ࡱ> *,&'()O[ B 5@ 8bjbj22 ,<XX...BCCCCL2H<BʏzI#mmmmnnnюӎӎӎӎӎӎ$RҒ.snnssmms8m.mюsюRm.mnI 0uOCw a<0ʏ׀<,BB.ɋnH[oEpqnnnBB?CBBCON THE CROSS-SECTION DISTORTION OF THIN-WALLED BEAMS WITH MULTI-CELL CROSS-SECTIONS SUBJECTED TO BENDING R. PAVAZZA and B. BLAGOJEVI Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Ruera Boskovia bb, 21000 Split, Craotia. Abstract This paper deals with distortion of the cross-section contour of thin-walled beams with simple multi-cell closed rectangular cross-sections. The cross-section distortion is considered in the limit case. It is assumed that beam plates are hinged together along their longitudinal edges. Double symmetric three and two-cell closed cross-sections are considered. The stresses and displacements are obtained in the closed analytical form. The additional stresses and displacements due to distortion with respect to the stresses and displacements of the ordinary theory of bending are obtained. The boundary conditions are given in the general form. Some illustrative examples are given. Keywords: Distortion of cross-sections; Bending of thin-walled beams; Three and two cell closed sections; Hinged walls; Analytic method. Introduction The reliability of the assumption, in the theory of bending of thin-walled beams, that the shape of the cross-section is maintained depends on the stiffness of the beam transverse framing, or the cross-section itself (Vlasov, 1961); in some cases, also on the load distribution in the transverse direction (Kollbrunner and Basler, 1969). The thin-walled beams are assembled of a number of thin plates that are restrained along their longitudinal edges. The plates can be stiffened by frames, usually in the transverse direction; sometimes, also by transverse bulkheads (diaphragms). Transverse framing, or the cross-section itself, cannot prevent the cross-section distortion entirely. In the limit, it may be assumed that plates are hinged along their longitudinal edges. These two different types of structural behavior (with the rigid cross-sections and hinged cross-sections) may be considered as two limiting cases of stresses and deformations of the actual structure (Kollbrunner and Basler, 1969). The theory that deals with such idealized beams, with hinged plates, is the folded plate theory (Schwyzer, 1920, Kollbrunner and Basler, 1969). By the theory, the longitudinal deformations and therefore the longitudinal normal stresses are equal at hinged connections. Hence, the normal stresses are linearly distributed over the cross-sections of each plate. The end cross-sections are assumed rigid. Each plate is at most preceded and followed by one plate only; the method is no longer applicable if one hinge belongs to three or more plates. The problem of the cross-section distortion can be solved in an exact way, in general, by using three-dimensional models (Hughes, 1983; Bull, 1988). However, in the case of ordinary beams, with small dimensions of the cross-section contour with respect to the length of the beam, simpler analytical as well as numerical methods may also be applied (Boytzov, 1972; Boswell and Zhang, 1984; Boswell and Li, 1995; Hsu et al., 1995; Kim and Kim, 1999, 2000, 2001; Pavazza and Matokovi, 2000; Kim et al., 2002; Pavazza, 2002). In this paper, an analytical approach to the problem of the cross-section distortion of prismatic beams with closed rectangular thin-walled cross-sections subjected to bending with influence of shear will be considered. The double symmetric cross-sections with three and two closed cells will be analysed. The results will be given in the analytical closed form, suitable for parametric studies. 2. Thin-walled beams subjected to bending with influence of shear The forces-displacements relations for a prismatic thin-walled beam subjected to bending, by ordinary beam theories that include the shear influence (Filin, 1975), are governed by the following differential equations  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (1) where  EMBED Equation.DSMT4  is the displacement in the z-direction due to bending,  EMBED Equation.DSMT4  is the angular displacement with respect to the y-axis due to bending, EMBED Equation.DSMT4  is the cross-section moment of inertia with respect to the y-axis,  EMBED Equation.DSMT4  is the bending moment with respect to the y-axis,  EMBED Equation.DSMT4  is the shearing force with respect to the z-axis,  EMBED Equation.DSMT4  is the line load per unit length in the z-direction, Oxyz is the rectangular co-ordinate system, where the y and z-axis coincide with the principal axes, where the z-axis is the axis of symmetry, and E is the modulus of elasticity (Fig.1);  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (2) where  EMBED Equation.DSMT4  is the displacement in the z-direction due to shear,  EMBED Equation.DSMT4  is the angular displacement with respect to the y-axis due to shear,  EMBED Equation.DSMT4 is the cross-section shear area;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (3) where  