In the name of God
To knowing behavior of the buildings during the earthquake motions, one of the best ways is numerical analysis; modeling the frame, wall, foundation and in general the whole structure in software then put loading on the structure and finally analyzing it.
For masonry structures it’s not that easy to build a numerical model, because of the type of masonry materials such as brick, adobe. The masonry materials are not homogenous and specially for bricks and blocks it have to be modeled discrete to gain a real results. And for discrete modeling lots of time should be passed. However there is a better way and that is the concrete plasticity property. Referring to brick and mortar material based on the laboratory tests a small discrete model should be built and calibrate with the real test specimen, therefore continuum model would be built base on the discrete results; finally the whole model could be assembled in the software. Since deficiencies of strength and ductility can be seen in the analysis results, a retrofitting method could be suggested.
In this report the FRP materials selected for retrofitting modeling with the masonry structure.
This numerical analysis took much less time than discrete ones; by the way most of the discrete models could not be run in computer due to the huge amount of equations and matrices.
This method has an accurate result with many advantages as stated above.
Ancient buildings are the treasure of the times with no measurable value. Save them from any disaster is the most important and serious responsibility. Most of the disasters are the natural ones such as hurricane, flood, earthquake etc. Iran (the author nationality) has thousands of historical buildings and heritages that unfortunately most of them are being destroyed, bam citadel was an example of these structures that recently destroyed in bam earthquake in December 2003 and it was the largest adobe structure in the whole world.
To save the masonry structures against disasters, we need information about the lacks of the strength in structures to preventing retrofitting them in a wrong way that may cause more damages to the structures.
In this report the author efforts to express a way for modeling and analyzing the masonry structures and find the lacks of strength; and finally retrofitting modeling and analyzing could be established to understanding the behavior of the structure after then.
At first, types of the materials and mortar used in the structure need to be defined and tested in the laboratory with a simple test specimen; therefore based on the laboratory results the discrete model could be assembled with respect to the test results and loadings; after then the continuum model would be assembled and calibrate with discrete model, and when the parameters were defined, the whole structures could be modeled with different loading and boundary conditions.
The flow chart bellow represents the whole project in brief.
At this time let’s look after the discrete modeling of a brick wall (Figure 1) with uniform laterally loaded and fixed bottom (with assuming that this model was tested in laboratory and all the information prepared to compare with numerical specimen). The author chose ABAQUS/CAE FEA software for analyzing the models. The elasticity modulus, Poisson ratio and density have been defined for brick as elastic behavior and Drucker-Prager with compressive hardening for plastic behavior. Explicit dynamic analysis performed in all models. Explicit dynamic Satisfies the dynamic equilibrium equations at the beginning of the increment, t; the accelerations calculated at time t are used to advance the velocity solution to time t + Dt/2 and the displacement solution to time t + Dt. The use of small increments (precept by the stability limit) is advantageous because it allows the solution to proceed without iterations and without requiring tangent stiffness matrices to be formed. It also simplifies the treatment of contact.
The extended Drucker-Prager models are used to model frictional materials, which are typically granular like soils and rock, and exhibit pressure-dependent yield (the material becomes stronger as the pressure increases); also used to model materials in which the compressive yield strength is greater than the tensile yield strength, such as those commonly found in composite and polymeric materials; and allow a material to harden and/or soften isotropically; generally allow for volume change with inelastic behavior, the flow rule, defining the inelastic straining, allows simultaneous inelastic dilation (volume increase) and inelastic shearing. Figure 2 and Figure 3 presents the Yield surfaces in the meridional plane and flow surfaces of the linear model.
Sometimes experimental data are not directly available. Instead, the user is provided with the friction angle and cohesion values for the Mohr-Coulomb model. The Mohr-Coulomb failure model is based on plotting Mohr’s circle for states of stress at failure in the plane of the maximum and minimum principal stresses. The failure line is the best straight line that touches these Mohr’s circles and could be useful for Drucker-Prager parameters.
Mohr-Coulomb model is defined by
Where s is negative in compression. From Mohr’s circle,
And 
(Figure
4)
Substituting for t and s, multiplying both sides by cos f, and reducing, the Mohr-Coulomb model can be written as
(Figure 5)
Where
After defining the material, defining contact property need to be performed. Since the mortar can not resist in tension therefore in tangent behavior of contact we may define the hard contact with allowing the separation in tension. There are three ways to define the tangent behavior of a contact, friction, shear stress and elastic slip; depend on the test result user may choose between them, anyway the author decided to use the frictional behavior. There are several friction types that ABAQUS/CAE support in explicit dynamic analysis; however the types that could be used in this particular job are penalty type, exponential and tabular. In penalty type only the time of the fracture of the mortar could be modeled with no possibility for post behavior modeling; and for tabular type, test results may be needed to applied to the contact property.
After modeling the geometry the analyze process may be started. By changing the coefficient of friction the result may be varying, for example Figure 6 presents the wall with no friction; it’s so true! Since there is no friction the whole wall will move uniformly except the first row that has a fixed boundary condition. Figure 7 presents the wall displacement with 1.0 friction coefficient and Figure 8 presents the model with friction coefficient of 5.0, diagonal cracks were appear in this condition. Figure 9 is the same as Figure 8 with scale 5 times greater then before. Figure 10 and Figure 11 present the wall displacement when the coefficient of friction is 10. By comparing between two coefficient of 5.0 and 10, no significant difference could be located.
