구조 역학
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Overview
출처: 로베르토 레온, 버지니아 공대, 블랙스버그, 버지니아 토목 및 환경 공학부
일년 내내 전 세계 어딘가에 혼란을 일으키고 있는 주요 지진 사건없이 진행되는 것은 드문 일입니다. 인도네시아에서 2005년 반다 아쉬 지진과 같은 경우에 피해는 6건의 지역에서 큰 지리적 지역과 사상자를 포함시켰습니다. 일반적으로 지진의 수와 강도는 증가하지 않고 있지만, 건설 환경의 취약성이 증가하고 있습니다. 일주-태평양 ‘불의 벨트’, 저층 해안 지역의 해저 상승, 취약지역의 에너지 생산/유통 및 디지털/통신 네트워크 중요 노드의 농도가 증가함에 따라 지진 방지 설계가 미래 지역사회 회복의 핵심이라는 것은 분명합니다.
지진 피해에 저항하기 위한 구조물 설계는 1964년 니가타 지진 이후 일본과 1971년 산 페르난도 밸리 지진 이후 미국에서 주로 작업을 통해 지난 50년 동안 크게 발전했습니다. 이 작업은 세 개의 병렬 트랙을 따라 진행되었습니다 : (a) 개선 된 건설 기술을 개발하기위한 실험 작업은 손상과 인명 손실을 최소화하기 위한 것입니다. (b) 고급 기하학적 및 비선형 재료 모델을 기반으로 하는 분석 연구; 및,(c) 결과의 합성은 (a) 및 (b) 예기치 않은 하중에 저항하는 구조의 능력을 향상시키는 설계 코드 프로비저프로 한다.
실험실 환경에서의 지진 테스트는 종종 어렵고 비용이 많이 듭니다. 테스트는 주로 다음 세 가지 기술을 사용하여 수행됩니다.
- 구조의 일부가 이상화된 경계 조건과 함께 천천히 적용되고 이에 상응하는 미리 결정된 측면 변형을 사용하여 테스트되는 준 정적 테스트(QST). 이 기술은 구조의 특정 부분의 인성과 변형 능력에 대한 구조적 세부 사항의 효과를 평가하는 데 특히 유용합니다.
- 하중도 느리게 적용되는 의사 동적 테스트(PSDT)는 테스트가 진행됨에 따라 모션 방정식을 해결하고 직접 테스트 피드백(주로 즉각적인 강성)을 활용하여 구조의 실제 강성과 감쇠 특성을 평가함으로써 동적 효과를 고려합니다.
- 전체 구조의 스케일 모델이 유압 작동 식 베이스 또는 기초를 사용하여 입력 모션을 받는 경우 테이블을 흔들어줍니다. 쉐이크 테이블은 구조가 인위적으로 억제되지 않고 입력이 진정한 지면 모션이며 실제 지진에서 기대하는 것처럼 실제로 관성 힘이기 때문에 보다 충실한 테스트 기술을 나타냅니다. 그러나 전력 요구 사항은 엄청나며 거의 본격적인 작업할 수 있는 몇 개의 쉐이크 테이블만이 전 세계에 존재합니다. 전 세계적으로 1985년 고베 지진의 여파로 지어진 일본 E-Defense 시설의 쉐이크 테이블인 본격적인 구조물에 대한 테스트를 수행할 수 있는 대형 쉐이크 테이블은 하나뿐입니다.
이 실험에서는 작은 쉐이크 테이블과 모델 구조를 활용하여 일부 구조 모델의 동적 동작 특성을 연구할 것입니다. 이러한 동적 특성, 주로 자연 주파수 및 댐핑뿐만 아니라 구조 적 세부 사항 및 구조의 품질로 인해 구조물이 지진에 다소 취약합니다.
Principles
Procedure
Results
First, determine the frequency (ω) at which the maximum displacement occurred for each model. The original simple formula discussed above, 
, needs to be modified because the mass of the beam itself (mb = Wbeam/g), which is distributed over its height, is not negligible compared with the mass at the top (m = Wblock/g). The equivalent mass for the case of a cantilever beam is (m+0.23mb), where m is the mass at the top and mb is the distributed mass of the beam. The stiffness k is given by the reciprocal of the deformation (
) caused at the top of the cantilever by a unit force:
(Eq. 11)
where L is the length of the beam, E is the modulus of elasticity, and I is the moment of inertia. I is given by 
, where b is the width and h is the thickness of the beam. Thus, the natural circular frequency of a cantilever beam, including its self-weight is:
(Eq.12)
Based on this equation, the predicted natural frequencies are calculated in the Table 1.
