Efficient three-dimensional modelling of high-rise building structures.
Jameel, Mohammed ; Islam, A.B.M. Saiful ; Khaleel, Mohammed 等
Introduction
Incorporation of wall openings and staircase providing slab
openings are essential in building structures, which should be precisely
investigated. Furthermore, the inclusion of shear walls and slabs
improves the lateral stiffness, and thus the structural performance of a
building. Shariq et al. (2008) mentioned that the adequate lateral
stiffness in buildings may be achieved by providing shear walls, which
resist the lateral forces primarily due to their high in-plane
stiffness. In multi-storey buildings, shear walls placed in the form of
elevator cores, enclosed stairways, shear boxes or facade shear walls
are capable of providing considerable lateral stiffness to the structure
to enable it to resist horizontal loadings such as earthquakes and wind
(Madsen et al. 2003). These components of structures bear wall/slab
opening and staircases are additionally added. These corresponding
studies have been carried out by several researchers such as Kim and Lee
(2003, 2005), Kim et al. (2005), Kim and Foutch (2007) and Lin et al.
(2011). Combining a frame system and a shear wall system is appropriate
for a multi-storey building with shear walls arranged around the lift
shafts and stair wells. Both shear walls and frames participate in
resisting the lateral loads resulting from earthquakes or wind or
storms, and the portion of the forces resisted by each one depends on
its rigidity, modulus of elasticity and its ductility, and the
possibility to develop plastic hinges in its parts (Wang et al. 2001;
Islam et al. 2012a, b; Balkaya, Kalkan 2003, 2004). Simulation-based
assessment and multiple criteria assessment works on multi-storey
buildings have been carried out by Jameel et al. (2011)
In structural modelling, it is required to consider the wall
openings and slab opening since openings commonly exist in multi-storey
buildings, for functional reasons (such as doors, windows, air ducts,
etc.) as well as to accommodate staircases and lift shaft. These wall
openings may reduce the lateral stiffness of a building structure.
Experimental tests indicated that slender shear walls containing
openings are susceptible to unpredictable failure characteristics due to
buckling and excessive cracking around the openings (Guan et al. 2010).
Realising the importance of considering wall openings in structural
design and analysis, more researchers have incorporated wall openings
into their research studies (Shariq et al. 2008; Dolsek, Fajfar 2008a).
Moreover, RC frame with masonry infill is a popular structural
system in many parts of the world. Infill panels can change the overall
resistance and stiffness of buildings (Borzi et al. 2008). According to
findings of Kose (2009), RC frames with infill shear walls had a shorter
period of approximately 5-10%, compared with RC frames without infill
shear walls regardless of whether they had shear walls or not. Dolsek
and Fajfar (2008a, b) have shown that 'masonry infill highly
increases the stiffness and strength of a structure as long as the
seismic demand does not exceed the deformation capacity of the infills;
after that, both the global stiffness and the global strength strongly
deteriorate' and 'the infills can completely change the
distribution of damage throughout the structure'.
Along with the shear walls and slabs of a multi-storey building,
staircases can also significantly increase the building resistance
(Borzi et al. 2008). However, very limited studies have been done on the
effect/role of staircase on the lateral stiffness or structural response
of multi-storey buildings. Experimental assessment of vibration
serviceability of stair systems has been performed by Kim et al. (2008),
involving steel stair system and RC stair system. Theoretically,
staircase, which is similar to an inclined slab that is connecting slabs
of adjacent storeys, may acts as a diagonal brace between floors, thus
contributing to both vertical and horizontal stiffness of the structure.
This may help to limit the lateral deflection of the building subjected
to wind load and/ or seismic force.
Although numerous studies have been performed on multi-storey
high-rise buildings, proper research on efficient design which includes
accurate effect of wall/slab openings and staircases is still lacking.
It is detected that shear walls (both RC shear wall and masonry wall)
and slabs offer some structural strength, thus contributing to the
lateral stiffness of the structure, which might lead to economical
design and material savings. Simultaneously, wall openings needs to be
taken into consideration in structural analysis to avoid overestimating
the structural stiffness of a designed building. Studies can be done to
investigate the effect of considering shear walls, slabs and wall
openings in modelling and analysis, as compared to the conventional
frame structure concept. Furthermore, it is still unknown to what extent
the staircase contributes to the lateral stiffness of the structure.
So the objectives of the study are:
--To carry out the analysis of a multi-storey building, considering
the effect of a masonry infill wall;
--To study the effect of shear wall and slabs on the response of a
multi-storey framed structure;
--To investigate the effect of a wall opening on the response of a
multi-storey frame-shear wall system structure;
--To investigate the influence of staircases on the behaviour of
the multi-storey frame structure.
1. Structural model
Due to computational complexity and time-consuming nature of
analysing shear walls and slabs, the analytical method is almost
impossible without any assumptions, approximations and simplifications.
As for the experimental method, it involves prohibitively high expenses
and material wastages, thus not economically feasible in real-world
engineering design. Therefore, the finite element analysis is applied in
this study, with the help of sophisticated engineering software and
modern high-end workstation.
1.1. Finite element modelling
For each research objective, several multi-storey building
structures with the same plan view configuration have been modelled and
analysed in ETABS. However, these modelling cases might be different in
terms of the number of storeys and/or type of modelling concept applied.
All the modelling cases are done in three dimensional instead of two
dimensional, to obtain a more accurate analysis result. The material
properties and the sectional properties for structural elements are
standardised/made consistent in this research. Relevant analysis results
are then presented in figures and tables for an easy comparison. A total
of 61 modelling cases have been analysed in this study.
