SM 2 03 Bending BENDING M Chrzanowski Strength

SM 2 -03: Bending BENDING M. Chrzanowski: Strength of Materials /14

SM 2 -03: Bending Formal definition: the case when set of internal forces reduces solely to the moment vector which is perpendicular to the bar axis Mx=0, My ≠ 0, Mz=0 N=0, Qy=0, Qz=0 My Example: a straight bar loaded by concentrated moments applied at its ends. M M Qz(x)=0 y „Pure” bending My = M x z M. Chrzanowski: Strength of Materials My(x)=M=const or Mx=0, My =0, Mz ≠ 0 N=0, Qy=0, Qz=0 2/14

SM 2 -03: Bending Remarks on terminology M. Chrzanowski: Strength of Materials 3/14

SM 2 -03: Bending NORMAL (proste) INCLINED (ukośne) 90 o <90 o y y x x 90 o z Mx=0, My ≠ 0, Mz=0 M. Chrzanowski: Strength of Materials <90 o z Mx=0, My ≠ 0, Mz ≠ 0 4/14

SM 2 -03: Bending PURE (czyste) IMPURE („nie-czyste”) NON-UNIFORM (poprzeczne) y y x z Mx=0, My ≠ 0, Mz=0 N=0, Qy=0, Qz=0 M. Chrzanowski: Strength of Materials x z Mx=0, My ≠ 0, Mz=0 N=0, Qy=0, Qz ≠ 0 5/14

SM 2 -03: Bending End of remarks M. Chrzanowski: Strength of Materials 6/14

SM 2 -03: Bending EXPERIMENTAL approach Galileo (1564 -1642) M. Chrzanowski: Strength of Materials E. Mariotte (1620 -1684) 7/14

SM 2 -03: Bending EXPERIMENTAL approach Jacob Bernoulli (1654 -1705) Galileo (1564 -1642) P u. D D h D’ z l M=M(x) Mx=0, My ≠ 0, Mz=0 Q=Q(x) N=0, Qy=0, Qz ≠ 0 x w. D u is linear function of z ! is linear function of z and does not depend on x if M=const|x For h<<l shear forces can be neglected N=0, Qy=0, Qz = 0 M. Chrzanowski: Strength of Materials 8/14

SM 2 -03: Bending Continuum Mechanics application y tension x compression z Hooke law: M. Chrzanowski: Strength of Materials 9/14

SM 2 -03: Bending Continuum Mechanics application z z x My M. Chrzanowski: Strength of Materials y ? 10/14

SM 2 -03: Bending Equilibrium conditions z z x My y ? y-axis is the central inertia axis of crosssection area y-z axes are central principal inertia axes of cross-section area M. Chrzanowski: Strength of Materials 11/14

SM 2 -03: Bending Continuum Mechanics application Axes x – which coincides with bar axis y, z – which are central principal inertia axes of the bar cross-section area are principal axes of strain and stress matrices M. Chrzanowski: Strength of Materials 12/14

SM 2 -03: Bending Pure plane bending z Neutral axis z x y My Wy is called where For z=0 (i. e. along y-axis ) there is section modulus and section of y-axis within bar cross-section is called neutral axis (for normal stress and strain) Neutral axis coincides with only non-zero bending moment component My M. Chrzanowski: Strength of Materials 13/14

SM 2 -03: Bending Important remarks 1. All above formulas are valid only for principal central axes of cross-section inertia 2. If moment vector coincides with any of two principal axes we have to deal with plane bending. If this is not the case – we have to deal with inclined bending and derived formulas cannot be used. 3. Bar axis (x-axis) is one of the principal axis of strain and stress matrices. As two remaining principal stress and strains are equal therefore any two perpendicular axes lying in the plane of bar cross-section are also principal axes. 4. The neutral axis for normal stress and strain coincides with bending moment vector. M. Chrzanowski: Strength of Materials 14/14

SM 2 -03: Bending stop M. Chrzanowski: Strength of Materials 15/14