The wall structures may undergo a reduction in stiffness and strength if and when subjected to cycles of alternating loads. The magnitude of deformation / damage primarily depends upon the type of reinforcement (if any) and or the mode of failure. (Meli, 1973) In the following, general remarks on the seismic behaviour of different types of masonry shear walls are given based both on the experimental results and on the direct observation of the effect of earthquakes. Structures of unreinforced masonry have a completely brittle type of failure and linear elastic behaviour has to be considered for the analysis of seismic effects. The use of unreinforced masonry in seismic zone is objectionable due to the experience of catastrophic failures. The behaviour of interior reinforced walls is nearly elasto-plastic with significant ductility and very small deterioration if and when subjected to the alternating loads; especially where failure is governed by bending.
Types of Wall Failure
Loads, previously not considered during the design either in regard to their magnitude or frequency of occurrence may subject a structure to additional loading. These loads are either accidental or are applied as a result of some other mechanical action. When dealing with accidental loads (i.e. seismic, blast), the structures may face two kinds of loading situations, in-plane and out-of-plane load application. Walls behave / perform very poorly if and when subjected to out-of-plane load patterns / displacements of any kind, and thus fail due to flexure. Whilst, similar wall structures when subjected to in-plane load patterns / displacements may perform better overall but will often fail in Shear Stress. Since, masonry having a relatively high compressive strength, is very weak in resisting bending and shear. It is thus that a masonry wall may be prone to collapse or any other immediate mode of failure if and when subjected to flexure or shear. The different modes of failure for masonry are as under:
In-Plane failure occurs as a result of a laterally induced ground motion in a direction parallel to the plane of the wall. This (parallel to the wall-plane) laterally-induced motion causes the wall to experience stresses that exceed the strength of its masonry material, thus initiating a failure mechanism more commonly known as In-Plane failure. This failure comes about due to shear overstress and or flexural instabilities; and is more often identified by the presence of X-shaped cracks in the walls, these may further be categorized as flexure or shear cracks (Meli, 1973).
Generally, in-plane failure alone is insufficient to cause a structural failure / collapse, however, it may contribute to a progressive collapse mechanism or at least help initiate one if and when it occurs. Meaning, a wall may develop cracks following / due to this type of failure, but will still have sufficient strength to continue supporting the structure itself, or at least itself alone. This failure does not bring any reduction to the wall’s compressive strength, but only very little.
Solid walls are significantly more resistant to in-plane failure as compared to the walls with openings for doors and windows. In order to inhibit the occurrence of in-plane associated failures, an increase in the lateral strength of the load-bearing masonry walls can be advised, which in turn can be achieved by building additional walls that can adequately / sufficiently resist shear loads if and when they occur.
Out-of-plane failure may either occur as a result of out-of-plane inertia force triggered by the considerable mass of the masonry wall if and when shifted away from its center of gravity or as a result of the out-of-plane action of the floor on the wall. (Maheri & Najafgholipour, 2012) This mode of failure more commonly occurs in walls with inadequate wall-diaphragm anchorage and dynamic instability of slender walls, and is of great interest in unreinforced masonry walls (URM), as it is often much more destructive and damaging than in-plane failures.
Out-of-Plane failures are prompted / instigated when the shockwaves (resulting either due to earthquake or a blast) in the ground travel in any direction other than the plane of the wall. Unsupported (i.e. un-buttressed) or unreinforced masonry walls are at far greater risk of failure if and when subjected to ground motion perpendicular to the face of the wall than from the motion parallel to the face of the wall. Out-of-Plane failure is dominated by the flexure form of cracking. The wall begins cracking at the mid-height eventually resulting in either an Inward-Outward fall or simply an Overturning failure.
The ground motion as caused by an earthquake or a blast is usually not unidirectional, thus causing a wall to often fail due to the combination of the in-plane and out-of-plane failure modes. In most cases, the in-plane failure associated cracks may appear or occur first, thus weakening the wall considerably, which is then usually followed by the out-of-plane cracking / failure.