1.2. Configurations of building components
Out of 61 models in this research, which studied the effect of
shear walls, slabs, wall openings and a masonry wall, 56 have the same
plan view as the basis for comparison. The common plan view of these 56
modelling cases is shown in Figure 1. In this illustration, the thin
line indicates concrete beams, whereas the thicker line indicates shear
walls. The smallest square at the intersection of the gridlines
indicates concrete columns. The lift core is placed at the very centre
of the plan view of the modelled building.
This simple and symmetrical plan view has been adopted, such that
the analysis result in this research is not affected by uncertainties or
factors such as a complex building shape. As such, the analysis results
of the modelling can be interpreted more readily. The modelled
multi-storey building structure is symmetrical about both x-axis and
y-axis, with the total planned area of 42x42 m. The dimension of each
panel is 6x6 m, as all beams are 6 m in span length, supported by
columns. Thus, column-to-column distance is only 6 m. Each storey is 3-m
high, throughout the multi-storey building. The sectional properties of
the structural elements are standardised as mentioned in Table 1. It is
noted that a shear wall is considered as a concrete wall except for
cases when masonry is mentioned.
[FIGURE 1 OMITTED]
A shell element is used to model the shear wall and slab in ETABS,
to take into consideration both in-plane membrane stiffness and
out-of-plane plate bending stiffness of the section. Any unmeshed shell
element has an unrealistically high stiffness. Thus, the shear walls and
slabs in the modelling are meshed into finer elements in order to
improve the accuracy of the simulation result, and better reflect the
actual behaviour of a real structure. The major trade-off, however, is
the increase in the analysis time taken by the programme. Thus, the mesh
size of the element used is compromised between the computational time
and accuracy.
The common plan view for the remaining five modelling cases with
staircase is similar to that without a staircase, only with a slight
modification to accommodate the staircases. Panel dimension, storey
height, standardised sectional properties of structural elements, etc.
remain the same. The common plan view for modelling with staircase is as
shown in Figure 2a. To better illustrate the modelling detail,
three-dimensional view for a 5-storey building structure modelling is
shown in Figure 2b.
[FIGURE 2 OMITTED]
In the modelling, there are openings on the slab (3.0x4.5 m) to
accommodate the staircase. This, however, might greatly reduce the
stiffness as well as the stability of the multi-storey building
structure. Hence, in order to maintain the structural stability, some
beams and columns have been added at the edge and corner of a slab
opening, respectively, while supporting the staircase at the same time.
Some additional beams are added at each mid-storey height to partially
support the span of the staircase.
2. Finite element formulation
Based on Newton's Second Law of Motion and
D'Alembert's Principle (Fraser 1985) of Dynamic Equilibrium,
the equation of motion governing the deformation or displacement u(t) of
the idealised MDF (Multi-Degree-of-Freedom) system structure, assuming
to be linearly elastic and subjected to an external dynamic force, p(t),
is given by the following matrix equation:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)
where: the term m is mass matrix of the structure; c is the damping
matrix of the structure; k is the stiffness matrix of the structure; u
is the displacement matrix; [??] is the velocity matrix and [??] is the
acceleration matrix:
The problem with free vibration requires that the external dynamic
force, p(t), to be zero. In addition, for systems without damping, the
damping matrix, c, is also zero. Thus, by substituting p(t) = 0 and c =
0 into Eqn (1), the matrix equation governing the free vibration of
linear MDF systems becomes:
m[??] + ku = 0. (2)
The free vibration of an undamped system of its natural vibration
modes for a MDF system can be described mathematically by:
u(t) = [q.sub.n](t)[[phi].sub.n]; (3)
where: [q.sub.n](t) is the time variation of the displacements;
[[phi].sub.n] is the deflected shape, which does not vary with time. The
time variation of the displacements; [q.sub.n](t) can be described by
the following simple harmonic function:
[q.sub.n](t) = [A.sub.n] cos [[omega].sub.n]t + [B.sub.n] sin
[[omega].sub.n]t. (4)
[A.sub.n] and [B.sub.n] are the constant of integration that can be
determined mathematically based on the initial conditions of motion.
Substituting Eqn (4) with Eqn (3) leads to subsequent equations, in
which the natural circular frequency, [[omega].sub.n], and deflected
shape, [[phi].sub.n], are unknown.
Displacement matrix:
(u(t) = [[phi].sub.n]([A.sub.n] cos [[omega].sub.n]t + [B.sub.n]
sin [[omega].sub.n]t)). (5)
Velocity matrix:
[??] = [[phi].sub.n](-[[omega].sub.n] [A.sub.n] sin
[[omega].sub.n]t + [[omega].sub.n] Bn cos [[omega].sub.n]t). (6)
Acceleration matrix:
[??] = [[phi].sub.n](-[[omega].sup.2.sub.n] [A.sub.n] cos
[[omega].sub.n]t - [[omega].sup.2.sub.n] [B.sub.n] sin
[[omega].sub.n]t). (7)
Simplifying the acceleration matrix u gives
[??] = [[omega].sup.2.sub.n][[phi].sub.n]([A.sub.n] cos
[[omega].sub.n]t - [B.sub.n] sin [[omega].sub.n]t). (8)
This is similar to:
[??] = -[[omega].sup.2.sub.n][q.sub.n][[phi].sub.n]. (9)
Therefore, the matrix equation (Eqn 2) for free vibration of linear
MDF systems leads to:
[-[[omega].sup.2.sub.n]m[[phi].sub.n] + k[[phi].sub.n]][q.sub.n](t)
= 0. (10)
The following algebraic equation meets the solution, which is named
as matrix eigenvalue problem. Since the stiffness and mass matrices k
and m are known, the scalar eigenvalue [[omega].sup.2.sub.n] and the
eigenvectors or mode shapes [[phi].sub.n] are determined:
k[[phi].sub.n] = [[omega].sup.2.sub.n]m[[phi].sub.n]. (11)
To indicate the formal solution to Eqn (11), it is rewritten as:
[k - [[omega].sup.2.sub.n]m][[phi].sub.n] = 0. (12)
This can be interpreted as a set of N homogenous algebraic
equations for the N elements [[phi].sub.jn](j = 1, 2, ..., N).
