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Bending stresses
Outcomes
This section addresses aspects of the following syllabus outcomes:
A student:
H6.2 demonstrates skills in analysis, synthesis and experimentation related to engineering
Source: Board of Studies NSW (1999) Engineering studies,
preliminary and HSC courses.Board of Studies, Sydney.
Bending stresses
The behaviour of beams under load depends on the:
- the materials from which the beam is made
- the nature of the forces acting on the beam
- the size of the beam
- the shape of the beam
- the orientation of the beam with respect to the direction of the forces applied.
In this unit we are concerned only with rectangular beams but the concepts discussed apply to more complex shapes.
This unit aims to investigate the nature of internal forces involved in bending and to consider the relationship between beam geometry, bending moments and the resultant bending stress. This information helps engineers to determine suitable materials, cross-sectional shapes and orientations for civil engineering applications.
Internal forces
Consider a simple horizontal beam supported at each end and
with a load in the centre, sufficient to cause the beam to bend. The material
on the inside of the bend will be placed into compression while the material on
the outer side of the bend will be in tension. For a graphic illustration visit
Bending of beams 
where grid lines, drawn on a foam beam prior to bending are distorted in a clear
demonstration of compression and tension. If one side of a beam during bending
is in tension while the other is in compression it stands to reason that
somewhere between the two lies a neutral plane with no loading.
Also go to University of Wisconsin for additional information.
The centroid
The geometric centre of a two-dimensional area is known as the centroid. In relation to beams the centroid will be the geometric centre of the cross-sectional shape of the beam. It is significant because the neutral plane, discussed above, will pass through the centroid of the beam. When considering a rectangular beam, the centroid is simply found by joining the diagonals but becomes more complex for composite shapes.
Orientation of beams
Does a beam have the same strength properties irrespective of its orientation to the forces, that is whether we use it laying flat or on edge? Or, can we improve the strength characteristics of beams by using particular orientations? Activity 1 aims to investigate this concept.
Aim:
To investigate the significance of the geometry of the
cross-section and in particular to develop a relationship between distance from
the neutral axis and strength.
Preparation:
To carry out this experiment you will need:
- ice-cream sticks, about 8 x 1.5 mm to simulate a “beam”
- a bucket with a handle
- a second bucket with water in it
- a 100 mL measure
- two short lengths of 50 x 25 mm timber pieces with a 1.5 mm slot cut across them to allow the “beam” to be held on its edge (see Figure 1). These slots could be trenched by hand with a tenon saw or by your teacher using a circular saw.
Method:
Step 1. Place the handle of the bucket at the centre of a
“beam” and support the “beam” on its edge at both ends
with the timber pieces (as shown in Figure 1). Test the
“beam” to breaking point by adding water to the bucket 100 mL at a
time. Measure and record the water quantity when the “beam”
breaks.
Figure 1 Set up for experiment to test beam on its edge
Step 2. Repeat with the “beam” lying flat as shown in Figure 2.
Figure 2 Set up for experiment to test beam lying flat
Note: For greater accuracy this activity should be repeated several times and the results averaged.
Results
You will note a significant difference emerges in the load
capacity of the beam depending upon its orientation (see Results of Activity 1).
The “beams” are the same size, the same material and the same
length, but the load capacity is different for different orientations. In fact,
if you measure your ice-cream stick “beam” you should find that the
load capacity in each case is in proportion to half the depth of the beam. For
example, if your ice-cream stick was 8 mm x 1.5 mm, then the beam on edge should
have supported at least five times the load for when it was lying
flat.
Revision
It may be beneficial for you to review the concepts of shear
force and bending moments at this time, visit:
Bending moments and follow the link to 4.1 Shear Forces and Bending Moments I.
The result is best explained in terms of the maximum distance from the neutral plane to the top or bottom edge of the “beam”. When the ice-cream stick was lying flat there was a short distance between the bottom of the beam and the neutral axis, whilst when on edge this distance is significantly greater.
Bending moments
When external forces are applied to a beam they create internal shear forces and bending moments within the beam. The magnitude of these varies from one end of the beam to the other depending on the location and direction of the applied forces and the reactions.
For a beam to resist an applied bending moment, the material from which it is made must develop a moment over its cross-sectional area that resists the applied moment. An extremely thin beam has little ability to resist the applied moment (little material between the neutral axis and the top or bottom of the beam. The more material the beam has on either side of the neutral axis, the greater is its ability to resist the applied moment and therefore the applied load.
Moment of inertia
The nature and calculation of the resistive moment in a beam is very complex. As we have seen, it depends in part on the orientation of the beam and therefore the distance “y” from the neutral plane to the edge of the beam. It also depends on the size, shape and distribution of material about the centroidal axis.
This second component, the
moment of inertia ,
or second moment of area, varies according to the geometry of the beam and is
usually calculated or read from tables. The Engineering Studies syllabus
requires that the Moment of Inertia (I) is provided and therefore it is not
necessary for candidates to calculate the moment of inertia.
Bending stresses
Bending stresses are internal stresses caused by bending moments acting at a given distance from the neutral axis. Their magnitude is given by the formula:
Details of a beam under load are shown. Calculate the maximum bending stress for the beam.
Web links to Useful resources
Bending and Shear
Beam Shear Behaviour
Neutral Axis
Second Moment of Area
Pure Bending
Moment of Inertia
Material Property Search
Resources
Copeland, P.L. (2001) Engineering studies; the definitive guide. Volume 2. Star Printery Sydney.
Ivanoff, V. (1997) Engineering mechanics. McGraw-Hill, Sydney
Schlenker, B. R. & McKern, D. (1994) Introduction to engineering mechanics. Jacaranda Press, Sydney.
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