The susceptibility of a wall structure to an out-of-plane failure mode can be greatly decreased by reducing distances between two adjacent wall supports. Also, the floor and ceiling joists, if and when properly attached to the masonry walls can contribute to an increased overall structural stability and stiffness, thus providing an effective resistance against any impending out-of-plane failure. However, such attachments in older buildings are of little efficacy and provide only a very little resistance to the subject failure mechanism.
Shear is the governing mode of failure observed in masonry walls (structures) subjected to lateral loading conditions (i.e. earthquakes, blasts, and wind, etc) (Atkinson, Amadei, Saeb, & Sture, 1989). Such unusual loading conditions may tend to induce both horizontal bed-joint shear-failure and diagonal cracking failure modes. Shear-failures are brittle and catastrophic than flexure failures, and are therefore of much concern when dealing with URM walls.
The majority of masonry walls tend to bend in both the horizontal and vertical directions simultaneously; thus behaving as two-way bending plates essentially. Ideally, walls like those shown above should be built and tested in bending to failure. The failure mode will be a flexural crack that will occur along the bed joint, in the case of simple vertical bending at or near the position of maximum moment. When the wall is subject to simple horizontal bending, the flexural failure crack will develop through the perpendicular joints (perpends) and the bricks, as shown on the right.
The overturning failure is influenced by a building’s vertical profile, and is critical in nature due to the severity of its aftermath. Slender walls that are either too tall or too long and does not possess any anchorage or support of any kind are much vulnerable to shaking in their weak direction in particular. Thus, when shaken / pushed in its weak direction, such a wall may tend to topple causing an overturning failure. This type of failure can be prevented or at least the risk of its occurrence can be greatly reduced by limiting (managing) the length-to-thickness and or height-to-thickness ratios.
This failure is more commonly associated or occurs in the walls with two or more wythes, when exterior masonry wall portion drops and falls in a manner similar to that of a falling curtain if and when cut loose at the top. The failed wall portion falls and pounds down on the ground near to the base of the wall, becoming a pile of bricks and mortar. Masonry-backed stonewall and brick veneer collapse mostly occur in a curtain-fall manner.
Methods to Improve Lateral Resistance in Masonry Walls
Masonry structures being the most primitive form of construction are being used since ancient times. Some of the earliest known man-made structures are masonry structures. The Pyramids of Giza, the Great Wall of China, the Ancient City of Moen Jo Daro are only a very few examples of the masonry structures that are a testament to the efficacy of this form of construction. As with any other technology, masonry structures have developed and improved over time as well, only to be more efficient, stronger, durable, and more economical. Previously, masonry structures involved making hugely thick walls that will resist over-turning, allow building taller walls, and deter the enemy forces if and when needed. Despite being very efficient, the thick masonry walls were costlier, and required huge spaces to build. With rapid urban development – cities growing at alarmingly faster rates than ever, all, while construction costs are becoming more and more unaffordable every day, at times when threats like terrorism are localised and ever more present; cheaper, durable, and stronger systems have become ever more necessary.
Masonry Construction methods that offer increased lateral strength at economy rates have been around for little over two centuries now. However, not all are equally efficient and economical. Reinforced Walls, may although offer increased lateral resistance and durability, but are often very expensive and require skilled professional labour. Confined masonry requires structural frames that are expensive to build, require design effort and skilled labour for construction. Some of these methods are being discussed here.
Reinforced Masonry Walls
In reinforced masonry walls, entire panels of masonry (brickwork) are reinforced either to increase the lateral load resistance of wall panels or to aid in handling and installation of factory-made wall panels. This reinforcement may also be provided so as to resist or at least impede the potential cracking at the corners of openings, or to restrain the freestanding parapets, particularly at locations or areas with greater probability of seismic occurrences, or where the unreinforced masonry (URM) walls will be too unstable for any reason.
Openings in masonry can be provided with reinforcement at the opening-top to then help support any form of construction above. Special provisions must be made to support the lowest course of masonry forming the top of openings, that is hung below the reinforcement to support it.
The reinforcement can either be horizontal as provided in bed joints or wythes, or can be vertical if provided at either face of the wall.
The most elementary form of reinforcement in brick masonry walls is wire reinforcement provided in or at the bed-joints. This is commonly done above and below the openings, since the stress concentrations at the corners of openings allow for the initiation of the crack forming and propagation process, and are therefore more vulnerable to cracking even before the actual design load is applied.
