ME221 Lecture 15 1

ME 221 Statics

Lecture #15a

Sections 7.3 - 7.4

ME221 Lecture 15 2

Homework #6• Chapter 7 problems:

– 6, 19 & 26

• Chapter 6 problems– 3 & 6– Due Monday, June 28

• MatLab Group Problems– 7.19, 7.26 & 6.15 – Due Monday, June 28

ME221 Lecture 15 3

Last Lecture:Internal Forces in Structures

• Reviewed internal/external forces

• Found internal forces

• Started shear & moment diagrams

ME221 Lecture 15 4

Generate a shear / bending diagram as follows:

2. Take a section on each side of an applied force or moment and inside a distributed load

(take a new section whenever there is a change in the load or shape of the beam)

- draw a FBD and sum forces / moments

3. Repeat 2 along the length of the beam

1. Find reaction forces

w(x) distributed load

V(x) shear force

M(x) moment

Shear and Moment Diagrams using Sectioning Method

ME221 Lecture 15 5

V

M

V

M

Sign Convention

Positive Shear and Positive Moment

ME221 Lecture 15 6

Positive Shear

M M

Positive Moment

Effect of External Forces

ME221 Lecture 15 7

Relations Between w, V, and M

In balancing forces, we can come up with differential equations relating w, V, and M. These are as follows:

,

dM x dV xV x w x

dx dx

This means you can integrate the shear diagram to obtain the moment diagram.

dx

M+dMM V

V+dV

w(x)

0)()(0 dxxwdVVVF

0)(0 dMMVdxMM

Thus,

ME221 Lecture 15 8

Shear Forces• Area under load curve

x

duuwxV0

)()(

ME221 Lecture 15 9

Bending Moments• Area under shear force curve

x

duuwuxxM0

)()()(