| Beam Number | Length (in) |
Width (in.) |
Thick. (in.) |
I (in.4) |
E (ksi) |
Weight (lbs) |
Beam Weight (lbs.) |
Effective Mass (lbs-sec.2/in) |
Natural frequency (cycles per second) |
| 1 | 12.0 | 1.002 | 0.124 | 1.59E-04 | 10200 | 0.147 | 0.149 | 4.70E-04 | 2.45 |
| 2 | 16.0 | 1.003 | 0.124 | 1.59E-04 | 10200 | 0.146 | 0.199 | 4.97E-04 | 1.55 |
| 3 | 20.0 | 1.002 | 0.125 | 1.63E-04 | 10200 | 0.146 | 0.251 | 5.28E-04 | 1.09 |
| 4 | 24.0 | 1.003 | 0.125 | 1.63E-04 | 10200 | 0.148 | 0.301 | 5.63E-04 | 0.80 |
| 5 | 28.0 | 1.001 | 0.125 | 1.63E-04 | 10200 | 0.144 | 0.350 | 5.82E-04 | 0.62 |
| 6 | 32.0 | 1.000 | 0.124 | 1.59E-04 | 10200 | 0.146 | 0.397 | 6.15E-04 | 0.49 |
| 7 | 36.0 | 1.002 | 0.126 | 1.67E-04 | 10200 | 0.147 | 0.455 | 6.52E-04 | 0.41 |
| 8 | 40.00 | 1.000 | 0.125 | 1.63E-04 | 10200 | 0.148 | 0.500 | 6.81E-04 | 0.34 |
Table 1: Natural frequencies of the cantilever beams tested.
The measured and the theoretical values of the normal frequency for our model systems are compared in the Table 2. The actual natural frequencies were computed by carefully displacing the cantilever beam by 1 inch and then looking at the displacement vs. time response. The comparison below are made in terms of periods (Td , in sec.) as these were determined from Td = u0-u1, as shown in Figure 2(b). This requires care and patience to obtain reliable results. The demonstrations shown were only meant to give an overall illustration of the system behavior.
| Beam Number | Natural frequency (cycles per second) |
Predicted Period (sec.) |
Actual Period (sec.) |
Error (%) |
| 1 | 2.45 | 2.56 | 2.65 | -3.33% |
| 2 | 1.55 | 4.06 | 4.23 | -4.22% |
| 3 | 1.09 | 5.78 | 6.79 | -17.52% |
| 4 | 0.80 | 7.84 | 8.04 | -2.54% |
| 5 | 0.62 | 10.06 | 10.63 | -5.70% |
| 6 | 0.49 | 12.79 | 13.04 | -1.97% |
| 7 | 0.41 | 15.32 | 16.78 | -9.50% |
| 8 | 0.34 | 18.59 | 20.56 | -10.59% |
Table 2. Comparison of results.
The differences stem primarily from the fact that the beams are not rigidly attached to the wooden base, and the added flexibility at the base increases the period of the structure. Another source of error is that the damping was not accounted for in the calculations, because damping is very difficult to measure and amplitude dependent.
Next, from each of the displacement vs. time histories, extract the maximum value for each frequency and plot the magnitude of the displacement vs. normalized frequency like that in Figure 3. An example is shown in Figure 4, where we have normalized frequency versus the first natural frequency (Beam Number 1) and plotted the maximum displacement of that beam when the shake table was subjected to a varying sinusoidal deformation with amplitude of 1 in.
Figure 4: Deformation of Beam #1 vs. normalized table frequency.
Initially, when the ratio of ω/ωn is small, there is not much response as the energy input from the table motion does not excite the model. As ω/ωn approaches 1, there is a very significant increase in the response, with the deformations becoming quite large. The maximum response is reached when ω/ωn is very close to 1. As the normalized frequency increases beyond ω/ωn = 1, the dynamic response begins to die down; when ω/ωn becomes large we are in a situation where the load is being applied very slowly with respect to the natural frequency of the structure, and the deformation should become equal to that from a statically applied load.
The intent of these experiments is primarily to show the changes in behavior qualitatively, as shown in the demonstrations for the two frame structures. Obtaining results similar to those in Figures 3 and 4 requires great care and patience as sources of friction and similar will affect the amount of damping and thus shift the curves similar to those in Figure 3(c) to the left or right as the actual damped frequency, 
, changes.
Applications and Summary
In this experiment, the natural frequency and damping of a simple cantilever system were measured by using shake tables. Although the frequency content of an earthquake is random and covers a large bandwidth of frequencies, frequency spectra can be developed by translating the acceleration time history into the frequency domain through the use of Fourier transforms. If the predominant frequencies of the ground motion match that of the structure, it is likely that the structure will undergo large displacement and consequently be exposed to great damage or even collapse. Seismic design looks at the acceleration levels expected form an earthquake at a given location based on historical records, distance to the earthquake source, the type and size of the earthquake source, and the attenuation of the surface and body waves to determine a reasonable level of acceleration to be used for design.