Reasonable solution of the equation leads to:
det[k - [[omega].sup.2.sub.n]m] = 0. (13)
When the determinant is expanded, a polynomial of order N in
[[omega].sup.2.sub.n] is obtained. The vibrating system with N DOFs
contains N natural frequencies [[omega].sub.n](n = 1, 2, ..., N)
corresponding natural periods, [T.sub.n]; and natural modes
[[phi].sub.n]. Each of these vibration properties is natural or
intrinsic property of the structure in free vibration, which is
load-independent, but depends only on its mass and stiffness properties.
3. Numerical study
All of the 61 models have been configured as per the materials and
load assignment discussed in subsequent sections. Equivalent static
analysis has been performed for all the configurations. The maximum roof
displacement induced by wind loading has been thus determined as well.
Besides the equivalent static analysis, the free vibration analysis is
carried out for every individual model. The free vibration analysis is
used to determine the undamped free vibration mode shapes and natural
frequencies of a structure. The natural periods of the structure are
determined from the free vibration analysis. Natural frequency
([f.sub.n]), which is the reciprocal of the natural period ([T.sub.n]),
can thus be calculated. All the obtained results are then evaluated to
see the optimal structural modelling.
3.1. Material properties
Except for brick masonry wall in some modelling, almost all
structures in this research have been modelled as an RC building. The
general concrete properties applied in modelling include density ([rho])
of 2447 kg/[m.sup.3] or self-weight (SW) of 24 kN/[m.sup.3] and
Poisson's ratio ([upsilon]) of 0.2. Two types of concrete have been
used in the modelling, that is, concrete C40 and concrete C60. The
overall compressive strength of a masonry wall depends on the
compressive strength of the individual masonry units and the type of
mortar used, besides the quality of workmanship. In this study, it is
assumed that type S mortar and clay masonry units with compressive
strength of 4400 psi (or 30.3 N/[mm.sup.2]) are used for the modelled
brick masonry wall. Thus, the net area compressive strength ([f.sub.cu])
of the masonry shall be 1500 psi, that is, equivalent to 10.3
N/[mm.sup.2] For convenient allusion, information on all material
properties is summarised in Table 2.
3.2. Load assigned
The structural modelling includes dead load (DL) or SW of the
building in consideration. At the same time, the occupancy of the
modelled building is assumed to be general office, with live load (LL)
of 2.5 kN/[m.sup.2], which is in accordance with Table 1 in British
Standards Institution (1996).
Moreover, in cases where staircases are being considered in the
modelling and analysis, the LL applied on the staircase is 4.0
kN/[m.sup.2]. It is logical that the design LL applied on the staircase
is higher as compared to that of the office area, because there is a
probability that a staircase may be crowded with people in emergency
cases.
3.3. Equivalent static analysis
Based on the provision in design standards, equivalent static
analysis, which may also be known as the quasistatic analysis, is used
for applying the wind load. The main concept of this analysis is that
the kinetic energy of the wind is converted into an equivalent static
pressure, which is then treated in a manner similar to that for a
distributed gravity load. The major advantage of this analysis is its
simplicity, by using modification factors to account for the dynamic
effects. Wind load parameters, to be inputted into the program for
generating wind load, are determined with reference to British Standards
Institution (1997). The maximum roof displacement induced by wind
loading, can thus be determined as well.
The effective wind speed, [V.sub.e] (in m/s), depends on several
factors, such as the basic wind speed ([V.sub.b]), the altitude factor
([S.sub.a]), the direction factor ([S.sub.d]), the seasonal factor
([S.sub.s]), the probability factor ([S.sub.p]), and the terrain and
building factor ([S.sub.b]), as shown in the following equations
(British Standards Institution 1997):
Site wind speed, [V.sub.s] = [V.sub.b] x [S.sub.a] x [S.sub.d] x
[S.sub.s] x [S.sub.p]; (14)
Effective wind speed, [V.sub.e] = [V.sub.s] x [S.sub.b]. (15)
For all modelling, the basic wind speed, [V.sub.b] of 33 m/s has
been assumed. The altitude of the site is assumed to be the same as the
mean sea level; thus, the altitude factor, [S.sub.a], is 1.0. Assuming
that the orientation of the building is unknown or ignored, the
direction factor, [S.sub.d], is 1.0. Also, the multi-storey building is
assumed to be permanent and is exposed to the wind for a continuous
period of more than 6 months; therefore, the seasonal factor, [S.sub.s],
is 1.0.
