October 2014 Influence of residual stresses in the structural behavior of Abstract tubular columns and arches Nuno Rocha Cima Gomes Instituto Superior Técnico, Universidade de Lisboa, Portugal Contact: nuno.cima.gomes@tecnico.ulisboa.pt The present dissertation concerns the study of the effects of residual stresses in structures made from tubular hollow sections, specifically steel columns and an arch. It was performed a nonlinear geometrical analysis using the finite element software ABAQUS, in which the main purpose was to evaluate the effect of residual stresses caused by a longitudinal weld along the tube of the columns and the arch. This weld was made during the fabrication of the tube, in order to close the hollow section. The behavior of three columns was evaluated, with the same transversal section but different lengths (4 to 21 m). In this study, several different combinations of residual stresses and the geometrical imperfection were analysed, in order to find the worst cases and to compare them to Eurocode 3 equivalent imperfections. For the study of these effects on the arch was chosen a specific arch with a defined geometry where the variables were the position of the load in the arch, the symmetric and asymmetric loading, the position of the weld in the section and the residual stresses caused, and the geometrical imperfection. The results are compared with the Eurocode 3 equivalent geometrical imperfections. In the end, it was concluded that the residual stresses have an important role in the behavior of this kind of steel structures, and that specifically different positions of the longitudinal weld in relation to the loading and the geometrical imperfection can lead to different behaviors of the structure. Keywords: Column, Arch, Geometrical Imperfections, Residual Stresses. 1
1. Introduction and Objectives There are no structural components with a geometry that is exactly the same as the one that the designer conceived. Usually structural members that are assumed to be straight, have deviations from its conceptual axis. The existence of these deviations between the theoretical configurations of parts and their actual geometries justify that, under the current standards of design of steel structures, when assessing the conditions of security of any steel member whose sections are subject (wholly or partially) to compressive stresses, it has to be assumed that its geometry is not the theoretical one but a geometry that it is obtained by introducing the so-called initial geometric imperfections. By introducing such imperfections (or deviations between the theoretical geometry and the geometry of the part assessed), it is intended to take into account not only the possibility of the member being more sensitive to compressive stresses, as well as the existence of the residual stresses and any heterogeneities in the material properties. The geometric imperfections set forth in standards take into account these factors: deviations between the theoretical and the actual geometry of the member and the material imperfections (residual stresses and heterogeneity of the mechanical properties). The principle followed for the definition of the first factor is to seek the actual geometry that produces the most unfavorable effects on its behavior, for example, for a linear compressed member, 2 the "imperfect" axis should have the geometry of the first buckling mode. The amplitude of the imperfections was established based in the results of sampling conducted in a large number of members and taking into account the tolerances specified in the standards for metal structures construction, of which the most well-known is the EN1090. In regards to the residual stresses, they are converted into geometric imperfections that produce the same effects. Therefore, the standards set different amplitudes to the geometrical imperfection to be considered, according to the manufacturing process of the structural member, like hot-rolled or welded components. Within each of these sets, there are also different values depending on the direction of inertia of the section. Residual stresses affect differently the buckling in different directions. It turns out that the regulations do not identify the specific value of each of the two parts of the equivalent geometric imperfections specified. Sometimes it is possible to know the real geometry of the structures before they become subjected to the loading, including of course the permanent actions. This is the case of structures that have secondary structures supporting them only during the assembly. By performing a rigorous survey of the geometry of these structures it is possible to determine the first component of the geometric imperfections. In such cases, which geometric imperfection should be added to the real one, to take into account the effects of residual stress? How important the residual stresses really are?
The results presented in this work are intended to help answering these questions. It is not the same for all cross sections and for all structural configurations; so this study concerns only the circular hollow sections, for columns first, and later to the case of a particular tubular arch with a large span. 2. Imperfections Like it was stated in the introduction, there are 2 kinds of imperfections: i) Geometric Imperfections: - Imperfections in the member axis, from its theoretical position (global geometrical imperfections). - Imperfections of the cross section, such as deviations in size and shape geometry (local geometrical imperfections). ii) Material heterogeneity: - Heterogeneity in the yield stress of the steel. - Existence of residual stresses resultant from the manufacturing process. 2.1. Global geometrical imperfections The manufacturing process of the linear elements introduces small random deviations between the real member axis and the theoretical straight axis. These deviations in the theoretical axis can assume a lot of different shapes, but, for simulation it is assumed as a single curvature direction (first buckling mode), with a maximum deflection which produces equivalent effects to the real imperfection. The real imperfection in columns has been studied, and it is assumed that a common value for it is L/1000. The same principle is applied to arches, the first buckling mode is introduced in the analysis as the geometrical imperfection to simulate the deviations in the geometry. 2.2. Local imperfections There are several local imperfections that can appear in circular hollow sections. This work focused only on the main ones, which are listed below: - Out-of-roundness (Figure 1) Figure 1 - Out-of-roundness, [1] - Dimple imperfections (Figure 2) Figure 2 - Dimple caused by a weld, [1] 2.3. Residual stresses The residual stresses found in every type of metal structure can be classified by the phase of the production process in which they appear. Such stresses can arise in the metallurgy (hot-rolling, cutting, cold-forming, welding) or during assembly of the structure (mainly by welding of the components, in order to assemble them). The causes of the appearance of residual stresses in steel products are: a) time of uneven cooling along the areas of the section, in the case of hot-rolled profiles; b) high temperature (fusion of the steel) during the welding process and the temporary yielding of steel for members cold formed; 3
c) flame-cutting; d) mechanical forming processes; and e) any treatment of steel. During the installation of metal structures, the welds between the elements of the structure are also a source of residual stresses. In simple cases there are methods to estimate the shape and magnitude of these residual stresses. This work aims to study the effect of the longitudinal welds in circular hollow section. Here is presented a way of estimating the residual stresses caused by welding. In Table 1, it is presented the initial bow imperfections proposed in Eurocode 3 for the columns. Table 1 - Values of initial bow imperfections, [3] In the part 2 of Eurocode 3, the shape and amplitudes to be considered in plane buckling of arches in steel arch bridges, is presented. 3. Numerical analysis of columns Figure 3 - Residual stresses caused by welding, [2], mm (1) A w cross sectional of added weld metal (mm 2 ) Ʃt sum of the plate thickness meeting at the weld (mm) σ r yield stress (MPa) p process efficiency factor 2.4. Equivalent imperfections In the methodology used in Eurocode 3, geometrical imperfections and residual stresses are taken into account by using the buckling curves that depend on the section s geometry and manufacturing process of that structural component. These curves are part of the effects of reductions in strength due to all the imperfections mentioned above, but with emphasis on the geometric imperfections and residual stresses. 3.1. Simulation overview ABAQUS software was used to simulate the effects of residual stresses in a circular tube of Class 4, which allows the performance of geometrically and materially non-linear analyzes of finite element models. Several analyzes were performed using models with columns s r ss s r ss a r uc s r ss. It w s also carried out a stability analysis, in order to obtain the global and local modes of the columns to be introduced as the settings of the geometrical imperfections. The steel chosen for this analysis was the S355, which has a yield strength of 355 MPa, an elasticity modulus of 210 GPa and Poisson's ratio of 0.3. For simplicity, the perfect elastic-plastic behavior of the material was assumed, with a limit of 15% for the ultimate strain. 4
The main properties of the cross section are summarized in Table 2. Table 2 - Properties of the cross section Diameter D 500 mm Thickness t 8 mm Area A 12566.37 mm² Inertia I 3.93 mm 4 Radius of gyration i 176.8 mm Normal resistance N pl 4461.06 kn To simulate the columns, finite elements type "S8R5" were used. These are shell elements of reduced thickness with 8 nodes, 5 degrees of freedom per node and reduced integration. The boundary conditions of the column are simply supported in the bottom section with the vertical displacement restrained, and the same for the upper section, but not restrained in the vertical direction. The torsional rotation was also restrained in both ends. The load is concentrated in the upper section with the gravity direction. The selfweight was not considered to simplify the analysis. The amplitude of the geometrical imperfections considered is summarized in Table 3. The local imperfection was obtained from the dimple imperfections. Table 3 - Geometrical imperfections Equ v t I p. curv L/250 Equ v t I p. curv c L/150 Global imperfection L/1000 Local imperfection 3 mm The width of tensile residual stresses, caused by the weld, was estimated based on equation (1), and then introduced with the tensile stress of 355 MPa in the software that makes the equilibrium of the stresses, resulting in the diagram of the Figure 4. Figure 4 - Residual stresses introduced in the cross section For the simulation of the columns, three relative positions between residual stresses (weld) and the plane of the global imperfection, were considered, and are represented in Figure 5, 6 and 7. Figure 5 - Model A Figure 6 - Model B Figure 7 - Model C Several analyses were performed, combining the types of imperfections. These analyses are listed below: Model 0 No imperfection; Model 1 - Global imperfection only (L/1000) 5
Model 2 Residual stresses only Model 3 Global and local imperfections Model 4 Geometrical imperfections and residual stresses Model 5 Equivalent imperfections from Eur c 3 curv L 5 curv c (L/150) 3.2. Results The results are presented in form of the ultimate normal resistance supported by the columns, and is summarized in Table 4. Table 4 - Results of the analysis of columns L=4m L=14m L=21m Modelo N u (kn) N u (kn) N u (kn) Model 0 4437 4437 4437 Model 1 4331 3097 1657 Model 2 4319 2511 1518 Model 3 4276 3086 1657 Model 4 - A 4047 2666 1542 Model 4 - B 3151 2109 1334 Model 4 - C 3948 2223 1325 Model 5 - curve a 3981 2302 1341 Model 5 - curve c 3713 1958 1171 3.3. Conclusions It seems evident that the residual stresses in tubular columns influence their ultimate strength. This influence can be favorable (increases in resistance) which occurs generally in the models A, in which the longitudinal weld is executed on the same side where the curvature of the global geometrical imperfection is considered. However it can often be unfavorable (decrease in resistance), as it is shown in the other models, in which there was a combination of the effects of eccentricities resulting from global geometric imperfection with the one resultant from the process of welding (residual stresses). It is possible to conclude that in columns in which the bearing capacity is conditioned by the occurrence of overall buckling - long columns - the design curve "c" proves to be suitable for design, even having in consideration the size of the sample analyzed. The same cannot be said about short columns, in which local buckling precedes the global. Another possible conclusion is that, the less slender is the column, the lower is the significance of the relative position of the residual stresses over the plane of initial curvature. This can be important because it allows greater predictability of the effects on short columns. 4. Numerical analysis of an arch 4.1. Introduction To understand the importance of the residual stresses on the strength and behavior of an arch it was used the ABAQUS software, as for the columns analyzed in the previous chapter. The program performs the geometrically nonlinear analysis of the behavior of the arch subjected to a set of predefined loads. The geometry of the arch was defined such that it may simulate, by its dimensions, an arch suspending cover of a stadium bench. This is the typical case of an arch that is fully supported until nearly all of the permanent loads are applied. These arches are usually supported, by temporary structures, throughout its length until the whole structure of the cover is assembled. The loads were also representative of the loads existent in the cover of a stadium. 6
These loads are transmitted to the arch through hangers, consequently, the permanent loads have two parts: i) concentrated loads applied by vertical hangers; ii) the weight of the arch tube, that is a load distributed over its length. The variable action considered has an intensity of 0.3 kn/m 2. Finally it should be noted that the configuration chosen for the axis of the arc is such that it approximately corresponds to the anti-funicular of permanent loads. It is not exactly the anti-funicular of the permanent loads, because in the case of an arch made with 2.0 meters of diameter tube, it must be manufactured with straight segments which are then welded to each t r s. For this reason, small bending moments arise in the sections of the arch, even only with permanent loads. The analysis refers only to the behavior of the arc in his plan. 4.2. Geometry and model The arch analysis is only made for in-plane behavior. The axis of the arch is formed by a sequence of straight sections, with lengths from 6 m to 9.3 m. The properties of the arch are summarized in Table 5. Table 5 - Properties of the arch (geometry, material and cross section) Geometry span 221.5 m bow 31.4 m Material S355 Section (CHS) class 4 f y 355 Mpa D 2000 mm f u 490 Mpa t 30 mm E 210 GPa A 185668 mm 2 ε y 0.17% I 90.1x10 4 mm 4 ε u 2.54% N rd 65912.2 kn 0.3 M el,rd 31982.2 kn.m To simulate the arch, finite elements type "S8R5" were used, as in the columns. The arch boundary conditions consist of restraining the displacements in all directions, in both ends. The amplitude of the geometrical imperfections considered is summarized in Table 6. The local imperfections were not considered. Table 6 - Global geometrical imperfections Equivalent L/600 362 mm I p r ct curv Equivalent L/400 554 mm I p r ct curv c Global imperfection L/1000 221 mm Figure 8 - Pattern of residual stresses (MPa) The pattern of residual stresses originated during the longitudinal welding of the tube is represented in Figure 8. Five different positions of the weld in the cross section were considered, which are shown in Table 7. 7
Table 7 - Positions of the weld in the section Weld position Diagram Position 1 Position 2 Position 3 Figure 9-1 st buckling mode (global geometrical imperfection) The study consisted in combining the several positions of residual stresses in the section, global imperfection and position of the variable load, and find out the resistance in each combination. 4.3. Results It is shown in Figure 10 the results for the load parameter in each case. The load parameter is the multiplicative factor of the variable load, which could be resisted by the structure. Position 4 Position 5 Another variable in the study was the positioning of the variable load in the arch, for which three positions were considered.. In the first case, the variable load is distributed along the whole length of the arch, the second in the left half and the third in the right half of the arch. The global geometrical imperfection was always introduced as it is shown in Figure 9, which corresponds to the geometry of the first buckling mode, so the variable load introduced in the left part leads to increases in the resistance, so it is not shown here. Figure 10 - Results for the load parameter 4.4. Conclusions The first conclusion is that the stresses which produce the collapse of the tube are the compressions caused by bending. Whether bending is generated by asymmetric loads applied on the arch or the second order moments caused by compressive stresses in a geometry already deformed and asymmetric yield of the sections, with consequent variation of the neutral axis position. 8
