- Email: [email protected]
A trivial way to construct a tripartite response spectrum grid
Soil Dynamics and Earthquake Engineering 97 (2017) 182–183
Contents lists available at ScienceDirect
Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn
Technical Note
A trivial way to construct a tripartite response spectrum grid
MARK
Eduardo Kausel Professor of Civil and Environmental Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Room 1-271, Cambridge, MA 02139, USA
A BS T RAC T We demonstrate in this note a disarmingly simple and elementary way to construct a tri-logarithmic response spectrum grid, albeit restricted to the metric system. The reason is that we rely herein on the very close approximations for the acceleration of gravity g ≈ π 2 as well as π 2 ≈ 10 , whereas scales in the US Customary system of units do not benefit from these near coincidences. It is believed that the method proposed may be useful not only in an engineering office, but even more so in the context of a course on Structural Dynamics and Earthquake Engineering.
1. Introduction As anyone who has tried drawing a pseudo-velocity response spectrum in tripartite format will have discovered, that task it is not quite straightforward. Indeed, although ultimately doable, it requires some time and effort to write a program able to accomplish that goal. Of course, once the program is in place, one need not worry any further about it, but often —and especially in the classroom— it behooves to teach the students a way to quickly draw response spectra in trilogarithmic format without expending an inordinate effort in doing just that, or asking them instead to resort to a black box. This is especially true when the students rely on their own software and ad-hoc programming to complete home assignments. Moreover, this can also be true in an engineering office when that program is not readily available. Here we present a nearly trivial way to accomplish that goal and without much effort on the part of the user. The straightforward details are given in the ensuing. 2. Metric tri-partite response spectrum The relationship between the maximum relative displacement Sd , the pseudo-velocity Sv and the pseudo-acceleration as a fraction of gravity is
Sa(f ) =
ω Sv 2πf Sv = , g g
Sd (f ) =
Sv S = v ω 2πf
Now, in the metric system g = 9.80665 [m/s2]. In addition π 2 = 9.86960 ≈ 10 . Hence, the ratio of these two quantities is π 2 / g = 1.00641 ≈ 1.0 , so to an excellent fit we can make the approximation g = π 2 , at least when expressed in the metric system. It follows that
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.soildyn.2017.03.022 Received 21 August 2016; Accepted 14 March 2017 0267-7261/ © 2017 Published by Elsevier Ltd.
Sa(f ) =
2πf Sv 2πf Sv f ≈ = 1 Sv [g] g π2 π 2
Sd (f ) =
1 1 π Sv = 2 Sv ≈ 0.1 2πf π 2f 1
1 π 2
f
Sv [m]
In particular, when f = 2 π [Hz] and Sv = 1 [m / s] are chosen, then Sa = 1 [g] and Sd (f ) = 0.1 [m]. This means that a point exists on the main horizontal-vertical grid with tripartite coordinates 1 f = 2 π , Sv = 1, Sa = 1, Sd = 0.1 which can be used to define the origin of coordinates for the inclined logarithmic grid. Alternatively, we can scale everything down by a factor 100 and choose instead Sv = 1 [cm / s], Sd (f ) = 0.1 [cm] and Sa = 0.01 [g], as done in Fig. 1. Moreover, scaled repetitions of this pattern recur at ten-multiples of these numbers, so that the major lines of the grid at 45 degrees also passes though the 1 points at f = 2 π , Sv = 10, 100, ⋯[cm/ s], as identified in Fig. 1 by means of heavy dots. This means that the inclined grid has in both 1 directions a geometric size that is sin 45° = 2 2 times the horizontalvertical grid. Hence, we can easily construct a metric tripartite response spectrum as follows: Draw an empty logarithmic grid from, say 0.1–30 Hz for both axes. Make sure the grid is square, and not rectangular (i.e. horizontal and vertical axes must have the same scale). If necessary, scale or resize the plot so that both axes are equal. 1 Copy this grid, and scale down that copy by a factor 2 2 = sin 45°. Make sure the background of the latter is transparent and not opaque white. Rotate the copy counterclockwise by 45 degrees. 1 Overlay the rotated copy so that the point at f = 2 π , Sv = 1 becomes the origin Sa = 1, Sd = 0.1 of the rotated grid. Alternatively,
Soil Dynamics and Earthquake Engineering 97 (2017) 182–183
E. Kausel
Fig. 1. : Overlap of logarithmic plot with rotated, & scaled plot (scaling factor
1 2
1 2
2 ). Observe that the origin of the rotated plot lies on the horizontal axis at f = π .
