# Elastic design spectrum

Although the recorded ground acceleration and response spectra of past earthquakes provide a basis for the rational design of structures to resist earthquakes, they cannot be used directly in design, since the response of a given structure to past earthquakes will invariably be different from its response to a future earthquake.

Elastic design spectrum

Although the recorded ground acceleration and response spectra of past earthquakes provide a basis for the rational design of structures to resist earthquakes, they cannot be used directly in design, since the response of a given structure to past earthquakes will invariably be different from its response to a future earthquake. However, certain similarities exist among earthquakes recorded under similar conditions. These characteristics have been discussed in previous sections on response spectra. Earthquake data with common characteristics have been averaged and ŌĆśsmoothedŌĆÖ to create ŌĆśdesign spectraŌĆÖ.

This spectrum is normalized to a maximum ground acceleration of 1.0g. The maximum ground velocity is specified by 1219.2 mm/s and maximum ground displacement is 914.4 mm. The principal regions (acceleration, velocity and displacement) of the design spectrum are identified in which the response is approximately a constant, amplified value. Amplification factors are then applied to the maximum ground motion in these regions to obtain the desired design spectrum. The procedure is as follows:

1.       Plot the anticipated ground motion polygon on four-way logarithmic       paper.

2.       Apply the appropriate amplification factor presented in Table 17.2 with maximum ground motion to construct design spectrum for specific damping   values.

3.       Draw the amplified displacement bound parallel to maximum ground motion displacement.

4.       Draw the amplified velocity bound parallel to maximum ground velocity.

5.       Draw the amplified acceleration bound parallel to maximum ground acceleration. 6.     Below a period of 0.17 s the amplified acceleration bound approaches the maximum ground acceleration. Draw a straight line from the amplification acceleration bound at 0.17 s to the maximum ground acceleration line at 0.033 s.

7.     Below a period of 0.033 s the acceleration bound is the same as maximum ground acceleration.

In general, the spectral intensities for vertical motion can be taken as approximately two-thirds of horizontal motion when the fault positions are horizontal. Where fault motions are expected to involve large vertical components, the spectral intensity, vertical motion is assumed to be equal to horizontal.

Example 17.6

Construct a NewmarkŌĆōHall design spectrum for a maximum ground acceleration equal to that of Northridge earthquake 0.308g and for concrete buildings.

Solution

(a)  Determine maximum ground motion parameters Maximum ground acceleration | u╦Ö╦Ög (t ) | = 0.308g

Maximum ground velocity = 1219.2 ├Ś 0.308

| u╦Ö g ( t ) |max = 375.5mm/s

Maximum ground displacement = 914.4 ├Ś 0.308

|  Ug(t) |max = 281.63 mm

(b)Determine the amplified response parameters for p = 0.05

From table amplified Spa = 2.6 ├Ś 0.308g

= 0.801g m/s2

Amplified Spv = 1.9 ├Ś 375.5

= 713.45 mm/s

Amplified Spd = sd = 1.4 ├Ś 281.63

= 394.28 mm

(b) Construction of design spectrum

(i)        Draw the maximum ground motion polygon using | u╦Ö╦Ög ( t ) |max,

| u╦Ö g (t ) | max, | u g (t ) |max .

(ii)Draw the amplified displacement Sd bound parallel to the maximum ground displacement.

(iii)   Draw the amplified velocity Sv parallel to the maximum ground velocity line. It intersects displacement bound at T1 = 3.5 s.

(iv)   Draw the amplified acceleration Spa bound to maximum ground acceleration. It intersects velocity bound at 0.55 = T2. Extend the amplified acceleration bound downward left to the point corresponding to T3 = 0.17.

(v)     Draw the amplified acceleration bound linearly from the point corresponding to T3 = 0.17 so that it intersects a line at T4 = 0.033 s with maximum ground acceleration.

This spectrum is shown in Fig. 17.20.

Researchers have developed procedures to construct the design spectra. From the ground motion parameters the recommended values for firm ground are Ta = 1/33; Tb = 1/8; Te = 10; Tf = 33 s. The amplification factors for ╬▒a, ╬▒v, ╬▒d for Spa, Spv, Sd were developed for two different non-accedence probabilities 50% and 84.1% as given in Table 17.3. NewmarkŌĆōHall elastic spectrum construction is shown in Fig. 17.21.

Program 17.3: MATLAB program for drawing NemarkŌĆōHall design spectra

%********************************************************** % TO DRAW ELASTIC DESIGN NEWMARK-HALL SPECTRA %********************************************************** %give ip=1 for 50% mean and ip=2 for 84.1% median ip=1

%********************************************************** c=[3.21 4.38;2.31 3.38;1.82 2.73];

d=[-.68 -1.04;-.41 -0.67;-.27 -.45]; %**********************************************************

%give damping value rho %********************************************************** rho=5 %********************************************************** %**********************************************************

