"CESE 656 – Advanced Mechanicsfor Structural EngineersCivil Structural EngineeringUniversity of Alabama at BirminghamBending Non-symmetric BendingBending when loads are not applied in a plane of symmetry2 Bending CESE 656 Non-Symmetric Bending Formula for Arbitrary Axes? Assume plane sections remain plane? Neutral Axis is not parallel to axis of M? Stress varies linearly through cross-section with a slopeperpendicular to the neutral axis? ? = ? + ?? + ?? (General equation of a plane)? ? 3 Bending CESE 656 Equilibrium to Find a, b, and c? ? = 0 = ? ?? = ? ?? + ? ? ?? + ? ? ??? ? ? ? ? 2 ? ? = ? ? ?? = ? ? ?? + ? ? ? ?? + ? ? ??? ? ? ? ? 2 ? ? = - ? ? ?? = - ? ? ?? - ? ? ?? - ? ??? ? ? ? ? ? z passes through centroid, so? ? ? ? = ? ? ? = 0? ? ? By definition2 ? ? = ? ? ?? ? 2 ? ? = ? ? ?? ? z-axis is outward normal of cut-face? ? = ? ?? ? ? x- and y-axis form right-hand Cartesian system4 Bending CESE 656?? ?? ?? ?? ??Equilibrium to Find a, b, and c? ? = 0 (Since dA ? 0)? ? = ? ? + ? ?? ? ? ? ? ? = -? ? - ? ?? ? ? ? ? Solving for a and b we get? ? + ? ? ? ? ? ? ? ? ? = -2 ? ? - ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? =2 ? ? -? ? ? ? ? z-axis is outward normal of cut-facex- and y-axis form right-hand Cartesian system5 Bending CESE 656 Non-Symmetric Bending Formula for Arbitrary Axes? Substituting a, b, and c into ? = ? + ?? + ??? ? ? ? + ? ? ? ? +? ? ? ? ? ? ? ? ? ? ? ? ? ? = - ? + ?? ? 2 2 ? ? - ? ? ? -? ? ? ? ? ? ? ? ? ? ? = ? sin ? ? = -? cos ?? ? ? The sign of M is taken from the direction of x-projectionM6 Bending CESE 656 Location of the Neutral Axis? Along the neutral axis s = 0, sozz ? ? +? ? ? ? ? ? ? ? ? = ? = ? tan ?? ? +? ? ? ? ? ? ? ? ? +? ? ? ? ? ? ? ? tan ? = , and if ? ? 0? ? ? + ? ? ? ? ? ? ? ? -? co t ? ? ? ? ? tan ? =? - ? co t ? ? ? ? z-axis is outward normal of cut-facex- and y-axis form right-hand Cartesian system7 Bending CESE 656 More Convenient Form of the BendingFormula for Arbitrary Axes? ? + ? ? ? ? +? ? ? ? ? ? ? ? ? ? ? ? ? ? = - ? + ?? ? 2 2 ? ? - ? ? ? -? ? ? ? ? ? ? ? ? ? ? +? ? ? ? ? ? ? ? tan ? =? ? + ? ? ? ? ? ? ? ? Eliminating My from these two formulas yields? ? -? t an ? ? ? ? = (+a is CCW x toward y)? ? ? -? t an ? ? ? ? 8 Bending CESE 656 Example – Non-symmetric cross-sectionusing the bending formula for arbitrary axes? The Z-section shown is subjected to a bending moment of M = 20 kN·m directedalong the axis of the horizontal piece of the cross-sectiona. Determine the orientation of the NA w.r.t. the x-axisb. Determine the stress at point PyQxSR9 Bending CESE 656 Section Properties? First find Moments of Inertia about x- and y-axes3 3 600? ? 10 0? ? 100? ? 30 0? ? 2 ? ? = + 2 + 1 0 0 ? ? 3 0 0 ? ? 2 0 0 ? ?? 12 12 9 4 ? ? = 2.9 10 ? ?? 3 3 1 00? ? 600 3 00? ? 1 00 2 ? ? = + 2 + 1 0 0 ? ? 3 0 0 ? ? 25 0 ? ?? 12 12 9 4 ? ? = 5.6 10 ? ?? y? ? = 1 0 0 ? ? 3 0 0 ? ? -2 5 0 ? ? 2 0 0 ? ??? ? + 1 0 0 ? ? 3 0 0 ? ? 2 5 0 ? ? -2 0 0 ? ?9 4 ? ? = -3.0 10 ? ??? QxSR10 Bending CESE 656?? ??"