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 AXISLONGITUDINAL AXIS

Gottfried Wilhelm Leibniz  Philosopher  Mathematician  Scientist  Biology  Geology  Psychology  Computer science  Physicist  Engineer  Linguistician  Librarian  Lawyer  Philologist  Sinophile Made important contributions to:  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

Let’s start with math…  Discovered calculus (yeah, I know, I know, Newton discovered calculus, right? Well not really. Leibniz actually discovered calculus at the same time as, and independently of Newton)  Many of the notations Leibniz created are still used today  E.g the integral sign and using d for derivatives  Was the first to use integrals to find the area under a curve  Discovered the product rule for differentiation

Then onto science…  Was big in the world of statics and dynamics  Disagreed with Descartes and Newton on many subjects  He saw space, time, and motion as relative, whereas Newton thought them to be absolute  He recognized this xxx years before Albert Einstein  This led him to his Theory of Motion pertaining to kinetic and potential energy  He realized that the total energy in a system will be conserved

 Next we have engineering…  Leibniz believed greatly in applying theory to real world applications  “father of applied science”  Designed many useful items  Wind-driven propellers, water pumps, mining machines, hydraulic presses, lamps, submarines, clocks, steam engine (with Denis Papin)  Contributed to computer science  Documented the binary numeral system  While studying the system, he imagined a machine that could represent binary numbers  He actually envisioned the first computer  Brushed upon the concept of feedback

Academic Life

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 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  Force Resultant  Equilibrium

The Equation How much will it deflect in relation to the load placed on it? EI = w(x) d4ud4u dx 4 That’s how much Amount of deflection Young’s Modulus Area Moment of Inertia The Applied Load

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  The beam must be initially straight, and must have a cross section that is constant throughout.  The beam must have an axis of symmetry in the plane of bending  The beam must have a tendency to fail by bending rather than by crushing, wrinkling, or buckling And Another….

So what’s it good for then? Haha, 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

Dynamic Bending Put that beam in motion, and watch it go, go, go