International Institute of Welding A world of joining experience

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1 International Institute of Welding A orld of joining eperience XV , XV-F Special cases of the calculation of residual elding distortions J. Farkas K. Jármai University of Miskolc, Hungary ABSTRACT In the first part the calculation of residual deflection of a elded beam ith variable crosssection due to shrinkage of an eccentric longitudinal eld is presented. The Okerblom s method is used and the bending moment of the diagram of curvatures as a virtual load gives the deflection. The second part treats the calculation of the angular distortion of the flange of a elded I-beam due to transverse shrinkage of the fillet elds connecting the flange to the eb. The Masubuchi s diagram predicts the measure of this angular distortion. To eliminate this distortion prestraining can be used. The Kumoze s diagram helps the design of a prestraining jig. The calculation is illustrated by numerical eamples. KEYWORDS: Residual elding distortions; elded beam; beam deflection; angular distortion; prestraining. Paper presented at the IIW Congress Stellenbosch, South Africa, March 8-10, 006.

2 1 INTRODUCTION The main requirements of modern elded structures are the safety, fitness for production and economy. In the optimum design the safety and production requirements are fulfilled by design and fabrication constraints and the economy is achieved by minimization of a cost function. Fabrication aspects are very important for elded structures, since the distortions caused by shrinkage of elds should be limited. For fabrication constraints a relatively simple calculation method is developed, hich enables to predict the residual elding stresses and distortions The books of Okerblom [1], Vinokurov [], Masubuchi [] and Kuzminov [4] give suitable calculation methods, hich have been adapted and applied by the authors [5,6,7,8]. In the present paper to special cases of distortion calculation are detailed, hich have not been treated in our previous studies. RESIDUAL DEFLECTIONS DUE TO SHRINKAGE OF ECCENTRIC LONGITUDINAL WELDS IN THE CASE OF SIMPLY SUPPORTED BEAMS WITH VARIABLE CROSS-SECTION The basic Okerblom s formulae are as follos: the residual strain at the center of gravity and the curvature for steels can be calculated as [5,6] QT ε G = (1) A C Q y T T = () I here Q T 0 = v N [ ] UI 600U V ρ = η η0 A () α is the thermal impulse due to elding, U[V] arc voltage, I arc current, v speed of elding, η 0 thermal efficiency, A cross-sectional area, I moment of inertia of the beam cross-section, 6 A cross-sectional area of the eld, ρ = kg/m is the density of steel, = kg/ah (Amper-hour) is the coefficient of penetration.. α N The second epression of Eq.() can be derived in folloing ay. The mass of penetrated metal during a eld pass lasting t[s] time can be calculated as [ ] m kg [ A][ t s] α I = N (4) 600 This mass can also be epressed as m = A Lρ (5)

3 L is the eld length. Combining Eqs. (4) and (5) the elding speed can be calculated as v [ mm s] [ ] α N I t[] s 600ρA L mm / = = (6) Substituting Eq.(6) into Eq.() one obtains the second epression. With values of U = 7V, η = for butt elds Q T (J/mm) = 60.7A (mm ), (7) for SMAW (shielded metal arc elding) fillet elds Q T = 78.8A (8) and for GMAW (gas metal arc elding) of SAW (submerged arc elding) fillet elds Q T = 59.5A. (9) The maimum deflection due to shrinkage of a single eccentric longitudinal eld in the case of a simply supported beam of constant cross-section can be calculated using the correlation beteen the distributed load p, bending moment M and deflection d M ( z) dz = p( z) (10) and d ( z) dz = M ( z) EI = C (11) i.e. the deflection can be obtained by calculating the bending moment considering the bending moment diagram as a virtual loading (Fig.1) ma ML L ML CL =. = = (1) EI 4 8EI 8 In the case of a beam of variable cross-section (Fig.), instead of M-diagram the variable C- diagram should be considered as a virtual loading and its bending moment gives the deflection. In the case of the beam shon in Figure approimately an average height and the corresponding C 1 can be considered instead of complicated variable C in the beam parts ith linearly varied beam height. Thus ma L0 L0 L1 L0 L0 = C1L1 + C0 L1 + C1L1 C0. (1) 4 here

4 Q y C0 10 Q 4 y T 0 T 0 T1 T1 = C1 = (14) I 0 I 1 Figure 1-- Calculation of the maimum deflection for a elded beam of constant crosssection Figure -- Calculation of deflections for a beam of variable cross-section It should be noted that, for the more correct calculation, the beam parts of variable crosssection can be divided in more parts of constant cross-section. Numerical eample Calculate the maimum residual deflection of a beam of asymmetric cross-section ith parts of linearly varied eb height (Fig.) due to the shrinkage of to eccentric longitudinal elds connecting the to angles to an rolled I-section. The angles strengthen the I-beam against lateral torsional buckling e.g. in the case of a crane runay. The eld eccentricities and moments of inertia for the to beam parts of different eb heights are as follos:

