Integrated Mechanical Testing
Laboratory

College of Engineering and Applied Science

Arizona State University

## Objective

The purpose of this experiment is to determine the shear force, bending moment and the load from the strain measurements, of a cantilever beam, loaded in bending.

## Theory

The cantilever beam is a widely used structural element, for example in airplane wings, supports for overhanging roofs, the front spindles of automobiles etc. A cantilever is commonly defined as a beam which is built-in and supported at only one point , and loaded by one or more point loads or distributed loads acting perpendicular to the beam axis.

This experiment will study a cantilever beam in its simplest form- that is, a parallel-sided beam of a constant cross section , rigidly clamped at its fixed end and loaded by a single load on the beam centerline near the free end.

The cantilever beam is shown in the sketch , along with the associated shear force and bending Moment diagrams

.

Another characteristic of the cantilever beam used in this experiment is that the stress is uniaxial everywhere on the beam surface except in the immediate vicinity of the loading point and the clamped end.

The surface stress at any section, X, along the beam axis can be calculated from

where: = normal stress on the beam surface at section ,

X, psi (N/m^{2})

c = distance from neutral axis to the extreme fiber of the

beam surface, in (m)

I = moment of inertia of beam cross section, in^{4}(m^{4})

P = load, lbs (N)

B = beam width, in (m)

t = beam thickness, in (m)

Z = section modulus of beam, in^{3}(m^{3})(1) For uniaxial stress, Hooke’s law can be expressed as:

where: e = nornal strain, in/in (m/m)

E = modulus of elasticity, psi(2)

Therefore the longitudinal strain at any section , X , is

(3) The above equation demonstrates that the axial strain varies linearly along the beam from zero at the loading point to a theoretical maximum of 6PL/Ebt2 at the fixed end.

## General

## Procedure

This experiment can be performed by using a beam with three strain gages, installed uniformly spaced along the axis of the beam as shown in the gage installation diagram below.

Since the strain distribution along the beam is presumably linear, shear force can be written as

where : D M is the change in bending moment over an increment of length

defined by the corresponding change in distance,D X.(4) Solving equation (3) for M and substituting into equation (4),

(5) From eqn (5) the shear force can be obtained from the difference in strain indications of any pair of gages, divided by the distance between the gages

(6)

(7) The above eqns give the shear force and thus the load applied to the beam. Since the answers will generally differ slightly due to experimental error, their average is the best estimate of the load.

With the load known, the stress can be calculated from the following:

(8)

However, the stress can also be calculated directly from the measured strain at that point with Hooke’s law for uniaxial stress as follows:

(9)

The stresses calculated from Eqs. (8) and (9) can be compared as verification of the fundamental beam relationships used in this experiment.

## Dr. Kingsbury, Lab Manager, Integrated Mechanical Testing Laboratory.

Copyright © 2003 IMTL, Fulton School of Engineering, Arizona State University. All rights reserved.

Revised: May 25, 2005.