EMBED Equation.DSMT4 is the total displacement in the z-direction and  EMBED Equation.DSMT4  is the total angular displacement with respect to the y-axis. The stresses are given by  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (4) where  EMBED Equation.DSMT4 is the normal stress in the x-direction,  EMBED Equation.DSMT4 is the shear stress in the s-direction,  EMBED Equation.DSMT4  is the shear flow in the s-direction,  EMBED Equation.DSMT4 is the cross-section rectangular co-ordinate,  EMBED Equation.DSMT4 is the statical moment of the cut-off portion of the cross-section with respect to the y-axis,  EMBED Equation.DSMT4  is the wall thickness and s is the cross-section curvilinear co-ordinate, with the origins  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  (Fig.1); Fig.1 It is assumed that the shear influence on bending is small; it will be taken into account in the calculations of displacements only, by using (2) and (3). The cross-section contour is assumed rigid (neglecting the contour contraction). Thus, from Hookes law,  EMBED Equation.DSMT4 , (5) where EMBED Equation.DSMT4  is the strain in x-direction. The shear flow is defined by the normal stress  EMBED Equation.DSMT4 , by the equilibrium equation in the x-direction  EMBED Equation.DSMT4 . (6) The normal stress in the s-direction  EMBED Equation.DSMT4 , which is ignored in Eqn (5) as a small quantity, can be obtained by the shear flow, by using the equilibrium equation in the s-direction  EMBED Equation.DSMT4 . (7) where  EMBED Equation.DSMT4 . 3. Thin-walled beams subjected to bending with the cross-section distortion 3.1. Distortion of the cross-sections of thin-walled beams subjected to bending If the frames or diaphragms are omitted, or the cross-section is not stiff by itself, the cross-sections of thin-walled beams are no longer forced to maintain their shape. In the limit, it may be assumed that the beam walls are hinged along their longitudinal edges. Thus, the walls will be loaded by forces per unit length along their longitudinal edges, in their planes (Pavazza, 2002). Fig.2 In the case of symmetrical closed thin-walled cross-sections with three cells (Fig.2a), the inner vertical walls and the central horizontal walls may be treated due to symmetry as a unique beam with the rigid cross section (beam component 1). Two cell cross-sections may be treated as a special case of three cell cross-sections (Fig.2b). 3.2. Beam components The beam may be decomposed into four beam components (Fig.3). In the case of double symmetric beam cross-sections, it may be assumed that the beam components 1 and 3 are subjected to bending with influence of shear and the beam components 2 and  EMBED Equation.DSMT4  to tension and shearing due to distortion of the cross-section shape. Fig.3 For the beam components 1 and 3, the following differential equations may be written  EMBED Equation.DSMT4 (Filin, 1975; Pavazza, 1991)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (8) where  EMBED Equation.DSMT4  is the displacement in the  EMBED Equation.DSMT4 -direction,  EMBED Equation.DSMT4  is the angular displacement with respect to the  EMBED Equation.DSMT4 -axis,  EMBED Equation.DSMT4  is the cross-section moment of inertia with respect to the  EMBED Equation.DSMT4 -axis,  EMBED Equation.DSMT4  is the bending moment with respect to the  EMBED Equation.DSMT4 -axis,  EMBED Equation.DSMT4  is the shearing force with respect to the  EMBED Equation.DSMT4 -axis,  EMBED Equation.DSMT4  is the force per unit length in  EMBED Equation.DSMT4 -direction,  EMBED Equation.DSMT4  is the moment per unit length,  EMBED Equation.DSMT4  is the rectangular co-ordinate system, where the  EMBED Equation.DSMT4 , EMBED Equation.DSMT4 -axes coincide with principal axes of the corresponding cross-sections;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (9) where  EMBED Equation.DSMT4  is the displacement in the  EMBED Equation.DSMT4 -direction due to shear,  EMBED Equation.DSMT4  is the angular displacement with respect to the  EMBED Equation.DSMT4 -axis due to shear,  EMBED Equation.DSMT4 is the cross-section shear area;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (10) where  EMBED Equation.DSMT4 is the total displacement in the  EMBED Equation.DSMT4 -direction and  EMBED Equation.DSMT4  is the total angular displacements with respect to the  EMBED Equation.DSMT4 -axis . For the beam component 1, the moment per unit length may be expressed as  EMBED Equation.DSMT4 , (11) where  EMBED Equation.DSMT4  is the force per unit length along the longitudinal edge A and h is the height of the cross-section. The stresses may be expressed as follows  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (12) In this equation  EMBED Equation.DSMT4  is assumed to be distributed along the vertical wall of the beam components 1 only, as a constant (which is expressed symbolically by a vertical line); it represents the influence of the beam