Chapter 6 Bending

6.1 Shear and Moment Diagrams Beam Beam – slender bar, support transverse load

6.2 Graphical Method for Constructing Shear and Moment Diagrams Regions of Distributed Load * Differential equations:

6.3 Bending Deformation of a Straight Member Observations * Horizontal lines – curved * Vertical lines – remain straight, but rotated * Upper portion – compressed, lower portion – stretched * Neutral surface – no change in longitudinal length

Coordinate System * x-axis – longitudinal axis on neutral plane * y-axis – vertical axis * z-axis – transverse axis on neutral * x-z plane is the neutral plane

Assumptions * The longitudinal line does not change in length, but become a curve. * Cross sections remain planar, and perpendicular to longitudinal axis. * Deformation of cross sections is neglected.

Normal Strain At location x, cut a small segment of length dx

The strain reaches its maximum, at top or bottom. One is tension, and another is compression.

6.4 The Flexure Formula Normal Strain and Stress Both are linearly varying with respect to y (distance to neutral axis). Questions!!! Where is the neutral surface(axis)? How to determine the curvature of the deformed beam?

Where is the neutral surface(axis)? Location of Neutral Surface For the cross section, there is no longitudinal (normal) force,

From the first equation of statics, Centroid of cross section: * For symmetric shapes, the neutral plane is in middle, such as rectangular, I-beam.

Flexure Formula The internal moment M is caused by the normal stress. Note: (i) Above neutral axis, compressive stress (negative ) produces positive moment.

Relation between the curvature and the moment Define I as the moment of inertia of cross section about the neutral axis (z-axis): Note that this equation relates the curvature of the beam to the bending moment. Moments of inertia are always positive and have dimensions of length to the fourth power. For example, Then, the maximum stress can be rewritenn as the function of the moment. Substituting the maximum stress into the expression of stress, We can determine the stresses in terms of the bending moment, called as the flexure stress or bending stree. By now, it still has one question which doesn’t be answered? Here, we have known the maximum stress equal to MC/I. It also can be expressed in terms of maximum strain based on Hooke law. And the maximum strain is related to the curvature. Combining these three equations, we can get the relationship between the curvature and the moment, which called as the moment curvature equation. - Flexure formula

For composite shape (consisting of several simple shapes) * Centroid: * Moment of inertia: Use Parallel-Axis Theorem IC – moment of inertia about the axis passing through the centroid I– moment of inertia about neutral axis A – total area, dy – distance between two axes

* Procedure: (i) Divide the area into several parts of simple shapes. (ii) Determine the overall centroid (neutral axis) (iii) For each part, calculate the moment of inertia about axis passing through its own centroid. (iv) For each part, use parallel-axis theorem to calculate the moment of inertia about the neutral axis. (v) The total moment of inertia of the entire area is the algebraic sum (+ or -) of those of all parts.

EXAMPLE 6.13

EXAMPLE 6.14

EXAMPLE 6.14 (CONTINUED)

Chapter 7 Transverse Shear

7.1 Shear in Straight Members * A beam supports both shear (V) and moment (M). * The shear force V is the result of transverse shear stress distribution. * Since shear must appear in pair for balance, the transverse shear stress and longitudinal shear stress coexist.

There is relative displacement between adjacent layers.

There are shear forces between adjacent layers. Longitudinal shear stress are present.

7.2 The Shear Formula Cut the beam at location x, and take a small segment of length dx. Normal stress distribution Normal stress distribution

Consider area A (above y), and write equilibrium equation

 - shear stress at point y of the cross section at x V – internal shear force at this cross section I – moment of inertia of this cross section t – thickness at y Since V = V(x), Q = Q(y),  = (x,y) – function of x and y.