Bending Shear and Bending Moment Diagrams covered in

Bending Shear and Bending Moment Diagrams – covered in Statics and self-reviewed in Mechanics of Materials Know terminology: simply-supported versus fixed; cantilevered, overhanging. Know sign conventions for positive versus negative shear and moments. If you can’t remember, memorize the shear and bending moment diagram for a simply-supported beam with a point load.

Bending Deformation of a Straight Member Assumptions: straight, prismatic beam made of homogeneous material; cross-sectional area of beam is symmetrical with respect to an axis, and bending moment is applied perpendicular to axis of symmetry. Illustration of Bending for a Bar

Distortion of a Rubber Bar due to Bending

Assumptions Longitudinal axis does not charge length; Cross-sections remain plane; Any deformation of cross-section within its own plane can be neglected.

Bending Stress Variation

6. 32. Known: The ski supports the 180 lb weight of the man. The snow loading on its bottom surface is trapezoidal. Find: Determine the intensity, w, and draw the shear and bending moment diagrams for the ski. V, lb x M, ft∙lb x

6. 49. Known: A steel beam has the cross section shown. Find: Determine the largest internal moment the beam can resist if the moment is applied as follows: (a) About the z axis. (b) About the y axis. Data: The allowable stress for the steel is σallow = 24 ksi.

6. 54. Known: An aluminum strut has a crosssectional area in the form of a cross. Find: Determine the bending stress acting at (a) Point A, and (b) Point B. Show the stresses acting on volume elements at these points. Data: The moment M = 8 k. N∙m.

Unsymmetric Bending Moment Applied Along Principal Axis The flexure formula applies for sections that are unsymmetric if the moment is applied about one the principal axes. Note that one of the principal axes is along an axis of symmetry, and the other principal axis is perpendicular to it.

Unsymmetric Bending – (continued) Moment Arbitrarily Applied A moment M applied in the yz plane at an angle θ has components Mz = M cosθ and My = M sinθ as shown below. The flexure formula can be applied to each moment component to obtain the resultant normal stress at any point on the cross-section.

The stresses combine to form the resultant stress distribution below:

Orientation of Neutral Axis (N. A. )

6. 102. The box beam is subjected to a bending moment of M = 15 kip∙ft directed as shown. Determine the maximum bending stress in the beam and the orientation of the neutral axis. y B y C z A D B C z A D

6. 108. The 30 mm diameter shaft is subjected to the vertical and horizontal loadings of two pulleys as shown. It is supported on two journal bearings at A and B which offer no resistance to axial loading. Furthermore, the coupling to the motor at C can be assumed not to offer any support to the shaft. Determine the maximum bending stress developed in the shaft.