5 5 Figure -- Numerical eample of a elded beam of variable cross-section For the original height at the mid-span h 0 = mm y T0 = 7 mm, I 0 = mm 4, for the decreased height h 1 = 68.8 mm y T1 = 74.6 mm and I 1 = mm 4. The thermal impulse due to elding of to single bevel (1/V) longitudinal butt elds of size a = 1 mm (A = 170 mm ) is Q T = = 068 J/mm. With Eq.(14) C 0 = mm -1, C 1 = mm -1, and the maimum deflection for L 0 = 5000 and L 1 = 000 mm ith Eq. (1) is ma = 9.6 mm. For a beam of constant I 0 ith Eq. (1) ma = 0.7 mm. ANGULAR DEFORMATION DUE TO TRANSVERSE SHRINKAGE OF FILLET WELDS The angular deformation due to transverse shrinkage of fillet elds can cause a distortion of flanges of elded I-beams (Fig.4). This can result in difficulties in fabrication of these beams. e.g. in assembly of vertical stiffeners or in connection of transverse beams. Masubuchi [] has given a diagram for angular deformation in function of plate thickness for different values of the mass of consumed electrode per unit eld length (Fig.5). ρ = (15) ( g / cm) = a10 / η0 = a / a

6 6 Figure 4 -- Residual angular deformation due to shrinkage of fillet elds Figure 5 -- Angular deformation in the function of plate thickness and fillet eld size The values of lg are given in Table 1. The diagram in Figure 5 has been obtained using covered electrodes 5 mm in diameter for steels. The maimum angular changes ere obtained for plate thickness approimately 9 mm. When the plate as thinner than 9 mm, the angular deformation as reduced ith plate thickness. This is because the plate as heated more in the thickness direction, thus reducing the bending moment. On the other hand, hen the plate thickness is larger than 9 mm, the angular deformation decreases because of larger plate rigidity. Table 1 -- Values of lg for different fillet eld sizes a (mm) (g/cm) lg

7 7 The residual angular deformation can be decreased or eliminated by prestraining. Figure 6 shos a prestraining jig. The required diameter D of a prestraining round bar can be calculated comparing the stress in the flange plate initiated by prestressing ith the stress required to eliminate the distortion. L t F D L/ Figure 6 -- Prestraining a beam flange to eliminate the angular deformation The elastic deflection caused by a force F is ( L / ) F FL D = = ; EI Et t I = (16) 1 from hich Et D F = (17) L The stress from bending moment FL/ is FL / FL 6EtD σ = = = (18) t / 6 t L From Eq.(18) the required diameter of the prestraining round bar is D r σ r L = (19) 6Et Based on eperiments Kumose et al. [,9] have given a diagram for required prestraining stress σ in function of plate thickness and fillet eld size (Fig.7). r Numerical eample Flange plate thickness t = 10, idth L = 700 mm, fillet eld size a = 5 mm, elastic modulus for steels is E = MPa. From Table 1 lg = 0.46, using diagram of Figure 5 β = ( 0 ).

8 According to the diagram of Figure 7 σ = 90 MPa and Eq.(19) gives D r =.5 mm. r 8 Figure 7 -- Required prestress to eliminate the angular distortion 4 CONCLUSIONS To special cases of the prediction of residual elding distortions are presented. The Okerblom s calculation method of beam deflection due to shrinkage of longitudinal eccentric elds is etended to the case of beams of variable cross-section. The deflections of a simply supported elded beam can be calculated using the diagram of curvatures as a virtual load. The angular deformation of a elded I-beam flange due to transverse shrinkage of fillet elds can be predicted using the Masubuchi s diagram in function of plate thickness and fillet eld size. The measure of prestraining to eliminate the angular distortion can be predicted using the Kumose s eperimental data. Both cases are illustrated by numerical eamples. ACKNOWLEDGEMENTS The research ork as supported by the Hungarian Scientific Research Foundation grants OTKA T8058, T7941. REFERENCES [1] Okerblom N.O., Demyantsevich V.P., Baikova I.P.: Design of fabrication technology of elded structures (in Russian). Leningrad, Sudpromgiz, 196. [] Vinokurov V.A.: Welding stresses and distortion. (Translated from Russian). Boston Spa, Wetherby, England, The British Library, 1977.

9 9 [] Masubuchi K.: Analysis of elded structures. Pergamon Press, Oford, Ne York etc [4] Kuzminov S.A.: Welding deformations of ship structures (in Russian). Leningrad, Sudostroenie, [5] Farkas J.,Jármai K.: Analysis and optimum design of metal structures. Balkema, Rotterdam-Brookfield, [6] Farkas J., Jármai K. Analysis of some methods for reducing residual beam curvatures due to eld shrinkage, Welding in the World 1998, 41 (4) [7] Farkas,J. Thickness design of aially compressed unstiffened cylindrical shells ith circumferential elds, Welding in the World 00, 46 (11/1) 6-9. [8] Farkas,J., Jármai,K.: Economic design of metal structures. Rotterdam, Millpress 00. [9] Kumose T., Yoshida T., Abbe T., Onoue H.: Prediction of angular distortion caused by one-pass fillet elding, Welding Journal 1954, (10)