Reinforced Cavity Masonry
In reinforced cavity masonry walls the masonry units are provided with purpose-made hollow openings on their face, called core holes to accommodate the vertical reinforcement. The core holes in the body of the brick are located such that they offer proper alignment to all the core holes in the succeeding masonry courses in a normal stretcher bond, thus forming a hollow tube to later accommodate the reinforcement and the filler grout. They have to be large enough to allow some tolerance in laying, and also large enough to allow bars of the required diameter to be inserted, lapped where necessary, and grouted with mortar or fine-aggregate concrete.
Masonry walls (of clay brick or concrete block units) when confined using vertical and or horizontal RC confining members provided and built around all four (4) sides of a masonry wall panel (Brzev, 2007) are referred to as Confined Masonry Walls. Vertical members (i.e. tie-columns or practical columns) may although resemble columns in RC frame construction but they tend to be of far smaller cross-section when used for the masonry confinement. Horizontal elements (i.e. tie-beams), resemble beams in RC frame construction, and are also comparatively smaller in cross section. In order to differentiate these confinement components (elements) from the ordinary beams and columns, alternative terms horizontal ties and vertical ties are used instead of tie-beams and tie-columns. Confining members are effective in providing:
Enhanced stability and integrity to the masonry walls against in-plane and out-of-plane loads (as they contain the damaged masonry walls),
Enhanced strength (resistance) to masonry walls if and when subjected to earthquake or blast loads, by reducing their brittleness and thus improving their overall performance.
Mechanically Anchored Masonry (Lego) Block Wall Structures
The potential risks that may befall a certain civil engineering structure when exposed to dynamic actions have been argued and emphasized over the course of many studies, mostly purposed at studying the effects of the seismic activities and to mitigate the same. The out-of-plane vulnerability of the masonry envelop under dynamic loading is considered critical due to the potential risk of loss of lives and has been highly stressed by very many researchers, particularly when dealing with earthquakes and explosion debris. (Pereira, Campos, & Lourenço, 2014)
The concept of increasing masonry joint shear strength to improve the overall lateral resistance of the wall structure(s) is a tried, tested and proven method using the methods discussed earlier, however, this study attempts to do the same using mechanically anchored masonry units. The mechanical anchorage mimics and imitates the vertical reinforcement action in Reinforced Masonry Walls, all while these blocks are still very ordinary masonry units.
Dynamic loads have a significantly larger effect than the static loads of the same magnitude as they tend to accentuate a structure’s inability to respond / react quickly to the loading (by deflecting). The dynamic loads subject a structure to deformations accompanied with large displacements, which in turn destabilizes the structure and causes it to fail prior to reaching its static maximum strength. Dynamic displacements are the measure of the deflections as induced by these dynamic loads.
Time History Analysis
Time history analysis is used to obtain a structure’s reaction at various designated time points for a defined lasting interaction. This is contrary to other available analysis types that show the structure’s reaction in the form of amplitudes obtained for a single moment.
Time-history analysis provides for linear or nonlinear evaluation of dynamic structural response under loading which may vary according to the specified time function, and involves finding a solution of the following dynamic equilibrium equation of the time variable “t”:
K u(t) + C d⁄dt u(t) + M 〖 d〗^2⁄dt u(t) = r(t)
This equation is solved using either modal or direct-integration methods. Initial conditions may be set by continuing the structural state from the end of the previous analysis. Additional notes include:
Step Size – Direct-integration methods are sensitive to time-step size, which should be decreased until results are not affected.
HHT Value – A slightly negative Hilber-Hughes-Taylor alpha value is also advised to damp out higher frequency modes, and to encourage convergence of nonlinear direct-integration solutions.
Nonlinearity – Material and geometric nonlinearity, including P-delta and large-displacement effects, may be simulated during nonlinear direct-integration time-history analysis.
Links – Link objects capture nonlinear behavior during modal (FNA) applications.