Since the modelled building is for normal design application, the
probability factor [S.sub.p] is 1.0 as well. The terrain and building
factor ([S.sub.b]) shall be determined based on the assumptions that the
location of the multi-storey building in modelling is in a town area,
and the closest distance to a sea is greater than 100 km. Just as
additional information, it will be shown how the effective wind speed
[V.sub.e] is related to the dynamic wind pressure. The dynamic pressure
[q.sub.s] (in Pa) and the effective wind speed [V.sub.e] (in m/s) are
related as described in the following equation:
Dynamic pressure, [q.sub.s] = 0.613 [V.sup.2.sub.e]. (16)
This dynamic pressure will act on the surface of the multi-storey
building. The overall wind load exerted on the multi-storey building is
given as follows:
The overall horizontal loads,
P = 0.85([P.sub.front] - [P.sub.rear])(1 + [C.sub.r]). (17)
The factor 0.85 accounts for the non-simultaneous action between
the front and rear faces. [P.sub.front] is the horizontal component of
the surface load summed over the windward-facing shear walls and roofs.
[P.sub.rear] is the horizontal component of the surface load summed over
the leeward-facing shear walls and roofs. Besides the effective wind
speed, the main parameters to be inputted in program are the dynamic
augmentation factor, [C.sub.r], and size effect factor, [C.sub.a]. The
factor, [C.sub.r], depends on the building type factor, [K.sub.b] as
well as the actual height of the building above ground, H. The size
effect factor, [C.sub.a], depends on site exposure and the diagonal
dimension.
3.4. Free vibration analysis
Besides the equivalent static analysis, the free vibration analysis
has been performed. The free vibration analysis, which is also known as
eigenvector analysis, is used to determine the undamped free vibration
mode shapes and natural frequencies of a structure, which provide an
excellent insight into the behaviour of the structure (Computers and
Structures Inc. 1995). This is due to the fact that natural frequency is
load-independent; it only depends on the mass and stiffness of the
structure. Each structural member (e.g. beam, column, shear walls and
slab) of the actual building structure contributes to the inertial
(mass) property, elastic (stiffness) property, and energy dissipation
(damping) property of the building structure. However, in the idealised
system, each of these properties is categorised into three separate pure
components, that is, mass component, stiffness component and damping
component.
Free vibration is initiated by disturbing the structure from its
equilibrium position by some initial displacements and/or by imparting
some initial velocities. In the present analysis, natural periods and
mode shapes are the outputs of the free vibration analysis. The
fundamental period (or natural period), [T.sub.n], of a building depends
on the distribution of stiffness and mass along its height (Kose 2009).
Thus, the stiffness of the structure model is evaluated based on its
correlation of fundamental frequency and the maximum roof displacement.
4. Results and discussion
Natural frequency and lateral deformation are important parameters
in calculating the base shear and the base overturning moment for
structural element design. Thus, the maximum roof displacement and
natural frequency are used in evaluating the structural performance
under dynamic loadings.
For evaluating the effect of different modelling concepts of the
same multi-storey building structure, the analysis results are presented
in terms of the maximum roof displacement [D.sub.roof](mm),
[DELTA][D.sub.roof](%), natural frequency [f.sub.n] (Hz) and
[DELTA][f.sub.n](%). [DELTA][D.sub.roof](%) indicates the percentage
reduction of the maximum roof displacement based on that of the frame
structure modelling. [DELTA][f.sub.n](%) indicates the percentage
increment of natural frequency based on that of the frame structure
modelling.
In most structural design under lateral loadings, it is desirable
to limit/minimise the lateral deformation or increase the lateral
stiffness of the structure, within optimum construction cost. Thus, the
percentage reduction of the maximum roof displacement
([DELTA][D.sub.roof]) and the percentage increment of the natural
frequency ([DELTA][f.sub.n]) are used to assess the improvement on
structural performance of other types of modelling in comparison to that
of the frame structure modelling.
Intrinsically, the effect of considering particular structural
elements or details (e.g. shear walls, slabs, wall openings, staircases,
etc.) in modelling can be studied by the comparison of the relevant
plotted figures.
4.1. Effect of wall openings
In addition to the 'frame' and 'frame + shear
wall' modelling with 20% shear wall opening, 'frame + shear
wall + slab' modelling with 20% shear wall opening has been
performed and analysed. '20% O shown in the following figures in
this section indicates 20% shear wall opening. 20% shear wall opening,
in fact, means that 20% area of each shear walls refers to the openings
(e.g. windows), to more accurately reflect the real multi-storey
building structure.
Figure 3 gives a general overview of the maximum roof displacement
for those 5 modelling concepts. It can be observed that the maximum roof
displacement increases as the number of storeys increases. For higher
multi-storey building structure, the 'frame + shear wall'
modelling and 'frame + shear wall + slab' modelling, with and
without openings, have significantly less of the maximum roof
displacement, as compared to that of the ' frame modelling. The
percentage reduction in the maximum roof displacement (over that of the
conventional frame structure modelling) for 'frame + shear
wall' modelling and 'frame + shear wall + slab' modelling
is shown in Figures 4 and 5, respectively.
Through Figure 4 for 'frame + shear wall' modelling, the
higher percentage reduction of maximum roof displacement indicates
higher lateral stiffness of the structure. From observation, the effect
of shear wall opening becomes increasingly important for building
structures higher than 15 storeys. This is shown by the 'gap'
between the figures of 'with openings' and 'without
openings'. The 'gap' becomes increasingly larger as the
number of storeys increases.
[FIGURE 3 OMITTED]
Also, the figure for 'with openings' is generally lower
than that of 'without openings', indicating that all openings
in 'frame + shear wall' structure resulted in a lower lateral
stiffness of the building structure, as compared to those counterpart
without openings. However, even with the presence of wall openings, the
significance of considering shear walls in addition to the frame
structure modelling increases as the number of storeys increases.
Based on Figure 5, the effect of a shear wall opening on the
lateral stiffness of a multi-storey building is significant in all
cases, with the exception of the 5-storey building. This is most
probably due to round-up error since the maximum roof displacement value
for 5-storey 'frame + shear wall + slab' modelling is very
small.
