Beam Analysis
•
Structural members that offer resistance to bending caused
by loads applied along their length are known as beams.
•
The most important of all structural members.
•
How do beams behave:
1.
Determine the internal forces in beams.
2.
Understand how these forces are related to the shape of the
cross-section of a beam and the material they are constructed
from.
Universal I-beams
Universal beams in a steel
framed building
Steel box girders
RC beams and slabs
Wings and bones
Boeing 787
Dreamliner
Proxima femur
Types of beams
• Statically determinate beams
Sufficient support to prevent collapse and reactions can be
calculated from equations of equilibrium.
Simply supported
3 reactions
R1
R2
Cantilever
R3
3 reactions
R1
M1 R
2
Simply supported with overhang
3 reactions
R1
R2
R3
Types of beams
• Statically Indeterminate Beams (next year in detail)
More supports than are necessary to provide equilibrium
Continuous
Propped Cantilever
4 reactions
4 reactions
Double built-in
6 reactions
Beam Loading
• Loading can be point loads or distributed loads.
• The most common loadings are:
– Uniformly distributed line load (UDL):
(e.g. self-weight, building contents, wind loading)
w kN for each
metre span
e.g.
5 kN/m acting on 4m span  20 kN
resultant acts through centre of the UDL
Beam Loading
– Concentrated point load:
Never in reality but good approximation
Beam Loading
• Less common loading:
– Triangularly distributed line load:
e.g. hydrostatic loading, soil pressure
e.g.
Koffer Dam
subject to
hydrostatic
pressure
– Discrete couple:
e.g. due to connection with other structural member
Load effects
Tension
d/ 2
neut
ral
axis
Compression
d/ 2
hogging
Tension
y
Mz
z
d
x
Compression
Rectangular
X-section
b
Effect is internal couple
with a moment, M.
Called a Bending
Moment
Load effects
Shearing
V
V
Accompanied
by internal
shear force V
V
V
Take imaginary cut
through beam at Y-Y
BENDING
MOMENT
Resisting internal moment
to balance the moment
due to applied force
Providing an internal vertical
force to balance the
vertical applied force
What is the effect of
these two?
SHEAR
FORCE
Expedition
workshed …
Relationship between w, V and M
Fundamental relationship between loading and internal forces –
very important.
dx – small distance
dM – small change in bending
moment
dV – small change in shear force
F
FBD:
y
V  w  dx  V  dV   0
Therefore,
dV
 w
dx
A
åM
 0:
@A
(
) (
V   w dx
)
= 0 : - M + M + dM + w × dx × dx 2 -V × dx = 0
\ dM -Vdx = 0 Þ
dM
=V
dx
or M = ò V dx
Some common loading situations
1. No distributed load, w:
V   w dx  wx  c1
 V  c1
M   V dx  Vx  c2
since w  0
Constant
Linear
2. Uniformly distributed load, w:
V   w dx  wx  c1
Linear
wx 2
M   V dx    wx  c1  dx 
 c1 x  c2
2
Parabolic
Note: for maximum (and minimum) bending moment, M,
dM
V  0
dx
Therefore, maximum (and min) B.M. occurs at point of
zero shear force.
Shear force and
bending moment diagrams
• Bending moment and shear force vary along length of the beam.
• Therefore, required to produce longitudinal diagrams.
Shear force diagrams (SFD)
• Scaled, visual representation (graph) of shear force variation
along beam.
• Plotted with beam centreline as longitudinal axis.
+ ve
- ve
Positive plotted vertically up – just a convention!
Shear force and
bending moment diagrams
Bending Moment Diagrams (BMD)
•
Scaled, visual representation (graph) of bending moment
variation along beam.
compression
sagging
+ ve
tension
Sagging taken as positive bending and plotted on tension
side – just a convention!!
tension
- ve
compression
hogging