components 2 on the beam component 1. For the beam component 3, similarly, one has  EMBED Equation.DSMT4 , (13) where  EMBED Equation.DSMT4  is the force per unit length along the longitudinal edge B;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (14) Here is assumed that the origin of s is at B, i.e B (Fig. 1). For the beam component 2, it may be written (Pavazza, 2000, Pavazza, 2001)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (15) where EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4 , (16) where  EMBED Equation.DSMT4  is the cross-section area of the beam component 2,  EMBED Equation.DSMT4  is the displacement of the cross-section in the  EMBED Equation.DSMT4 -direction as a plane section and  EMBED Equation.DSMT4  is the normal force;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (17) where EMBED Equation.DSMT4   EMBED Equation.DSMT4 . (18) Then  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  is the displacement in the  EMBED Equation.DSMT4 -direction;  EMBED Equation.DSMT4  is warping due to shear  EMBED Equation.DSMT4 , (19) where EMBED Equation.DSMT4  is a function that can be obtained by assuming that EMBED Equation.DSMT4  when  EMBED Equation.DSMT4 , when according to (4)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  (20) where  EMBED Equation.DSMT4  is the coordinate of the shear flow zero point for the rigid cross-section,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are the statical moments of the cut-off portion of the rigid cross-section for the points A and B, respectively. Thus, from (19) and (20)  EMBED Equation.DSMT4 , (21) i.e.  EMBED Equation.DSMT4 , (22) and  EMBED Equation.DSMT4 . (23) The stresses then read  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (24) where  EMBED Equation.DSMT4 . The stresses given by (24) satisfy the equilibrium equation given by (6) only if  EMBED Equation.DSMT4 ; (25) otherwise the solution can be used only approximately. The beam component  EMBED Equation.DSMT4 will be under antisymmetric tension and shearing, with respect to the beam component 2. 3.3. Compatibility conditions The compatibility conditions, along the edges A and B, taking into account (23), may be expressed as  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (26) Hence  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (27) Then, referring to the first equation of (15) and the first and second equations of (8)  EMBED Equation.DSMT4 . (28) and, referring to the second equation of (15), (16), and the third equation of (8)  EMBED Equation.DSMT4 , (29) i.e.  EMBED Equation.DSMT4 . (30) The second equation of (27), taking into account the first equation of (8), may be written as  EMBED Equation.DSMT4 . (31) The functions EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  may then be obtained from (30) and (31):  EMBED Equation.DSMT4   EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   EMBED Equation.DSMT4 . (32) For the case  EMBED Equation.DSMT4 , one obtains  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (33) Substitution of (33) into (8), taking into account (11) and (13), yields  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (34) where  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (35) Since  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (36) from (34), taking into account (1), one obtains  EMBED Equation.DSMT4 , i.e.  EMBED Equation.DSMT4 . (37) Thus, referring to (1) and (4),  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , given by (32), become  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (38) where  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (39) 3.4. Internal forces and displacements By substituting (28) into the fourth equation of (8), taking onto account (11) and (13), the following differential equations may be written  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (40) where  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are given by (35). By multiplying the first equation from (36) by  EMBED Equation.DSMT4  and the second by  EMBED Equation.DSMT4 , and subtracting the second equation from the first, the following differential equation may be written  EMBED Equation.DSMT4 , (41) where  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (42) The following differential equations can then be plotted (see Appendix)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (43) where  EMBED Equation.DSMT4 . (44) Taking into account the first equation of (43), it may be written  EMBED Equation.DSMT4 , (45) i.e. referring to the second equation of (8) and the second equation of (43)  EMBED Equation.DSMT4 . (46) From the equilibrium, it follows  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (47) Substitution of (28) into the first equation of (47) yields  EMBED Equation.DSMT4 , (48) where  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (49) Here  EMBED Equation.DSMT4 . (50) From (46) and (48), taking into account (50), it may be written  