Seismic Analysis of RC Walls with Unequal Lengths (Analisi Sismica di pareti in CA di diversa lunghezza)

9 Seismic analysis of RC walls with unequal lengths 
1.INTRODUCTION 
When structural elements with different characteristics are forced to 
displace together, compatibility forces form in the connection between them 
and this influences mainly the shear distribution among the structural 
elements themselves.  
In the present study this phenomenon is investigated analyzing a 2D coupled 
wall structure composed by two walls of different lengths, which represents 
part of an 8-story building. Only the in-plane behavior of the system is 
considered in the present work. 
The main objective is to know whether the modeling of the structure 
influences the results obtained by static non-linear analyses. For that 
reason the pushover analysis results obtained with different modeling 
approaches are compared.  
The modeling approaches considered are the following: 
̻ 
Beam element models, which can be distinguished into: 
̻ 
Lumped plasticity model; 
̻ 
Models with nonlinear beam elements with fiber sections; 
̻ 
Shell element models. 
For the beam element models, the behavior of the coupled wall system is 
studied by means of the software “SeismoStruct” version 5.0.5 (SeismoSoft, 
2011) instead for the latter the program “Vector2” version 2.9 is used 
(Vecchio & Wong, 2002). 
Concerning in particular the sensitivity of shear force distribution at the 
base of the walls, the following aspects are also investigated: 
̻ 
Longitudinal reinforcement distribution, 
̻ 
Different assumptions concerning the shear stiffness, 
̻ 
Floor rigidity, 
̻ 
Effect of hysteretic parameters, 
̻ 
Effect of discretization of elements over story height. 
The present report shows, first of all, a brief literature review that 
summarizes the previous studies related to the topic (Chapter 2), then the
1. Introduction 10 
 
reference structure, used for the following analyses is described in 
Section 3. The differences between the modeling approaches are explained in 
Chapter 4. Beam models are considered first: the results obtained from the 
pushover analysis are shown in Section 5, and Chapter 6 proposes some 
sensitivity analyses regarding the aforementioned modeling parameters. 
Afterwards the shell element model is studied in Section 7, and the 
analysis results are compared with the ones obtained with the beam models. 
Once a reliable beam model to represent the structure is found, a 
parametric study can be carried out (Chapter 8) in order to evaluate the 
effect of the coupling also on different pairs of structural walls. In the 
end the conclusions of the whole study are summarized and reported in 
Section 9.
11 Seismic analysis of RC walls with unequal lengths 
2.LITERATURE REVIEW 
In 1998 Paulay and Restrepo published a paper about mixed structural 
systems and they noticed that “the traditional approach, based on the 
theory of elasticity is inappropriate for the purpose of quantifying 
displacement and ductility relationship” (Paulay & Restrepo, 1998). To 
explain the concept they considered a system composed by vertical lateral 
force-resisting elements with markedly different elasto-plastic force-
displacement characteristics as the one reported in Fig.2.1. 
             
Fig.2.1: Interconnected elements. (Paulay & Restrepo, 1998) 
 
Fig.2.2: Wall element response based on the theory of elasticity. (Paulay & 
Restrepo, 1998) 
In the usual design procedure, the lateral forces are assigned to the 
resisting elements proportionally to the element stiffness, in order to 
satisfy the compatibility of displacements and rotations at the same height
2. Literature Review 12 
 
