Introduction to beam bending

There are no circuits in what follows.I will not use the words voltage, current,

or op-amp today (well, maybe just once or twice if you count this slide).

Labs and work

• You need to let us know (while working, not after the fact) if labs are taking too long.

• You should be working, but it should be reasonable (3 credit course, ~9hrs/week total).

• Waiting to the night before, means you are up all night.

Lab etiquette

• Clean up, clean up, everybody let’s clean up. • Keep your op-amp. Do not take them out of

the breadboard and leave on the desk. • Throw away old wires, resistors, and scraps.• Cups, plates, and food? Really?• Solderless breadboard, www.digikey.com part

438-1045-ND. $8.98. 438-1046-ND is $13.82 and comes with a little box of wires.

Beam bending

Galileo, 1638 (though he wasn’t right)

DaVinci-1493

"Of bending of the springs: If a straight spring is bent, it is necessary that its convex part become thinner and its concave part, thicker. This modification is pyramidal, and consequently, there will never be a change in the middle of the spring. You shall discover, if you consider all of the aforementioned modifications, that by taking part 'ab' in the middle of its length and then bending the spring in a way that the two parallel lines, 'a' and 'b' touch a the bottom, the distance between the parallel lines has grown as much at the top as it has diminished at the bottom. Therefore, the center of its height has become much like a balance for the sides. And the ends of those lines draw as close at the bottom as much as they draw away at the top. From this you will understand why the center of the height of the parallels never increases in 'ab' nor diminishes in the bent spring at 'co.'

Normal stress (σ) and strain (ε)

P P

L

A

P

L

δ

Stress-strain

Yield stress in “ordinary” steel, 200 MpaHow much can 3 x 0.25 bar hold in tension?

Hooke’s law

steelfor GPa 200E

modulus sYoung'or modulus elastic is

E

E

What is the strain just before steel yields?

Shear stress

• P28 fig 1-24, 1-28

Shear stress in tension/compression

P P

P P

NV

-80 -60 -40 -20 0 20 40 60 80-0.5

0

0.5

1

Nor

mal

/She

ar

N

V

Beams in bending

Simply supported beam

LbLa

P

How do we find reaction forces?

LbLa

P

Ra Rb

)( baba

ba

LLRPL

PRR

Sum forces

Sum moments

L

LPR

L

LPR

ab

ba

State of stress inside the beam

LbLa

P

Ra Rb

L

LPR

L

LPR

ab

ba

State of stress inside the beam

LbLa

P

Ra RbImagine a cut in the beam

L

LPR

L

LPR

ab

ba

State of stress inside the beam

VM

Calculate shear and bending moment to hold at equilibrium

x

P Lb/L

State of stress inside the beam

VMx

P Lb/L

X

X

V

M

PLb/L x

P Lb/L

State of stress inside the beam

V

Mx

P Lb/L

X

X

V

M

P P Lb/L

P La/L

P La Lb/L

Shear and bending moment diagramLb

La

X

P Lb/L

P La/L

P La Lb/L

V

M

Shear and bending diagram examples

Beam in pure bending

MM

Beam in pure bending

Fig 5-7, page 304

Beam in pure bending

Lines, mn and pq remain straight – due to symmetry. Top is compressed, bottom expanded, somewhere in between the length is unchanged!

y

(1/radius) curvature theis

This relation is easy to prove by geometry

Neutral axis

Beam in pure bending

MM

“If a straight spring is bent, it is necessary that its convex part become thinner and its concave part, thicker. This modification is pyramidal, and consequently, there will never be a change in the middle of the spring.” DaVinci 1493

y

y=0

Normal stress in bending

yE

M

Take a slice through the beam

σ

0)( ydAEydAEdAy

Neutral axis is the centroid

y

Normal stress in bending

M

σ

interia ofmoment is

)( 22

I

IEM

MdAyEdAyEydAy

Flexure formula

EI

MyI

My

Moment of inertia, I

b

h

12

3bhI

CrossSection

Lab

• Calculate shear/moment diagram for your beam, now know M(x).

• Calculate I, look up E.• Calculate strain, ε(x).• Stand on beam, measure strain.

Strain gages