Response Spectrum Analysis
Response-spectrum analysis (RSA) is a linear-dynamic statistical analysis method which is employed to measure the contribution from each natural mode of vibration to indicate the likely maximum seismic response of an essentially elastic structure. It provides insight into dynamic behavior by measuring pseudo-spectral acceleration, velocity, or displacement as a function of structural period for a given time history and level of damping. It is practical to envelope response spectra such that a smooth curve represents the peak response for each realization of structural period.
Response-spectrum analysis is useful for design decision-making because it relates structural type-selection to dynamic performance. Structures of shorter period experience greater acceleration, whereas those of longer period experience greater displacement. Structural performance objectives should be taken into account during preliminary design and response-spectrum analysis.
Modal analysis, or the mode-superposition method, is a linear dynamic-response procedure which evaluates and superimposes free-vibration mode shapes to characterize displacement patterns. Mode shapes describe the configurations into which a structure will naturally displace. Typically, lateral displacement patterns are of primary concern. Mode shapes of low-order mathematical expression tend to provide the greatest contribution to structural response. As orders increase, mode shapes contribute less, and are predicted less reliably. It is reasonable to truncate analysis when the number of mode shapes is sufficient.
A structure with N degrees of freedom will have N corresponding mode shapes. Each mode shape is an independent and normalized displacement pattern which may be amplified and superimposed to create a resultant displacement pattern, as shown in Figure 2 2:
Figure 2 2 Resultant Displacement and Modal Components
Modes are inherent properties of a structure, and are determined by the material properties (mass, damping, and stiffness), and boundary conditions of the structure. Each mode is defined by a natural (modal or resonant) frequency, modal damping, and a mode shape (i.e. the so-called “modal parameters”). If either the material properties or the boundary conditions of a structure change, its modes will change. For instance, if mass is added to a structure, it will vibrate differently.
Dynamic Analysis (Numerical Approach)
Mathematical modeling, or idealization, is a process by which an engineer or scientist passes from the actual physical system under study, to a mathematical model of the system, where the term model is a symbolic device built to simulate and predict aspects of behaviour of a system. The process is called idealization because the mathematical model is necessarily an abstraction of the physical reality. The analytical or numerical results produced by the mathematical model are physically re-interpreted only for those aspects.
The modeling can further be extended to a three-level approach commonly used in materials science. Three levels of observations and modelling are are the micro-, meso- and macro-level.
At each subsequent level more or less detail is recognised in the material structure. The most global approach to modelling is the macro-level, where the material is regarded as an equivalent continuum. No material structure is distinguished, and all non-linear behaviour is included in the constitutive law. The advantage from a numerical point of view seems that the structure that is analyzed can be divided in relatively large finite elements, which tends to reduce the computational effort. A basic characteristic of a macro- (or simplified) model is that they try to encompass the overall (global) behavior of a structural element without modeling all the possible modes of local failure. (Asteris, 2008)
The second level of observations is the meso-level. At this scale (10-3 to 10-2 m), individual particles are distinguished and are assumed to be embedded in a matrix. Thus, the response of the material under mechanical loads can be explained and computed from inter-particle interaction.
Micro – (or fundamental) models, on the other hand, model the behavior of a structural element with great detail trying to encompass all the possible modes of failure. (Asteris, 2008)
At the macro-level, two different approaches can be adopted, namely a (smeared) continuum approach (crack band models) and a discrete approach. In the crack band models, the crack is smeared over a larger volume, generally extending over part of a finite element or sometimes several elements, whereas in the discrete approach crack appears as a discrete jump between element boundaries. Elementary to these models is a continuum based fracture law, which is in general derived from Linear Elastic Fracture Mechanics (mainly in the discrete models), or from Non-Linear Elastic Fracture Models like the Fictitious Model by Hillerborg et al.(1976), both for continuum and discrete approaches.
Mechanical Properties of Walls
Designing the Masonry Block(s)
Ordinary Block Lego Block
Length (in) Width (in) Height (in) Length (in) Width (in) Height (in) Cones (in)
Top Bottom Height
15 8 6 15 8 6 1.5 3.5
Compressive Strength of Masonry Block(s)
The compressive strength of the masonry blocks is the measure of the force that is resisted by the block till before it fails by crushing when subjected to a uniform axial compressive strength.
