afst.bundel 8 MEI 07 - Technische Universiteit Eindhoven

afstudeerbundel faculteit bouwkunde
58
J.H. van Zanten
The influence of a non-uniform
building cross-section on its
structural behaviour
An analytical verification method of numerical studies
Afstudeerrichting
Structural design
Afstudeercommissie
Dr. Ir. M.C.M. Bakker
Dr. Ir. J.C.D. Hoenderkamp
Prof. Ir. H.H. Snijder
Datum afstuderen
29 januari 2008
Samenvatting
In dit project heb ik geprobeerd om een eenvoudige handberekeningsmethode
op te stellen, waarmee men een complexe
hoogbouw constructie kan dimensioneren en ontwerpen in het
beginstadium van een project. Hierbij heb ik met name de
nadruk gelegd op een mogelijke verandering in plattegrond van
het gebouw. Ik heb geprobeerd om een benaderingsmethode te
ontwikkelen, waarmee in een vroeg stadium de horizontale verplaatsingen
van zo’n gebouw kunnen worden bepaald met relatief
eenvoudige handberekeningen. Deze berekeningen kunnen
in een later stadium gebruikt worden om de complexe eindige
elementen modellen te toetsen.
Trefwoorden
Shear
High-rise
Analytical
Deflection
Bending
Problem, Goal
This project is a derivative of the problems involved in verifying
the finite element method models of complex structures.
The goal for this project was to design a simple hand calculation
method to analyse the deflections of a complex high-rise
structure with shifts in the neutral axes of the structure.
Approach
The original structure consists of eight storeys, each with a
different cross-section. This structure is simplified by making a
finite element method model with only three different crosssections,
each three storeys high. Complicating factors of the
design are, among others, the shifts in neutral axes of the
structure, the misalignment of the shear centre en centroid of
the middle cross-section, the discrete character of the structure,
and the torsional moments on the structure due to wind
loading. In this research, the structure is examined, subjected to
a point load and a torsional moment at the top of the structure.
The structure is approximated by means of an “equivalent
non-uniform column”. However, in order to be able to
investigate this non-uniform structure, first three uniform
structures are examined, that later combine to form the nonuniform
structure. These uniform structures are examined with
“equivalent uniform columns”.
The properties of the individual cross-sections are implemented
in the non-uniform column.
All cross-sections are examined for shear, bending and torsional
deflections, where the top cross-section is rotated compared to
the other two cross-sections, so this cross-section is subject to
“double bending”.
Boundary conditions
A number of boundary conditions are used to enable the
comparison of the 3D truss structures to the equivalent
columns. For example, the top floors of the uniform structures
are not only infinitely stiff in plane, but also infinitely stiff out of
plane. This is also the case for the floors at the connecting levels
of the non-uniform structure. This ensures that plain sections
remain plane, which is a boundary condition necessary for
applying Steiners theorem. This theorem is at the basis of the
equivalent column theory, so it must be applied in order to
compare the two types of structures. Another condition that
must be met is that all connections of all elements in the truss
must be hinged connections. The analysis itself is linear
elastic only, and finally, warping for the middle cross-section is
not taken into account. This is accomplished by transforming
the middle cross-section into a “Neuber tube”. This ensures that
no warping occurs, only pure torsion. The difference between
the “Neuber tube” of the middle cross-section, and the original
middle cross-section is assumed to account for the warping
stiffness. This assumption is still subject to debate, and requires
further research in order to be shown valid.
Results
If the results of the uniform and non-uniform structures are
compared with the hand calculations and the equivalent
columns, the shear deflections are found to correlate completely.
Between the results of the bending deflections the differences
are minimal (the maximum difference is less than 2%), and the
torsional deflections correlate exactly as well.
All of these differences are within the boundaries set in the goal
for this project (maximum difference of 5%), and furthermore,
the differences can be explained. The most important reasons
for the differences between the different models are the
discretization errors. For a nine storey uniform structure, the
discretization error can be computed to be 0.309%.
This matches the differences found for the uniform structures.
For the non-uniform structures the discretization error is larger,
but the deformations are all within the goals set for this project.
Due to the positioning of the equivalent columns (in the shear
centres) the Z-deflections of the equivalent non-uniform column
deviate from the Z-deflection of the simplified non-uniform
structure. The X- and Y-deflections however correlate completely.
This is expected, and it is suggested that accurate X- and Y-
deflections are more important.