1 Introduction

Reinforced concrete frame buildings are the most popular buildings nowadays in city centres, especially as office spaces (e.g. Paulay and Priestley 1992). The shape of such buildings is more often than not, irregular. These are no longer simply cuboid in nature; architects are trying to give them more contemporary forms. The dynamic response of building structures to earthquakes is affected by irregularities in their arrangement. Increasingly, structures with cantilever elements are created and the upper storeys protrude from the building. Such forms can constitute a problem in earthquake areas or in areas when surface vibrations come from rockbursts originating from mining activities. Soft (first) storey effect could be the reason for collapse (Bencat 2016; Cai et al. 2015; Çelebi et al. 2014; Chandramohan et al. 2016; Dolsek and Fajfar 2001; Ruiz and Diederich 1989; Sezen et al. 2003). The other cause is that irregular building structures often have solid structural components located on one side. This is the reason why irregular buildings are sensitive to torsional vibrations (Gokdemir et al. 2013). The result is that irregular buildings often suffer higher levels of damage than regular structures (Kim and Hong 2011). Torsion is a complex problem which can be analysed by a nonlinear approach using the simplified SESA method based on the superposition of modal effects with the use of overdamped displacement response spectra (Köber and Zamfirescu 2016). In Bencat (2016), the prediction method for the assessment of induced vibrations in irregular buildings is discussed.

Earthquakes are becoming increasingly stronger and of longer durations. Such a situation occurred in 1952 during the Kamchatka earthquake (Ben-Menahem and Toksöz 1963), and in 1964 in Alaska (Kanamori 1970). These earthquakes took place in remote areas and this is why they did not result in large losses; however, however, long-duration earthquakes have recently occurred in populated areas such as the sizable earthquake that occurred in the east of Japan in 2011 (Çelebi et al. 2014; Takemura et al. 2011). The effect of earthquake duration is not easy to investigate because building behaviour during earthquake could influenced by other factors like amplitude or response spectral shape. This is a subject that has been studied in different papers in the past, often with unfortunately ambiguous results (e.g. Shome et al. 1998; Tremblay and Atkinson 2001; Chai 2005). Results presented in various papers (e.g. Hancock and Bommer 2006; Iervolino et al. 2006; Dogangun 2004) have proved a lack of dependence between the duration of an earthquake and interstorey drift in building structures and positive correlation has been identified between the duration of strong earthquakes and structural damage (Hancock and Bommer 2006). The impact of the duration of ground motion on the behaviour of a structure has been found to be significantly visible for earthquakes of high intensities which cause the collapse of buildings (e.g. Belejo et al. 2017). Previous studies which have taken into consideration the effect of the duration of the records (e.g. Belejo et al. 2017) have shown that: (1) duration seems to not play an important role in terms of peak displacement responses and torsion behaviour of non-ductile reinforced concrete building; (2) the ground motion duration effects should consider the peak responses as well as number of cycles, residual deformations and energy dissipation. In turn results of investigation presented in (Hou and Qu 2015) indicated significant influence of the duration of ground motion on that of hysteretic energy dissipation demands.

Poland belongs to regions with low and very low seismic hazard but from time to time, weak earthquakes occur such as an event in 2004 in the Podhale region (Guterch et al. 2005; Zembaty et al. 2007; Wiejacz and Debski 2009). Poland, especially the south of Poland, is exposed to random mining tremors occurring as earthquakes (Zembaty 2004); these events are beyond human control. The maximum magnitudes of underground mining tremors are about 4.0–4.6. Such a value of magnitude can be comparable with weak earthquakes. There are however many differences between earthquakes and mining tremors (Tatara et al. 2017). The major differences are:

  1. (i)

    Duration of intensive phase of vibrations—mining tremors last less than earthquakes. The duration of the intensive phase can even be ten times less.

  2. (ii)

    Peak ground accelerations—mining tremors and earthquakes differ with regard to the order of magnitude. PGA for mining tremors can be about 0.3 g (g—acceleration of gravity) while the PGA for earthquakes may be 1–2 g (Furumura et al. 2011).

