Hot-rolled steel members usually have significant residual stresses due to uneven cooling of different section parts during the manufacturing process or by fabrication such as cold- bending or welding. These stresses which are internal stresses in an externally unloaded member are in self-equilibrium at any cross-section. For a hot-rolled I-section the portions which are most exposed to air (e.g. the flange tips) cool and shrink first before the other portions such as the flange junctions and the web which have more material inside. As it is mentioned by Szalai and Papp (2005), “the most influential parameter of the distribution and amplitude of residual stresses (considering identical manufacturing processes) is the shape of the profile”. The residual stresses in a straight hot-rolled I- section are usually compression (−) at flange tips and tension at the web to flange junction (+).
When a member is in compression, those portions of the cross-section with compressive stresses yield first and can no longer support additional load. This may lead to a significant decrease of the member ultimate load. Figure 2-27 (a) schematically shows the load-displacement behaviour of a geometrically imperfect beam-column with and without residual stresses. The effect of residual stress is plotted as a function of column slenderness in Figure 2-27 (b). It can be seen that the effect of residual stresses is significant for columns with medium length and tends to be negligible when the column slenderness is small or large. Moreover, it is shown by Buonopane (2008) that residual stress may have a significant impact on the strength of steel frames by increasing lateral deflections and thereby second-order effects. Thus, since these internal longitudinal stresses can cause material nonlinearity and premature yielding, they should be incorporated into finite element analyses of steel structures.
Extensive measurements of residual stresses have been conducted over the last 40 years (Huber and Beedle 1954; Beedle and Tall 1962; Jez-Gala 1962; Lay and Ward 1969; Young 1975) and several residual stress distribution models have been developed based on these experimental data measured in different parts of the world, e.g. Europe, Japan, Australia and America. Generally, the actual residual stress profile is complex and depends on the material properties, section geometry and the manufacturing process.
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2.5.5 Warping torsion
Thin-walled steel frames with slender open-section members may experience significant torsion and therefore large warping displacement due to applied loads. The warping deformation is generally defined as the longitudinal displacement caused by torsion. For an I-section, the warping displacement consists of longitudinal displacements of the flanges in opposite directions. Past studies of the effects of warping on the behaviour of thin-walled structures may be divided into two distinct categorises: The effect of warping and end restraint (i) in an isolated member and (ii) in a joint. The boundary conditions for warping at the ends of an isolated member can be divided into three main groups: completely free, fully restrained or partially restrained. In some cases, a member is connected to a joint such that there is no restraint against warping. Such a condition applies, for example, when an I-section is connected only through its web and the flanges are free to warp (Figure 2-28 (a)). A member can also be rigidly connected to a joint which effectively prevents warping deformations. This situation may occur when the flanges of an I-section are rigidly connected to a stiffened joint (Figure 2-28 (b)). These restraints need to be taken into account in the analysis of thin-walled structures as the magnitude of warping stress can be relatively large and contribute to failure.
Figure 2-27: Effect of residual stresses (Szalai and Papp 2005) (a) on the ultimate strength of a column (b) on the buckling curve
(b) (a)
Most of the early research about warping focused on the response of a single member to warping deformations regardless of the influence of other members connected at the ends. The most important contributions in this context have been made by Timoshenko (1905; Timoshenko and Gere 1961), Wagner (1936) and Vlasov (1939) who studied the warping (or non-uniform) torsion of I-beams and derived a general theory for thin-walled members. In the numerical implementation of the theory, many researchers (Trahair 1966; Krahula 1967; Krajcinovic 1969; Rajasekaran and Murray 1973; Epstein and Murray 1976; Yoo 1980) introduced the first derivative of the twist rotation as the seventh degree of freedom to represent warping deformation. Toward this objective, conventional 12×12 stiffness matrices were replaced by the new ones with warping considered as an additional degree of freedom. In these studies, the end warping condition was assumed to be either completely free (Trahair 1966; Kitipornchai and Trahair 1971; Tebedge and Tall 1973) or fully restrained (Dinno and Merchant 1965; Razzaq and Galambos 1979; Chaudhary 1982) at both ends of the member.
The flexural-torsional behaviour of plane frames has been studied by numerous investigators, but in most cases either warping at joints was neglected by assuming six degrees of freedom for beam elements (Orbiso 1982; Liew et al. 2000; Chen et al. 2001; Kim et al. 2001; Jiang et al. 2002) or considered to be fully prevented (Renton 1962; Hartmann and Munse 1966; Trahair 1969; Chu and Rampetsreiter 1972; Vacharajittiphan and Trahair 1975). Baigent and Hancock (1982) investigated the behaviour of a joint in which the webs of two C-profiles were joined by a flat plate. In that case, the members could warp freely and independently (see Figure 2-29) and there was no need to address the problem of transmission of warping through the joint.
Figure 2-28: Joints configuration (a) warping free (b) warping fully prevented
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In subsequent studies, the focus shifted from isolated members to frames by realising that at joints warping displacements in one member may redistribute and produce warping and twisting in other connected members. This implies that when one member warps, the flanges of adjoining members must rotate, causing a distortion of the cross-section. Thus, the resistance of adjoining members to distortion provides a level of restraint on the warping torsion of the loaded member. Several experimental works were conducted to determine the warping restraint at member ends (Dinno and Gill 1964; Ojalvo and Chambers 1977; Beerman 1980). All experiments reported the difficulties of restraining warping and demonstrated that even very stiff end connections do not provide full torsional warping restraint. Consequently, the concepts of continuous warping and partially restrained warping were introduced into research.
Austin et al. (1957) studied the subject of elastic end warping restraint but no information was given to evaluate the degree of warping restraint. Trahair (1968) introduced the ratio between the elastic flange and the fixed-ended flange moments as the degree of warping restraint. Ettouney and Kirby (1981) proposed a warping restraint factor, which is the ratio between the bimoments of the partially and fully restrained cases, similar to the warping “spring” concept introduced by Yang and McGuire (1984). For both studies, static condensation was used to eliminate undesired degrees of freedom. Although the basic idea of the two methods was same, the Yang and McGuire’s procedure seems to be more representative of partial warping restraint between two members as it operates with the warping deformation which is easier to measure than the bimoment. The model featured a hypothetical warping rigidity applied as an internal spring at the joint.
The transmission of warping torsion through the joint was investigated for two special connections by Renton (1974). It was shown that the bimoment of two members at the common node are in equilibrium and equal warping occurs between adjacent elements with the same sign for first joint (a) and opposite sign for second joint (b) (see Figure 2-30).
Some attempts can be found in the literature to define different warping magnitudes for different members at joints. Bazant and El Nimeiri (1973) considered a unique warping deformation for all elements sharing a node and warping displacement continuity between adjacent elements was imposed on the model. A six DOF, zero-length connection element was presented to model warping deformation discontinuity at the member joints by Blandford (1990) and Chandramouli et al. (1994) (see Figure 2-31). Three moments ( ,
and ), three rotations ( , and ) and a linear moment-rotation relationship were considered for the joint element. Since multiple nodes were provided for different members in same location, it was possible to have different warping displacements in members connected to the joint.
Figure 2-30: Joints details considered by Renton (1974)
(a) (b)
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The warping and distortion at angle joints composed from two steel I-section members of the same cross-section were investigated by Vacharajittiphan and Trahair (1975). Four types of joints were considered in the study (see Figure 2-32) which calculated numerical values of end warping restraint.
The study concluded that warping and distortion are interdependent and depend on the joint configuration details. Subsequently, these particular types of joints composing two or more channels or I-sections were studied by Sharman (1985), Krenk and Damkilde (1991), Morrell et al. (1996), Masarira (2002), Tong et al. (2005), Camotim et al. (2010) and Basaglia et al. (2009; 2010) to determine the effect of warping transmission through the joints. To investigate the behaviour of these joints, Krenk and Damkilde (1991) considered the continuity conditions for the flanges to express the distortion in terms of warping parameters of the two beams at the joint. They developed a simple method to formulate two elastic-energy components associated with warping and distortion at beam ends. It was concluded that the unstiffened joint has two independent warping parameters while the joints with one and two stiffeners impose equal magnitudes of warping in the connected beams. This result was also proved by Sharman (1985). The joint with three stiffeners provides full warping restraint.
An important contribution to the research on the transfer of warping through joints was presented by Basaglia et al. (2009; 2010) and Camotim et al. (2010) using a numerical model which considered transmission of warping torsion and local displacement
(a)
Uns
One pair of stiffener
Two pairs of stiffeners Three pairs of stiffeners Unstiffened
(b)
Figure 2-32: (a) Angle joint under load (b) joint stiffening arrangements (Vacharajittiphan and Trahair 1975)
compatibility at frame joints of various configurations. The results of the model were compared with shell element FE analysis using ANSYS and excellent agreement was achieved.