ASTM Designation: C 140
Procedure: In order to determine the compressive strength of the subject block(s), three samples representing each type of block (i.e. Ordinary, and Lego) were placed / laid individually in the UTM on their stretcher face(s) followed by proper curing and surface drying. The samples were then loaded in compression till they failed.
Elastic Modulus of the Block(s) Material
Elasticity is the property of solid materials to return to their original shape and size after the forces deforming them have been removed. Whilst, the elastic modulus can be defined as the change in stress with an applied strain.
ASTM Designation: E 111
Procedure: A test cylinder representing the Block Material was prepared and cured for 28 days and tested under uniformly applied compressive load in the UTM to its ultimate strength. The test specimen was fixed with strain sensitive proving rings which were in contact with the strain-gauges (to measure strain value). The ratio of ultimate load to the ultimate strain is the elastic modulus.
Compressive Strength of Masonry Prisms
ASTM Designation: C 1314
Procedure: Prepare masonry prism(s) using Ordinary Blocks (4 nos.) and cement-sand mortar (1:4) as the bonding agent. The thickness of the mortar at joints must not exceed 0.75 inches and should not be less than 0.35 inches in any case. The specimen(s) must be properly cured and hardening of the mortar must be ensured. The specimen(s) be measured for their length at least four times in each direction and average length be considered for the purpose of calculation(s). The specimen should be placed in the UTM (if in case prepared elsewhere) gently with extreme care and the centroidal axes of the platen and the prism must coincide in all cases. Capping of the prism be done (to ensure uniform load distribution) using either sulfur based capping or sand/silt based, whichever appropriate and or available. Extensometers / Strain Gauges (3 nos. on either side) be attached to the short and long side(s) of the blocks for separate calculation(s). The experiment should be repeated at least five times given the availability of the blocks else three times at least.
Elastic Modulus for the Masonry Walls
Shear Strength of the Masonry Joints (Direct Shear Test)
Procedure: Prepare masonry prism(s) using Ordinary Blocks (3 nos.) and cement-sand mortar (1:4) as the bonding agent. The thickness of the mortar at joints must not exceed 0.75 inches and should not be less than 0.35 inches in any case. The specimen(s) must be properly cured and hardening of the mortar must be ensured. The specimen(s) be measured for their length at least four times in each direction and average length be considered for the purpose of calculation(s). The specimen should be placed horizontally in the UTM gently with great care and the centroidal axes of the platen and the center block must coincide in all cases. Extensometers / Strain Gauges be attached on the loading face of the block to determine the actual vertical displacement of the center block. The experiment should be repeated at least five times given the availability of the blocks else three times at least.
Shear Strength of the Mechanical Anchors (Direct Shear Test)
Shear Strength of the Masonry Walls (Diagonal Wall Test)
ASTM Designation: E 519
Relation Between Masonry Compressive Strength and Brick & Mortar Compressive Strength
A general failure hypothesis for masonry under compression is based on the different material characteristics of units and mortar (Francis et al. 1971, Hilsdorf 1969). In general, the compressive strength of a unit is higher than the compressive strength of mortar whereas the coefficient of lateral expansion (Poisson’s ratio) is higher for the mortar. Thus, under compression the mortar, i.e. bed joint, tends to expand laterally at a somewhat greater rate than the unit. In order to fulfil the compatibility conditions this lateral deformation of mortar is restrained (by the unit). Due to this confinement the mortar is able to resist the compressive stresses that are higher than its uniaxial strength. On the other hand, in order to maintain equilibrium, tensile stresses must be introduced into the unit. These tensile stresses induce the vertical cracking of the units and finally lead to the failure of the specimen.
Shaking Table Test
Several different experimental techniques have been used by various researchers to test the behaviour / response of structures if and when subjected to dynamic lateral loads of cyclic / oscillating / variable nature. The testing is done to assess and or verify the dynamic and seismic performance of these structures. One such technique is the use of an Earthquake Shaking Table (or simply a shake table). It is an instrumental setup that is employed for shaking structural models or building components (often to their ultimate strength) which simulates a wide range of ground motions, including reproductions of previously recorded earthquake time-histories.
The earliest shake table was invented at the University of Tokyo in 1893 using a simple wheel based mechanism to help categorize types of building construction. Modern shake tables typically consist of a rectangular platform that can be driven in up to six (6) degrees of freedom (DOF) by a servo-hydraulic or other type(s) of actuators.