3. Comparison with the true grid
if the vertical axis were to be in [cm/s], then the displacements would be in [cm] while the accelerations would be scaled down by a factor 100, as shown in Fig. 1. Annotate additional decimal multiples as needed. The beauty of this approach is that along vertical lines, one half of the major ticks of the rotated axes coincide with decimal multiples on 1 the main grid, and the inclined axes share the same 2 2 scaling. Thus, both grids are square and it is very easy to accomplish the overlapping. Fig. 1 shown herein depicts a schematic view of the proposed method, and without any truncation or further modification of the plot. Of course, one could easily add additional copies of the grid to extend the range of variables, as may be desired, in which case the original grid would best be defined from, say, 0.1–10, and not 30. Finally, if what the user wanted was to plot the actual response spectrum for a real or synthetic earthquake, it would suffice to plot that spectrum in doubly-logarithmic Cartesian coordinates, then save that plot and proceed to delete the spectrum so that only the bi-logarithmic grid remains (and making sure that it is transparent). Thereafter, proceed as described previously.
To obtain the true location f0 , Sv 0 of the origin in the exact inclined grid, it would suffice to consider the following identities:
Sd 0 =
Sv 0 = 0.1, 2πf0
which,
when
1
Sa0 = Sv 0 solved
2πf0 g
for
=1 both
Sv0
and
f0
would
yield
π
f0 = 2π 10g = 1.57609 = 1.006 2 and Sv0 = 0.9934 . These are so close to the approximate origin proposed —the difference is smaller than 1%— that the true and approximate grids are nearly indistinguishable. 4. Conclusion A simple, graphical method is suggested for the construction of a tripartite grid in the metric system. The two approximations used are sufficiently close that a true exact grid can hardly be distinguished from the approximate grid if the two were drawn together.
183
Contents lists available at ScienceDirect
Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn
Technical Note
A trivial way to construct a tripartite response spectrum grid
MARK
Eduardo Kausel Professor of Civil and Environmental Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Room 1-271, Cambridge, MA 02139, USA
A BS T RAC T We demonstrate in this note a disarmingly simple and elementary way to construct a tri-logarithmic response spectrum grid, albeit restricted to the metric system. The reason is that we rely herein on the very close approximations for the acceleration of gravity g ≈ π 2 as well as π 2 ≈ 10 , whereas scales in the US Customary system of units do not benefit from these near coincidences. It is believed that the method proposed may be useful not only in an engineering office, but even more so in the context of a course on Structural Dynamics and Earthquake Engineering.
1. Introduction As anyone who has tried drawing a pseudo-velocity response spectrum in tripartite format will have discovered, that task it is not quite straightforward. Indeed, although ultimately doable, it requires some time and effort to write a program able to accomplish that goal. Of course, once the program is in place, one need not worry any further about it, but often —and especially in the classroom— it behooves to teach the students a way to quickly draw response spectra in trilogarithmic format without expending an inordinate effort in doing just that, or asking them instead to resort to a black box. This is especially true when the students rely on their own software and ad-hoc programming to complete home assignments. Moreover, this can also be true in an engineering office when that program is not readily available. Here we present a nearly trivial way to accomplish that goal and without much effort on the part of the user. The straightforward details are given in the ensuing. 2. Metric tri-partite response spectrum The relationship between the maximum relative displacement Sd , the pseudo-velocity Sv and the pseudo-acceleration as a fraction of gravity is
Sa(f ) =
ω Sv 2πf Sv = , g g
Sd (f ) =
Sv S = v ω 2πf
Now, in the metric system g = 9.80665 [m/s2]. In addition π 2 = 9.86960 ≈ 10 . Hence, the ratio of these two quantities is π 2 / g = 1.00641 ≈ 1.0 , so to an excellent fit we can make the approximation g = π 2 , at least when expressed in the metric system. It follows that
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.soildyn.2017.03.022 Received 21 August 2016; Accepted 14 March 2017 0267-7261/ © 2017 Published by Elsevier Ltd.