%give peak ground acceln, peak ground velocity peak ground disp %********************************************************** pga=981.0;

pgv=121.92;

pgd=91.44;

ca=c(1,ip)+d(1,ip)*log(rho);

cv=c(2,ip)+d(2,ip)*log(rho);

cd=c(3,ip)+d(3,ip)*log(rho); for k=.00001:.00001:.0001 x=0.01:1:100 t=log(2*pi*k)-log(x) y=exp(t) loglog(x,y,ŌĆśkŌĆÖ),grid on

hold on t=log(k*9.81/(2*pi))+log(x) y=exp(t)

loglog(x,y,ŌĆśkŌĆÖ) hold on

end

for k=.0001:.0001:.001 x=0.01:1:100 t=log(2*pi*k)-log(x) y=exp(t) loglog(x,y,ŌĆśkŌĆÖ),grid on hold on

t=log(k*9.81/(2*pi))+log(x)

y=exp(t)

loglog(x,y,ŌĆśkŌĆÖ) hold on

end

for k=.001:.001:.01 x=0.01:1:100 t=log(2*pi*k)-log(x) y=exp(t) loglog(x,y,ŌĆśkŌĆÖ),grid on hold on

t=log(k*9.81/(2*pi))+log(x)

y=exp(t)

loglog(x,y,ŌĆśkŌĆÖ) hold on

end

xlabel(ŌĆś period in secsŌĆÖ)

ylabel(ŌĆś spectral velocity sv in cm/secŌĆÖ) for k=.01:.01:.1

x=0.01:1:100 t=log(2*pi*k)-log(x) y=exp(t) loglog(x,y,ŌĆśkŌĆÖ),grid on hold on

t=log(k*9.81/(2*pi))+log(x)

y=exp(t)

loglog(x,y,ŌĆśkŌĆÖ) hold on

end

for k=.1:.1:1 x=0.01:1:100 t=log(2*pi*k)-log(x) y=exp(t) loglog(x,y,ŌĆśkŌĆÖ),grid on hold on

t=log(k*9.81/(2*pi))+log(x)

y=exp(t)

loglog(x,y,ŌĆśkŌĆÖ) hold on

end

for k=1:1:10 x=0.01:1:100 t=log(2*pi*k)-log(x) y=exp(t) loglog(x,y,ŌĆśkŌĆÖ),grid on hold on

t=log(k*9.81/(2*pi))+log(x)

y=exp(t)

loglog(x,y,ŌĆśkŌĆÖ) hold on

end

for k=10:10:100 x=0.01:1:100 t=log(2*pi*k)-log(x) y=exp(t) loglog(x,y,ŌĆśkŌĆÖ),grid on hold on

t=log(k*9.81/(2*pi))+log(x)

y=exp(t)

loglog(x,y,ŌĆśkŌĆÖ) hold on

end

for k=100:100:1000 x=0.01:1:100 t=log(2*pi*k)-log(x) y=exp(t) loglog(x,y,ŌĆśkŌĆÖ),grid on

hold on t=log(k*9.81/(2*pi))+log(x) y=exp(t)

loglog(x,y,ŌĆśkŌĆÖ) hold on

end

for k=1000:1000:10000 x=0.01:1:100 t=log(2*pi*k)-log(x) y=exp(t) loglog(x,y,ŌĆśkŌĆÖ),grid on hold on

t=log(k*9.81/(2*pi))+log(x)

y=exp(t)

loglog(x,y,ŌĆśkŌĆÖ) end

axis([0.01 100 0.02 500]) text(0.2,0.02,ŌĆś0.001ŌĆÖ); text(0.6,0.1,ŌĆś0.01ŌĆÖ); text(2,0.3,ŌĆś0.1ŌĆÖ); text(7,1,ŌĆś1ŌĆÖ); text(20,3,ŌĆś10ŌĆÖ); text(80,10,ŌĆś100ŌĆÖ) text(20,1,ŌĆśSd in cmŌĆÖ) xlabel(ŌĆś period in secŌĆÖ) ylabel(ŌĆś Sv in cm/secŌĆÖ) text(0.01,200,ŌĆś100ŌĆÖ) text(0.01,20,ŌĆś10ŌĆÖ) text(0.01,2,ŌĆś1ŌĆÖ) text(0.02,0.4,ŌĆś0.1ŌĆÖ) text(0.07,0.1,ŌĆś0.01ŌĆÖ) text(.02,0.8,ŌĆśSa/gŌĆÖ) xc(1)=0.01; xc(2)=0.0303; xc(3)=0.125;

xc(4)=cv*pgv*2*pi/(ca*pga);

xc(5)=cd*pgd*2*pi/(cv*pgv);

xc(6)=10;

xc(7)=33.0

xc(8)=100.0;

yc(1)=pga*0.01/(2.0*pi);

yc(2)=pga*0.0303/(2*pi);

yc(3)=ca*pga*0.125/(2*pi);

yc(4)=cv*pgv;

yc(5)=cv*pgv;

yc(6)=cd*pgd*2*pi/10;

yc(7)=pgd*2*pi/33;

yc(8)=pgd*2*pi/100;

line(xc,yc,ŌĆślinewidthŌĆÖ,3,ŌĆścolorŌĆÖ,ŌĆśkŌĆÖ);

title(ŌĆś Newmark- Hall Design Spectrum 50% Median and rho=5%ŌĆÖ)

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