A significant amount of research using shake tables has been undertaken by the researchers in recent years, the primary focus of which has mainly been oriented towards assessing the ultimate behaviour of steel and RC building structures, and or other structural elements (i.e. Walls, Frames with infills etc). Among the most prolific examples of the use of shaking tables include the tests performed at the Tsukuba facility on 1:1 scaled models (Minowa et al, 1996).
The test involves fixing the specimen(s) to a rectangular platform and then (often) shaking them to their point of failure. While video records and data from string potentiometers (transducers), and accelerometers are used to interpret the dynamic response and or behaviour of the test specimen(s). The shake tables have found a distinct place in instruments used for seismic research, and are being extensively used for their ability to help simulate distinct and complex patterns of ground / lateral motion when exciting the structures in such a way that they are subjected to conditions representing true earthquake-ground motion or accidental / blast wave motions.
Test Setup and Instrumentation
The test setup involved making four block masonry walls (two Ordinary and two Lego block walls) over the shake table platform placed in in-plane and out-of-plane directions as shown in Figure 4 1.
The instrumentation of involved fixing of string potentiometers (transducers) / LVDT (Linear Variable Differential Transformers) coded as 9-A, 9-B, 10-A, and 10-B for signal acquisition when lateral movement / sinusoidal displacement wave is applied. These string potentiometers (transducers) are helpful in detecting and or measuring any linear position change (displacement) and velocity using a flexible cable and spring-loaded spool. In the subject test, the transducers were fixed on each of the walls (near the top end) to record the displacement signal with respect to time using data-loggers attached to a computer when to and fro motion was applied to shake table platform using servo-hydraulic actuator.
The displacement signal thus acquired helped in determining the stiffness of wall structures and in assessing their individual performance if and when subjected to in-plane and out-of-plane lateral displacement(s).
Four separate walls including two ordinary (ordinary block) walls and two mechanically anchored (Lego block) walls were made over the shake table platform at / near its edges as shown in Figure 4 1.
The ordinary block walls were cured for a period of 6-7 days in order for the mortar joints to attain the requisite strength before testing.
The walls were fixed with accelerometers and LVDTs on / near the top ends for data recording purpose.
The test was initiated with an input amplitude (lateral displacement) of 20 mm at a frequency of 0.1 Hz, and lasting for 20 cycles. The frequency was then increased to 0.5 Hz for the second run, with an increment of 0.5 Hz for every subsequent run.
Data was recorded in terms of acceleration and displacement at a sampling rate of 500 readings per second.
Photographs were taken to collaborate with the numerical findings from the data recorder.
Input (Displacement) Data
A sine wave or a sinusoid (a smooth repetitive oscillation) was applied to the test specimen(s) as input (displacement) data when shaking the shake table platform. This sine displacement data is a function of time (t) and can be mathematically expressed as:
y(t)=Asin(2πft+ φ)= Asin(ωt+ φ)
A: the amplitude.
f: the frequency.
ω=2πf: the angular frequency.
φ: the phase (in radians).
The input parameter values used for the subject test are as under:
Amplitude (A) : ± 20 mm
Phase (φ) : 0 rads
Frequency (f) : 0.5 Hz increments
Sampling Rate : 500 Samples / Second (0.002 s)
No. of Cycles : 20
Shaking Table Test Results
The test results were obtained / acquired from a computer connected accelerometers and LVDTs using The Shore Western Control System (SWCS) – manufacturer provided data acquisition software. The results were obtained in a text based file.
Digital Seismic Data Acquisition Systems
The modern digital seismic data acquisition systems (i.e. MARS88, Quanterra, RefTek, Shore Western Control System, STL, Titan etc.) utilize the oversampling and decimation techniques; and in order not to violate the sampling theorem, each digital sampling rate reduction must include a digital anti-alias filter. To achieve maximum resolution during oversampling, the filters must be maximally steep. In addition, they should be stable and cause no distortion of the input signal, at least not within the filter’s passband. This requires linear-phase filters which are passing signals without phase changes, causing only a constant time shift. Digital anti-alias filters are generally implemented as zero-phase FIR filters. From practical point of view, it is important to know that they can “generate” precursory signals to impulsive seismic arrivals because of their symmetrical impulse response. These artifacts lead to the severe problems for the determination of onset times and onset polarities (i.e. they can be easily misinterpreted as seismic signals). Different methods to suppress them have been reported (e.g., the zero-phase filter can be changed into a minimum-phase one, prior to any analysis of onset polarities).
