Supplement Cantilever Method Page 1 of 15 Supplement: Statically Indeterminate Frames Approximate Analysis - Cantilever Method In this supplement, we consider another approximate method of solving statically indeterminate frames subjected to lateral loads known as the “Cantilever Method.” • Like the “Portal Method,” this approximate analysis provides a means to solve a statically indeterminate problem using a simple model of the structure that is statically determinate. • This method is more accurate than the “Portal Method” for tall and narrow buildings. Statically Indeterminate Frames Assumptions for the analysis of statically indeterminate frames by the cantilever method include the following. • A tall building is conceptualized as a cantilever beam as far as the axial stresses in the columns are concerned. • Axial stresses in the columns of a building frame vary linearly with distance from the center of gravity of the columns. - The linear variation is on stress, not the axial force. • Points of inflection are located at the mid-points of all the beams (horizontal members). - A “point of inflection” is where the bending moment changes from positive bending to negative bending. - Bending moment is zero at this point. • Points of inflection are located at the mid-heights of all the columns (vertical members). • The axial stresses in the columns at each story level vary as the distances of the columns from the center of gravity of the columns. • It is usually assumed that all columns in a story are of equal area, at least for preliminary analysis.
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Supplement
Cantilever Method
Page 1 of 15
Supplement: Statically Indeterminate Frames
Approximate Analysis - Cantilever Method
In this supplement, we consider another approximate method of solving statically
indeterminate frames subjected to lateral loads known as the “Cantilever Method.”
• Like the “Portal Method,” this approximate analysis provides a means to solve a
statically indeterminate problem using a simple model of the structure that is
statically determinate.
• This method is more accurate than the “Portal Method” for tall and narrow
buildings.
Statically Indeterminate Frames
Assumptions for the analysis of statically indeterminate frames by the cantilever
method include the following.
• A tall building is conceptualized as a cantilever beam as far as the axial stresses in
the columns are concerned.
• Axial stresses in the columns of a building frame vary linearly with distance from
the center of gravity of the columns.
- The linear variation is on stress, not the axial force.
• Points of inflection are located at the mid-points of all the beams (horizontal
members).
- A “point of inflection” is where the bending moment changes from positive
bending to negative bending.
- Bending moment is zero at this point.
• Points of inflection are located at the mid-heights of all the columns (vertical
members).
• The axial stresses in the columns at
each story level vary as the distances
of the columns from the center of
gravity of the columns.
• It is usually assumed that all columns
in a story are of equal area, at least
for preliminary analysis.
Supplement
Cantilever Method
Page 2 of 15
Example Problem: Statically Indeterminate Frame
Given: The 3-story frame, loaded as
shown.
Find: Analyze the frame to determine
the approximate axial forces, shear
forces, and bending moments in each
member using the cantilever method.
Comments
If we take a cut at each floor, we
expose three actions (axial force,
shear force, and bending moment) in
each of the 3 columns.
• For each cut, there are only 3 equations of equilibrium, but 9 unknowns.
• For each floor, there are 6 more unknowns than equations of equilibrium.
For the three cuts (three floors), the frame is statically indeterminate (internally)
to the 18th degree: 27 unknowns and only 9 equations of equilibrium for 3 free-
body diagrams.
• By introducing the hinges at the
mid-point of each beam and at the
mid-height of each column, we
make 15 assumptions.
• In addition, for each floor we make
assumptions regarding the
variation of the column stress
(relating 3 unknowns to a single
unknown) - an additional 6
assumptions.
• In total, we made 21 assumptions,
which allow the frame to be
analyzed using the equations of
equilibrium.
Supplement
Cantilever Method
Page 3 of 15
Solution
Top story
Find the “center of gravity (centroid)” of the columns in the top story.
• Assume in this problem that all columns in the top story have equal areas (and let