Nonlinear Buckling of Prestressed Steel Arches
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1 Nonlinear Buckling of Prestressed Steel Arches R. Giles-Carlsson and M. A. Wadee Department of Civil and Environmental Engineering, Imperial College London, Exhibition Road, London SW7 2AZ, UK June 22 Abstract This paper looks into the application of prestress technology to steel arches used in long span steel structures. The use of prestressing has been implemented in previous engineering applications to reduce the deflections and the amount of material that is required thereby reducing the energy impact. Therefore the capacity of the arch and the post-buckling behaviour is investigated for different initial geometries and different loading conditions. The models used evolve from simple limit point systems and other single degree of freedom systems to more sophisticated multiple degree of freedom systems. The results have shown that the initial geometry of the system governs the buckling mode, with a slender arch resembling the behaviour of a single degree of freedom tied arch. This is compared to a deeper arch with a greater load carrying capacity which behaves similar to an individually prestressed element. The analysis of uneven loading conditions has shown that uneven loading causes an asymmetric response of the structure and uplift loads cause compression of the bottom chord of a truss that could cause failure if prestressed sufficiently. Keywords: Prestress, Steel Arches, Nonlinear, Buckling.. INTRODUCTION Modern construction places new constraints on designers; shape and size of structures playing a key role. Greater spans and lower costs are always being demanded of modern designs, examples include aircraft hangars or factories where the roof is required to have a large span. The answer is simple; shells and arches allow longer spans but at the cost of high vertical rises which in turn lead to high material and manufacturing costs. Prestess technology can be used to create shallow arches that can span long distances with reduced deflections. Aircraft hangars are a typical example of prestressed tubular steel trusses in the shape of a shallow arch; highlighting the importance of space constraints, such as the requirement for no internal columns. (Ellen et al., 2). A patent by Ellen (987) details the application of this prestress technology to a long span tubular arch; this specific application has been implemented successfully in Australia. However prestressing technology is by no means a new technology with many earlier applications in other areas of the world and other branches of engineering. For a steel arch truss structure, the bottom chord of the truss will be in tension and the top chord will be in compression; further, the elements are tubular and therefore can readily house a steel cable. Therefore the prestressing can be applied to the steel arch leading to an increased capacity and reduced deflection of the bottom chord. The construction of the arch can also be greatly enhanced by the use of prestressing; with the structure being raised into its elevated position by tensioning of the bottom chord, thus removing the need for cranes in certain situations, (Ellen, 987). The construction can be carried out in three key stages: ) construction of the frame on the ground, 2) prestressing of the arch into position and then 3) prestressing the columns, which are on rollers, as to raise the entire structure into position and then fix. Figure shows these three stages of the construction, (Ellen, 987). Figure : An application of prestressing technology in a steel truss arch (Ellen, 987) A great deal of previous literature is already present in the field of investigation of steel arches; in particular, the modes of failure have been researched thoroughly by Pi and Bradford (24) in addition to other sources. The inplane capacity of shallow arches and out of plane capacities have been studied by Bradford and Pi (26); more sources exist but these are purely given as examples.
2 2. METHODOLOGY Failure of a structural system can be described as either a snap-through failure, or it can buckle; the difference being described in Thompson (963). According to Thompson, if the loss of stability of a system occurs at the local maximum of the equilibrium path, then the failure can be categorized as a snap-through failure mechanism, similar to that of a Von Mises Truss; whereas if the loss of stability occurs at a point of bifurcation (when two equilibrium paths intersect), then we class it as buckling failure. For shallow arches, the behavior is commonly described as unstable and can exhibit either symmetrical snap-through failure or it can fail by antisymmetric bifurcation depending on the arch geometry and loading conditions (Thompson and Hunt, 983). To analyse the models analyticaly, the total potential energy method is used. The total potential energy of the system can defined as the sum of the gain in potential energy and subtraction of the work done by the load, V = U t +U c P,. () In this case, the gain in potential energy has contributions from both the tube and the cable. In this equation, the parameter U t is the gain in potential energy by the tube, U c is the gain in potential energy from the cable and P is work done by the load in the direction of loading. The first axiom from Thompson (963)refers to the first derivative of this equation equating to zero, thereby allowing the system to be in a state of equilibrium; the second axiom describes that a minimum point of this equation is required for the system to be stable, (Wadee, 2). Once the first derivatives of the total potential energy are formulated, numerical software can be implemented to find the equilibrium path(s) of the system and therefore describe the mode of failure and load carrying capacity, (Thompson, 963). 