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (51) and according to third equation of (8) and the third equation of (43)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (52) According to (51), taking into account the second equations of (8), (1) and (43), it may be written  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (53) where  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (54) From (28) and (51), one has  EMBED Equation.DSMT4  EMBED Equation.DSMT4 . (55) From (53), it follows  EMBED Equation.DSMT4 . (56) Thus, taking into account the third equations of (1), (41) and (43),  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  given by (32) may be written as  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (57) where  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are given by (39). By substituting (57) into (52), taking into account (12) and (14), (35), (39) and (49), where  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , the following relation can be obtained  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (58) Hence, taking into account the second equation of (47)  EMBED Equation.DSMT4  . (59) The displacements due to shear  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 may be obtained, according to the second equations of (2) and (9), as follows  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (60) where  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 are the bending moments of the beam and the beam components 1 and 3 at the beam left end. From (60) and (51), it follows  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (61) The unknown shear areas  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 can be obtained from the condition: when  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . Thus, from (61) and (60)  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4 . (62) By substituting (62) into (61), one may finally write  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (63) where  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (64) 3.4. Stresses The stresses for the beam component 1 may finally be obtained by substituting the first equation of (51) and the first equation of (52) into (12):  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (65) where  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (66) The stresses for the beam component 3 may be obtained by substituting the second equation of (51) and the second equation of (52) into (14):  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (67) where  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (68) The stresses for the beam component 2 may be obtained by substituting (55) and (57) into (24), taking into account the first equations of (1) and (43), and (39):  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (69) where  EMBED Equation.DSMT4   EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  (70) The force per unit length  EMBED Equation.DSMT4  can be obtained from (17) and (18), taking into account (57):  EMBED Equation.DSMT4 , (71) where  EMBED Equation.DSMT4 . (72) Here  EMBED Equation.DSMT4 , (73) where  EMBED Equation.DSMT4  is the normal stress in the  EMBED Equation.DSMT4 -direction for  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  is the thickness of the beam component 2. The first part of the expressions for stresses represent the stresses for the case of rigid cross-sections. The second part ( EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 ) represent the additional stresses due to cross-section distortion, defined by the load  EMBED Equation.DSMT4  and internal forces  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , and the moment  EMBED Equation.DSMT4  (see Appendix). The same holds for the displacements given by (53) and (54), and (63) and (64), where the additional displacements due to distortion are defined by the displacements  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , and the moment  EMBED Equation.DSMT4 (see Appendix). 3.5. Boundary conditions The boundary conditions at the ends of a beam may be formulated as follows  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (74) Then, according to the first equation of (41),  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ; (75) and according to (56)  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ; (76) and according to (27)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (77) Thus, the boundary conditions are defined by the components  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . In terms of forces, the boundary conditions may be formulated as follows  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (78) Then, according to (59) and (46)  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ; (79) and from (58) and (48),  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 . (80) Here the boundary conditions are defined by the components  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . The simply supported ends may be defined by  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 . (81) Then, according to the first equation of (75) and the second equation of (79)  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ; (82) and according to the first equation of (76) and the second equation of (80)  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 . (83) Referring to (65), (67), (69) and (4), it follows  EMBED Equation.DSMT4 . (84) For the built-in end, it may be written  EMBED Equation.DSMT4   EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (85) for the left end (A);  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (86) for the right end (B). Then, according to (75),  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (87) and according to (76),  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (88) for the left end (A);  EMBED Equation.DSMT4 . (89) for the right end (B). The free end may be defined as  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (90) Then, according to (79),  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (91) and according to (80),  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (92) Referring to (65), (67), (69) and (4), it follows  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 . (93) 4. Illustrative examples In order to illustrate the application of the expressions for the stresses and displacements, a thin-walled beam with the double symmetric three-cell closed cross-section with the constant thickness is considered (see Appendix). Geometrical properties and the material  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 Equation.DSMT4 ;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . Here l is the length of the beam. The shear areas are calculated as pure geometrical properties, for Poisson`s ratio equal to zero (Bhat and de Oliveira, 1985; Cowper, 1966)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 . Simply supported beam under uniformly distributed load The internal forces at  EMBED Equation.DSMT4 (Example 1, Appendix):  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . The internal forces and displacements at  EMBED Equation.DSMT4 (Example 1, Appendix):  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . Additional stresses due to distortion at  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 . Additional stresses and displacements due to distortion at  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 . Built-in beam under uniformly distributed load. Components of the internal forces at  EMBED Equation.DSMT4 (Example 2, Appendix):  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . Components of the internal forces and displacements at  EMBED Equation.DSMT4 (Example 2, Appendix):  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . Additional stresses due to distortion at  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 . Additional stresses and displacements due to distortion at  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 . The additional stresses and displacements due to distortion are calculated for  EMBED Equation.DSMT4 , i.e.  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  Fig. 4. Fig. 5. Fig. 6. Fig. 7. Fig. 8. The stresses and displacements are calculated for the beam subjected to uniformly distributed load  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  ( EMBED Equation.DSMT4 ), both for simply supported and built-in beam ends, for  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . The results are given in respect to the stresses and displacements for rigid cross-sections (Fig.4-10). In order to check accuracy of the obtained results, the normal stresses and displacements at  EMBED Equation.DSMT4  for the simply supported beam are also calculated by the finite element method; the shear stresses are calculated for EMBED Equation.DSMT4 . The 3D membrane model of one-quarter of the beam is investigated. A high mesh density with 3200 quadrilateral elements and 12880 nodes is used. The displacements of the nodes are restricted at  EMBED Equation.DSMT4  in the cross-section contour direction and at  EMBED Equation.DSMT4  in the beam longitudinal direction; at  EMBED Equation.DSMT4  (in the plane of symmetry) the displacements are restricted in the beam transverse direction. 