  3. (iii)

    Content of predominant frequencies—generally in the case of earthquakes, the predominant frequencies are low; the predominant frequencies of ground acceleration vibrations in the case of rock bursts are in the bands up to 6.5 Hz, from 6.5 to 12 Hz and higher than 12 Hz (sometimes even up to 40 Hz) (Maciag et al. 2016).

  4. (iv)

    Frequency of occurrence—strong earthquakes that cause structural damages occur in the same area with a frequency of the tens to hundreds of years. By contrast, mining tremors with a high value of PGA caused by mining activity and underground explosions occur much more often, within a few years of each other.

  5. (v)

    Depths of hypocenter—mining tremors occur at much shallower depths than earthquakes; the depth of the later could be as much as 180 km (Kufner et al. 2017).

The most precise way to estimate the dynamic response of irregular structures subjected to earthquakes or paraseismic actions is to create 3D model and conduct a nonlinear response history analysis (RHA). However, this approach has been described as being complicated, time consuming and is not useful in practice, for example, by Erduran 2008. These shortcomings resulting in the use of non-linear static analysis procedures (NSP) which are user friendly and faster to implement (Gupta and Krawinkler 2000; Chopra and Goel 2002). Combination of pushover analysis of multi degree of freedom model (MDOF) with the response spectrum method (RSM) for an equivalent single-degree-of-freedom (SDOF) system is another nonlinear method used in earthquake analysis named as the N2 method (Fajfar and Fishinger 1988; Fajfar et al. 2002). The proposed N2 method used both acceleration-displacement formulation and inelastic spectra resulting in the possibility of a visual interpretation of the procedure. The results of the N2 method are sufficiently accurate providing that the structure vibrates predominantly with the first natural frequency. The simplified nonlinear analysis procedures used an equivalent single-degree-of-freedom (SDOF) system have also been developed and proposed by several researchers, such as Kuramoto et al. (2000) and Fajfar (2000).

Numerical models are also very useful for studying the dependence between strong-motion duration and structural damage (e.g. Barbosa et al. 2017; Chandramohan et al. 2016); this is due to providing the opportunity to separate different components of ground motion and investigating interesting aspects of the dynamic response of the structure in question.

In this article, numerical studies are presented for five-storey irregular shaped RC building models subjected to five different types of excitation in the form of recorded surface horizontal vibrations; this is due to the differences between mining tremor and earthquake phenomenon. Although many numerical studies have recently been performed to investigate structural behaviour under earthquake and mining tremor excitation (Naderpour et al. 2016; Papaloizou et al. 2016; Lu et al. 2016; Dulinska and Nowakowska 2016; Kozlowska et al. 2016), there is still a need to investigate and compare results of numerical analysis made on the same object, in the same soil conditions. This paper is concerned with the duration of earthquakes and mining tremor excitations and the resultant structural responses.

2 FEA model

The building selected for analysis is located in Cracow. It is a reinforced concrete frame, five-storey building with a system of columns and floors that is typical for such a structure. The building dimensions are 17.6 by 17.9 m. The height of the building is 23.2 m. The dimensions of the structural components are as follows: columns are from 25 by 25 cm up to 35 by 35 cm, floor slabs are 25 cm thick. It is a irregular building with concrete staircases located on the north side of the building. Views of the building are presented in Fig. 1. Plans of the building and its cross section are presented in Figs. 2 and 3, respectively.

Fig. 1
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View of the building: a north-east elevation, b south-east elevation

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Fig. 2
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Plan view of the building: a ground floor, b 4th floor

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Fig. 3
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Cross section of the building

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The dynamic analysis of the building requires the construction of a spatial model using for example FEM (finite element method). The choice to construct a spatial model resulted from the irregularity of the position of load-bearing elements and the inability to apply simplifications in the construction of the model. The analyses were performed using DIANA FEA software (Diana 2017). The building model includes all the elements important from the point of view of stiffness and mass distribution, which determine the dynamic properties of the model. Support of the model at the foundation level is applied as fixed in translation and rotational degrees of freedom. Figure 4 shows the analysed structural model.

Fig. 4
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FEA model of the building

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For the reinforced columns, 3-node curved beam-column elements were used. For the walls and floor slabs, a layered shell element was used. This layered shell element is composed of an 8-node quadrilateral. The 8-node quadrilateral allows the inclusion of drill rotation degree of freedom and ensures better approximation of the results.