Similar to the cases for 'frame + shear wall' modelling,
the wall openings tend to reduce the lateral stiffness of the structure,
thus allowing a greater maximum roof displacement. However, the effect
of wall openings is more pronounced in the cases of 'frame + shear
wall + slab' modelling. This is shown by the figure of percentage
for 'with openings' is much lower than that for 'without
openings'.
By observing the general trend in Figure 6, the mode 1 natural
frequency (or fundamental frequency) for the 'frame + shear wall +
slab' modelling without opening is the highest, followed by the
'frame + shear wall + slab' modelling with 20% opening, then
'frame + shear wall' modelling without opening, 'frame +
shear wall' modelling with 20% opening and, lastly, the
'frame' modelling. This indirectly implies that
'frame' modelling has the lowest lateral stiffness, while the
'frame + shear wall + slab' modelling without an opening has
the highest lateral stiffness.
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
The difference between 'frame + shear wall' modelling,
with and without openings, is quite close together as compared to that
for the 'frame + shear wall + slab' modelling, with and
without opening. This indicates that the effect of a shear wall opening
is more pronounced in the cases of ' frame + shear wall +
slab' modelling.
In addition, it can be observed that fundamental frequencies for
all types of modelling concepts decrease as the number of storey
increases. In other words, higher buildings have lower fundamental
frequencies and vice versa. The percentage increment of fundamental
frequency for 'frame + shear wall' modelling. The percentage
increment in fundamental frequency for the modelling with 20% opening
ranges from 36.8 to 134.9% (from a 5-storey building to a 40-storey
building), whereas that for the modelling without a shear wall opening
ranges from 36.7 to 178.6% (from a 5-storey building to a 40-storey
building). The maximum difference in the percentage is 43.7%, in the
case of 40-storey buildings. For a 15-storey building and lower, the
difference in the percentage is less than 3.0%.
The effect of wall openings becomes increasingly significant,
especially for building structures higher than 15 storeys. In other
words, the significance of considering wall openings increases as the
number of storey increases. This matches well with the result for
percentage reduction of the maximum roof displacement in Figure 4. The
effect of a shear wall opening is significant for 'frame + shear
wall + slab' modelling, regardless of the number of storeys in a
building. The percentage increment in the fundamental frequency for the
modelling with 20% opening ranges from 173.4 to 229.5%, whereas that for
the modelling without opening ranges from 280.5 to 383.8%. The minimum
difference in the percentage is 107.1%, in the case of 40-storey
buildings, whereas the maximum difference in the percentage is 159.1%,
in the case of 15-storey buildings.
[FIGURE 6 OMITTED]
4.2. Effect of a masonry wall
In this part of the research, brick masonry is used to model the
exterior shear wall, instead of the RC with C40 concrete. However, the
lift core shear wall is still being modelled as an RC shear wall with
C60 concrete. In the following figures, the 'shear wall'
refers to the RC exterior shear wall, whereas the 'masonry shear
wall' refers to the brick masonry exterior shear wall. The
'frame' modelling and 'frame + shear wall' modelling
are shown in figures for the purpose of comparison only.
By observing Figure 7, it is found that, for all modelling
concepts, the maximum roof displacement increases as the building height
increases. Also, for the same building height, the 'frame'
modelling has the highest maximum roof displacement followed by the
'frame + masonry wall', while the 'frame + shear
wall' modelling has comparatively the lowest maximum roof
displacement. Based on Figure 8, the effect of the brick masonry wall is
compared with that of the RC shear wall.
From Figure 8, the figures for the percentage reduction in the
maximum roof displacement for 'frame + shear wall' modelling
and 'frame + masonry wall' modelling have a similar trend,
that is, the effect of considering the shear walls is, in general, more
significant for higher buildings. It is also observed that the brick
masonry wall provides less structural stiffness to the building as
compared to that of the RC shear wall. This may be due to the fact that
brick masonry and RC not only differ in terms of material type, but also
their compressive strengths. The C40 RC has the compressive strength of
40 N/[mm.sup.2], whereas the brick masonry only has the compressive
strength of 10.3 N/[mm.sup.2].
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
With reference to Figure 9, all modelling concepts have decreasing
fundamental frequencies as the number of storey increases. This is
because the lower buildings are generally stiffer than their higher
buildings counterpart. For the same building height, the
'frame' modelling has the lowestfundamental frequency,
followed by the 'frame + masonry wall' modelling, while the
'frame + shear wall' modelling has the highest fundamental
frequency.
From Figure 10, the significance of considering RC shear wall on a
multi-storey building increases as the number of storeys increases
(36.7-178.6%), as discussed in earlier sections. As for the masonry
wall, the percentage increment in fundamental frequency is around
71.2-77.6%, with the exception of the 5-storey building (33.1%). The
difference in structural performance of these two modelling concepts
increases as the number of storeys increases; this is shown by the
'gap' between the figures.
The similar trend of figures allows for the same figure
interpretation and deduction. Although mode 1 and mode 2 natural
frequencies are, in general, the same, but the mode shape is different.
The corresponding natural frequencies in the second higher mode have
been illustrated in Table 3. This is most probably due to the
symmetrical arrangement of the building's plan view.
For higher modes, natural frequencies for the 'frame + masonry
wall' modelling decrease as the number of storeys increases.
However, this decrement is not as obvious as that of the lower modes.