of the structure. In the elastic range, the base shear and the top 
displacement ∆
e
 depend on the fundamental period T, which is based on the 
total stiffness of the system.  
When a structure is designed to behave in a ductile manner, the design 
forces for Ultimate Limit State (U.L.S.) are reduced proportionally to the 
displacement ductility factor adopted, µ
∆
, and the yield displacement of 
the system is computed as: 
∆
�� =
∆
�� �� ∆
                                                                                             (2. 1) 
It is normally assumed that each element i commences yielding when the 
displacement ∆
y
 is reached, thus the corresponding force-displacement 
relationships are those represented by the lines in Fig.2.2. 
The moment-curvature analysis of typical rectangular sections of reinforced 
concrete walls showed that the yield curvature is not much sensitive to the 
amount of reinforcement used and the compression axial force applied and 
thus it can be estimated as (Paulay & Restrepo, 1998): 
Φ
�� =
�� �� �� �� ��                                                                                       (2. 2) 
Where �� �� is the tensile yield strain, l
w
 is the wall length and λ is a 
constant that depends on the position of the neutral axis at the onset of 
yielding. The same equation can be found in (Priestley, Calvi, & Kowalsky, 
2007), where the authors recommend to use λ = 2 for rectangular walls, 
because a series of analysis on different reinforced concrete walls showed 
that it is a sufficient accurate value to be used for design purposes.  
As a consequence, the corresponding yield displacement, that is 
proportional to the yield curvature, is also inversely proportional to the 
wall length and that means that elements with different lengths have 
different yield displacements, hence the elements do not yield 
simultaneously. 
Starting from that simple observation, Rutenberg wrote a paper in 2004 
about a cantilever wall system where he studied the behavior of coupled 
walls of different lengths (Rutenberg, 2004). He noticed that with yielding 
the force distribution among the walls will change. This is because “the
13 Seismic analysis of RC walls with unequal lengths 
formation of a plastic hinge in a given wall diverts the incremental shear 
and moment to the unyielded ones, and hence leads to their redistribution” 
(Rutenberg, 2004). That effect can be observed when there are rigid floor 
diaphragms which force the deformation compatibility and allow the 
formation of coupling forces through them.  
To explain the concept, he considered the two-story system supported by two 
flexural walls, shown in Fig.2.3, where the Wall 2 is much stiffer than the 
Wall 1. Wall 1 is fully fixed at its base, whereas Wall 2, is hinged. The 
two horizontal pin-ended rigid members model the floor slabs connecting the 
two walls. For simplicity he assumes that the external horizontal force 
acts only at the top story. The hinge at the base of Wall 2 represents a 
plastic hinge which has formed at a horizontal force level H. Under the 
additional force increment ΔH, Wall 2 will enforce on Wall 1 its straight-
line deflected shape at every floor level. The resulting forces on the 
system and the deflected shapes are also represented in Fig.2.3 on the 
right.  
 
Fig.2.3: Two-story wall system. (Rutenberg, 2004) 
It can be seen that the shear force on Wall 1 is increased by 2.5ΔH, while 
the shear on Wall 2 is reduced by 1.5ΔH and an important aspect of the 
moment transfer to the unyielded walls is the large force acting on the 
first floor diaphragm ΔN =3ΔH. Yet, this simple model does show that the 
shear demand on the shorter wall may be much larger than that evaluated on 
the basis of the usual assumptions. In other words, it may not be correct 
to assume, as is often done in practice, that shear force demand is 
proportional to bending moment demand.
2. Literature Review 14 
 
When the concept is extended to a more complicated structure, i.e. when a 
multi-story structure composed by walls of unequal length is considered, 
the behavior of the system is qualitatively similar to the two story case. 
Figure 2.4 shows the results of a pushover analysis with inverted 
triangular loading pattern on the wall structure previously studied by 
(Priestley & Kowalsky, 2000) where the plastic hinges are allowed to form 
at ground level only. It can be observed that “the large shear force 
transferred from the long wall to the short ones starts when the first 
plastic hinge forms at the base of the stiffest wall” (Rutenberg, 2004). 
Moreover, Rutenberg noticed that the shear force redistribution is 
influenced also by the fact that after the formation of a plastic hinge at 
the stiffest wall base, at the upper stories the wall is still elastic and 
can carry part of the incremental forces applied. Also for that reason the 
results are different from the ones obtained considering the elastic 
stiffness allocation. 
 
