结构动力学
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Overview
资料来源: 布莱克斯堡弗吉尼亚理工大学土木与环境工程系罗伯特. 里昂
现在很少有一整年没有发生重大地震事件在世界的某处肆虐。在某些情况下, 像印度尼西亚的2005的震后疼痛地震一样, 这一损失涉及了六个数字中的大片地理区域和伤亡人数。总的来说, 地震的数量和强度并没有增加, 但是, 建筑环境的脆弱性正在上升。随着地震活动区周围的无管制城市化, 如环太平洋 “火带”, 低洼沿海地区的海平面上升, 以及能源生产/分配和数字/电信的日益集中网络关键节点在脆弱地区, 很明显, 抗震设计是未来社区恢复能力的关键。
在过去的50年里, 设计结构来抵御地震破坏的进展很大, 主要是通过1964的新泻地震后的日本工作, 以及1971圣费尔南多河谷地震后的美国。这项工作沿着三条平行轨道前进: (a) 旨在发展改进的施工技术以尽量减少损害和生命损失的实验工作;(b) 基于先进几何和非线性材料模型的分析研究;(c) 将 (a) 和 (b) 中的结果综合成设计代码规定, 以提高结构抵御意外负荷的能力。
在实验室环境中进行地震测试往往是困难和昂贵的。测试主要是使用以下三技术进行的:
- 准静态测试(QST), 其中部分结构测试使用缓慢应用和等价预定的横向变形与理想化的边界条件。这项技术特别有助于评估结构细节对结构特定部分的韧性和变形能力的影响。
- 伪动态测试(PSDT), 当载荷也缓慢地应用时, 通过求解运动方程作为测试进展, 并利用直接测试反馈 (主要是瞬时刚度) 来评估实际刚度, 从而考虑到动态效应结构的阻尼特性。
- 震动表, 其中完整结构的尺度模型受输入运动使用液压驱动的基础或基础。震动表代表了一种更忠实的测试技术, 因为结构不是人为约束的, 输入是真实的地面运动, 而由此产生的力是真正的惯性, 正如人们在实际地震中预期的那样。然而, 电力需求是巨大的, 只有少数能够在几乎完全规模工作的震动表在世界各地存在。在全球范围内, 只有一个大型的震动台能够进行全面的结构测试, 这是在1985神户地震之后建立的日本电子防御设施的震动台。
在本实验中, 我们将利用一个小的震动台和模型结构来研究一些结构模型的动态行为特征。正是这些动态特性, 主要是固有的频率和阻尼, 以及结构细节和构造的质量, 使结构更容易受到地震的影响。
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!