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Lecture 11
1
Lecture 11
1
Outline:
Shear Centre
External Work and Strain Energy
1. Shear Centre
Shear center is the point through which when a force is applied it will cause the beam to bend but not
twist.
The beam in Fig. 1 below twists and bends at the same time under the vertical load applied
through the centroid, however as shown in Fig.2 when the vertical load is applied through the shear
centre the beam only bends but does not twist.
Fig. 1. Combined twisting and bending behaviour
Fig. 2. Bending behaviour only
The location of the shear center is dependent on the geometry of the cross section only and does not
depend on the magnitude of the applied load. Previously, as in Fig. 3, we applied the shear force along
a principal centroidal axis which was the
axis of symmetry
for the crosssection. The vertical force
applied through the centroid was causing bending behavior only. This means the shear centre was on
the symmetry axis. In this lecture we are investigating the effect of the shear force along an axis that is
not the axis of symmetry of the crosssection as in Fig. 4.
Centroid
Axis of symmetry
Fig. 3. Bending behaviour only
Fig. 4. Combined twisting and bending behaviour
1
Mechanics of Solids
Dr Emre Erkmen
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Shear flow distribution in thinwalled beams
We will limit our analysis to opensection thinwalled beams. A beam can be characterized as thin
walled when the flange and/or web thickness is relatively small comparison to the depth and/or width
of the section. As we discussed in Lecture 5, the shear flow act in both longitudinal and transverse
planes, however for thinwalled sections the shear flow along the thickness direction can be considered
negligible. For instance we showed in Lecture 5 that for an Ibeam under vertical shear force, the web
carries most of the acting shear force because of the thinness of the flanges in vertical direction.
Therefore, only the shear flow that acts parallel to the walls of the members will be considered.
Under the applied shear force, the shear flow distribution along the walls can be calculated by isolating
segments of the crosssection and considering the first moment area
Q
of the isolated segment about
the neutral axis in the previously developed formula.
VQ
q
I
=
Shear flow on Ibeam
For the shear flow on the top right flange an arbitrary isolated segment is shown with the shaded area
in Fig. 5. For the shear flow on the web the isolated segment is shown with the shaded area in Fig.6.
Fig. 5. Isolated segment on the top right flange
Fig. 6. Isolated segment on the web
In this case the first moment of the areas for the segments shown in Figs 5 and 6 can be calculated as
For the top right flange
For the web
2 2
d b
Q
x t
=
−
2
2
1
2
2
4
db
d
Q
y
t
=
+
−
The shear flow distributions on the other flanges can be calculated in a similar manner. The shear flow
distribution across the section is as shown in Fig. 7 below.
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This note was uploaded on 08/22/2011 for the course ENG 48331 taught by Professor Brown during the Three '11 term at University of Technology, Sydney.
 Three '11
 BROWN
 Shear, Strain

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