Hence this could be the theory of discrete modeling of a masonry building in FEA software. After approving the discrete model, the important thing is to replace the discrete model with the continuum one.
The main purpose of the continuum modeling is to reach the minimum modeling and analyzing time, however many computers can not handle the giant discrete models due to the big amount of contacts, functions and degrees of freedom.
The plasticity picked for this kind of modeling is the concrete damaged plasticity. The concrete damaged plasticity model assumes nonassociated potential plastic flow. The flow potential used for this model is the Drucker-Prager hyperbolic function, so that would be no changing in the flow rule comparing to the discrete model.
Figure 12 presents the counters of damage on the brick wall. It should be noticed that the number of the parameters for this type of plasticity is more than the Drucker-Prager method; hence some parameters must be defined separately from Drucker-Prager method to get the most accurate result comparing the discrete model. Comparing Figure 12 and Figure 9 represents the cracks on two models are perfectly mach. By the way Figure 13 presents the displacement counters in discrete model, there are some irregularity in counters that effected by appearing the cracks on the wall, if the curve that follows the fracture of each counter would be drawn, the curve will be same as the red counter in Figure 12. Also in Figure 14 where the displacement in continuum model is shown, a perfect mach could be found with Figure 13.
Thus this could be the how to build this specific masonry structures in FEA software. For further investigation the author suggested to build a simple masonry dome with acceleration load at the bottom that described in Figure 15. The dome could be the original structure that needed to be considered.
Figure 16 and Figure 17 present the damage on masonry dome wall after applying load in axis#1. The cracks appeared on columns, corner of the opening and the back wall reasonably, and so in Figure 18 and Figure 19 with loading in axis#3. This analysis took a few minutes to be completed. In this type of analyze the only thing that control the time of the analyze is loading manner and using contacts, that means using contact and time history loading that contains a big number of data will significantly increase the run time.
After realizing the lacks of strength and ductility in the structure, it’s now the time to do some retrofitting program. Knowing the behavior of the structure after retrofitting is important, the retrofitting might do in a wrong way and cause more damages to the structure due to the strong motion during the earthquake. Thus it’s very essential to build the retrofitted model.
There are several ways to retrofit a masonry building such as steel column and plate, steel stud partition, elastomeric spray, geo-textiles, internal concrete skin and FRP materials.
The FRP materials are the modern materials and newer than the others and used widely nowadays, hence it would be a substantial structure model.
One of the advantages of using concrete damaged plasticity model is that it could be used with rebar definition. As the FRP behave in only one direction (due to its polymeric reinforcement in one direction) thus it could be modeled like the reinforced embedded on sides of the wall. In ABAQUS/CAE the rebar should be defined in surfaces, that means to model the FRP material as a rebar, first the surface must be embedded and then the rebar as the property of surface should be defined. The parameters of the rebar are so simple: material, area of bars, spacing and the orient of bars as presented in Figure 20.
For the dome structure in previous section the rebar modeled on surface at the corner of the main opening as Figure 21. The same analysis and loading were applied, after running the model the result was like Figure 22 to Figure 24.
The modeling FRP has a significant effects on behavior of the structure. That means selecting the rebar option for retrofitted model would consequence reasonable behavior. Figure 25 presents the displacement contour. This is obvious that by modeling the FRP materials, displacement contours that represent the appeared cracks would be changed and became more regular.
The time spending with FRP and in general with contacts would be very much more than the simple model with no contact between parts, thus it should be notice that using contacts should be done in a well consideration.
Concrete damaged plasticity model results very accurate reasons, by the way it spends much less time than the other methods such as Drucker-Prager method.
Implicit models could not handle the large deformations and cracks during analysis due to the small time increment.
Using the contact between parts causes the more analyzing time.
FRP modeling needs to define surfaces embedded to the structure, hence they cause increasing the analyze time. That means FRP modeling needs more considerations, unless there would be very much wasting of time.
Figure 1- Brick wall for discrete model
Figure 2- Yield surfaces in the meridional plane
Figure 3- Typical yield/flow surfaces of the linear model in the deviatory plane.
Figure 4- Mohr-Coulomb failure model
Figure 5- Mohr-Coulomb model in the deviatory plane
Figure 6- wall displacement with friction coefficient = 0.0
Figure 7- wall displacement with friction coefficient = 1.0
Figure 8- wall displacement with friction coefficient = 5.0
Figure 9- wall displacement with friction coefficient = 5.0 scale = 5/1
Figure 10- wall displacement with friction coefficient = 10.0
Figure 11- wall displacement with friction coefficient = 10.0 scale = 5/1
Figure 12- Damage counters in the continuum model
Figure 13- wall displacement counters in discrete model
Figure 14- wall displacement counters in continuum model
Figure 15- load applied to the masonry dome
Figure 16- Damage on dome structure, load from axis 1
Figure 17- Damage on dome structure, load from axis 1
Figure 18- Damage on dome structure, load from axis 3
Figure 19- Damage on dome structure, load from axis 3
Figure 20- Rebar modeling as surface property
Figure 21- Rebar modeling as surface property on dome wall
Figure 22- damage parameter on models at t=1.0
Figure 23- damage parameter on models at t=2.0
Figure 24- cracks appear on models at t=2.5
Figure 25- displacement contours on models at t=2.5