Signal Processing (Data Filtering)
In signal processing, a filter is a device or process which helps in suppressing / reducing / removing certain unwanted component(s), aspect(s) or feature(s) from the obtained signal data. This is usually accomplished by removing some frequencies from the obtained / acquired signal data in order to suppress / reduce the interfering signals (or background noise). Meaning, any frequencies above the Nyquist or any other noise types e.g. low frequency swell noise from the data are removed or at least suppressed prior to re-sampling.
There are very many types of filters. The classification of filters is largely based on their underlying mathematical functions. Filters may be classified as follows:
Linear or Non-Linear
Time-Invariant or Time-Variant (Shift Invariance)
Causal or Not-Causal
Analog or Digital
Discrete-Time (Sampled) or Continuous-Time
Passive or Active type of Continuous-Time Filter
Infinite Impulse Response (IIR) or Finite Impulse Response (FIR) type of Discrete-Time or Digital Filter.
In seismic signal processing, linear continuous-time filters are used for filtering the acquired signal data if and when required, thus making them a quintessential tool required for this study. As a nonlinear form of filter may potentially result in the output signal containing frequency components not present in the input signal. Some important filter families designed in this way are:
Butterworth filter: It is the most nascent form of linear continuous-time filters and has a maximally flat frequency response.
Chebyshev filter: It is helpful in providing the best approximation to the ideal response of any filter for a specified order and ripple.
Bessel filter: It is helpful when a maximum flat phase delay is required.
The fundamental difference between these filter families is that they all use a different polynomial function to approximate to the ideal filter response. This results in each having a different transfer function.
It is a type of signal processing filter introduced and described by a British Engineer and Physicist Stephen Butterworth in his renowned paper entitled “On the Theory of Filter Amplifiers (1930)”; and is designed to have a maximum possible flat frequency response in the passband. It is also generally referred to as a maximally flat magnitude filter. The Butterworth filter offers better attenuation rate than Bessel and has an attenuation of -3dB at the design cutoff frequency, while the attenuation beyond the cutoff frequency is a moderately steep -20dB/decade/pole. The pulse response of the Butterworth filter is better than Chebyshev, however it has moderate overshoot and ringing in step response.
A notch filter is a band-stop filter with a narrow stopband. A “notch filter” rejects a narrow frequency band and leaves the rest of the spectrum little changed. The most common example is 60-Hz noise from power lines. Another is low-frequency ground roll. Such filters can easily be made using a slight variation on the all-pass filter.
Narrow-band filters and sharp cutoff filters should be used with caution. An ever-present penalty for using such filters is that they do not decay rapidly in time. Although this may not present problems in some applications, it will certainly do so in others. Obviously, if the data-collection duration is shorter than or comparable to the impulse response of the narrow-band filter, then the transient effects of starting up the experiment will not have time to die out. Likewise, the notch should not be too narrow in a 60-Hz rejection filter. Even a bandpass filter has a certain decay rate in the time domain which may be too slow for some experiments. In radar and in reflection seismology, the importance of a signal is not related to its strength. Late arriving echoes may be very weak, but they contain information not found in earlier echoes. If too sharp a frequency characteristic is used, then filter resonance from early strong arrivals may not have decayed enough by the time the weak late echoes arrive.
Figure 4 2 Notch Filter
A curious thing about narrow-band reject filters is that when we look at their impulse responses, we always see the frequency being rejected! For example, look at Figure 4 2. The filter consists of a large spike (which contains all frequencies) and then a sinusoidal tail of polarity opposite to that of the frequency being rejected.
Nyquist’s Theorem (Sampling Theory)
The Nyquist’s Theorem, also known as the sampling theorem, is a principle that engineers follow in the digitization of analog signals. For analog-to-digital conversion (ADC) to result in a faithful reproduction of the signal, slices, called samples, of the analog waveform must be taken frequently. The number of samples per second is called the sampling rate or sampling frequency.
