THE EFFECT OF SELF-EQUILIBRATING STRESSES DUE TO MULTI-LINE SPOT WELDED STIFFENERS ON THE NATURAL FREQUENCIES OF PLATE

In this paper an investigation has been made into the effect of residual stresses on the vibration characteristics of a thin rectangular stainless steel plate with multiline spot welded stiffeners. A new general frequency equation with and without the effect of residual stresses due to multiline spot welding along the length and width of the plate for different boundary conditions were obtained. The results give that the free ends tries to increase the natural frequencies while the clamped edges try to decrease the natural frequencies; also the central position weld line has the great influence on the natural frequenc ies. ةصلاخلا ثحبلا اذه يف مت صخ ىلع ةيقفاو هيدومع طوطخ ةدعل يطقنلا ماحللا ةجيتن ةيقبتملا تاداهجلاا ريثأت ةسارد حئافصلا ىلا ةبسنلاب تازازتهلاا صئا أدصلل ةمواقملاو ةاوقملا ةقيقرلا . تلاداعم قاقتشا مت ةديدج ل ددرتل تا يعيبطلا ة ماحللا طوطخ نم ددعل ةيقبتملا تاداهجلاا ريثأت دوجو مدعو دوجوب ب يطقنلا لأا يببتلا دودد تاا تلااحلا عبل ةاوقملا ةقيقرلا حئافصلل ييرعلاو يلوطلا اجت ةفلتخملا . جئاتنلا ترهظأ ثيد ةرحلا تاياهنلا نا ةيعيبطلا تاددرتلا نم ليلقتلا ىلع لمعت يتلاو ةديقملا تاياهنلا نم سكعلا ىلعةيعيبطلا تاددرتلا ةدايز ىلع لمعت , ةفايأ " تاداهجلاا ةميق نا طخ عقومو ةيقبتملا ا ماحلل ةيعيبطلا تاددرتلا ىلع ريبكلا ريثأتلا هل .


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On the other hand, residual stresses are induced at each stage of the life cycle in most engineering component, from original material production to final disposal. Residual stresses are created by welding, forging, casting, rolling, machining, surface treatment, heat treatment etc.
Resistance spot welding is a process used for joining faying surfaces. Major advantages of resistance spot welding are high speed and suitability for automation. Many researches have been published regarding joining strength and residual stresses of spot welds, Berglund 2002, Bae 2003, and Xin 2005. Faiz 2006, studied the effect of spot distributions and residual stresses induced from spot welding, on the vibration characteristics of plates with one array of spot welded stiffeners on the longitudinal centerline of thin rectangular plate. Theoretical method based on the theory of bending of thin plate was used to obtain the governing differential equation; expressions of the exact frequency equation were derived. Finite element modeling was adopted to predict the tendom force produced due spot welding, finding the natura l frequencies and mode shapes.

-VIBRATION ANALYSIS OF THE ORTHOTROPIC PLATES.
Due to the existence of stiffeners, the fundamental equation for small deflection theory of bending of thin plates is used to give the details of the theoretical analysis of residual stresses that result from welding and its effect on the natural frequencies and mode shapes. The governing differential equation of deflection for an orthotropic plates, subjected to a force ( x N per unite length) acting on the edges of the orthotropic plate, Timoshenko 1961, can be written as x This function is composed of the following elements: ½*(x-direction bending moments*rotation); The strain energy stored in a complete plate is obtained by integration eq. (4) over the surface.