Another important conclusion is that the position of the longitudinal weld is relevant to the strength of the arch. When the welding causes compressive residual stresses on the upper and lower parts of section, there are significant decreases in the overall resistance of the arc, since the bending moments yield these areas where there are compressive stresses, reducing considerably the rigidity and bending resistance, and increasing the eccentricities of second order. Despite this, it appears that the arch almost always behaves within the elastic range, except locally and for load levels near the ultimate load. Therefore, the collapse is always due to the loss of stability or the arch. Finally, it appears that the residual stresses represent a significant part of the initial imperfections that Eurocode 3 recommends for arches. The most adverse case found for the resistance is equivalent to a geometric imperfection of L/290 (763.8 mm). This means that the actual geometrical imperfections (L/1000) part is only 29%, while the effect of residual stresses is 71%. 5. Analysis of an arch with a stiffness beam 5.1. Introduction In the previous section we analyzed the structural behavior of a single arch. It is known that the load carrying capacity of the arch is very sensitive to factors causing bending moments, such as the asymmetric loads and/or the existence of asymmetric geometrical imperfections. 9 In general, the purpose of an arch is to support other secondary structures that work together with it. The stiffness of these structures, however minimal, can improve the behavior of the arches, mainly because it decreases the asymmetric displacements. Typical cases of the importance of flexural stiffness elements are the arch bridges. The higher the stiffness of the bridge deck, the smaller bending moments in the arch. The same situation happens in the coverage of large areas. The secondary structure can decisively contribute to the reduction of displacements in the arch where it is suspended. The objective of this part of the study is to evaluate the benefits that an element with relatively high bending stiffness can bring to the arch behavior and especially to its ultimate capacity. As a result of some research in projects carried out in arched covers of stadiums, it was decided to choose a stiffness beam whose bending stiffness is equal to half the bending stiffness of the arch. 5.2. Results The results for the load parameter are shown in Table 8. Table 8 - Results for the load parameter In the Table 9, a comparison is made between the parts of each imperfection for the arch with and without the beam, with
the Eurocode recommended imperfections for the curv s c. Table 9 - Importance of the two main imperfections in the equivalent imperfection 5.3. Conclusions The introduction of the stiffness beam benefits the behavior of the arc in terms of gaining strength and ductility. This benefit results from the redistribution of applied loads provided by the beam, which results in a considerable reduction of the values of the maximum bending moment. With the beam, the compressive stress increases and this causes larger yielded areas of material. The behavior changes from the elastic regime as it happened in general when the arch was simulated separately, to a more plastic behavior. This increase of the compressive stresses causes yield and local instability in the tubes, the undulation of the walls of the tube and the out-roundness of the section. These effects turn out to be the cause of the failure for some of the cases studied. With regard to the load capacity of the arc, the residual stress pattern that simulates a longitudinal welding, along the lateral part of the section (positions 2 and 5), is more favorable, opposing to what was found when the arch was modeled isolated. The most unfavorable situation, with respect to the bearing capacity of the arch, is the weld made on top of the section of the tube (position 1), also the opposite of 10 what happened when the analysis was made without the beam. In general, the equivalent geometric imperfection established for the curve "c" in Part 2 of Eurocode 3 is a good simulation of the effects of real imperfections (geometrical and residual stresses) corresponding to an initial geometric imperfection of L/1000, plus the effects of residual stresses considered. Therefore, if the deformed geometry of the bow is known precisely, in theory, to simulate the effect of residual stress should be added to this an imperfection of L/667, which represents the difference between L/400 (equivalent imperfection) and L/1000 (real geometrical imperfection). An exception to the above stated is made, considering the results obtained in the model with residual stresses in the position 1, which has a continuous longitudinal weld at the top of the arch, in which the load parameter is lower than that obtained from the analysis results of the model with the equivalent geometrical imperfection corresponding to curve "c". For this case it is concluded that the appropriate equivalent imperfection is equal to L/365 (less than L/400 value of the curve "c"). Overall, it is confirmed that the part of global equivalent geometrical imperfections proposed by the Eurocode 3, which corresponds to the residual stresses, is higher than the part of real geometric imperfection.