atw - International Journal for Nuclear Power | 08/09.2019

atw Vol. 64 (2019) | Issue 8/9 ı August/September
432
AMNT 2019
State of the art in science and technology
The out-of-plane behavior of URM walls has been the
subject of various research studies over recent decades in
several countries (e.g. [5–10]). Different influencing
parameters as well as construction and brick types were
studied. The experimental investigations of Dafnis et al.
[11], Meisl et al. [12] and Dazio [13] focused, in particular,
on the effect of the boundary conditions on the
out-of-plane behavior of URM walls. One of the conclusions
of their research was that the connection at the top of
the wall should be considered as one of the most important
boundary conditions.
In practical applications, the models of Paulay and
Priestley [14] and Griffith et al. [15–17] as well as the
models from KTA 2201.3 [1] and DIN EN 1996 [2] are
often used.
In analytical and numerical investigations conducted at
the TUK, those simplified analytical models have been
evaluated. For this purpose, the dimensions of the wall as
well as the properties of masonry and mortar from [11]
were used while the height as well as vertical load has been
varied. Additionally, a numerical model to investigate the
behavior of a URM wall under earthquake loads was
developed. For the numerical model, a reinforced concrete
ceiling was considered at the top of the wall in the nonlinear
dynamic time-history analyses. The results of the
considered simplified analytical models and the numerical
model showed a wide range of estimations of the loadbearing
capacity, especially in case of very low vertical
loads. In all cases, the numerical analyses led to higher
| | Fig. 2.
Experimental dynamic seismic testing of the AAC block wall on the shaking table
at the TUK / analytical and experimental capacity acceleration.
| | Fig. 3.
Geometric dependencies / Pivot point at the bottom.
capacities than the simplified analytical models. In case
of a vertically loaded wall, the factor was 2-3. In case of
vertically unloaded walls up to 8.
As one of the reasons for the higher capacities in the
numerical simulation, the vertical stiffness due to the
concrete ceiling was identified. During the rocking process,
the wall center moves horizontally, leading to a rotation of
the bricks. This results in an axial elongation of the wall,
which increases the axial load acting on the wall and has a
stabilizing effect. For more details on the analytical and
numerical models and the conducted investigations, see
[18].
To verify the results obtained from analytical and
numerical analyses, shaking table tests have been conducted
at the TUK with URM walls from heat insulating
clay bricks [19] and autoclaved aerated concrete (AAC)
blocks [20]. The investigations confirmed the large
influence of the vertical stiffness of the upper boundary on
the out-of-plane capacity. While the analytical model of
Griffith et al. led to good estimations of the capacity in case
of no vertical stiffness, all models underestimate the
capacity in case of vertical stiffness present at the upper
support, since this stiffness is neglected in those methods
(Figure 2).
Development of analytical model
The rocking of masonry walls under earthquake loads
represents a non-linear time-dependent process. It is
therefore ideally represented by a non-linear, dynamic
time-history simulation. However, a discrete modelling of
complete wall systems including bricks, mortar joints,
boundary conditions etc. is usually not feasible in practice.
To determine the out-of-plane capacity more precise than
simplified analytical methods without using complex
models, an idealization to an equivalent single-degree- offreedom
(SDOF) system can be used. Since the vertical
stiffness at the upper support was identified as a significant
influencing factor in the described investigations and it is
not considered in existing models, an analytical model to
determine the out-of-plane force-displacement relationship
of URM walls is developed. For this purpose, the wall
is simplified similar to the models of Griffith and Paulay
and Priestley by two rigid slabs. The support conditions,
the crack height, vertical loads and the vertical stiffness at
the top boundary are considered.
From the geometric relations, the deformations of the
wall can be determined and the work can be determined
using the principle of virtual work. The external work d A a
consists of the external load and the inertia of mass. The
earthquake load at the base causes the displacement of the
mass in horizontal and vertical direction. The inner work
d A i consists of the vertical displacement of the mass
against gravity acceleration, the axial load and the vertical
stiffness of the top support.
By incrementally increasing the displacement of the
wall and determining the associated work, the forcedisplacement
relationship can be determined. Since the
real deformation of the wall is not an infinitesimal small
deformation, the actual geometric relations have to be
considered (Figure 3).
Instead of the simplified assumption of a crack at half
the height of the wall as in the models by Paulay and
Priestley and Griffith et al., the crack height is determined
by means of elastic beam theory. It is assumed that the wall
bends at the point where the first cracks form.
After the wall cracked, rocking of the wall around
its resting position is generated by the constant change of
AMNT 2019
Analytical Model for the Investigation of the Out-of-Plane Behavior of Unreinforced Masonry Walls
ı Moritz Lönhoff, Lukas Helm and Hamid Sadegh-Azar