3. MODEL As mentioned before, the total potential energy of a system can be summarised by the summation of the strain energies minus the work done by the force applied in the direction of loading. This prestressed element system has two parts, the cable and tube, both of which have their own expression for strain energy and can be linearly superimposed. During stage one, the tube and cable are both undergoing elastic deformation and therefore both expressions for strain energy are increasing linearly with tube and cable displacement (x t and x c ) as: U t = 2 k tx 2 t, U c = 2 k cx 2 c. (2) Once the element has been displaced by a sufficient amount, the tube will yield and undergo plastic deformation (atx t = x ty ) and the cable continues to behave elastically described as stage 2 of loading. As a result, the strain energy of the tube becomes U t = 2 k tx 2 ty +k tx ty (x t x ty )+ 2 k ts(x t x ty ) 2, (3) which now incorporates strain hardening stiffness coefficient, k ts. The total potential energy is formulated in the same way as before, summing the strain energy and subtracting the work done. With further extension, the cable will reach its yield stress (at x c = x cy and fail initiating stage 3 of loading whereby both tube and cable are deforming plastically. In a similar manner, the expression for the strain energy of the cable changes upon yielding, U c = 2 k cx 2 cy +k cx cy (x c x cy )+ 2 k cs(x c x cy ) 2, (4) with another strain hardening coefficientk cs for the cable. To try and model the concept of prestressing, consider a single steel tube with a single steel cable housed within the tube, both will be subjected to tension during loading, but before loading or prestressing they will both be in equilibrium. The member that yields first will most likely be the tube since its yield strength is considerably less than the cable. Therefore it could be advantageous to pre-compress the tube before the tensile load is applied thereby increasing its load carrying capacity and reducing its deformation. To compress the tube, the cable is highly tensioned and then grouted within the tube where upon it is unable to return to its initial equilibrium state and therefore cause the tube to compress. Upon loading the tube will initially be in compression and the cable will be in tension; as tensile loading increases, both will go into tension and eventually yield. Figure 2 attempts to show this process, whereby the tube is displaced x t in compression and the cable is displacedx c in tension as a result of prestressing; x is then the displacement after prestress and during loading, (Ellen et al., 2). So if tension is considered to be the positive direction of force, figures 3 and 4 can represent the stress diagrams for the tube and cable mentioned previously; the stress distribution for the tube shows an initial state of compression at a position x t and the stress distribution of the cable in an initial state of tension. The cable would commonly have a very high yield stress and would therefore be able to withstand moderate prestress tensile forces, (Ellen et al., 2). The truss design can then be extended to a multi degree of freedom system as shown by figure 5, with two vertical or diagonal struts and an additional link along the top chord. The length of the top chord can be varied by the use of a parameter,γ, as well as the other two parameters, β and α. Variation of this three parameters change the geometry of the arch model and allow in-depth analysis of the behaviour for different values for the parameters. There are two prestressed elements incorporated and two applied loads at the apexes. In addition, this design allows for asymmetric failure modes. As a result, the analysis becomes more sophisticated and more combinations are possible with regard to 2
3 with respect to each generalised coordinate. Unlike simpler models, there are two expressions for the equilibrium path for each stage and therefore need the implementation of solver software AUTO c to find the system response. Figure 2: Coordinate system to help illustrate prestress, adapted from Ellen et al. (2) Figure 5: Multiple degree of freedom model to analyse shallow arch response to loading. 4. RESULTS 4. Geometric Variations Figure 3: Stress distribution for the cable after prestressing. Figure 4: Stress distribution for the tube after prestressing. the prestressed stages. There are a total of nine possible combinations, the limits for the stages are governed by the extension of the tubes. Therefore expressions for the total potential energy can be formulated in a similar method as previously. The equilibrium paths are then obtained by differentiation of the total potential energy The results of this paper were based upon tube and cable cross-sectional areas of 733mm 2 and 2mm 2 respectively, with yield strengths of 355N/mm 2 and 2N/mm 2 respectively. Further the strain hardening coefficient for both was 2N/mm 2.The analysis was carried out for ranges of initial elevation fromβ =.75 to β =.4, the latter of which corresponding to quite shallow