The Main Ideas
Beams that strengthen a structure are subject to stresses put upon
them by the weight of the structure and by external forces such as
wind. How does an engineer know that the beams will be able to
withstand such stresses? The answer to this question begins with
the linear analysis of static deflections of beams, which is the topic
of this module.
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
The strategy is to mathematically describe the quantities that affect the
deformation of a beam, and to relate these quantities through
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
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.
- 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 .
- 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
Next: Centers of Mass and Centroids
Return to: Outline
The Geometry Center Calculus Development Team
Copyright © 1996 by The Geometry Center.
Last modified: Fri Apr 12 15:50:48 1996