A nonlinear total strain rotating crack model was assumed for concrete elements like walls, slabs, and columns. For the infill walls of cellular concrete, the same material model with different parameters was also adopted. The properties of the materials are shown in Table 1.

Table 1 Material properties
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3 Model verification and dynamic analysis

3.1 Natural frequencies of the model

The natural frequencies were used to verify the model of the analysed building. The range of natural frequencies is relatively dense. The values of the first ten calculated natural frequencies with their mass participation factors are shown in Table 2. The mode shapes obtained from modal analysis are shown in Fig. 5.

Table 2 First ten natural frequencies of the FEM model
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Fig. 5
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First three mode shapes corresponding to the natural frequencies of the model

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The designed 3D model of the building has been subjected to experimental verification in order to compare the theoretical values of the natural frequencies with the values obtained from in situ experimental research of the building. Low values of damping in reinforced concrete buildings (Ciesielski et al. 1995) are the reason for identifying frequencies of free vibrations with their natural frequencies. The tests used several methods of generating building excitation—a modal hammer, traffic on a nearby street and controlled rides of light truck through the threshold release were used as sources of excitation for the investigated building.

Characteristics of the modal hammer are as follows:

  1. 1.

    PCB Model 086D50 large-sledge impulse hammer

  2. 2.

    sensitivity: (± 15%) 0.23 mV/N

  3. 3.

    measurement range: ± 22,240 N

  4. 4.

    hammer mass: 5.5 kg.

Preliminary results of numerical tests have shown that the values of fundamental lateral frequencies in ‘x’ and ‘y’ directions are low resulting in further the selection of appropriate measuring equipment. The measuring system consisted of PCB acceleration sensors and appropriate instrumentation. The error of the measured accelerations did not exceed 11.61%. The details of the measurement method have been described in the paper by Tatara et al. (2017).

The PCB sensors were located in selected points on three levels of the building. The sensors recorded vibrations in three directions corresponding to two perpendicular horizontal directions and vertical direction ‘z’. An example of a selected point with PCB accelerometers mounted to the bearing element of the building is shown in Fig. 6.

Fig. 6
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PCB sensors located at the ground level of the structure

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The analysed building is characterised by low natural vibration frequencies which resulted in the adoption of the low-pass filter with cut-off frequencies equal to 10 and 50 Hz for the recorded vibrations. FFT analysis of the filtered vibration records was then performed. Example results of FFT analysis of horizontal building vibrations recorded on the fourth floor due to impact of the modal hammer at the bottom of the structural column located in 1/F′ axis is shown in Fig. 7.

Fig. 7
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Result of the FFT of the horizontal components ‘x’ and ‘y’ of the structure on the 4th floor due to the modal hammer impact in the column

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The records of free vibrations and results of their FFT analysis were the basis for determining the values of natural flexural vibrations in the horizontal directions and the frequency of torsional vibrations. The calculated and measured values of natural frequencies are compared in Table 3. Differences between the values of the calculated and measured fundamental horizontal frequencies do not exceed 3%. Similar differences were observed for the second natural frequencies of the examined building. This was the basis for accepting the adopted model for further seismic analysis. Note that the fundamental frequencies are significant in the determination of wind load (PN-EN 1991-1-4 (191) Eurocode 1; Riera and Davenport 1998) and in results of the dynamic calculation of surface structures subjected to earthquakes and mining shocks (Tatara et al. 2017).

Table 3 Natural frequencies of the model based on dynamic measurements (Hz)
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In dynamic analysis, records of the horizontal components of ground acceleration vibrations caused by underground mining-related exploitation of mineral resources and earthquakes were used (see Figs. 8–12 respectively). Records of earthquakes which occurred in the district of Sitka, Alaska (1972) and in El Centro (1940) presented in Figs. 11 and 12, respectively were also applied. In Poland, mining-related surface vibrations mainly originate from the underground exploitation of hard and lignite coal and copper ore. A detailed description of the phenomenon of mining shakes in Poland is given by Tatara et al. (2017).

Mining-related wave propagation and amplification in soils have been the research interests of many authors (Driad-Lebeau et al. 2009; Semblat et al. 2000, 2005). The problem of the accumulation of the dynamic response of a rock mass disturbed by mining activities has been studied in many papers (e.g. Cai et al. 2015). In Poland, surface mining-related vibrations have a significant impact on surface and underground structures.