Fig.2.4: Plan of 8-story wall-supported building. (Priestley & Kowalsky, 2000) 
 l
w
 (m) A
w
 (m
2
) EI (kNm
2
) 
I/ΣI 
M
y
 (kNm) 
M
y
/ ΣM
y 
2xWall 1 3 1.50 4.40x10
6 
0.216 6614 0.356 
Wall 2 6 1.50 15.95x10
6
 0.784 11956 0.644 
Total  3.00 20.35x10
6
 1.000 18570 1.000 
Table 2.1: Properties of the two walls. (Rutenberg, 2004) 
After two years, in 2006, Rutenberg and Nsieri completed the study 
publishing another paper which focused more on design suggestions that have 
to be considered for ductile cantilever wall system (Rutenberg & Nsieri, 
2006). Concerning the seismic shear force demand on flexural walls, they
15 Seismic analysis of RC walls with unequal lengths 
reported the results obtained by parametric studies which confirmed that 
“non-simultaneous yielding at the bases of walls has a significant effect 
on the base shear distribution among them”, in particular “the shear demand 
on the flexible walls is likely to be much larger than is commensurate with 
their relative stiffness, or even with their relative flexural strength” 
(Rutenberg & Nsieri, 2006). In order to consider this fact, they proposed a 
simple pushover procedure to estimate the base shear in each wall. The 
procedure comprises the following steps: 
̻ 
The base shear demand of the system is determined for an equivalent 
“super-wall”, whose properties (stiffness and moment capacity) are equal 
to the sum of the ones of the individual walls, using the following 
proposed formula which also accounts for the effect of higher vibration 
modes: 
�� �� = [0.75 + 0.22( �� + �� + ���� )] �� ��                                                              (2. 3) 
Where V
d
 is the design shear force obtained for a triangular 
distribution of forces over the height of the building, T is the 
fundamental period of the structure, q is the force reduction factor. 
̻ 
Depending on the base shear demand of the system, the equivalent height 
of the system can be computed as: 
ℎ
����
=
�� �� �� ��                                                                                         (2. 4) 
Where M
y
 is the yield moment of the equivalent structure. 
̻ 
A cyclic static analysis with the resultant force at the equivalent 
height is carried out up to the peak base shear V
a
. 
̻ 
It is also possible to “follow the post first-yield shear force 
distribution among a number of walls” by means of successive 
applications of the following analytical expressions derived for multi-
story system composed by walls of unequal length, which have constant 
properties throughout the height, where the floor diaphragms are 
infinitely rigid. 
∆ �� �� , �� (0) = � ∆ �� (0) + �� ∙ ∆ �� (0) ∙
∑ �� ℎ
� ℎ ∙ ∑ �� �� �
� ∙
�� ����
∑ ��                                                   (2. 5) 
∆ �� �� , ℎ
(0) = � ∆ �� (0) − �� ∙
∆ �� (0)
ℎ
� ∙
�� �� ℎ
∑ ��                                                           (2. 6)
2. Literature Review 16 
 
In which ΔV
if
(0) and ΔV
ih
(0)are the respective shear forces and I
if
 and I
ih
 