arches; values of γ ranging from unity, thus a similar length to the other rigid links, down to /2 to represent very short bar lengths; and many different values of α. Before dealing with the results directly, it is worth mentioning that the individual values of parameters α and γ did not necessarily dictate the shape of the equilibrium path but instead it seemed that the ratio of these two parameters was of more importance. Therefore the ratio of γ/α, shall be referred to as the aspect ratio of the this model and denoted η from now on. In order to make a good judgment of the effects of prestressing, three different levels of prestress force were considered: zero prestress force, half the optimal prestress force and the optimal prestress force. The variation of the geometric parameters showed a transition from a limit point system to equilibrium paths resembling an individual prestressed element under axial tensile loading. Figure 6 shows a system with a high aspect ratio, η 2, whereby the shape of the curve has a local maximum and looks like a snap-through failure mode. Then as the aspect is decreased through figures 7 and 8, the equilibrium path begins to change its shape showing a sharper initial stage and then a more gradual decline. The load carrying capacity of the system also increases with decreasing aspect ratio. Further reduction of the aspect ratio causes continued changes to the shape of the equilibrium path and a continued increase of the load carrying capacity. Figures 9 3
4 .2 25 P r = kn P r = 9 kn 2 P r = kn P r = 9 kn Figure 6: Equilibrium path type, η 2 Figure 8: Equilibrium path type P r = kn P r = 9 kn 5 4 P r = kn P r = 9 kn Figure 7: Equilibrium path type 2 Figure 9: Equilibrium path type 4 and show the equilibrium paths as a result of low aspect ratios, whereby the arch becomes relatively deep, the paths now show similarities to the equilibrium paths of the SDOF prestressed tied arch and individually prestressed element. With very clear distinction between the three stages of loading. In particular, figure has a monotonically increasing path that suggests that the load carrying capacity of the system continues to increase beyond the data shown. The variation of the shapes of the equilibrium paths can be found in further tests carried out varying the other geometric parameters. This transition of the shape of the equilibrium is believed to be caused by the variation of extension of the prestressed elements during loading. To simplify the matter, consider only a symmetric response, assuming that the extension of the prestressed elements and displacements of the apexes are the same. Figure shows the variation of the lengths of the prestressed elements for different aspect ratios. For a system with a low aspect ratio, η, the length of the prestressed element increases monotonically without seemingly reaching a maximum. Then as the aspect ratios tends to a relatively large value, η 2, the length of the prestressed element initially increases, then reaches a maximum and then afterwards decreases again. Therefore, if the aspect ratio is sufficiently large, it can be deduced that the prestressed element does not extend much and therefore does not reach the yield displacements of the tube or cable at any level of prestress, therefore no higher stage of loading is reached and the equilibrium path does not vary its shape. In comparison, the prestressed element present in a system with a low aspect ratio will extend a sufficient amount to cause yielding of the tube and cable, further the magnitude of extension is much greater than the yield displacements and therefore prestress ceases to have a significant impact and the equilibrium path can monotonically increase without sufficient changes to the shape. The variation of all three geometric parameters effect the initial geometry and therefore extension of the prestressed element. 4.2 Variation of Loading Parameters Wind loading is likely to be an issue when considering shallow, long span arches, especially when used to in aircraft hangars; the wind loading causes uplift which can lead to either uneven loading or full uplift loading on the arch structure. To simulate uneven loading, load param- 4
5 P r = kn P r = 9 kn 6 4 Pr = kn Pr = 9 kn Pr = 79.9 kn 5 Load capacity, P, [kn] LT Figure : Equilibrium path type 5 LT LT Figure : Length of prestressed elements, L T due to apex displacement by gravity loading for different aspect ratios of η, η =.2 and η 2 respectively. eters are applied to the loads,µandλ, with both equal to representing a purely symmetric uplift load case. It is important to consider the reversed loading case whereby the bottom chord of the truss goes into compression or tension as a result of asymmetric loading. First considered was the case of full uplift force, causing pure compression in both prestressed elements, the results in figure 2 show the load carrying capacity going through the three distinct stages of loading, the second stage being strongly affected by the prestress force magnitude. Application of a prestress force in the compressive loading case causes the load capacity at the beginning of stage 2 to decrease; therefore the prestress load could cause problems during high winds. The response due to full uplift loading was completely symmetric. An asymmetric load case can be applied to the model by variation of one of the load parameters so they have non-equivalent magnitudes. For example, figure 3 shows