What the general public often does not realize is that current seismic design provisions are only intended to minimize the probability of collapse and loss of life in the case that a maximum credible earthquake occurs to an acceptable level (around 5% to 10% in most cases). While structural designs to obtain lower probabilities of failure are possible, they begin to become uneconomical. Minimizing losses and improving resilience after such an event are not explicitly considered today, although such considerations are becoming more common, as many times the contents of a building and its functionality may be much more important than its safety. Consider for example the case of a nuclear power plant (like Fukushima in the 2011 Great Kanto Earthquake), a residential ten-story building in Los Angeles, or a computer chip manufacturing facility in Silicon Valley and their exposure and vulnerability to seismic events.
In the case of the nuclear power plant, it may be desirable to design the structure to minimize any damage given that the consequence of even a minimal failure can have very dire consequences. In this case, we should try to locate this facility as far away as possible from earthquake sources to minimize exposure, because minimizing vulnerability to the desired level is very difficult and expensive. The reality is that it is prohibitively expensive to do this given the public's desire to avoid not only a Fukushima-type incident, but also even a more limited one, like the nuclear disaster on Three Mile Island.
For the multi-story building in Los Angeles, it is more difficult to minimize exposure because a large network of seismic faults with somewhat unknown return periods is nearby, including the San Andreas Fault. In this case, the emphasis should be on robust design and detailing to minimize the structure's vulnerability; the owners of the residences should be conscious that they are taking a significant risk should an earthquake occur. They should not expect the building to collapse, but the building may be a complete loss if the earthquake is of a large enough magnitude.
For the computer chip plant, the problems may be completely different because the structure itself may be quite flexible and outside the frequency range of the earthquake. Thus, the structure may not suffer any damage; however, its contents (chip manufacturing equipment) may be severely damaged, and chip production could be disrupted. Depending on the specific set of chips being manufactured at the facility, the economic damage both to the owner of the facility and to the industry as a whole can be tremendous.
These three examples illustrate why one needs to develop resilient design strategies for our infrastructure. To reach this goal we need to understand both the input (ground motion) and output (structural response). This issue can only be addressed through a combined analytical and experimental approach. The former is reflected in the equations listed above, while the latter can only be achieved through the experimental work done through quasi-static, pseudo-dynamic, and shake table approaches.
Transcript
Structural dynamics, or the analysis of structure’s behavior when subjected to dynamic forces, is critical both for designing buildings able to resist earthquake and fatigue loads, and for providing occupant comfort in structures subjected to wind and other types of cyclic loads.
In order to develop resilient design strategies for our cities’ infrastructures, we need to understand both the input, for example, the ground motion during seismic activity, and the output, or the structural response of the buildings. This issue can only be addressed through a combined analytical and experimental approach.
Seismic testing in a laboratory setting is carried out using shake tables, where scale models of complete structures are subjected to input motions using an electrically or hydraulically actuated base. This method represents a more faithful testing technique, as the structure is not artificially restrained, and the input is true ground motion.
This video will illustrate the principles of dynamic analysis by using a shake table and model structures to study the dynamic behavior characteristics of different structural models.
The usual self weight loads acting on a structure are quasi static because they change very slowly or not at all with time. In contrast, loads produced by hurricanes and blasts, for example, are extremely dynamic in nature.
During an earthquake, the ground moves with certain acceleration while the structure tends to stay still. As a consequence, the dynamic loads acting on a structure are inertial, and they depend on the mass, stiffness, and damping of the structure. To solve this problem analytically, we employ basic physics laws and simplified models of the actual structures.
For example, both a bridge and a frame with rigid beam can be simplified to a single degree of freedom system, consisting of an elastic cantilever with length L and mass m, stiffness k, and damping c. Alternatively, another model system can be represented by a mass attached to a spring of elastic constant k, as well as a dash pot with a damping coefficient c. These components can be combined in parallel and in series to model different structural configurations.
For our mass and spring model system, if the ground is moving the external force acting on this system is proportional with the ground acceleration. The other forces in the system are the elastic force in the spring, proportional to the displacement, as well as the reaction force in the dash pot, proportional to the velocity.
Using Newton’s Second Law, we can write the equation of horizontal equilibrium of forces for this system. In the absence of external forces, and assuming the damping effects as negligible, this simplified equation has the following solution:
Here, wn is the undamped natural frequency of the system, and u0 is the initial displacement. If we add the effect of damping, the solution of the equation of motion is the following. Here the damped natural frequency of the system is expressed using the natural frequency and the damping coefficient.
The effective damping on the free oscillations of the system results in the decrease of the amplitude of vibrations with every cycle. Considering the displacements in two successive cycles, we can use the logarithmic decrement delta to calculate the damping constant zeta.
