ࡱ> oJrsIr s 5@xbjbj22 <~XXml'       ( PBPBPB8BTE( ܟF"Bi(jijiji}jWn,o$RޣV9] &tyj}j&t&t9  jijiHHH&tX ji jiH&tHHfݗ  jiF )$PB~6\0ܟ4~4(( (     4 pqHqirppp99( ( )2( ( )4th International Congress of Croatian Society of Mechanics September, 18-20, 2003 Bizovac, Croatia EMBED Word.Picture.8 ON THE STRESS DISTRIBUTION IN THIN-WALLED BEAMS SUBJECTED TO BENDING WITH INFLUENCE OF SHEARRadoslav Pavazza and Branko BlagojeviKeywords: Thin-walled beams, bending, influence of shear, stresses1. Introduction The ordinary theories of thin-walled beams assume that cross-sections remain plane during transverse bending. These theories are restricted to relatively long beams, where warping of the cross-section due to shear can be neglected. The theories can be applied in the analysis of relatively short beams, where the shear effect cannot be neglected, by taking into account deformation due to shear. Equations that consider this effect are known as Timoshenkos equations (1(. Considerable attention in the past is received to determine the shear factor, by which the deflection due to shear is defined. In the Timoshenkos equations this factor is given as the ratio between the effective shear area and total area of the cross-section (or between the total area and shear area). Thus, it is dependent on the shape of the cross-section only. The recent definitions of the shear factor, includes the Poisons effect. This effect gives for the shear factor two to ten per cent greater values (or lower values) (2, 3, 4(. In this paper, the shear factor will be assumed as a pure geometrical property. The analytical solution for stress distributions will be investigated. The cross-sections with the general shape will be considered. Some solutions of the problem can be found in literature, for simple rectangular cross-sections (5( as well as for simple thin-walled cross-sections with one axis of symmetry (6(. 2. Stresses in thin-walled beams subjected to bending with influence of shear2.1 Strains and displacements Assuming that the contours of the cross-sections of thin-walled beams with open cross-section maintain their shape it may be written (Fig1.)  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 1) where  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are the displacements of an arbitrary point S of the cross-section contour in y and z-directions,  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are rectangular coordinates,  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are coordinates of an arbitrary pole P, s is the curvilinear coordinate of the point S from the starting point M,  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are displacements of the pole P in y and z-directions, i.e. displacements of the cross-section contour as rigid contour, and  EMBED Equation.DSMT4  is angular displacement of the contour as a rigid contour with respect to the pole P;  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 2) where EMBED Equation.DSMT4 is displacement of the point S in the  EMBED Equation.DSMT4 -direction,  EMBED Equation.DSMT4 is the rectangular coordinate system with respect to the point S,  EMBED Equation.DSMT4 is the angle between the tangent  EMBED Equation.DSMT4  and the axis y and Oxyz is the rectangular coordinate system. By substituting  GOTOBUTTON ZEqnNum618386 \* MERGEFORMAT  REF ZEqnNum618386 \! \* MERGEFORMAT (1) into  GOTOBUTTON ZEqnNum288188 \* MERGEFORMAT  REF ZEqnNum288188 \! \* MERGEFORMAT (2), it may be written  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 3)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 4)  EMBED Equation.DSMT4 is the sectorial coordinate with respect to the pole P and the point M.  EMBED CorelDRAW.Graphic.9  Figure 1. Displacements of an arbitrary point of the cross-section contour The relation between displacements and the shear strain  EMBED Equation.DSMT4 can be expressed as  EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 5) Then, after substitution of  GOTOBUTTON ZEqnNum212065 \* MERGEFORMAT  REF ZEqnNum212065 \! \* MERGEFORMAT (4)  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 6)  EMBED Equation.DSMT4 ;  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 7)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ; EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 8)  EMBED Equation.DSMT4 is longitudinal displacement of the point  EMBED Equation.DSMT4 , where EMBED Equation.DSMT4 , i.e. displacements of the cross-section contour in longitudinal direction as a rigid contour,  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are the integration constants  EMBED Equation.DSMT4 ;  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 9)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are the components of the shear strain in the middle surface, with respect to the displacements EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 and EMBED Equation.DSMT4 , respectively;  EMBED Equation.DSMT4 ;  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 10)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 11)  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 are the angular displacements of the cross-section contour as a rigid contour with respect to y and z-axis, respectively,  EMBED Equation.DSMT4 is the relative angular displacement of the cross-section contour as a rigid contour with respect to the pole P. The strain in the longitudinal direction can be expressed as:  EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 12) 2.2 Stresses in terms of displacements By ignoring the normal stresses in the cross-section contour direction, the Hookes low for the state of plane stress can be written as follows  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 13) where  EMBED Equation.DSMT4 is the normal stress in the longitudinal direction and  EMBED Equation.DSMT4 is the shear stress in the beam middle surface, E is the modulus of elasticity and G is the shear modulus. Thus  