Intuitively, the strength of a beam is proportional to "the amount of force that may be placed upon it before it begins to noticeably bend." One of the goals of this lab is to quantify the factors that determine the strength of a beam.

The strategy is to mathematically describe the quantities that affect the
deformation of a beam, and to relate these quantities through
a *differential
equation * that describes the bending of a beam.
The quantities that we will meet in this module are:

- Material Properties
- The amount by which a material stretches or compresses when subjected to
a given force per unit area is measured by the modulus of
elasticity. For
small loads, there is an approximately linear relationship
between the force per area (called
*stress*) and the elongation per unit length (called*strain*) that the beam experiences. The slope of this stress-strain relationship is the modulus of elasticity. In intuitive terms, the larger the modulus of elasticity, the more rigid the material. - Load
- When a force is applied to a beam, the
force is called a
*load*, since the force is often the result of stacking or distributing some mass on top of the beam and considering the resulting force due to gravity. The shape of the mass distribution (or, more generally, the shape of the load) is a key factor in determining how the beam will bend. - Cross section
- The cross-section of a beam is determined
by taking an imaginary cut through the beam perpendicular to
the beam's bending axis. For example, engineers sometimes use
"I-beams" and "T-beams" which have cross-sections that look like
the letters "I" and "T". (The IT sections are especially popular
at the University of Minnesota.) The cross-section of a beam
determines how a beam reacts to a load, and for this module we
will always assume that the beam is a so-called
*prismatic beam*with a uniform cross-section. The important mathematical properties of a cross-section are its*centroid*and*moment of inertia*. - Support
- The way in which a beam is
supported also affects the way the beam bends. Mathematically,
the method by which a beam is supported determines the
*boundary conditions*for the differential equation that models the deflection of the beam. The next sections study*centroids*,*moments of inertia*, and*boundary conditions*in order to mathematically describe beam cross-sections and beam supports. We also study how these factors come together as constituent parts of the*differential equation*modeling the static deflection of beams.The final section provides a "beam simulator" for interactively experimenting with beam deformations and for comparing the theoretical model with experimental data.

**Next:**Centers of Mass and Centroids

**Return to:**Outline

The Geometry Center Calculus Development TeamCopyright © 1996 by The Geometry Center. Last modified: Fri Apr 12 15:50:48 1996