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 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 Team