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 14)  EMBED Equation.DSMT4 ;  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 15)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are the components of the shear stress with respect to displacements EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 and EMBED Equation.DSMT4 , respectively. For the equilibrium of a portion of the beam wall in the longitudinal direction, it can be written  EMBED Equation.DSMT4 ,  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 16) where  EMBED Equation.DSMT4 is the wall thickness. If  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 17) taking into account  GOTOBUTTON ZEqnNum896903 \* MERGEFORMAT  REF ZEqnNum896903 \! \* MERGEFORMAT (14)and  GOTOBUTTON ZEqnNum384923 \* MERGEFORMAT  REF ZEqnNum384923 \! \* MERGEFORMAT (15), it may be written  EMBED Equation.DSMT4 ;  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 18)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 19)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are the shear flows where EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and EMBED Equation.DSMT4 , respectively. This can also be written as follows:  EMBED Equation.DSMT4  EMBED Equation.DSMT4 ;  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 20)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 21)  EMBED Equation.DSMT4 is the cut-off portion of the cross-section area with respect to the coordinate s, i.e. EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are static moments of the cut-off portions of cross-section area with respect to the z and y-axes,  EMBED Equation.DSMT4 is the sectorial static moment of the cut-off portions of the cross-section area. Here, taking into account  GOTOBUTTON ZEqnNum498067 \* MERGEFORMAT  REF ZEqnNum498067 \! \* MERGEFORMAT (18) and  GOTOBUTTON ZEqnNum965306 \* MERGEFORMAT  REF ZEqnNum965306 \! \* MERGEFORMAT (20):  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 22) 2.3 Equlibirium equations It is assumed that the beam is loaded by transverse forces per unit length  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  that pass through the shear axis, parallel to the x-axis through the pole P,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 23) where  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are the surface loads with respect to the y and z-axes. For a portion of the beam wall the following equilibrium equations can be written (Fig. 2)  EMBED Equation.DSMT4 , EMBED Equation.DSMT4 , EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 24) Taking into account  GOTOBUTTON ZEqnNum212065 \* MERGEFORMAT  REF ZEqnNum212065 \! \* MERGEFORMAT (4), this can be written as follows  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 25) By integrating by parts, this may further be written as  EMBED Equation.DSMT4 , EMBED Equation.DSMT4  , EMBED Equation.DSMT4  EMBED Equation.DSMT4  . MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 26)  EMBED Visio.Drawing.6  Figure 2. Equilibrium of a portion of the beam wall By substituting (14) and (18) the following equations can then be written  EMBED Equation.DSMT4 ;  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 27)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 28)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 29) If y, z and  EMBED Equation.DSMT4 are the principal coordinates,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 =0, S( =0  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 30) then equations (27) take the form  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 31) 2.4 Internal forces and shear stresses Integration of the shear stresses over the cross-sections gives  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 32) where  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  are the shear forces with respect to y and z-axis, respectively, and  EMBED Equation.DSMT4  is the moment of torsion with respect to P. Substitution of (20) into  GOTOBUTTON ZEqnNum430534 \* MERGEFORMAT  REF ZEqnNum430534 \! \* MERGEFORMAT (32) gives  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 33) Here EMBED Equation.DSMT4 , EMBED Equation.DSMT4 , EMBED Equation.DSMT4 ,and integrating by parts:  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ; EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . Referring to  GOTOBUTTON ZEqnNum326530 \* MERGEFORMAT  REF ZEqnNum326530 \! \* MERGEFORMAT (31) and  GOTOBUTTON ZEqnNum337737 \* MERGEFORMAT  REF ZEqnNum337737 \! \* MERGEFORMAT (33) the following relations can be written  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 34) 2.5 Shear stresses in terms of internal forces By substituting  GOTOBUTTON ZEqnNum337737 \* MERGEFORMAT  REF ZEqnNum337737 \! \* MERGEFORMAT (33) into  GOTOBUTTON ZEqnNum965306 \* MERGEFORMAT  REF ZEqnNum965306 \! \* MERGEFORMAT (20), the shear stresses can finally be written as  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 35) 2.6 Internal forces and normal stresses By integrating the normal stresses over the cross-sections one may write  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 36) where EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are the bending moments with respect to y and z-axes and B is the bimoment. By substituting (14) into (36), it may be written  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 37)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 38) Substitution of  GOTOBUTTON ZEqnNum852658 \* MERGEFORMAT  REF ZEqnNum852658 \! \* MERGEFORMAT (35) into  GOTOBUTTON ZEqnNum539989 \* MERGEFORMAT  REF ZEqnNum539989 \! \* MERGEFORMAT (38) yields  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 39) Integrating by parts, it follows  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 40)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 41)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 42) Here  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . The second and third equations of  GOTOBUTTON ZEqnNum307253 \* MERGEFORMAT  REF ZEqnNum307253 \! \* MERGEFORMAT (37), taking into account  GOTOBUTTON ZEqnNum403146 \* MERGEFORMAT  REF ZEqnNum403146 \! \* MERGEFORMAT (40) and  GOTOBUTTON ZEqnNum712533 \* MERGEFORMAT  REF ZEqnNum712533 \! \* MERGEFORMAT (41), can be written as follows  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 43)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 44)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 45) Here Ary, and Ar,yz are the cross-section shear areas with respect to  EMBED Equation.DSMT4  displacements, Arz, Ar,zy are the cross-section shear areas with respect to  EMBED Equation.DSMT4  displacements; (y, (yz and (z, (zy are the corresponding shear factors. 