CHAPTER 7 EARTHQUAKE RESPONSE OF INELASTIC SYSTEMS Base shear force in a linearl elastic sstem due to ground excitation is Vb = ( A/ g) w where A is the pseudo-acceleration corresponding to natural period and damping ratio of the linear sstem. Most buildings are designed to remain elastic for much lower base shear. Therefore buildings designed to resist earthquake will still be damaged during a strong earthquake. However, the damage should be controlled to an acceptable level, e.g. no collapse of building. The response of structures deforming into their inelastic range is therefore of importance in earthquake engineering. 9-1
Force deformation relation Cclic tests to simulate inelastic condition under earthquake have been conducted. Results show that cclic force-deformation behavior depends on the structural materials and structural sstem. Plot of force versus deformation under cclic loading is called hsteresis loop showing inelastic behavior. The shape of these loops depends on structural materials and structural sstems. 9-2
Elastoplastic idealization For convenience, the force-deformation relation obtained from experiment testing is simplified b an elastic-perfectlplastic or elastoplastic relation. The idealization is such that the area under the approximate curve is the same as under the actual curve at a selected maximum displacement. On initial loading, this idealized sstem is linearl elastic with stiffness k as long as force does not exceed f, the ield strength. Yielding begins when force reaches f, at which deformation equal to ield deformation u. For elastoplastic sstem, the stiffness equal to zero while ielding occurs Unloading of a ielding sstem will follow a path with slope (or stiffness) equal to elastic stiffness of the initial loading branch. Maximum and minimum resisting force is f and f. Yielding can occur in both directions when force reaches ield strength. 9-3
For elastoplastic sstems after some ielding, the relation between force and deformation is no longer singled value as it depends on prior histor of motion and whether the deformation is currentl increasing (velocit>0) or decreasing (velocit<0). Corresponding linear sstem It is interesting to evaluate the peak response of an elastoplastic sstem due to ground excitation and compare to the peak of the corresponding linear sstem due to the same ground motion. Corresponding linear sstem is the linear sstem having stiffness equal to the stiffness of initial loading path of the elastoplastic sstem. Both sstems have the same mass and damping. Therefore, the natural vibration periods of both sstems are the same when the elastoplastic sstem oscillates with a small u < u. amplitude ( o ) 9-4
Normalized ield strength Normalized ield strength is defined as f f = = fo where f o and u o are the peak values of the earthquake-induced resisting force and deformation, respectivel, in the corresponding linear sstem. We ma interpret f o as strength required for the structure to remain elastic during ground excitation. For sstem that remains elastic ( f Yield strength reduction factor u u o > f ), f = 1 Yield strength reduction factor is defined b R fo uo 1 = = = f u f o R = 1 for linear sstem and is greater than 1 for sstems that deform into inelastic range. Ductilit factor is defined b µ = u u m where u m is the maximum deformation. Ductilit factor µ = 1 for linear sstem. The ratio between peak deformation of inelastic and um µ elastic sstem is = µ f = u o R 9-5
Equation of motion and controlling parameters The governing equation of motion is S (, ) ( ) mu + cu + f u u = mu t where the resisting force f (, ) as shown earlier. S g uu for an elastoplastic sstem is This equation has to be solved b numerical algorithm and we will use the average acceleration method for results presented in this chapter and t = 0.02 sec. Divide both sides of equation of motion b m where ω = n ( ) ( ) u + ζω u + ω u f u u = u t k m 2 2 n n S, g ζ c m = f ( u, u ) 2 ωn S = f S ( uu, ) f The natural frequenc and damping ratio is the same as for the corresponding linear sstem. For a given u ( t), () g u t depends on three sstem parameters: ω n, ζ, and u, as well as the form of force-deformation relation. Here, elastoplastic form will be used. 9-6
Effect of ielding To understand how the response of sdf sstem is affected b inelastic action or ielding, we compare response of an elastoplastic sstem to that of its corresponding linear sstem due to El Centro ground motion. Response histor First we compute the response of a linear sstem with T n =0.5 sec and no damping. The peak response of this linear sstem due to El Centro ground motion is u o=3.34 in. Then we analze another sstem, which is inelastic and has normalized ield strength f = 0.125 So, ( ) f = f f = 0.125 1.37w = 0.171w o The response of this inelastic sstem is plotted together with resisting force, time that ielding occurs and forcedeformation relation. 9-7
When deformation reaches ield deformation, ielding occurs. During ielding, deformation ma increase while resisting force is constant and equal to ield strength. When the velocit becomes zero and deformation begins to decrease, unloading occurs and ielding stops. The forcedeformation relation during unloading is parallel to initial loading branch. The sstem vibrates about a new static equilibrium position, which is permanent deformation after vibration stops. Yielding ma occur in the negative direction in a similar manner. 9-8