Sa(f ) =
2πf Sv 2πf Sv f ≈ = 1 Sv [g] g π2 π 2
Sd (f ) =
1 1 π Sv = 2 Sv ≈ 0.1 2πf π 2f 1
1 π 2
f
Sv [m]
In particular, when f = 2 π [Hz] and Sv = 1 [m / s] are chosen, then Sa = 1 [g] and Sd (f ) = 0.1 [m]. This means that a point exists on the main horizontal-vertical grid with tripartite coordinates 1 f = 2 π , Sv = 1, Sa = 1, Sd = 0.1 which can be used to define the origin of coordinates for the inclined logarithmic grid. Alternatively, we can scale everything down by a factor 100 and choose instead Sv = 1 [cm / s], Sd (f ) = 0.1 [cm] and Sa = 0.01 [g], as done in Fig. 1. Moreover, scaled repetitions of this pattern recur at ten-multiples of these numbers, so that the major lines of the grid at 45 degrees also passes though the 1 points at f = 2 π , Sv = 10, 100, ⋯[cm/ s], as identified in Fig. 1 by means of heavy dots. This means that the inclined grid has in both 1 directions a geometric size that is sin 45° = 2 2 times the horizontalvertical grid. Hence, we can easily construct a metric tripartite response spectrum as follows: Draw an empty logarithmic grid from, say 0.1–30 Hz for both axes. Make sure the grid is square, and not rectangular (i.e. horizontal and vertical axes must have the same scale). If necessary, scale or resize the plot so that both axes are equal. 1 Copy this grid, and scale down that copy by a factor 2 2 = sin 45°. Make sure the background of the latter is transparent and not opaque white. Rotate the copy counterclockwise by 45 degrees. 1 Overlay the rotated copy so that the point at f = 2 π , Sv = 1 becomes the origin Sa = 1, Sd = 0.1 of the rotated grid. Alternatively,
Soil Dynamics and Earthquake Engineering 97 (2017) 182–183
E. Kausel
Fig. 1. : Overlap of logarithmic plot with rotated, & scaled plot (scaling factor
1 2
1 2
2 ). Observe that the origin of the rotated plot lies on the horizontal axis at f = π .
3. Comparison with the true grid
if the vertical axis were to be in [cm/s], then the displacements would be in [cm] while the accelerations would be scaled down by a factor 100, as shown in Fig. 1. Annotate additional decimal multiples as needed. The beauty of this approach is that along vertical lines, one half of the major ticks of the rotated axes coincide with decimal multiples on 1 the main grid, and the inclined axes share the same 2 2 scaling. Thus, both grids are square and it is very easy to accomplish the overlapping. Fig. 1 shown herein depicts a schematic view of the proposed method, and without any truncation or further modification of the plot. Of course, one could easily add additional copies of the grid to extend the range of variables, as may be desired, in which case the original grid would best be defined from, say, 0.1–10, and not 30. Finally, if what the user wanted was to plot the actual response spectrum for a real or synthetic earthquake, it would suffice to plot that spectrum in doubly-logarithmic Cartesian coordinates, then save that plot and proceed to delete the spectrum so that only the bi-logarithmic grid remains (and making sure that it is transparent). Thereafter, proceed as described previously.
To obtain the true location f0 , Sv 0 of the origin in the exact inclined grid, it would suffice to consider the following identities:
Sd 0 =
Sv 0 = 0.1, 2πf0
which,
when
1
Sa0 = Sv 0 solved
2πf0 g
for
=1 both
Sv0
and
f0
would
yield
π
f0 = 2π 10g = 1.57609 = 1.006 2 and Sv0 = 0.9934 . These are so close to the approximate origin proposed —the difference is smaller than 1%— that the true and approximate grids are nearly indistinguishable. 4. Conclusion A simple, graphical method is suggested for the construction of a tripartite grid in the metric system. The two approximations used are sufficiently close that a true exact grid can hardly be distinguished from the approximate grid if the two were drawn together.
183