The mode 4 natural frequencies decrease from 2.8082 Hz (5-storey
building) to 0.9252 Hz (40-storey building). The percentage increment in
higher modes natural frequency for 'frame + masonry wall'
modelling is no longer dependent on the building height. It is observed
that the effect of the brick masonry wall is most significant for a
10-storey building, and least significant for a 30-storey building.
[FIGURE 9 OMITTED]
[FIGURE 10 OMITTED]
By and large, it is observed that the consideration of a brick
masonry wall in addition to the conventional frame modelling is
significant, especially for higher buildings. However, the lateral
stiffness provided by the brick masonry wall is not as significant as
compared to the RC shear wall.
4.3. Effect of a staircase
To investigate the effect of a staircase on the response of the
multi-storey building, staircases have been added to the 'frame +
shear wall' + slab modelling instead of the 'frame + shear
wall' modelling. The logic behind this is due to the fact that
slabs are much easier to be modelled as compared to a staircase; thus,
it is most probable that a practicing engineer would model slabs even
before considering a staircase.
In this research, there are slab openings to accommodate the
staircase. To maintain structural stability at the slab opening as well
as to support staircases, additional beams and columns have been added
at an appropriate location. Theoretically, addition of structural
elements such as staircases and additional beams and columns should
result in a much higher lateral stiffness of the structure, thereby
reducing the lateral deformation induced by lateral load (e.g. wind load
and seismic load). However, at the same time, the presence of slab
openings may reduce the stiffness of slab to a considerable degree.
Thus, it is important to investigate the resultant effect of considering
staircases in structural analysis.
Buildings of 5-, 10-, 20-, 30- and 40-storey were modelled with
staircases. The results are shown in Figures 11-14. The
'frame' modelling, 'frame + shear wall' modelling
and 'frame + shear wall + slab' are shown in figures for the
purpose of comparison only.
Figure 11 shows that the maximum roof displacement increases as the
number of storeys of a building increases. This applied to all modelling
concepts above. Therefore, the same interpretation or deduction can be
made. It is noted that the displacement variation for 'frame +
shear wall + slab + Stair' modelling and that of the 'frame +
shear wall + slab' modelling are extremely close to each other, to
the extent that both figures almost merge together (Figs 10-12). The
maximum roof displacement for 'frame + shear wall + slab +
Stair' modelling ranges from 0.1 to 8.6 mm (from a 5-storey
building to a 40-storey building), whereas that for ' frame + shear
wall + slab' modelling, it ranges from 0.1 to 8.8 mm (from a
5-storey building to 40-storey building) which are also mentioned in
Table 4.
[FIGURE 11 OMITTED]
The percentage reduction in the maximum roof displacement for
'frame + shear wall + slab + Stair' modelling ranges from 85.7
to 92.6% (from a 5-storey building to a 40-storey building), whereas
that for 'frame + shear wall + slab' modelling, it ranges from
85.7 to 92.4% (from a 5-storey building to a 40-storey building). This
indicates that consideration of staircases with slab openings (which
accommodate the staircases) in addition to the 'frame + shear wall
+ slab' modelling does not provide significant improvement on the
lateral stiffness of a multi-storey building structure. It can thus be
deduced that the additional lateral stiffness provided by the staircase
is offset by the adverse effect of slab openings.
[FIGURE 12 OMITTED]
As the building height increases, all the patterns of the
percentage reduction in the maximum roof displacement come closer
together. This might lead to a prediction/hypothesis that for a
50-storey building or higher, the consideration of slabs and a staircase
does not provide much improvement in structural performance over the
'frame + shear wall' modelling.
The fundamental frequency decreases as the number of storeys
increases, for all types of modelling (Figure 13). Also, the figure of
'frame + shear wall + slab + Stair' modelling is very slightly
lower than that of the ' frame + shear wall + slab' modelling.
The fundamental frequency for 'frame + shear wall + slab +
Stair' modelling ranges from 8.8106 Hz (5-storey building) to
0.8204 Hz (40-storey building), whereas that for the 'frame + shear
wall + slab' modelling, it ranges from 8.9366 Hz (5-storey
building) to 0.8295 Hz (40-storey building). This indicates that the
prior modelling is slightly less stiff as compared to the latter
modelling.
The percentage increment in fundamental frequency for 'frame +
shear wall + slab + Stair' modelling and that for the 'frame +
shear wall + slab' modelling are extremely close to each other,
where the former is very slightly lower than the latter. The percentage
increment in fundamental frequency for 'frame + shear wall + slab +
Stair' modelling ranges from 276.3% (40-storey building) to 376.0%
(10-storey building), whereas that for the 'frame + shear wall +
slab' modelling, it ranges from 280.5% (40-storey building) to
383.8% (10-storey building). This leads to the deduction that the
adverse effect of a slab opening (to accommodate the staircase) is
slightly more significant than the additional lateral stiffness provided
by staircases.
The higher mode natural frequencies (Table 5) for 'frame +
shear wall + slab + Stair' modelling range from 12.7551 Hz
(5-storey building) to 2.7716 Hz (40-storey building), whereas that for
the 'frame + shear wall + slab' modelling, they range from
12.7551 Hz (5-storey building) to 2.8193 Hz (40-storey building).
[FIGURE 13 OMITTED]
[FIGURE 14 OMITTED]
It is perceived that both 'frame + shear wall + slab +
Stair' modelling and 'frame + shear wall + slab'
modelling have achieved the maximum percentage increment of the higher
mode natural frequencies for a 10-storey building. For any building
higher than a 10-storey building, the percentage increment of the higher
mode natural frequencies starts to decrease.