If the ground motion is taken as sinusoidal function, the solution for the equation of motion is given by the following function. Here phi is the phase lag, and R is the amplification response factor.
Let’s plot this factor versus frequency ratio for different values of the damping coefficient zeta. For low values of damping, as the frequency of the forcing function approaches the natural frequency of the system, the response of the system becomes unstable, a phenomenon that is commonly referred to as resonance.
Now that you understand the theoretical concepts regarding the behavior of a linear elastic system to dynamic loads, let’s investigate these concepts using a shake table.
First, construct several structures using very thin, strong, rectangular, T6011 aluminum beams, 1/32 of an inch in width, and having different lengths. To build the first model, insert one single cantilever with length of sixteen inches to a very rigid wood block. Place a mass of 0.25 lb on the tip of the cantilever.
Similarly, build three other model structures by attaching three cantilevers with lengths of 24, 32, and 36 inches to the same rigid wood block. Attach a 0.25 lb mass to the tip of each cantilever. Using thin steel plates and rigid acrylic floor diaphragms instrumented with accelerometers, prepare two other specimens simulating simple frame structures with flexible columns and rigid floors.
For these demonstrations, a table top electrically actuated shake table with a single degree of freedom will be used. A computer digitally controls the table displacement and generates periodic sine waves or random accelerations. The input forcing function can be checked by comparing it to the output of an accelerometer attached to the table.
First, carefully mount the four cantilever structures to the shake table using bolts attached to the model’s base. Then turn on the shake table, and using the software, slowly increase the frequency, until the maximum response of the structure is obtained. Record in a notebook the value of this frequency. Continue increasing the frequency until the displacements of all the cantilevers reduce significantly.
Now, mount the one-story model structure to the shake table and repeat the procedure. Slowly sweep through frequencies until resonance is reached. Next, reset the software to run a typical ground acceleration time history to show the random motions that occur during an earthquake. Replace the one-story model on the shake table with the two story structure, and repeat the procedure. Note that two natural frequencies occur in this case. Record in a notebook the values of these frequencies.
Now let’s perform the data analysis and discuss our results.
First, determine the frequency at which the maximum displacement occurred for each model. For the case of a cantilever beam the equivalent mass is given by the mass at the top, and the distributed mass of the beam. The stiffness k is the reciprocal of the deformation delta, caused at the top of the cantilever by a unit force, where L is the length of the beam and E is the modulus of elasticity.
Here, I is the moment of inertia that can be easily calculated if the width b and the thickness h of the beam are known. Place data in a table and then calculate the natural circular frequencies. With these values calculate the predicted periods of motion for the cantilever beams tested.
Next, look at the displacement versus time response recorded in this experiment, and determine from these plots the corresponding periods of motion of the cantilever beam. Add these measured periods to the table and compare them with the theoretical values.
The differences between the theory and experiment are due to several sources of errors. First, the beams are not rigidly attached to the wooden base, and the added flexibility at the base increases the period of the structure. Second, the damping was not accounted for in the calculations because damping is very difficult to measure and amplitude-dependent.
In this experiment we recorded the displacement versus time histories of the beam when the shake table was subjected to a varying sinusoidal deformation with an initial one inch amplitude. From these graphs, extract the maximum value for each frequency, and plot the magnitude of the displacement versus normalized frequency.
Now take a look at your plot. Initially there was not much response, as the energy input from the table motion does not excite the model. As the normalized frequency approaches one, there is a very significant increase in the response with the deformations becoming quite large. The maximum response has reached very close to one. As the normalized frequency increases beyond one, the dynamic response begins to die down. A large value of the normalized frequency corresponds to the situation where the load is applied very slowly with respect to the natural frequency of the cantilever and the deformation should become equal to that from a statically applied load.
Structural dynamics is widely used in the design and analysis of buildings, products, and equipment across many industries.
Designing structures resilient to earthquake damage has progressed greatly in the last 50 years. Nowadays the results from the experimental work, as well as from the analytical studies, are corroborated into design code provisions that improve the ability of structures to resist unexpected loads during a seismic event.
One easily observable dynamic response of a structure to wind loads is that of cantilevered traffic lights. As the wind flows over the structure, the wind regime is disturbed and vortices are generated through a phenomenon known as vortex shedding. These vortices induce forces perpendicular to the wind direction, resulting in a cyclic vertical displacement of the cantilevered arm, and as a consequence, potential fatigue damage of the structure.
You’ve just watched JoVE’s Introduction to the Dynamics of Structures. You should now understand the theoretical principles governing the behavior of a structure subjected to dynamic loads. You should also know how to use a shake table to perform a dynamic analysis of a model structure.
Thanks for watching!