Beams and Strain Gages

Cantilever Beam One End Fixed , One End Free

Everything is a Spring Beams Bend When Loaded

Tip Load Creates Internal Moments and Forces

V

x

x

M(x)

F

V(x)

L-x

At any point along the beam, there is a reaction force and moment (from contact with material to the left of x-section) to maintain static equilibrium.

Isolate a section of the beam (NOT physically separated)Fx-section

Distribution of moment, M, and (shear) force, V, along beam

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.'

How Does the Force Deform the Beam?

Galileo-1638 Incorrectly claimed that the root of beam was in tension.

(give him a break-Newton was born a year after he died)

Consider a Free-Free Beam Loaded by a Moment

MM

neutral surface

For long, thin beams loaded by a transverse force, deformation is predominately due to internal moment as compared to internal force. (Euler-Bernoulli Beam Assumption)

The upper portion is in compression.The lower portion is in tension.

‘neutral surface’ does not change in length

Transverse Force Causes Axial Deformation

How do we relate Load to Deformation?

Load Stress Strain Deformation

Stress is a Quantitative Measure of Load Acting on a Material

In general stress is a tensor quantity-> represented by a matrix of scalars

stress matrix

For Simple Cases, Stress Can Be Represented by a Single

Scalar Component*

FAo

F

=sF

Ao

Force applied normal to a cross-section (axially)

average normal stress

over thecross-section

9

*In the stress matrix, only one term will have significant nonzero magnitude. The other eight terms ~ 0. [ ] s s11 = F/A0

Strain = Normalized Deformation

e =Lo

x Average normal engineering

strainover the

cross-section

10

F F

xLo

Ao

How are Normal Stress and Strain Related?

test specimen

extensometer(measures strain)

11

Uniaxial Tension Test

load cell(measures force)

Linear Elastic Behavior

yield stress

e

sy

linear elastic deformation

s

12

slope = E (Young’s Modulus)

When loading is released, material returns to its original length.

Aluminum (6061-T6): E = 40 ksi = 275 MPa

property of a material

Typical Ductile Material

yield stress

e

sy

linear elastic deformation

ultimate tensile strength

uniform plastic deformation

su

nonuniform plastic

deformation

fracture stress

s

“necking”

“strain hardening”

13

Bending (Normal) Stress in a BeamWill derive this in Mechanics of Solids and Structures

Mz

σ (-compression)

Neutral surface passes through the centroid of cross-section

y

x Mz

zz

z

I

yM

Mz = Moment about the z-axis (out of page) acting on the x-section. Constant over the x-section.

y = distance from neutral surface (in y-direction).

Izz = area moment of inertia of the x-section about z-axis.

Specified cross-section

σ (+tension)

Stress distribution is linear over x-section

O

Area Moment of InertiaConcept-measure of how hard a cross-section is to rotate.

O = centroid(always lies on lines of symmetry)

y

z

A

zz dAyI 2

y

z

12

3bhI zz

h

b

O = centroid

*effect from changing h >> b

Strain Gages Measure Axial Strain

6.4 x 4.3 mm

Strain gages must be well attached to the surface of a material so that it deforms as the material deforms.

The resistance of gage changes as it stretches or compresses.

Strain is Proportional to the Change of Resistance of the Gage

R is nominal resistanceGF is gage factor-a calibration constant. Ours have GF = 2.1

Change in resistance is very small-need a circuit to measure.

Wheatstone Bridge

R4

R 1

Vmeas

VCC

R2

R3

+ -

Invented by Samuel Hunter Christie ~1833Improved by Sir Charles Wheatstone ~1843

A Possible Setup

120

1 2 0

Vmeas

5 V

120

120+dR (strain gauge)

Strain gage

Proportional to strain !

R4

In Practice, Why Do We Want Variable R?

115

1 2 0

Vmeas

10

5 V

120

120+dR (strain gauge)

Strain gage

Now That You Know About Beams and Strain Gages

You Can Make a Calibrated Scale

Ezz

z

I

yM

Relate the force applied to the strain*Compare analytical predictions to measurements

* depends on both geometry and material

F