5. Conclusion An analytical method has been applied to estimate the additional stresses and displacements due to distortion of the cross-sections of thin-walled beams subjected to bending. Simple rectangular cross-sections with three and two closed cells with double axes of symmetry are considered. It is assumed that beams are long enough that warping due to shearing may be ignored in the stress calculation (in the case of rigid cross-sections, i.e. when the cross-sections maintain their shape). The cross-section distortion is considered in the limit, by assuming that beam walls are hinged along their longitudinal edges. The additional stresses and displacements due to the cross-section distortion are given in the analytical closed form and compared to the stresses and displacements of the ordinary bending theory (where the cross-sections maintain their shape). It is shown that the additional stresses and displacements due to the cross-section distortion can be significant (compared to the stresses and displacements of the ordinary bending theory), particularly, the additional shear stresses. A typical cross-section with three cells is analysed, where the ratio of the beam length and the cross-section breadth was equal 8 (the beam length to the cross-section height ratio equal 16) is analysed. The loads were distributed along the inner vertical walls only. The ends of the beam were simply supported and built-in, respectively. The comparison for the simply supported beam under uniformly distributed load along the inner vertical walls to the finite element solution of the problem has shown acceptable agreements of obtained results. Although the hinges between plates do not occur in actuality, it is important to analyse such conditions, together with the ordinary beam theory (where the cross-section is assumed rigid), as limits of the actual behaviour of the structure. In fact, there are no enough stiff cross-sections in actual thin-walled beam structures to guarantee the cross-section shape, especially under non uniform load distributions in the transverse direction. The real stiffness of the cross-section structure can be easily included in the consideration. The torsional stiffness of the plates together with bending stiffness of transverse framing (if the structure is framed) and the shear stiffness of transverse bulkheads (if there are any) should be taken into account. The same approach may be used in the case of thin-walled curved beams. The cross-sections of the components beams may be approximate by rectangular cross-sections; or be considered as curvilinear cross-sections. In the compatibility conditions and equilibrium equations, the cross-section properties of the beam components will be changed only. It should be noted that relative vertical displacements are small, in comparison with vertical displacement of the beams. In that case, in the finite element analysis the shell elements must be used. Acknowledgements This research was supported by Ministry of Science and Technology of the Republic Croatia Contract No. 0023025. References Boitzov, G.V., 1972. Analiz Poperechnoi Prochnosti Krupnotonnaznih Tankerov [Analysis of the Transverse Strength o Large Tankers]. Sudostroenie 9, 17-24. Boswell, L.F. Zhang, S.H., 1984, The effect of distortion in thin-walled beams, International Journal of Solids and Structures 20 (9/10), 845-862. Boswell, L.F., Li, Q., 1995. Consideration of the Relationships Between Torsion, Distortion and Warping of Thin-Walled Beams, Thin-Walled Structures 21, 147- 161. Bull, J.W., 1988. Finite Element Analysis of Thin-Walled Structures, Elsevier Applied Science, London-New York. Filin, A.P., 1975. Prikladnaya Mehanika Tverdovo Deformiruemovo Tela, II [Applied Mechanics of Solids, II]. Nauka, Moscow. Hsu, Y.T., Fu, C.C., Scheling, D.R. , 1995. EBEF method for distortional analysis of steel box girder bridges, Journal of Structural Engineering 121 (3), 557-566. Hughes, O.W., 1983. Ship Structural Design, John Wiley, New York. Kim, J.H., Kim, Y.Y., 1999, Analysis of thin-walled closed beams with general quadrilateral cross-sections, ASME Journal of Applied Mechanics 66 (4), 904-912. Kim, J.H., Kim, Y.Y., 2000. One dimensional analysis of thin-walled beams having general cross-sections. International Journal for Numerical methods in engineering 49, 653-668. Kim, J.H., Kim, T.S., 2000. Topology optimization of beam cross-sectiom, International Journal of Solids and structure 37, 477-493. Kim, J.H., Kim, Y.Y., 2001. Thin-walled multi-cell beam analysis for coupled torsion, distortion and warping deformation, ASME Journal of Applied mechanics 68, 260- 269. Kim, T.S., Kim, Y.Y., 2002. Multi-objective topology optimization of a beam under torsion and distortion. AIAA Journal 40 (2), 376-381. Kim, J.H., Kim, H.S., Kim, D.W., Kim, Y.Y., 2002. New accurate efficient modelling technique for the vibration analysis of T-joint thin-walled box structures, International Journal of Solids and Structure 39, 2893-2909. Kollbrunner, C.F., Basler, K., 1969. Torsion in Structures, Springer-Verlag, Heildeberg, New York. Pavazza, R., 1991. Saviane i Uviane stapova otvorenog tankostienog presieka na elastichnoi oodlozi [Bending and torsion of thin-walled beams on elastic foundation], Thesis, Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb. Pavazza, R., Matokovi, A., 2000. On the shear flow due to distortion of hull cross- sections of tankers with two longitudinal bulkheads. Proceedings, IMAM 2000, Vol II, Ischia, Italy, pp. F32-39. Pavazza, R., Plazibat. B., Matokovi, A., 2001. On the effective breadth problem of deck plating of ships with