are the respective moments of inertia of the i
th
 fixed and hinged walls, 
ΣI and ΣI
h
 are the moment of inertia sums of all walls and the hinged 
walls respectively. ΔM(0) is the base moment increment following the 
plastic hinge formation in the strongest wall due to the load increment, 
ΔV(0) is the total base shear increment and α=(3−√3) for walls having 
uniform flexural stiffness over the height. 
̻ 
If the purpose is to determine the peak base shear forces acting on wall 
i, it is not necessary to carry out a cyclic static analysis because 
“the full load reversal is commensurate with doubling the base shear at 
yield V
iy
 on each wall” (Rutenberg, 2004). Therefore, an estimate of the 
peak shear V
i,max
 on wall i can be obtained from a uni-directional 
pushover analysis: 
�� �� ,������
= 2� �� ����
� − � �� ���� ��
�                                                                    (2. 7) 
Where V
iy
 is the base shear at yield of the wall i, and V
ity
 is the base 
shear of the wall i when all walls have yielded. It can be noted that using 
this equation the maximum shear forces in the walls which yields first are 
expected after the change of the loading direction. 
In the end Rutenberg recognizes that real systems are much more complicated 
than those considered in the numerical examples included in the paper, thus 
he identifies aspects which might reduce the shear demand on the shorter 
wall: 
̻ 
Formation of plastic hinges at other locations than the base; 
̻ 
Rotation at foundation level, rocking of the foundation and yield 
penetration into the foundation; 
̻ 
Shear deformation of the walls; 
̻ 
Flexibility of floor diaphragms; 
̻ 
Spread of plasticity; 
̻ 
Hysteresis model. 
The authors also briefly investigated the shear deformability and the 
flexibility of the connection. They concluded that the consideration of 
these two parameters can reduce the shear demand on the shortest wall. A 
similar effect is observed also when plastic hinges are allowed to form not 
only at the base of the walls.
17 Seismic analysis of RC walls with unequal lengths 
The influence of the factors identified above on the base shear demand of 
the individual walls has been studied in a master thesis carried out in 
2005 by Beyer (Beyer, 2005). In that study, an 8-story R.C. coupled wall 
structure with two walls of different lengths was considered and the effect 
of various analysis types and modeling assumptions was investigated. The 
main conclusions are reported in the following: 
̻ 
The base shear demand of the short wall and the floor diaphragm forces 
are the quantities influenced the most by the modeling assumptions made. 
̻ 
The elasto-plastic and the modified Takeda rules were compared. It was 
found that with the first model the maximum value of the base shear in 
the short wall is obtained after the first load, when the structure is 
reloaded in the opposite direction, in accordance with Eq. (2.7). 
Conversely, considering the Takeda hysteresis rule, the maximum base 
shear is obtained within the first load branch. For that reason, it is 
argued that the proposed formula can be applied only if the elasto-
plastic model is implemented. 
̻ 
Comparisons between the rigid and the flexible connection between the 
walls showed that if the axial deformability is taken into account, the 
base shear demand of the short wall and the coupling forces are reduced, 
whilst the displacement drift and the total base shear are not 
influenced. 
̻ 
The shear flexibility of the walls does not produce any significant 
changes on the moment and shear envelopes of the long wall but it 
influences the shear demand of the short wall which is increased, as a 
consequence of the reduction in the total stiffness ratio of the long to 
the short wall. 
̻ 
Lumped plasticity and distributed plasticity models were compared: the 
capacity curves obtained from the first model were bilinear while the 
ones obtained from the latter are smooth, in accordance with the 
constitutive formulation of the elements. 
The objective of this project is to complete the sensitivity analysis 
concerning different modeling assumptions. For that reason, a similar 
structure to that used in (Beyer, 2005) is assumed as a reference structure 
as described in the following chapters.
19 Seismic analysis of RC walls with unequal lengths 
3.STRUCTURAL SYSTEM 
Figure 3.1 shows a sketch of the coupled wall system: the 2D model analyzed 
in the following chapters, belongs to a symmetric structure that represents 
an 8-story building where the vertical and the lateral load capacity is 
provided in total by 9 walls, 5 placed in the x-direction and 4 in the y-
direction, connected at the story level by concrete slabs. 
The total floor plan area is 25mx15m=375m², the story height is h
s
 =3m thus 
the building has a total height of 24m.  
The coupled wall system considered is composed of two concrete walls with 
constant section over the height. The wall lengths are 6m and 4m, 
respectively, the distance between the centre lines of the walls is 10m and 
the thickness of the walls is 0.2m. 
The following paragraphs describe in detail the characteristics of the 
structural elements. The geometrical and material properties are summarized 
in Paragraph 3.1 and 3.2, respectively; Section 3.3 shows the 
correspondence of the assumed characteristics to the Eurocode 
prescriptions, then Paragraphs 3.4 and 3.5 present the computed moment-
curvature analysis and the consequent force-displacement relationship that 
will be the fundamental parameter of comparison for the next analyses.
3. Structural System 20 
 
 
Fig.3.1: Elevation and plan view.