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The maximum kinetic energy of the element, Len 1989, is: It is assumed in the following analysis that any cross-section (x and y = constant), the plate is under longitudinal compression distribution uniformly a cross the breadth of the plate, with the equilibrating tension concentrated on the line of spot welding at ) ( The strain energy due to the mid-plane forces may therefore be written, by eq. (8) as In this case xi N and yi N is a negative constant, Golf, 1976. In the case of the mechanical system shown in Fig. 3. The total strain energy and kinetic energy are: Then eq. (7) can be written as: In general the form of w is not know. However, if assumed form, normally chosen to satisfy the boundary conditions, is substituted in eq. (10). The displacement function ) , is approximated by means of the expansion, Kaldas 1981: Let the plate be placed in a coordinate system with the origin at its center and the edge (a) is parallel to X-axis and the edge (b) is parallel to Y-axis.   8  7  6  5  4  3  2  1  3  2  1   ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  , , and K are constants depending on the boundary conditions as tabulated in Table. 2.
For the stress free condition (no residual stresses 0   yi xi N N ), then eq. (14) reduced to: , this meaning the lines of welding along the length and width of the plate are at the center, and all constant in Table. 2, will be zero excepts , , , and k .
-RESULTS AND DISCUSSIONS. An easy procedure was used in this paper to predict the effect of residual stresses due to multi-line of welding along the length and width of a spot welded stiffened plate on the natural frequency. A stainless steel plate of dimensions (120*100*0.6 mm) and a stiffener of (120*20*0.6 mm) were connected by spot welding. The plate and stiffener were assumed of the material with (E=207*10 3 N/mm^2 , G=80*10 3 N/mm^2 and 3 . 0   ) . In this study three different boundary conditions with one edge is free were discussed (C.C.C.F., C.C.S.F. and S.C.S.F.). Fig. 4,5 and 6. shows the variation of the fundamental frequencies with welds self-equilibrating stresses on the longitudinal and perpendicular center lines in a thin stiffened rectangular plate with edge s C.C.C.F. . As shown in Fig. 3, the mid-plane forces N X exist at any position along the width of the stiffener will decrease the natural frequency by a percent of ( 21.5% max.) beyond that of free-stress. The frequency change may be F. F. Mustafa The Effect of Self-Equilibrating Stresses Due to K. N. Salloo mi Mult i-Line Spot Welded Stiffeners on the S.M. Saliman Natural Frequencies of Plate 4398 understood in terms of energy. For given amplitude of vibration the variation of strain energy in a cycle, largely due to the basic flexural and torsional stiffness of the plate is modified through the work done by the inplane stresses under mid-plane extension. In the present case the in-plane stresses reduce the total exchange between potential and kinetic energy, and a reduction in frequency therefore follows. On the other, hand the mid-plane forces N y exist along the length of the plate will increase the natural frequency by a percent of (33.3% max) beyond that of free stress except at the edges in which the natural frequency will decrease beyond that of free stress case. This discussion may be valid also to other cases of boundaries (i.e C.C.S.F. and S.C.S.F. ) as shown in Fig. 7,8,9,10,,11 and 12, these results gives a good agreement with the analytical results obtained by Dickinson, 1978, without including the residual stress and Golf, 1976, Al-Ammir, 2004, with including the residual stresses. Fig. 5,8 and 11, shows that the position of spot weld line have a great influence on the natural frequencies when it lays parallel to the clamped edge and it have a little influence a long the free edge parallel to the length of the plate. The changing of residual stress have a significant effect on the magnitude of natural frequency, It is clear that the natural frequency decreased with increasing the magnitude of residual stress spatially near the clamped edge and less pronounced near the free edge as shown in Fig. 13,15 and 17. On the other hand Fig. 6,9 and 12, indicate that the central position spot weld line increased the natural frequency while the natural frequency decreased when it lays near the edges along the width of the plate. Fig. 14,16 and 18, gives the effect of magnitude of residual stresses on the natural frequency along the width of the plate, it clear that the natural frequency increased with increasing the magnitude of residual stresses when the spot weld line lays at the middle of plate and decreased with increasing of residual stresses at the two edges of the plate with a little different for case 2 as shown in Fig. 16, which have an identical edges along the width of the plate.

CONCLUSIONS
An important conclusion is that the boundary condition and amount of residual stresses have important factors and significant influence on the natural frequency, the free ends tries to increase the natural frequency while the clamped edges try to decrease the natural frequency, this is so clear in Fig. 5,8 and 11, therefore in case of C.C.C.F. spot welded plate the free end increased the natural frequency by a percent of 48.39% max., while in case of C.C.S.F. spot welded plate by a percent of 29.05%max., and in case of S.C.S.F. spot welded plate by a percent of 24.28%max.. Also the results give an indication that the central position weld line has the great influence on the natural frequency and this is shown well in Fig. 6,9 and 12, so, in case of C.C.C.F. spot welded plate the increase was 48.91% max., while in case of C.C.S.F. spot welded plate by a percent of 35.94%max., and in case of S.C.S.F. spot welded plate by a percent of 26.37%max .   Table 2 Values of the constants C 1 , C 2 , ……,K