external effects on beams internal effects sign...
TRANSCRIPT
Outline:
Review – External Effects on Beams
Beams – Internal Effects
Sign Convention
Shear Force and Bending Moment Diagrams
(text method)
Relationships between Loading, Shear Force
and Bending Moments (faster method for
SFD & BMD)
ENGR 1205 1 Chapter 10
Beams are bars of material that support
Beams are common structural members
Beams can support both concentrated and
distributed loads
We will only analyze statically determinate
beams in this class
ENGR 1205 2 Chapter 10
Typical Beam Configurations
ENGR 1205 3 Chapter 10
Beams can resist:
Tension / Compression
(happy face / frown face)
Shear force
Bending moment
Torsional moment
The amount of each kind of
internal load can change
throughout the length of
the beam depending on
external loads ENGR 1205 4 Chapter 10
We can ‘cut’ the beam at any point along its
length to analyze the internal forces at that
point
It is often difficult to tell the direction of the
shear and moment without calculations so
represent Vy and Mbz in their positive directions
and let the result tell you (+/-) whether you
drew it the right way
CONVENTION
ENGR 1205 5 Chapter 10
In North America, if the
moment tends to cause the
beam to curve
it is positive; if the moment
tends to cause it to curve
it is negative.
ENGR 1205 6 Chapter 10
The bending moment
is positive if
the upper side is in
it
it’s a
ENGR 1205 7 Chapter 10
A positive shear force at the
cut causes the beam to
rotate clockwise and
matches the “smiley face” shape
Positive Shear and Moment on a whole Beam
Positive Shear and Moment on a cut-out Beam
Note: Coming from the left or the right makes
a difference, so always start from left.
ENGR 1205 8 Chapter 10
The maximum bending moment (magnitude
– could be negative or positive) is often the
primary consideration in the design of a
beam
Variations in shear and moment are best
shown graphically
ENGR 1205 9 Chapter 10
Use equilibrium of whole beam to find
external reactions and supports
Then cut each section of the the beam and
replace any distributed loads with equivalent
concentrated loads and solve for unknown
axial, shear and moment at point x (your cut)
Plot Vy (shear force vs x) and Mbz (bending
moment vs x) along the beam
ENGR 1205 10 Chapter 10
Solve for R1 and R2
Isolate left section and solve for Vy
and Mbz between the left support
and 4kN load
Isolate right section and solve for
Vy and Mbz between the right
support and 4 kN load
Isolate a section between every
change in an external load, but do
not cut the section at the
concentrated load
ENGR 1205 11 Chapter 10
Draw the shear and bending moment diagrams
for the diving board, which supports the 80 kg
man. Determine the location and magnitude
of the maximum bending moment.
ENGR 1205 12 Chapter 10
ENGR 1205 Chapter 10 13
ENGR 1205 Chapter 10 14
Draw the shear and bending moment diagrams
for the beam and loading shown and
determine the location and magnitude of the
maximum bending moment.
ENGR 1205 15 Chapter 10
ENGR 1205 Chapter 10 16
Draw the shear and bending moment diagrams
for the beam and loading shown.
ENGR 1205 17 Chapter 10
Diagrams illustrate the value of the internal
forces (axial/shear/moment) that occur at
each point along a structure (beam).
Additionally,
ENGR 1205 18 Chapter 10
The shear diagram is the graphic
representation of the shear force at
successive points along the beam.
The shear force (Vy) at any point is equal to
the algebraic sum of the external loads and
reactions to the left of that point.
Since the entire beam must be in equilibrium
(the sum of Fy = 0), the shear diagram must
close at zero.
ENGR 1205 19 Chapter 10
The moment diagram is the graphical
representation of the magnitude of the
bending moment at successive points along the
beam.
The bending moment for the moment diagram
(Mbz) at any point equals the sum of moments
of the forces on the beam to the left about
that point.
Since the entire beam is in equilibrium (Sum of
M = 0), the bending moment diagram must
close to zero at right side.
ENGR 1205 20 Chapter 10
Cut the beam between each concentrated
load.
For each section solve for the unknown shear
force and bending moment
Equation for each in terms of x (the
distance along the beam
Sub in endpoint values of x to get numerical
values of Vy and Mbz at each cut
Plot Vy (shear force vs x) and Mbz (bending
moment vs x) along the beam
ENGR 1205 21 Chapter 10
The “text” method works every time, but it is
time consuming
You can draw the diagrams from the FBD if
you know the relationships between w (load
intensity), V (shear force) and M (bending
moment).
the negative slope of the slope of the
the shear diagram at moment diagram at the
a given point equals given point is shear
the load at the point at the point
ENGR 1205 22 Chapter 10
dx
dMV
dx
dVw
If there is no change in the load along the length
under consideration, the shear curve is a straight
horizontal line (or a curve of zero slope).
If a concentrated load exists, then there is a
vertical jump in the SFD ( force = drop down on
SFD, force = step up on SFD)
If a load exists, and is uniformly distributed, the
slope of the shear curve is constant and non-
horizontal.
If a load exists, and increases in magnitude over
successive increments, the slope of the shear
curve is positive (approaches the vertical); if the
magnitude decreases, the slope of the shear curve
is negative (approaches the horizontal). ENGR 1205 23 Chapter 10
If the slope of the SFD is zero, then the moment curve
has a constant slope that is equal to the value of the
shear for that increment.
If the slope of the SFD is positive, then the slope of
the moment curve is getting steeper.
If the slope of the SFD is negative, then slope of the
moment curve is getting flatter.
Changes in the shear diagram will produce changes in
the shape of the moment curve.
The area under the shear curve between two points is
equal to the change in bending moment between the
same two points.
ENGR 1205 24 Chapter 10
The area of the shear diagram to the left or to
the right of the section is equal to the moment
at that section.
The slope of the moment diagram at a given
point is the shear at that point.
The maximum moment occurs at the point of
zero shear. When the shear (also the slope of
the moment diagram) is zero, the tangent
drawn to the moment diagram is horizontal.
ENGR 1205 25 Chapter 10
ENGR 1205 26 Chapter 10
ENGR 1205 27 Chapter 10
Draw the shear and bending moment diagrams
for the beam and loading shown and
determine the location and magnitude of the
maximum bending moment.
ENGR 1205 28 Chapter 10
ENGR 1205 Chapter 10 29
ENGR 1205 Chapter 10 30
Draw the shear and bending moment diagrams for
the beam and loading shown and determine the
location and magnitude of the maximum bending
moment.
ENGR 1205 31 Chapter 10
ENGR 1205 Chapter 10 32
ENGR 1205 Chapter 10 33
ENGR 1205 Chapter 10 34
Draw the shear and bending moment diagrams
for the beam and loading shown.
ENGR 1205 35 Chapter 10