Any analog signal consists of components at various frequencies. The simplest case is the sine wave, in which all the signal energy is concentrated at one frequency. In practice, analog signals usually have complex waveforms, with components at many frequencies. The highest frequency component in an analog signal determines the bandwidth of that signal. The higher the frequency, the greater the bandwidth, if all other factors are held constant.
Suppose the highest frequency component, in hertz, for a given analog signal is fmax. According to the Nyquist’s Theorem, the sampling rate must be at least 2fmax, or twice the highest analog frequency component. The sampling in an analog-to-digital converter is actuated by a pulse generator (clock). If the sampling rate is less than 2fmax, some of the highest frequency components in the analog input signal will not be correctly represented in the digitized output. When such a digital signal is converted back to analog form by a digital-to-analog converter, false frequency components appear that were not in the original analog signal. This undesirable condition is a form of distortion called aliasing.
One of the most powerful and popular operations in vibration analysis is the decomposition of the most arbitrarily complicated waveform into a set of sine functions of different frequencies and phase angles. This is the exact reverse of the superposition of the sine functions.
DATA Processing Packages
Modern digital seismology may also involve the use of various data processing software packages (i.e. SeismoSignal etc.) or other numerical computing environments (i.e. MATLAB, Mathematica etc.) for the post data acquisition processing and analyses. SeismoSignal, a dedicated signal processing software package is efficient in processing strong-motion data and has the capability of deriving a number of strong-motion parameters often required by seismologists and earthquake engineers. In the course of this study various different set of filter configurations were used to evaluate different parameters. MATLAB, a numerical computing environment however, is contains numerous tools for digital signal processing and routine analysis including but not limited to filtering functions (Butterworth, Gaussian bandpass, notch filters, etc.), estimation of earthquake magnitude, baseline correction, instrument correction, simulation of arbitrary instruments characteristics, and so forth. (Mollova, 2007)
SeismoSignal constitutes an easy and efficient way to process strong-motion data, and is capable of deriving a number of strong-motion parameters often required by engineer seismologists and earthquake engineers, such as:
Elastic and constant-ductility inelastic response spectra
Fourier and Power spectra
Root-mean-square (RMS) of acceleration, velocity and displacement
Sustained maximum acceleration (SMA) and velocity (SMV)
Effective design acceleration (EDA) Acceleration (ASI) and velocity (VSI) spectrum intensity
Predominant (Tp) and mean (Tm) periods
Bracketed, uniform, significant and effective durations
SeismoSignal also enables the filtering of unwanted frequency content of the given signal. Three different digital filter types are available, all of which capable of carrying out highpass, lowpass, bandpass and bandstop filtering.
Introduction to Numerical Macro Modelling in SAP2000
Numerical FEM modeling of masonry structures is a very computationally demanding task and requires a careful consideration of the masonry structure characteristics, the non-linearity of the (masonry) material behaviour, and a substantial amount of reliable relevant experimental data to characterize the material. Masonry consists of individual masonry units called bricks / blocks which are either of ceramic origin or are stone / concrete blocks held together in the masonry formation by a sufficiently strong bonding material (mortar). Given the severely complex and intricate geometric nature of the masonry, it becomes overwhelmingly necessary to assume a certain convenient material model (stress-strain behaviour) and thus continue further with analysis using a reasonable FEM method or approach in order to then yield the global response of the structure. On the other hand, when a single element behaviour is studied, two types of approximations seem most effective namely, finite element method with discontinuous line elements and the plane element method.
In SAP2000 the Shell element is a three- or four- node formulation that combines membrane and plate- bending behaviour. The shell element can be of two types:
Homogeneous is the most commonly used type of shell. It combines membrane and plate behaviour. The membrane behaviour uses an Iso-Parametric formulation that includes translational in-plane stiffness components and a “drilling” rotational stiffness component in the direction normal to the plane of the element. Plate-bending behaviour includes two-way, out-of-plane, plate rotational stiffness components and a translational stiffness component in the direction normal to the plane of the element.