Bending
BEAMS... RODS... STRESS...SHELLS . LONG AGO, THE FOUR ELEMENTS LIVED
TOGETHER IN HARMONY. THEN EVERYTHING CHANGED WHEN THE STRESS
BECAME APPLIED PERPENDICULARLY TO A LONGITUDINAL AXIS
Gottfried Wilhelm Leibniz

Philosopher

Mathematician

Scientist

Biology

Geology

Psychology

Computer science

Physicist

Engineer

Linguistician

Librarian

Lawyer

Philologist
Made important contributions to:
1646-1716
Mathematics
Metaphysics
Epistemology (the investigation of
what distinguishes justified belief from
opinion)
Logic
Philosophy
Physics
Geology
Jurisprudence (philosophy of law)
History
Technology
Ethics
Probability Theory
What do we care about?
3 big Topics:
-Math
-Science (physics in particular)
-Engineering
Then
Let’s
Next
start
onto
wewith
science…
have
math…
engineering…
Was
Leibniz
Discovered
big
believed
in the
calculus
world
greatly
of
(yeah,
statics
in applying
I know,
and dynamics
theory
I know,toNewton
real world
discovered
applications
calculus,


right? Well not
really.
Disagreed
“father
Leibniz
of applied
actually
with Descartes
science”
discovered
and Newton
calculuson
at many
the same
subjects
time as, and independently of
Designed
Newton)
He
many
saw useful
space,items
time, and motion as relative, whereas Newton thought them to be

Wind-driven
propellers,
water
pumps,
mining machines,
hydraulic
Many
absolute
of the
notations
Leibniz
created
are still used
todaypresses,
lamps,
submarines,
clocks,
steam
engine
(with
Papin)Einstein
 E.g
He
therecognized
integral
sign
this
and
200
using
yearsd
before
for Denis
derivatives
Albert
Contributed
tofirst
computer
 Was the
found
the
energy
to use
ofintegrals
ascience
systemto
tofind
be mv
the2area under a curve
 Documented the binary numeral system
 Discovered
This led him
theto
product
his Theory
ruleof
forMotion
differentiation
pertaining to kinetic and potential energy
 While studying the system, he imagined a machine that could represent binary
He realized that the total energy in a system will be conserved
numbers


He actually envisioned the first computer
Brushed upon the concept of feedback
Academic Life

His father was a professor of history

When he died, Gottfried inherited his library

First studied math when he was 17

Never formally learned physics

Went to Law School
Hermann von Helmholtz

German physicist

CONSERVATION OF ENERGY!!!

He theorized a relationship between mechanics, heat, light,
electricity, and magnetism

Put them all into one category under a single force, or energy

Went to school for medicine

Never attended any math courses

Was completely self-taught
1821-1894
The Art and Science of Bending
Basics:
Definition: The change in form of an object when a load is applied perpendicularly to
a longitudinal axis of the object
Example:
The Types of Setups for Bending

Simple

Cantilever
The Quasistatic Scenario
Don’t worry, it just means that the amount of
bending, and the forces, don’t change over time
Euler-Bernoulli Theory of Bending
Applies to simple bending
only

The theory was constructed using a combination of four different
aspects of beam theory

Kinematic


Constitutive


Describes how the direct stress and direct strain within the beam are related
Force Resultant


The motion of the deflection
Help track important forces in the beam
Equilibrium

How the internal stresses of the beam balance out an external load
The Equation
Amount of deflection
How much will it deflect in relation
to the load placed on it?
4
dw
The Applied Load
EI dx4 = q(x)
That’s how much
Young’s Modulus
Area Moment of Inertia
The Components of that Equation

Young’s Modulus
 Mechanical
property of linear elastic solid materials
 Measures
the force needed to stretch or compress a
material sample

Area Moment of Inertia
 Shows
how an area’s points are distributed with
respect to an arbitrary axis
Bending Stress in a Beam
Some Prior Knowledge

Basis: It is assumed that plane sections in the beam will remain plane

This means that no shear forces on a section are taken into account

Therefore the theory does not deal with shear forces on the object
Another Drawback

Also relies on the fact that the yield stress of the object is greater
than the maximum applied stress

Yield stress is the amount of stress an object can take before it deforms
plastically rather than elastically

A material that deforms elastically returns to its original shape

A material that deforms plastically will remain deformed permanently to
some extent and is irreversible
And Another…

The material must follow Hooke’s law
And Another….

The beam must be initially straight, and must have a cross section that is constant
throughout.
And Another….

The beam must have an axis of symmetry in the plane of bending
And Another….

The beam must have a tendency to fail by bending rather than by crushing,
wrinkling, or buckling
So what’s it good for then?
Well that’s a good question

It does really well to explain simple cases of
bending

Provides a wonderful basis for the understanding
of bending

Gave other scientists a platform to work off of

Engineers use it to analyze simple beams under
an applied transverse load
Timoshenko to the Rescue!!


Stephen Timoshenko

Russian Engineer

Contributed immensely to the subjects of engineering mechanics,
elasticity, and the strength of materials
Improved the Euler-Bernoulli theory in 1921

How, you ask? Well, he accounted for the effect that shear forces had
on a beam

This made the theory much more accurate and applicable
The New, Improved, Equation

Kinematic assumptions

Normals to the axis stay straight even after
deformation

Beam thickness is the same before and
after deformation
Shear Correction Factor
Cross-sectional Area
Shear Modulus (The ratio of the shear stress
to the shear strain)
Dynamic Bending
Put that beam in motion, and watch it go, go, go
It’s not that bad, actually

Includes the mass and the second derivative of deflection with
respect to time
Oh…Never mind, I guess

Includes the effect rotational inertia of the cross-section of the beam

Also includes the shear stresses and accounts for shear deformations
Applications of Bending


Airplane wings “flutter”

Basically are bending dynamically

http://www.numeca.com/sites/numeca/files/wing.gif
Many structures have materials under a longitudinal load, which
bend

By analyzing and understanding bending, these structures can be
designed to be safer and better
Airplane Wing Bending
Bibliography

"Beam Defelction." (n.d.): 396-419. Www.me.berkeley.edu. Web. 30 July 2015.
<www.me.berkeley.edu/~lwlin/me128/BeamDeflection.pdf>.

"Bending." Wikipedia. Wikimedia Foundation, 1 July 2015. Web. 30 July 2015.

"Elastic Bending Theory." Elastic Bending Theory. N.p., 25 Jan. 2013. Web. 30 July
2015.

"Euler-Bernoulli Beam Equation." Euler-Bernoulli Beam Equation. N.p., n.d. Web.
30 July 2015.

"Gottfried Wilhelm Leibniz." Gottfried Wilhelm Leibniz. N.p., n.d. Web. 30 July
2015.

"Hermann Von Helmholtz." Wikipedia. Wikimedia Foundation, n.d. Web. 30 July
2015.

"Leibniz, Gottfried (1646-1716) -- from Eric Weisstein's World of Scientific
Biography." Leibniz, Gottfried (1646-1716) -- from Eric Weisstein's World of
Scientific Biography. N.p., n.d. Web. 30 July 2015.

Smith, Kevin Randall. "Psyography: Biographies on Psychologists." Psyography:
Biographies on Psychologists. N.p., n.d. Web. 30 July 2015.