Emily Bondank

MAT 266 Brewer

Honors Project May 2011

Calculating Bending Moment for Beams

A structural beam is designed to support a number of forces over its span. The sum of all of the forces on the beam is called the load. The individual forces that add up to create the load can be distributed in a variation of ways but no matter the distribution, the load applies the most stress to only one point in the beam. The point that the stress is directed to is at the fixed end of the beam. In engineering, the amount of stress applied to this part of the beam is called the ‘Bending Moment’ and is used as an indicator to tell how likely the beam is to break. The Bending Moment can be calculated for any variation of load distribution on a beam using the knowledge of functions, substitution, differential equations, and integration.

Step One

The equation for the Bending Moment must be determined through finding its variables and determining the relationship between the Bending Moment and those variables.

  • The greater the load on the beam and the farther the distance the load is from the beam denotes a greater stress applied to the fixed end and a larger Bending Moment.
  • From this knowledge, it can be determined that the variables in the equation of Bending moment are load and distance from the fixed end of the beam. It can also be determined that the relationship between the Bending Moment and both of these variables is linear.

Therefore the equation for Bending Moment is:

, whereM denotes Bending Moment, l denotes load, and d denotes distance from the load to the fixed end of the beam.

Step Two

It is necessary to determine the equation for load by finding the variables and constants that determine load and the relationship those variables and constants have with the load.

  • Load is represented as a constant with a value that depends on the magnitude of the distributed load, with units of load divided by units of distance, multiplied by the distance from the load to the fixed end of the beam.
  • There is a relationship between the constant and the distance along the beam defined by the function c(d).

Therefore the equation for load can be written as:

Step Three

The load equation now needs to be substituted into the Bending Moment Equation and a differential equation needs to be written as the product of the constant function and an infinitely small change in the independent variable, distance.

Two equations:

Through substitution one equation is made:

Now the small change in d is made in the substituted load function:

Step Four

The function of change in Bending Moment is integrated in order to determine the sum of Bending Moments and the sum of the individual forces that the load consists of.

The equation is rearranged to prepare for integration:

and

The equation is then integrated from length zero to length d, to determine the sum of the forces:

Once the function for c(d) is imputed into the integral the exact value of the Bending Moment can be found.

Example

Determining the Bending

Moment for a Parabolic

type of load

Step One

Step Two

Step Three

Step Four

  • A parabolic load is a load whose force varies quadratically along the beam's length and which is zero at one of the beam's ends. The load of the beam is:

Reference: