ME221 Lecture 25 1

ME 221 Statics

Lecture #25

Sections 7.1 – 7.4

ME221 Lecture 25 2

Homework #9Chapter 5 problems:

– 53, 54, 56, 62, 64, 69, 71 & 73

• Due Friday, October 31

ME221 Lecture 25 3

Homework #10Chapter 7 problems:

– 2, 5, 6, 8, 19, 21, 24, 26 & 35

• Due Friday, November 7

ME221 Lecture 25 4

Chapter 7:Internal Forces in Structures

• Review internal/external forces

• How to find internal forces

• Sample problems

ME221 Lecture 25 5

External/Internal Forces• External forces arise from contact or

gravitational attraction– Point and distributed loading

– Weight

• Internal forces are forces arising to hold bodies together– Internal stress is a form of an internal force

ME221 Lecture 25 6

Exposing Internal Forces• To analyze the stress at a given location in a

part, we need to know the forces at that particular section.

100 lb

a a

b b

c cAt any given section a-a, b-b, orc-c, there is an internal force arisingfrom the 100 lb external force.

ME221 Lecture 25 7

Method for Finding Internal Forces• Determine reaction forces

100 lb 100 lb

Ay=100 lb

Ax=0

Mz=0

– Use equilibrium equations• Section and solve second equilibrium problem

to find internal forces

a a

100 lb

Fy

Fx

Mz

ME221 Lecture 25 8

General Internal Forces• In general, there is a force and moment

component for each coordinate direction at a given section

– 6 possible unknowns

Sample problems:

ME221 Lecture 25 9

Example:

Determine the internal forces and moments in the bar built into the foundation as shown in the figure.

x

y

z

O

l

h

P

ME221 Lecture 25 10

(l-x)P

x

y

z

O

l

h

P

x R

M

pr

kPixlMM

kPRF

ˆx ˆ)(0

ˆ0

jxlPM

kPR

ˆ)(

ˆ

Horizontal Portion

ME221 Lecture 25 11

x

y

z

O

l

h

P

y

R

M

l

(h-y)

P

jyhilrp ˆ)(ˆ

iyhPjPlM

kPR

ˆ)(ˆ

ˆ

Vertical Portion

ME221 Lecture 25 12

Shear and Bending Diagrams (Secs. 7.3, 7.4)

• Topic is also called transversely loaded beams

• Beam classifications and boundary conditions

• Internal forces and the components’ specific rolls

• Relation between shear and bending

• Generation of shear and bending diagrams

• Sample problems

ME221 Lecture 25 13

Types of Beams by Supports• Transversely loaded beams have several

standard configurations• Determinate beams have the same number

of reactions as nontrivial equilibrium eqns.IndeterminateDeterminate

Simple

Overhanging

Cantilever

ME221 Lecture 25 14

Internal Force Component Rolls• Force components

• Moment components

– Axial is along beamP

– Shearing forces are transverse componentsVy

Vz

– Torsion along beam T

– Bending for transverse componentsMy

Mz

ME221 Lecture 25 15

Shear and Moment Diagrams

-Sectioning Method

-Integration

-Singularity Functions

ME221 Lecture 25 16

What is expected for shear and bending diagrams?

1. Show FBD and statics for each section

2. Determine equation for V(x) and M(x)

3. Draw shear and bending diagrams indicating linear or parabolic

4. Label end points of diagram as well as every region endpoint

ME221 Lecture 25 17

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 / moments3. 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 25 18

V

M

V

M

Sign Convention

Positive Shear and Positive Moment

ME221 Lecture 25 19

Positive Shear

M M

Positive Moment

Effect of External Forces

ME221 Lecture 25 20

20 lb/in 125 lb 125 lb

9 in. 12 in. 12 in. 12in.

V

x

M

x-ve

tension up

+ve

Tension down

ME221 Lecture 25 21