When time histor of man inelastic sstems with different values of f are plotted and compared. Peak deformations depend on normalized ield strength. Sstems having lower normalized ield strengths ield more often and for longer duration. The peak deformation of inelastic sstem can be smaller or larger than its corresponding elastic sstem. 9-9
Ductilit demand, peak deformation and normalized ield strength The peak deformation normalized b PGD (peak ground displacement) um/ u go for different value of normalized ield strength are plotted versus period. Peak deformation of elastic sstem u / u is also included. o go The normalized ield strength f has little influence in the velocit and displacement-sensitive spectral regions (medium and long period range). It has great effect on short period sstems. Smaller normalized ield strength leads to significant increase in peak deformation. 9-10
In the velocit and displacement-sensitive region, peak deformation u m of inelastic sstems can be larger or smaller than that of its corresponding elastic sstem u o. For ver long period region, sstem is ver flexible. The mass stas still while the ground is shaking. Whether the sstem ields or not, peak deformation equals to peak ground displacement regardless of strength of the sstem. This is known as the equal displacement rule. When the ratio between peak deformation of inelastic and corresponding elastic sstem um/ u o are plotted, it shows the effect of inelastic action. As observed previousl, inelastic sstem with normalized ield strength less than 1 has significantl larger deformation. 9-11
Next, ductilit factor µ = = ( )( ) = ( ) is plotted versus period. u / u u / u u / u u / u / f m m o o m o 1 Note that R =. Sstems with larger R has larger ductilit f factor µ. And for ver long period sstem, µ = R because u / u = 1. m o 9-12
Response spectrum for ield deformation and ield strength Define D 2 = u V = ωnu A = ωnu Note that D is the ield deformation of the sstem, not the maximum deformation. A plot of D against T n for fixed values of ductilit factor µ is called the (constant-ductilit) ield-deformation response spectrum and plots of V and A are called (constant-ductilit) pseudo-velocit and pseudo-acceleration response spectra, respectivel. These quantities are related in a similar wa as D V and A. Yield strength of an elastoplastic sstem is f = The peak force in its corresponding linear sstem (or the strength required for the sstem to remain elastic) is f o A g w A = w g 9-13
Yield strength for specified ductilit In the analsis to determine deformation response of an inelastic sstem due to a ground motion, numerical procedure is implemented to obtain the response time histor. The ield deformation u (ield strength), damping ratio, and ground motion are assumed to be known, then the maximum deformation u m is obtained from the procedure and ductilit factor µ can be calculated ( µ = u / u ). m Based on this approach, given a ductilit factor, we can not know u directl. Therefore, iteration must be done b assuming a ield strength value, and determining the response of inelastic sstem and then ductilit factor. 9-14
Construction of constant-ductilit response spectrum To plot a spectrum, such iteration must be done for each value of period. When a ductilit factor corresponds to more than one values of ield strength, the largest ield strength should be used for plotting response spectrum. Once ield strength D is determined, pseudo-velocit and pseudo-acceleration can be calculated and plotted for use in structural design. Yield strength of an elastoplastic sstem is f A = g w 9-15
Constant-ductilit response spectrum for D, V, and A can be plotted in the four-wa logarithmic form as the elastic response spectrum The maximum deformation of inelastic sstem can also be determined from such spectrum as u m = µ u 9-16
Relationship between ield strength and ductilit A structure is usuall not designed to have ield strength f o enough to remain elastic during an earthquake because that would be prohibitivel expensive. If we allow it to damage or ield and deform into inelastic range with a certain level of ductilit factor, the ield strength required to limit the ductilit demand of the sstem can be much lower than f o. In the building code, the reduction from elastic strength demand f o to required ield strength f is specified b ield strength reduction factor R. 9-17
R is equal to the ductilit factor in the ver long period range but is smaller and close to 1 in the short period range. The more ductile the structure is, the more reduction factor we can use. Relative effect of ielding and damping We plot the constant-ductilit ield strength spectrum for man damping values together. The curve for µ = 1 is the elastic response spectrum. Effect of damping is significant onl in the velocit-sensitive spectral region (medium period range) for both elastic and inelastic sstems. We can observe that effect of damping is less for inelastic sstem ( µ > 1) compared to elastic sstem ( µ = 1). 9-18
Inelastic design spectrum Simplified values of f or R have been proposed for use to construct a ield-strength design spectrum. where u = f u = o u R o 9-19
or The constant-ductilit ield strength spectrum is obtained b dividing the elastic design spectrum b the ield-strength reduction factor R. 9-20
9-21
9-22
9-23
Maximum inelastic deformation spectrum can be obtained b multipling the elastic response spectrum b um/ u o ratio which is equal to µ / R. 9-24
9-25
9-26
9-27