multiple equilibrium paths, the value of λ kept constant and the value of µ decreased from unity towards uplift. To simplify the analysis, the prestress force is set to zero. This asymmetry of load parameters cause the system follow a different shape of equilibrium path; the paths now converge onto the asymmetric paths present in figure 6 for the case of symmetric gravity loading. As Figure 2: The model response to a full uplift loading case, whereby the upward force is resembled by the positive load applied µ is decreased, the load carrying capacity of the system is increased as expected as the load applied is decreased, further the point at which the maximum load occurs moves towards zero. Finally, it seems the equilibrium paths intersect the asymmetric gravity loading case at a point of zero load carrying capacity (a point other than zero displacement). P, [kn] µ decreasing λ = and µ = λ = and µ =.5 λ = and µ = λ = and µ =.5 λ = and µ = Figure 3: Load carrying capacity against the displacement of the left apex for varying uneven loading conditions with λ =. As the other load parameter, λ is reduced whilst µ is has a maintained value of, the curves continue to move upwards and away from the case of symmetric gravity loading and eventually separate from the set of curves. At this point it would seem that the uplift now dominates over the gravity force and the system continues to move upwards and therefore has an equilibrium path similar to the full uplift load case. To investigate the effects further, the displacements of the apexes are compared and plotted in figure 4. The blue diagonal line represents the symmetric full gravity loading case where the displacements of the two apexes are the same and progress downwards. In ad- 5
6 dition, the curved blue line in the shape of a semi-circle is present for the full gravity loading showing the asymmetric branches that were seen in the previous figures. The other loading cases are superimposed upon this full gravity case and the results for the same first set as before are shown in figure 4. The two different groups that refer to the two different types of loading procedures seem to be symmetric. As the parameters are reduced towards uplift, the curves tend away from the symmetric case and become more asymmetric. However, it should be observed that the cases for both groups of curves all intersect at the same point, which is the point at which the curves return to zero load carrying capacity. A similar kind of behaviour has been reported previously by Thompson and Hunt (983) by analysis of a pinned arch, the equilibrium path showed a similar behaviour of a purely symmetric load path or asymmetric load paths that form a semicircle around the symmetric case. β2 Q λ decreasing µ decreasing Figure 4: Close up of the displacement of the two apexes for symmetric and asymmetric loading conditions. Figure 5 shows a transition from figure 4 whereby the curves cease to move upwards on this figure, but instead begin to move downwards in the figure, representing negative displacements and then proceed upwards again. This motion signifies an initial upward motion of the apex followed by the dominating downward movement as seen previously. However, in some cases, the lines again separate from this pattern of curving round and split off into negative values of Q and Q 2, seemingly to continuously decrease therefore representing the fully upward motion caused by the pure uplift case. 5. CONCLUSIONS AND FURTHER WORKS The use of the total potential energy method has been applied numerically to a two degree of freedom system with prestressed elements present in the bottom chord. Different initial geometries have been found to dictate the failure mode, slender arches have an equilibrium path P, [kn] µ decreasing λ decreasing Figure 5: Displacement of the two apexes for symmetric and asymmetric loading conditions of more substantial uplift forces. similar to a snap-through system and deeper arches behave similar to an individual prestressed element. In addition, uneven loading and uplift forces have been considered, pure uplift causing compression in the prestressed elements therefore the load carrying capacity for compression is reduced when a prestress force is applied. Further, the presence of asymmetric loads cause asymmetric failure modes that may cause the structure to displace downwards or upwards, depending upon the magnitude of uplift force. Further works should investigate the effect of prestress on the uneven loading case. References Bradford, M. A. and Pi, Y.-l. (26). Elastic Flexural- Torsional Buckling of Circular Arches under Uniform Compression and Effects of Load Height. Materials and Structures, (September). Ellen, M. E. (987). Post-tensioned steel structure. Ellen, M. E., Gosaye, J., Gardner, L., and Wadee, M. A. (2). Design and construction of long-span posttensioned tubular steel structures. Pi, Y.-L. and Bradford, M. A. (24). In-Plane Strength and Design of Fixed Steel I-section Arches. Engineering Structures, 26(3):29 3. Thompson, J. (963). Basic Principles in the General Stability Theory of Elastic Stability. :3 2. Thompson, J. and Hunt, G. (983). On the buckling and imperfection-sensitivity of arches with and without prestress. International Journal of Solids and Structures, 9(5): Wadee, M. A. (2). MSc Cluster in Advanced Structural Engineering,. Struct 24: Structural Stability, (): 9. 6
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