In Poland, the area of the Legnica-Glogowski Copper Region (LGCR) is one of the most exposed areas with regard to mining tremors. Mining tremors are characterised by a high frequency of occurrence in this area. The most numerous forms are rockbursts with energies not exceeded E7 (J). Over recent years, the intensity of mining shocks with energies exceeding E7 (J) is increasing. Such shocks generate surface vibrations with large maximum values of free-field vibrations. To date in the LGCR, mining shocks with high energy values have not resulted in fatalities as in the case of earthquakes. Mining shocks have caused material damage. Figure 10 presents, as an example, horizontal components of mining vibration caused by the most intense rocbursts with energy values of 1.9E9 (J) in the LGCR area and results of FFT analysis. The maximum value of horizontal components ‘x’ and ‘y’ (PGAx, PGAy) of acceleration mining-related vibration are greater than corresponding components of the analysed Sitka earthquake (Figs. 10, 11 and Table 4).

Table 4 Characteristics of the analysed components of vibrations
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The content of dominant frequencies of the analysed mining origin records is shown in Figs. 8, 9 and 10. Analysis of the FFT results of vibration records from Figs. 8, 9 and 10 indicates that the predominant frequencies of ground acceleration vibrations in all cases of rockbursts in the LGCR area are below 12 Hz and the highest frequencies do not exceed 25 Hz. In the dynamic analysis of the model, we also used other records of surface mining origin corresponding to rockbursts with smaller energy values of 4.2 E6 and 1.7 E8 (J)—see Figs. 8 and 9.

Fig. 8
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The horizontal components of mining tremor no. 1

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Fig. 9
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The horizontal components of mining tremor no. 2

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Fig. 10
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The horizontal components of mining tremor no. 3

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Fig. 11
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The horizontal components of the Sitka earthquake

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Fig. 12
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The horizontal components of the El Centro earthquake

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The duration of the intensive phase of mining origin vibrations is relatively short and does not exceed 10 s. In the case of earthquakes, these durations are much longer and amount to several dozen seconds. For example duration of intensive phase based on Arias intensity for Sitka earthquake is 3.5 times greater than in the case of mining tremor no. 3 (see Table 4).

Values of the coefficient (PGA/PGV) indicate that the strongest mining tremors correspond to weak, shallow earthquakes. The strongest analysed mining tremor (see Fig. 10) is characterised by higher acceleration values than the analysed Sitka earthquake. As a rule, the maximum values of mining origin free-field vibration accelerations do not exceed the value of 0.30 g (Maciag et al. 2016). Table 4 contains data characterising the basic parameters of surface horizontal components of the analysed earthquakes and vibrations of mining origin.

The dynamic numerical analysis was performed in two stages. In the first stage, gravity loads were considered as quasi static. The nonlinear static equations of gravity were solved using the iterative Newton method available in Diana code (DIANA user manual release 10.2). In the second stage of analysis, a nonlinear THA (time history analysis) was made with earthquake records as the excitation.

The Newmark method, with a constant time step of 0.005 s, was used to integrate the equations of motion in time. The constant time step of 0.005 s was selected based on a preliminary convergence study with respect to the integration time step size to ensure the accuracy of the results. The quasi-Newton (secant) method based on the Broyden–Fletcher–Goldfarb–Shanno (BFGS) stiffness update method was employed as the iterative method to solve the nonlinear dynamic equilibrium equations (DIANA user manual release 10.2). At the end of each time step of analysis, the last obtained secant stiffness matrix was stored and used as the initial stiffness matrix at the first iteration of the next time step. The convergence criterion was based on the relative norm of the last displacement increment vector.

4 Results of dynamic analysis

Generally, in the dynamic analysis combination of the dead load, the dynamic load (kinematic excitation) is considered. In the present investigation the global response for the kinematic excitations was analysed. Figure 13 shows an example character of deformation at the maximum deflection of the tower due to mining shock from Fig. 10. It may be noted that the model with its high irregularity has a tendency to rotate with significant rotational deformation above the highest floor—on roof level (see Fig. 13). A similar phenomenon is observed in the case of other kinematic excitations.