two longitudinal bulkheads, International shipbuilding progress 48, 51-85. Pavazza, R., 2002. On the load distribution of thin-walled beams subjected to bending with respect to the cross-section distortion, International Journal of Mechanical Sciences 44, 423-442 Schwyzer, H., 1920. Statische untersuchung der aus ebenen tragflchen zusammengesetzten rumlichen tragwerke, Diss. ETH Zqrich. Timoshenko, S. P. , Goodier, J. N., 1983. Theory of elasticity, McGraw-Hill, New York. Vlasov, V.Z., 1961. Thin-walled elastic beams, Oldbourne Press, London. APPENDIX The solution of the equation (41) may be presented, by the method of initial parameters, as follows (Pavazza, 1991)  EMBED Equation.DSMT4 , (A1) where  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (A2) where  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  . (A3) For the uniformly distributed load,  EMBED Equation.DSMT4 :  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4   EMBED Equation.DSMT4 . (A4) In the case of many load distribution fields, it may be written  EMBED Equation.DSMT4 , ( EMBED Equation.DSMT4 ) (A5) where  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (A6) For the partly distributed uniform load, along  EMBED Equation.DSMT4  ( EMBED Equation.DSMT4 ):  EMBED Equation.DSMT4 = = EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4 . (A7) For the concentrated force,  EMBED Equation.DSMT4 , at  EMBED Equation.DSMT4 :  EMBED Equation.DSMT4 . (A8) Example 1: Simply supported beam under the uniformly distributed load; boundary conditions by (81):  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (A9)  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4   EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ; (A10)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (A11) Example 2: Built-in beam under the uniformly distributed load; boundary conditions by (85):  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , (A12)  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ; (A13)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (A14) Example 3: Simple supported beam under the concentrated force  EMBED Equation.DSMT4  at  EMBED Equation.DSMT4 ; boundary conditions by (81):  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (A.15) EMBED Equation.DSMT4  Example 4: Built-in beam under the concentrated force at  EMBED Equation.DSMT4 ; boundary conditions given by (85):  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ; (A16)  EMBED Equation.DSMT4 . (A17) Figures Fig. 1. Rectangular closed thin-walled cross-sections with coordinate systems: a) three cell section; b) two cell-section. Fig. 2. Rectangular closed thin-walled sections with hinged walls: a) three cell section; b) two cell-section. Fig. 3. Beam components of the beam with double symmetrical three cell thin-walled cross-section. Fig. 4. Geometrical properties of the rigid cross-section: a) z-coordinate; b)  EMBED Equation.DSMT4 -coordinate (statical moment of the cut-off portion of cross-section area). Fig. 5. Stresses for the simply supported beam: a) additional normal stresses due to distortion at  EMBED Equation.DSMT4 ; b) normal stresses due to bending with distortion at  EMBED Equation.DSMT4 (FEM in brackets); c) additional shear stresses due to distortion at  EMBED Equation.DSMT4 ; d) shear stresses due to bending with distortion at  EMBED Equation.DSMT4 . Fig. 6. 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Fig. 7. Stresses for the built-in beam: a) additional normal stresses due to distortion at  EMBED Equation.DSMT4 ; b) normal stresses due to bending with distortion at  EMBED Equation.DSMT4 ; c) additional shear stresses due to distortion at  EMBED Equation.DSMT4 ; d) shear stresses due to bending with distortion at  EMBED Equation.DSMT4 (FEM in brackets). Fig. 8. Stresses and displacements for the built-in beam: a) additional normal stresses at  EMBED Equation.DSMT4 ; b) stresses due to bending with distortion at  EMBED Equation.DSMT4 ; c) additional displacements due to distortion at EMBED Equation.DSMT4 ; d) displacements due to bending with distortion at EMBED Equation.DSMT4  (FEM in brackets). Dr. Radoslav Pavazza Faculty of Electrical Engineering, Machanical Engineering and Naval Architecture University of Split Ruera Boakovia, bb 21000 Split Croatia E-mail  HYPERLINK mailto:Radoslav.Pavazza@fesb.hr Radoslav.Pavazza@fesb.hr Fax +38521563877 Mr. Branko Blagojevi Faculty of Electrical Engineering, Machanical Engineering and Naval Architecturef.g.~..............////,/0/M/N/e/f/g/h//~00000ڭ뜋rnfn[fjhQ)UjhQ)UhQ)!jh\ hQ)EHUmH sH hBcmH sH !jh\ hQ)EHUmH sH !jhsC hQ)CJUVnHtH!jh\ hQ)EHUmH sH h<<hQ)mH sH !j0h\ hQ)EHUmH sH !jnhsC hQ)CJUVnHtHhQ)mH sH jhQ)UmH sH !0 1 101214128888888hXJhQ)mH sH Uhynh \hQ)jhQ)U hQ)0J University of Split Ruera Boakovia, bb 21000 Split Croatia E-mail Branko.Blagojevi@fesb.hr Fax +38521563877 *8T8l8|88888888888 $ 7na$gdQ) $ Sna$gdQ) $ a$gd: $ a$gdyn *:pF. 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