4.4. Assessment of outcomes
It is common to have wall openings on multi-storey buildings. Based
on the maximum roof displacement and fundamental frequency figures for
'frame + shear wall' modelling, it is observed that the effect
of a shear wall opening becomes increasingly important for building
structures higher than 15 storeys. However, even with the presence of
wall openings, the significance of considering shear walls in addition
to frame structure modelling increases as the number of storeys
increases.
For the 'frame + shear wall' modelling, the trend of
fundamental frequency increment is similar to that of the maximum roof
displacement reduction, that is, the significance of considering shear
wall in addition to the frame structure modelling increases as the
number of storeys increases. This is not the case for the 'frame +
slab' modelling and 'frame + shear wall + slab'
modelling.
It has also been observed that the consideration of slabs alone in
addition to the frame modelling may have negligible improvement on
structural performance. However, when the slabs are combined with shear
walls in addition to the frame structure modelling, the reduction in the
maximum roof displacement and the increment in fundamental frequency are
significant.
However, the effect of considering slab is more significant in
higher modes, especially in the cases of lower buildings. Also, for
higher modes, natural frequency in 'frame + shear wall'
modelling, the significance of the shear wall is no longer dependent on
the building height. In terms of percentage increment in natural
frequency based on that of the 'frame' modelling, 'frame
+ shear wall + slab' modelling is generally performing much better
than considering shear walls or slabs alone, throughout all modes.
For 'frame + shear wall + slab' modelling, the effect of
shear wall opening on the lateral stiffness of a multi-storey building
is, in general, significant, regardless of the building height. Also,
the effect of wall openings is more pronounced in the cases of
'frame + shear wall + slab' modelling, as compared to the
'frame + shear wall' modelling. Based on all observations, it
is shown that wall openings do reduce the lateral stiffness of a
multi-storey building.
Brick masonry wall itself does possess some structural strength,
whether it is being considered in the structural design or not.
Generally, the effect of a brick masonry wall in addition to the
conventional frame modelling is significant, especially for higher
buildings. However, the lateral stiffness provided by the brick masonry
wall is not as significant as compared to the RC shear wall.
In this study, staircases have been added in addition to the
'frame + shear wall + slab' modelling. Theoretically, the
consideration of staircase in modelling and analysis result in a stiffer
structure. However, when considering the staircase in the modelling, the
lateral stiffness is slightly lower than that without a staircase. This
is most probably due to the fact that the effect of a slab opening to
accommodate the staircase is more significant than the additional
stiffness provided by staircases, at least for the modelled structure in
this study. Slab openings will adversely affect the lateral stiffness of
the structure.
Thus, the 'frame + shear wall + slab' modelling needs to
consider both wall openings and slab openings, in order to avoid the
overestimation of the structural capacity, which is highly undesirable
in structural design. Otherwise, a more conservative approach would be
using just the 'frame + shear wall' modelling, which requires
less computational resources. However, the trade-off would be
overdesign, within an acceptable limit.
Conclusions
Besides the traditional deign strategy existence of a shear wall,
wall openings, masonry wall and slab openings/ staircase have been
incorporated in this study by means of numerous structural modelling.
Meticulous reckoning on structural responses of dissimilarly configured
high-rise buildings sorts out the succeeding conclusions:
1) Wall openings and slab openings, which would reduce the lateral
stiffness of a structure, should be taken into the consideration in
structural analysis and design, especially in the case of a highrise
building, to prevent unsafe design;
2) For safety reasons, it is generally not recommended to model
staircases in addition to 'frame + shear wall + slab'
modelling unless the effect of wall openings and slab openings are
adequately considered in the analysis. This, however, might considerably
increase the modelling effort as well as computational time;
3) If the strength and stiffness provided by shear walls and slabs
are used for the advantage of structural design, there would be no
additional cost incurred. Yet, it is even possible to come up with an
effective and more economical design;
4) By considering shear walls and slabs in the modelling and
analysis, the structural elements of a multi-storey building, which are
subjected to the lateral load, may experience a lower shear, moment and
lateral deformation. Thus, the size of the structural member or the
steel reinforcement could be reduced to save cost while satisfying the
safety and serviceability requirement/provisions from the local design
codes;
5) To better understand the effect of wall openings, which are
common in multi-storey buildings, modelling with different percentage of
wall openings can be performed, for example, with 5%, 10%, 25% and 50%
wall openings. Also, the effect of a shear wall opening location shall
be investigated as well;
6) In this research, besides the free vibration analysis, only the
equivalent static analysis has been performed. Static pushover analysis
and response spectrum analysis could be performed to further investigate
the response of a multi-storey building under seismic loading.
doi: 10.3846/13923730.2013.799096
Acknowledgement
The authors would like to gratefully acknowledge the University of
Malaya (UM), for their constant support through the grant RG140-12AET
provided to fund the research work.
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Mohammed JAMEEL (a), A. B. M. Saiful ISLAM (a), Mohammed KHALEEL
(a), Aslam AMIRAHMAD (b)
(a) Department of Civil Engineering, University of Malaya, Kuala
Lumpur, Malaysia
(b) Department of Civil Engineering, Salman Bin Abdulaziz
University, Al-Kharj, Saudi Arabia
Received 5 Jan 2012; accepted 16 Jan 2012
Corresponding author: Mohammed Jameel
E-mail: jameelum@gmail.com
Mohammed JAMEEL did his PhD from Indian Institute of Technology
Delhi (IIT Delhi), India. He has successfully completed various
sponsored projects involving non-linear analysis of TLPs, Spar, FPSO
platforms, deep and shallow water mooring lines and risers. The projects
were supported by several government and private funding agencies. His
research area includes non-linear dynamics, earthquake engineering,
reliability engineering, offshore structures, artificial neural network
and non-linear finite element analysis. Presently, he is associated with
Department of Civil Engineering, University of Malaya, Malaysia.
