lateral loading increases is an important contributor to the instability of the arch. The thickness is considered to be variable but still small compared to other dimensions of the arch and also symmetric with respect to / line. This justifies that only the effect of bending moment be considered in formulating the deflection of the arch centerline. This study aims at presenting the methodology for obtaining critical snap loading and hence, the loading is considered to be a simple uniformly distributed one with intensity P. The deflection of the arch is considered to be small compared to initial curved shape. Although by snap through instability the structure undergoes considerable displacement, but this last assumption is reasonable enough for obtaining the critical snap load. The post buckling behavior is not considered here. Figure shows the free body diagram of a section of the arch in the deformed shape. From the equilibrium of the whole arch and of the segment shown, the left support reactions F and M, and the internal shear and moments V and M are found to be k F = V = ( w ) M = k ( w ) M = Fx Qw + k ( w ) k ( w ) Px P + () Therefore, based on the moment-curvature relations, the governing equation of the deflection of the arch is Qw Px( k w + = + w ( ) EI( EI( EI( x k ( ) ( w ) EI( x = x () z,w k k x, Figure : Initial Geometry of pin-ended shallow arch with torsional spring supports

3 P M Q M F V Q w x Figure : Free body diagram of a section of the arch in which E is the modulus of elasticity of the arch material (which is considered to be linear elastic), I( is the variable area moment of inertia of the arch cross section, and prime denotes differentiation with respect to x coordinate. Q is the unknown axial force resultant in the cross section to be found by the following procedure. Considering the axial strain of the arch, it can be written (see for example []) ε = ε + zκ (4) in which ε is the axial strain of a point on the cross section with vertical coordinate z measured from the centerline, and ε and κ are the axial strain of the centerline and change in its curvature respectively which can be expressed by the following two relations. ε = u + ( w ) / (5) w κ = ( w w) (6) In relation (5), u is the axial displacement and the second term on the right is due to the variation in arc length due to the deflection of the centerline []. The internal axial loading Q is constant throughout the centerline and is the resultant of the axial stress σ=eε, i.e. Q Q = σda = EA( ε = [ u + ( w ) / ] EA( (7) A in which A( is the variable area of the arch cross section. Since the displacements of the arch at the ends are zero, integration of the above equation over the span of the arch and making use of symmetry yields Q E / / = ( w dx A( ) dx Equations () and (8) are the coupled governing equations for finding the lateral deflection w and the axial force Q in the arch. As it is seen, equation (8) makes the set nonlinear and hence needing a numerical procedure for solving it. But fortunately, as it is seen in the next section, since Q is constant, it is possible to solve equation () for deflection using an innovative method and substitution of the solution into (8), gives a relation for Q to be used for obtaining the snap load. It should be mentioned that in this study the cross section of the arch is considered to be rectangular and only its thickness (in z direction) is varied and the depth (in y direction) is considered to be constant and taken as unity. In order to parameterize these equations and help with the nondimensional investigation of the phenomenon, the following parameters are introduced xˆ = + x, w wˆ =, R = e, k =, EI k =, EI P Q B =, C =, D = EI I EI in which I is the area moment of the inertia at. The parametric axial coordinate xˆ is introduced in such a way to be used in the power-law representation of the variation of moment of inertia in the following form (8) (9)

4 n xˆ xˆ.5 I ( xˆ) = I, = n ( xˆ).5 xˆ () and its evaluation be meaningful for the end. The variation of I in () corresponds to a symmetric one with respect to arch mid point. ater on, the spring stiffnesses will be considered as equal and therefore, a symmetric response for the arch deformed shape would be expected. From this, the variation of cross sectional area along the arch can be written as ˆ) / A ( x = I [ f x () ( ˆ)] Therefore, the governing equations in the nondimensionalized form can be written as w&& ˆ + Dwˆ B( xˆ )( xˆ) = 8R + ( wˆ & 4R)( xˆ) ( wˆ & + 4R)( xˆ ) () D 6C dxˆ [ ] / 6R = w& ˆ dxˆ + in which dot represents differentiation with respect to xˆ. The boundary conditions for () correspond to pinended supports. Once the deflected curve is solved from (), it can be substituted back into () and the relation between D and B (or Q and P) can be obtained. The snap load is obtained as a limiting case (sometimes it s called the limit load) and just before the jump, the internal mechanical parameters of the arch, including Q, reach an extremum. And therefore the equation from which the snap load can be obtained is db dd = Although the analytical solution for () has been presented in [7] for constant thickness case and in [6] for the arch without springs, it is not possible to solve it analytically for the general case under investigation and even it was possible to do so, its substitution into () would result in a very complicated nonlinear equation (4) that it could not be solved unless a numerical procedure would be utilized. Since the limit load is obtained numerically one way or another, a numerical procedure for obtaining the limit load is utilized here. Using a finite difference (FD) scheme, equations () and () are transformed into simultaneous equations in terms of arch centerline positions ŵ i at discrete nodal points xˆ i. The set of nonlinear equations are solved using built-in fsolve procedure in MATAB software and the deformed shape of the arch is obtained. This is done for each value of the loading P and by gradually increasing it, the load-deflection curve for a typical point along the arch (for example, the middle point) is closely monitored until the point that approximately satisfies (4) is obtained and thus the limit point load is estimated. The number of division points and the difference formula for FD are selected such that some verifying cases can be duplicated with acceptable accuracy. In the next section, several numerical examples based on the explained procedure are presented. NUMERICA RESUTS AND DISCUSSION In this section, some results based on the numerical procedure outlined in the previous section are presented. The cases discussed here all correspond to equal stiffnesses k ˆ = = and hence a symmetric response for the arch deflection. As the first case, the effect of power-law exponent on the critical snap-loading for a typical value of the support stiffness is investigated. The arch data are taken from [] which considers the constant thickness case. They correspond to R=.54, C=9.76, and =.66. Figure shows the result. The results are obtained both by finite difference (FD) and finite element (FEM). For the FEM =, a commercial FEA software with suitable element type and meshing was utilized. As it can be seen, there is good agreement between the two methods and as expected, the critical loading increases as the power-law exponent increases. The result given in [] is also shown as a verifying data, which shows good agreement with FD. Next, the effect of variation of both n and is examined. Figure 4 shows the result as a D surface. It is seen that the greater the power-law exponent n, the larger the critical loading, at it was seen in fig.. Also, with increase of, the critical loading increases but beyond a certain stiffness, it almost levels out. This corresponds to almost clamped supports. For this graph, the arch properties are R=. and C=. A D view of this surface which shows the effect of variation of stiffness is also shown in fig. 5. () (4) 4

