Jiangyu Li, University of Washington Lecture 2-4 Beam Mechanics of Materials Laboratory Sec. 3-4 Jiangyu Li University of Washington Mechanics of Materials Lab

Jiangyu Li, University of Washington Beam

Jiangyu Li, University of Washington Type of Supports Beam supported on a wall Beam-to-column connection Pole anchored to a concrete pier

Jiangyu Li, University of Washington Types of Beams Simply supported beam Cantilever beam Simply supported beam with overhang

Jiangyu Li, University of Washington Shear Force & Bending Moment

Jiangyu Li, University of Washington Sign Convention

Jiangyu Li, University of Washington Shear Force & Bending Moment Diagram Negative q Negative P Positive M 0

Jiangyu Li, University of Washington Shear Force & Bending Moment Diagram

Jiangyu Li, University of Washington Shear Force and Bending Moment Diagram

Jiangyu Li, University of Washington Deflection in Beam

Jiangyu Li, University of Washington Normal Stress in Beam How to identify the neutral axis?

Jiangyu Li, University of Washington Normal Stress Go through centroid !

Jiangyu Li, University of Washington Shear Stress

Jiangyu Li, University of Washington Distribution of Shear Stress

Jiangyu Li, University of Washington Shear Stress

Jiangyu Li, University of Washington Shear Stress

Jiangyu Li, University of Washington Deflection of Beam

Jiangyu Li, University of Washington Deflection of Curve

Jiangyu Li, University of Washington Boundary Condition

Jiangyu Li, University of Washington Continuity Condition

Jiangyu Li, University of Washington Deflection by Bending Moment Equation

Jiangyu Li, University of Washington Deflection by Loading Equation

Jiangyu Li, University of Washington Deflection by Superposition

Jiangyu Li, University of Washington Strain Energy of Pure Bending

Jiangyu Li, University of Washington Strain Energy of Bending

Jiangyu Li, University of Washington Strain Energy of a Beam in Shear Rectangular: 1.2 Circular:1.11 Thin-walled tubular, round:2.00 Box section:1.00 Structural section:1.00

Jiangyu Li, University of Washington Strain Energy of Bending

Jiangyu Li, University of Washington Castigliano’s Theorem When forces act on a elastic system subject to small displacements, the displacement corresponding to any force, collinear with the force, is equal to the partial derivative of the total strain energy with respect to that force. It can also be used to find the displacement when no force is applied at that point.

Jiangyu Li, University of Washington Modified Castigliano’s Theorem

Jiangyu Li, University of Washington Application

Jiangyu Li, University of Washington Inclined Load Notice the sign convention: positive Mz compress upper part, negative stress; positive My extend front part, positive stress!

Jiangyu Li, University of Washington Inclined Load Stress Neutral axis

Jiangyu Li, University of Washington Asymmetrical Beam The origin of y and z axes must be placed at centroid C; orientation is arbitrary.

Jiangyu Li, University of Washington Sign Convention for Curvature Similar equation apply to Bending toward z axis Note difference with sign convention in bending moment

Jiangyu Li, University of Washington Asymmetric Beam If z is a principal axis, M y =0, bending in x-y plane, analogous to a symmetric beam When z axis is the neutral axis;

Jiangyu Li, University of Washington Asymmetric Beam If y is a principal axis, M z =0, bending in x-z plane, analogous to a symmetric beam When y axis is the neutral axis;

Jiangyu Li, University of Washington Asymmetric Beam When an asymmetric beam is in a pure bending, the plane in which the bending moments acts is perpendicular to the neutral surface only if the y and z axes are principle centroidal axes and the bending moment acts in one of the two principle plane. In such case, the principle plane in which bending moment acts becomes the plane of bending and the usual bending theory is valid

Jiangyu Li, University of Washington Analysis of Asymmetric Beam Locating the centroid, and constructing a set of principal axes Resolving bending moment into M y and M z Superposition

Jiangyu Li, University of Washington Principle Axes

Jiangyu Li, University of Washington Analysis of Asymmetric Beam A channel section C 10x15.3 c=0.634 I y =2.28 in 4, I z =67.4 in 4 y A =5.00 in, z A = = in Calculating bending stress Locating neutral axis

Jiangyu Li, University of Washington Analysis of Asymmetric Beam

Jiangyu Li, University of Washington Normal Stress in Beam

Jiangyu Li, University of Washington Curved Beams Neutral axis is no longer the centroidal axis Positive M

Jiangyu Li, University of Washington Curved Beam

Jiangyu Li, University of Washington Curved Beams Curvature is large, e is small, r n is cloase to r c Recover to straight beam

Jiangyu Li, University of Washington Curved Beam Pay attention to the sign of s

Jiangyu Li, University of Washington Curved Beam Pay attention to the sign of s

Jiangyu Li, University of Washington Read Mechanics of Materials Lab Sec (a,b,c), 4.26(a,e) posted online Assignment