Dr. A. B. M. Saiful ISLAM He is a PhD and is working as a Research
Fellow at the Department of Civil Engineering, University of Malaya,
Malaysia. He completed his BSc in Civil Engineering and MSc in
Structural Engineering from Bangladesh University of Engineering and
Technology (BUET), Bangladesh. He is a member of Institution of
Engineers, Bangladesh and American Society of Civil Engineers (ASCE).
His research interests include offshore structures, non-linear dynamics,
finite element modelling, seismic protection, base isolation and
pounding and special tall buildings.
Mohammed KHALEEL He did his Masters in Structural Engineering from
Jawaharlal Nehru Technological University, India. He is currently
pursuing doctorate from the University of Malaya, Malaysia. His research
interests include non-linear finite element analysis, reliability
analysis, tall structures, offshore structures and pre-cast concrete
structures.
Dr. Aslam AMIRAHMAD is an accomplished academician who had joined
the Civil Engineering Department of Salman Bin Abdulaziz University,
Al-Kharj, as a Professor. Assuming charge on 1 December 2008, Dr. Aslam
had set his goals to develop the Department of Civil Engineering as a
team of talented and dedicated Faculty members and to take the Civil
Engineering Department to new heights, among the top Departments of
Salman Bin Abdulaziz University. He has done his U.G. and P.G. Degrees
in Civil Engineering from Aligarh Muslim University Aligarh and PhD from
Indian Institute of Technology Delhi.
Table 1. Modelling and design information summary
Parameter Rating
Plan area 42x42 m
Storey height 3 m
Beams Concrete C40; 400x600 mm
Columns Concrete C40; 800x800 mm
Slab Concrete C40; 120-mm thick
Exterior wall Concrete C40; 200-mm thick
Brick masonry wall; 203.2-mm (8-in.) thick
Lift core wall Concrete C60; 250-mm thick
Table 2. Material properties summary
Materials Properties
Concrete C40 [f.sub.cu] = 40 N/[mm.sup.2]
E = 28 GPa
[rho] = 2447 kg/[m.sup.3]
SW= 24 kN/[m.sup.3]
Poisson's ratio, u = 0.2
Concrete C60 [f.sub.cu] = 60 N/[mm.sup.2]
E = 32 GPa
p = 2447 kg/[m.sup.3]
SW= 24 kN/[m.sup.3]
Poisson s ratio, u = 0.2
Steel reinforcement [f.sub.y] = 460 N/[mm.sup.2]
Brick masonry wall [f.sub.cu] = 10.3 N/[mm.sup.2]
E = 2.10 GPa
[rho] = 2000 kg/[m.sup.3]
SW = 19.6 kN/[m.sup.3]
Poisson s ratio, u = 0.13
Table 3. Comparison of frequency considering
masonry wall (second higher mode shape)
Eigenvector analysis results for mode 2
Number of Frame Frame + wall
storeys
[f.sub.n] [f.sub.n] [DELTA]
(HZ) (HZ) [f.sub.n] (%)
5 2.0730 2.8604 38.0
10 0.9646 1.7544 81.9
15 0.6246 1.2133 94.3
20 0.4600 0.9663 110.1
25 0.3626 0.8340 130.0
30 0.2982 0.7452 149.9
35 0.2524 0.6729 166.7
40 0.2180 0.6074 178.6
Number of Frame + masonry wall
storeys
[f.sub.n] [DELTA]
(HZ) [f.sub.n] (%)
5 2.7886 34.5
10 1.6513 71.2
15 1.0988 75.9
20 0.8141 77.0
25 0.6441 77.6
30 0.5286 77.2
35 0.4440 75.9
40 0.3792 73.9
Table 4. Maximum roof displacement in static analysis
Number of Frame Frame + wall
storeys
[D.sub.roof] [D.sub.roof] [DELTA]D
(mm) (mm) (%)
5 0.7 0.5 28.6
10 4.4 1.7 61.4
20 22.6 7.4 67.3
30 58.6 13.0 77.8
40 116.0 19.5 83.2
Number of Frame + wall + slab Frame + wall + slab
storeys + staircase
[D.sub.roof] [DELTA]D [D.sub.roof] [DELTA]D
(mm) (%) (mm) (%)
5 0.1 85.7 0.1 85.7
10 0.2 95.5 0.2 95.5
20 1.1 95.1 1.1 95.1
30 3.5 94.0 3.4 94.2
40 8.8 92.4 8.6 92.6
Table 5. Comparison of frequency considering staircase
(second higher mode shape)
Eigenvector analysis results for mode 2
Number of Frame Frame + wall
storeys
[f.sub.n] [f.sub.n] [DELTA]
(Hz) (Hz) [f.sub.n] (%)
5 2.0730 2.8604 38.0
10 0.9646 1.7544 81.9
20 0.4600 0.9663 110.1
30 0.2982 0.7452 149.9
40 0.2180 0.6074 178.6
Number of Frame + wall + slab Frame + wall + slab
storeys + staircase
[f.sub.n] [DELTA] [f.sub.n] [DELTA]
(Hz) [f.sub.n] (%) (Hz) [f.sub.n] (%)
5 8.9366 331.1 8.8106 325.0
10 4.6664 383.8 4.5914 376.0
20 2.1487 367.1 2.1155 359.9
30 1.2631 323.5 1.2467 318.1
40 0.8295 280.5 0.8204 276.3