Fig. 13
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Example deformation model corresponding to the maximal inclination of the structure due to mining shock–mining tremor no. 3

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After nonlinear analysis of the model, the failure modes and crack propagation can be seen in Figs. 14, 15, 16, 17 and 18. As can be observed, the crack patterns (shown using the normal principal strains) are concentrated in the staircase shaft. The walls of the staircase together with the external columns transfer the seismic horizontal forces. Major cracks propagated at the last floor where the weak and soft closing of the staircase shaft is, and around the connection with the foundation slab.

Fig. 14
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Crack patterns for mining tremor no. 1

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Fig. 15
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Crack patterns for mining tremor no. 2

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Fig. 16
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Crack patterns for mining tremor no. 3

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Fig. 17
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Crack patterns for Sitka earthquake record

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Fig. 18
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Crack patterns for El Centro earthquake record

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In the case of mining shock from Fig. 10, despite the short time of the intensive phase, the range of damage defined by the cracks is significant but does not lead to significant damage of the model. While in the case of the earthquake from Fig. 11, the damage range is similar, despite half the values of PGA, but much longer duration of intensive phase (Figs. 16, 17). This leads to the conclusion that the duration of the intensive phase of seismic phenomenon is very important in the dynamic response analysis of structures. The same conclusion may be drawn from the analysis of mining shock no. 2 (Fig. 9). This is a shock with a very low duration of intensive phase and PGA values greater than the earthquake shock from Sitka (Sitka record—see Fig. 11). Only local cracks of very small widths can be observed in the model (Fig. 15). A very similar cracking scenario was observed during mining tremor no. 1 with a small PGA value and a short duration of intensive phase (Fig. 14). This indicates that even relatively weak mining shocks can cause the appearance of cracks in the irregular structure.

Damage to the infill walls and reinforced concrete walls of the model can be seen in Fig. 18, after kinematic excitation from Fig. 12 (the El Centro earthquake 1940). None of adopted kinematic excitations have such PGA values, only the Sitka earthquake record have a similar duration of the intensive phase. This earthquake was one of the most intensive in history. Major cracks occurred in the concrete walls of the staircase in every floor as well as in the columns and infill walls made of cellular concrete. The crack widths exceed 1 cm; the range of the damage is also very large. If the building was designed in the area of such interactions, its design would have to be significantly modified to withstand an earthquake of such intensity. Table 5 summarises the qualitative results of numerical modelling for all five records used for excitation.

Table 5 Summary of damage scenarios
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5 Conclusions

In the article, five types of seismic excitation were analysed—three were mining tremors and two were earthquakes. Excitations were selected due to basic characteristics such as intensive phase duration and peak ground acceleration (PGA values).

An irregular building with a stiff staircases on its north-east side was selected for analysis and it was verified by measurement results (in situ tests) and then subjected to all five kinematic excitations. Nonlinear time-domain calculations were made. Based on the size of cracks that appeared during excitation, construction stresses were determined.

The results of calculations show a significant influence of intensive phase duration on the dynamic response of the irregular building. Comparing the first two mining tremors (the first with low PGA and the second with high PGA values), it can be seen that there are only very small local cracks (Figs. 14, 15). However, when the third form of mining tremor is considered, it can be seen that significant cracks in structural elements appeared, but they did not lead to a sudden collapse of the model (Fig. 16). This excitation differs from first two mining tremors in relation to the intensive phase duration.

The thesis of the significant influence of the intensive phase duration on the structural response of irregular building is also confirmed when comparing numerical results obtained from mining tremors no. 3 and Sitka earthquake. The records referring to Sitka earthquake have a long-term duration of the intensive phase, but the PGA values are around half that of mining tremor no. 3. The size of damage to both of these cases is similar (see Figs. 16, 18). The literature overview confirms this relationship between long-term vibrations and damage occurring in building structures (Hancock and Bommer 2006); however, in this article, it is confirmed for mining shocks. This parameter is used in the intensity scale of seismic mining events in Poland (Mutke et al. 2008; Mutke and Dubinski 2016).

In the case of very high PGA values and long duration of the intensive phase (the El Centro earthquake), the range of damages in the model is large. For such an intensive phenomenon, the structure of this irregular building should be designed according to the seismic standard.