Theory of bending, torsion and buckling of thin-walled members of open cross section

✍ Scribed by Stephen P. Timoshenko

• Publisher: Elsevier Science
• Year: 1945
• Tongue: English
• Weight: 819 KB
• Volume: 239
• Category: Article
• ISSN: 0016-0032
• DOI: 10.1016/0016-0032(45)90013-5

✦ Synopsis

Buckling byFlexure and Torsion under Central Thrust.--Let us consider now the general case of buckling where, under central thrust, not only torsion but also bending of the axis of the compressed bar occurs. We assume that x and y are the principal centroidal axes of the cross section of the bar and xo, y,, are the co6rdinates of the shear center. The deflections of the shear center axis in the x-and y-directions, we denote by u and v, respectively, and denote, as before, by ¢ the angles of rotation of cross sections with respect to the shear center axis. Then the deflections of the centroidal axis during buckling, as can be seen from Fig. 26, are u + yo¢ and v -Xo¢.

Assuming that only a thrust P is acting at the ends as in the case of simply supported bars, we find that bending moments with respect to principal axes at any cross section are

The differential equations (I7) for the deflection curve of the shearcenter axis become d 2IA _ El,, dz 2 P(u + yo¢), d2u EI. dz 2 -P(v -Xo¢).

(56)

(Note---The Franklin Institute is not responsible for the statements and opinions advanced by contributors in the JOURN-~aL.) 343 * The system of equations equivalent to eqs. ( 56) and ( 57) was first obtained by Robert Kappus; see "Jahrbuch der deutsehen Luftfahrtforschung," I937 and "Luftfahrtforsehung," vol. I4, p. 444, 1938. * This solution can be greatly simplified by the use of nomogram as shown in the previously mentioned paper by Kappus. * Equation (57) was developed from consideration of an element of.the bar between two adjacent cross sections and is not affected by changes in the end conditions. * These equations were first obtained by V. Z. Vlasov in his book, loc. cit., J. F. I., Apr. x945, p. 256.