5 4.E-6.5E-6.E-6 Pcr/E.5E-6 FEM FD Ref. [].E-6.5E-6.E n Figure : Critical load for several power-law exponents using different methods 7.E-6 6.E-6 5.E-6 Pcr/E 4.E-6.E-6.E-6.E-6.E+.E+.E+.E+4 k^.e+6.e+8.e+.e+ 4 n 6 8 Figure 4: D plot of parameter effects on critical loading As an interesting case, the effect of swapping the moment of inertia between the ends of the arch and the middle point, i.e. one arch narrowing and the other arch thickening toward the center, on the critical snap load at different values of the power-law exponent is investigated. Figure 6 shows the result. The indices and correspond to the thickening and narrowing arches respectively. For arch number, the moment of inertia increases from I at the ends to (.5) n n I at the middle point by the power-law I = I xˆ. For arch number n n the order is reversed by the power-law I =.5 I xˆ. For arch number, there are no springs. For arch number, the spring constant is adjusted at several values. The case k ˆ = verifies against fig. 7 of [6]. It can be seen that by proper choice of stiffness, the effect of arch narrowing can be nullified and the critical loading of thickening arch be obtained. 4 CONCUSION A mixed analytical-numerical method is presented here for obtaining snap load of power-law variable thickness shallow arch with spring supports. The agreement between the results from this method and those from other procedures is quite satisfactory. Based on these results, the increase of power-law exponent and the 5

6 spring stiffness each increases the critical loading. Beyond a certain point, the increase of stiffness has little effect and the case of clamped supports is obtained. With proper choice of spring constant, the effect of narrowing the arch on decrease of critical loading can be eliminated. 6.E-6 5.E-6 Pcr/E 4.E-6.E-6.E-6 n=4 n= n=.e-6.e+ E+7 E+8 ^ k Figure 5: D plot of effect of variation of stiffness at the supports.5. (Pcr)/(Pcr)k.5.95 k^= k^=8.5e+ k^=9e+ k^=47e n Figure 6: comparison of critical loading for thickening unstiffened and narrowing stiffened arches REFRENCES - Timoshenko, S.P. 95,"Buckling of Curved Bars with Small Curvature" Journal of Applied Mechanics (ASME), (): 7 - Biezeno, C.B. 98, "Das Durchschlagen eines schwach gekriimmten Stabes" Zeitschrift Angew. Math, und Mech. (ZAMM), 8: - Marguerre, K. 98, "Die. Durchschlagskraft eines schwach gekrummten Balken" Sitz. Berlin Math. Ges., 7:9 4- Fung, Y.C.& Kaplan, A. 95, "Buckling of ow Arches of Curved Beams of Small Curvature" NACA TN Simitses, G.J.& Rapp, I.H. 977, "Snapping of ow Arches with Non-Uniform Stiffness", Eng. Mech. (ASCE), (): Atai, A.A.& Naei, M.H.& Eghtefari, R., A mixed analytical-numerical investigation of snap-through of low arches with a power-law variable thickness, J. Mech. Sci. Tech. (JMST), In Press 7- Plaut, R.H. 99, Buckling Of Shallow Arches With Supports That Stiffen When Compressed, Eng. Mech. (ASCE), 6(4): Pi, Y.. &Bradford, M.A.& Tin-oi, F. 7, Nonlinear analysis and buckling of elastically supported circular shallow arches, International Journal of Solids and Structures, Vol. 44, pp Pi, Y.. &Bradford, M.A.& Tin-oi, F. 8, Non-linear in-plane buckling of rotationally restrained shallow arches under a central concentrated load, International Journal of Non-inear Mechanics, Vol. 4, pp Pi, Y..& Bradford, M.A. 9, Non-linear in-plane postbuckling of arches with rotational end restraints under uniform radial loading, International Journal of Non-inear Mechanics, Vol. 44, pp Bazant, Z.P.& Cedolin,., Stability of Structures, Dover Publications Inc. - Eghtefari, R. 9, Analysis of snap-through in variable thickness arches, M.Sc. Thesis, Islamic Azad University, Karaj Branch, Summer (in Persian) 6