2.7 Normal stresses in terms of internal forces By substituting  GOTOBUTTON ZEqnNum307253 \* MERGEFORMAT  REF ZEqnNum307253 \! \* MERGEFORMAT (37) into  GOTOBUTTON ZEqnNum896903 \* MERGEFORMAT  REF ZEqnNum896903 \! \* MERGEFORMAT (14), taking into account  GOTOBUTTON ZEqnNum108689 \* MERGEFORMAT  REF ZEqnNum108689 \! \* MERGEFORMAT (43) and  GOTOBUTTON ZEqnNum852658 \* MERGEFORMAT  REF ZEqnNum852658 \! \* MERGEFORMAT (35) it may be written  EMBED Equation.DSMT4   EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 46) 2.8 Cross-sections with one axis of symmetry In the case of the cross-sections with one axis of symmetry (for example, z-axis)  EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 47) where due to the symmetry EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , EMBED Equation.DSMT4 . 2.9 Cross-sections with two axes of symmetry In the case of the cross-sections with two axes of symmetry  EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 48) where due to the symmetry EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  3. Illustrative example Static moments of the cut-off portions of the cross-section area of the I-section are (Fig. 3)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , where due to simplicity  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are denoted as  EMBED Equation.DSMT4  and EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4   EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 .  EMBED CorelDRAW.Graphic.9  Figure 3. a) I-section section with curvilinear coordinates; b) static moment of the cut-off portions of the cross-section area EMBED Equation.DSMT4 ; c) cut-off portions of the cross-section area  EMBED Equation.DSMT4  For a T-section EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  ( EMBED Equation.DSMT4 , EMBED Equation.DSMT4 ). Thus  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , where EMBED Equation.DSMT4 . For bending in the vertical plane the normal stresses given by  GOTOBUTTON ZEqnNum286788 \* MERGEFORMAT  REF ZEqnNum286788 \! \* MERGEFORMAT (47), taking into account  GOTOBUTTON ZEqnNum403146 \* MERGEFORMAT  REF ZEqnNum403146 \! \* MERGEFORMAT (40) and  GOTOBUTTON ZEqnNum155597 \* MERGEFORMAT  REF ZEqnNum155597 \! \* MERGEFORMAT (45), may be written as  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 49)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 50) For EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  ( EMBED Equation.DSMT4 ):  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ; EMBED Equation.DSMT4 , EMBED Equation.DSMT4 . Thus EMBED Equation.DSMT4 ; EMBED Equation.DSMT4  and EMBED Equation.DSMT4 , for EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  for EMBED Equation.DSMT4 . For a simply supported beam under uniformly distributed transverse load,  EMBED Equation.DSMT4 , EMBED Equation.DSMT4 :  EMBED Equation.DSMT4 ; EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ; EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , where l is the beam length. Thus, for  EMBED Equation.DSMT4 and EMBED Equation.DSMT4 , respectively:  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 51)  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 .  MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 52) 4. Comparison with finite element method The beam is modelled by quadrilateral membrane elements; the case  EMBED Equation.DSMT4  by 1440 elements, and the case  EMBED Equation.DSMT4  by 1536 elements;  EMBED Equation.DSMT4 and EMBED Equation.DSMT4 . Table 1. Comparison of the results of the analytic solution and FEA   EMBED Equation.DSMT4  FEA  EMBED Equation.DSMT4   EMBED Equation.DSMT4  FEA  EMBED Equation.DSMT4 h/l=1/51,047 1,045 0,2 1,047 1,046 0,1h/l=1/31,131 1,145 1,2 1,130 1,135 0,4 Here  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . 5. Conclusion The stress distributions for bending with influence of shear of thin-walled beams are obtained in the analytical closed form. The expressions for the stress distributions are approximate; it is assumed that normal stresses in the transverse direction are small compared to the normal stresses in the longitudinal direction. The shear factor is determined as a pure geometrical property. The expressions are obtained for the beams with general thin-walled cross-sections. The comparison with the results of the finite element analysis of a thin-walled beam with T-section subjected to uniform transverse loads has shown a high agreement of the obtained results. ReferencesTimoshenko, S. P., On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine, Vol. 41, 1921, pp 744-746. Cowper, G. R., The shear coefficient in Timoshenkos beam theory, Journal of Applied Mechanics, Vol. 33, 1966, pp 335-340. Bhat, U. and de Oliveira, J. G., A formulation for the shear coefficient of thin-walled prismatic beams, Journal of Ship Research, Vol. 29, No. 1, 1985, pp 51-58. Senjanovi, I. and Fan, Y.,  The bending coefficients of thin-walled girders , Thin-Walled structures, Vol. 10, 1990, 31-57. Gastev,  Kratkij kurs soprotivlenija materialov , Gosuizdat, Moskva 1959. Boytzov G. 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