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Introduction and equations of motion

Book: Dynamics of structure, theory and application to earthquake engineering. By, Chopra.

Structural Dynamics is the structural response (time varying displacement, stresses) induced by time varying loads (magnitude, position, direction of load). Earthquake is a kind of dynamic load. The response of structure is in form of relative displacement (u), relative velocity (u ?) and relative acceleration (u ?). We need to calculate Umax.

The two basic dynamic loads are: Deterministic (Time variation of loading is fully known.

E.g. Vibrations due to machine, Rotatory machinery i.e. earthquake recorded {induced motion non-periodic with irregular pattern}), and, Non-deterministic (Response of structure to next earthquake, Wind loads). Moreover, probabilistic and statistic approach is used to forecast and predict the source of loading (Probabilistic displacement frequency ~ velocity, acceleration).

For Deterministic loads, two approaches are use:

  • Periodic sine waves
  • Fourier Analysis.

The reason to choose simple sine waves is;

  • Equations analyzed easily
  • The solution for simple case helps in calculation of irregular case by using Fourier series.

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    On the other hand, Fourier series is complicated and irregular history can be analyzed using it.

Fourier analysis i.e. P ~ t (for irregular case) is calculated by summing (individual plots of P~t) P1 +P2 + P3 · Pn which are all regular cases. In the end, Non-periodic can be either short duration or long duration. These can be evaluated by Duhamel integral, Frequency domain analysis (if fn is known we can determine natural velocity from plot of V ~ t), and Time integration methods.

Mathematical models:

In Disturbed Parameter system, displacement changes with time.

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Different positions have different displacements. A PDE i.e. Partial Differential Equation can be used, which is very complicated.

Figure 20 Famous model used in Structural dynamics

In Single Degree of Freedom System (SDOFS), if p(t) is removed then the mass will keep on vibrating “Forever” in case there is no dashpot available. There is only one (u), (u ?) and (u ?). The u1 at the top and the u2 at the middle of beam, respectively, are not independent.

mu ?+ku=p(t) Where, p(t) is the external force.

In Multiple Degree of Freedom System (MDOFS), the u1 at the top and the u2 at the middle of beam, respectively, are independent.

The first term is inertial force, second one is Dashpot force, third one is spring force and fourth one is External force.

Figure 21 Numerically evaluating the dynamic response, Model implied

The problem with above equation is how to solve it. This equation is for SDOFS, whereas another set of equations is used to determine MDOFS. The umax, (u_max ) ?,and (u_max ) ? are calculated, and then used in design to prevent such failures. In 1995, Hanshin Earthquake JPN occurred, wherein fire occurred after shaking and overhead road / elevated road turned upside down. Another earthquake of Wenchuan hit China back in 2008.

Numerical Evaluation of equation:

At time ti , Pi is the external force and ti+1 is another equation of motion, i.e. Acceleration , Velocity and fs. As we have 3 number of unknowns, so we need 3 equations. What is the procedure to obtain 2 more equations, we need to have some relationship between these 3 parameters or with some other parameters or we will use certain assumptions.

The three important requirements for numerical procedure are:

  • Convergence (Exact solution means theoretical solution)
  • Stability (Stable: Following smooth increasing or decreasing trend, Unstable: Not following smooth increasing or decreasing trend)
  • Accuracy. For any assumptions made, the above 3 requirements should be met.

Direct Integration Methods: Assumptions:

  • Slope of u ~ t (i.e. Displacement history) , is approximately velocity at time ti.
  • For acceleration and time ti, Acceleration = (Slope increment)/(Time increment).

The additional equations for velocity and acceleration, respectively, are developed as a result. (i) Central Difference method (ii) Average Acceleration methods: It seems to be a simple but very strong assumption.

3 different equations are developed for 3 unknowns. In addition, acceleration, velocity and displacement terms are calculated.

(iii) Linear Acceleration methods: Assumption; Linear relation between acceleration and time. Same procedure is followed as the Average acceleration method. In numerical methods, the philosophy is that at step I, all parameters are known.

Moreover, at next step, many parameters are unknown. U (t) can be calculated, which yields Umax, and on this basis predictions can be made. (iv) Newmark’s ? Method: Related to previous methods. Two assumptions related to acceleration and velocity were incorporated. It also used Integration methods. The Newmark’s sliding block method employs 3 main equations of acceleration, velocity and displacement. It is mostly employed in Geotechnical engineering. Whereas, the Newmark’s ? method is mostly used in Structural engineering.

Table 3 Final equations used in Newmark’s ? method and Newmark’s sliding block method

In case of horizontal earthquake force, the column acts like a beam. If the column is made up of brick, then its ability to prevent moment is VERY LOW. It is not considered good for earthquake. The S-waves (horizontal earthquake force) cause more damage and therefore these are studied in detail than the P-waves (vertical earthquake force) because the column caters the vertical acceleration well.

The vertical acceleration of 0.1g, 0.2g is considered OK whereas the Horizontal acceleration of 0.2g is immensely dangerous. Japan is located on Median tectonic line (major seismic fault). The 2011 earthquake of 9 magnitude causing Tsunami left 22000 people disappeared and dead, causing net loss of almost 200 US $.


In my viewpoint, Civil engineering works are among the chief carriers of human civilization and development. However, these are also the cause of different disasters. Firstly and most importantly, we need to keep soil at controlled stress level, with limiting deformation and such that Pore water pressure (PWP) will not increase. Experience suggests that, the lab test Results are reliable to be used.

As depicted in the report above, the civil engineering structures designed on basis of such results exhibit relatively good performance. In addition, we cannot know 100% about the properties of a material (soil, in particular) but have to accept the tests results.

Moreover, the geologists know that rock is ‘viscous’, and has creep (especially at higher stresses, it is much significant). According to Laboratory test results, the strain decreases with time.

However, some of the researchers do not accept this. Various aforementioned equations are used for rough evaluation of deformation in structures, and the result is not bad. We can certainly predict settlement and deformation with the help of such plots, to an extent of acceptable accuracy. Thus, it proves that civil engineering is reliable for such structures.

Other than that, I was able to grasp the idea that preventive measures in form of mathematical calculations and designing structures is immensely important. The understanding of roots for an equation, its physical interpretation, and above all the application are vital.

Lastly, our ability to improve thinking by comparing various research works, finding research gaps (as my esteemed instructor, Prof. Dr. Deng Jianliang kept on driving our minds towards it, and it was the best thing ever I came across), and brain storming will lead towards safety of living beings, sustainability, and a better civil engineering future.

Cite this page

Chopra - Dynamics of Structures. (2019, Dec 01). Retrieved from

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