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On the dynamic behaviour of elastic–plastic structures equipped with pseudoelastic SMA reinforcements
Computational Materials Science 25 (2002) 1–13 www.elsevier.com/locate/commatsci
On the dynamic behaviour of elastic–plastic structures equipped with pseudoelastic SMA reinforcements A. Baratta *, O. Corbi Department of Scienza delle Costruzioni, University of Naples Federico II, Via Claudio 21, 80125 Naples, Italy
Abstract In the paper the dynamics of a structural elastic–plastic frame, endowed with pseudoelastic shape memory alloy (SMA) tendons and subject to some vertical and horizontal loading conditions, are analysed. A wide numerical investigation is carried out in order to show how SMAs are able, more than ordinary or other special materials, to give to the structure good dissipative skill and high re-centering capacity. At this aim, some comparisons are presented with the cases of a simple oscillator, of a frame with elastic–plastic tendons and of a frame with unilateral elastic–plastic tendons unable to sustain compression. 2002 Elsevier Science B.V. All rights reserved. PACS: 81.05.Bx; 43.40.At; 43.40.Tm Keywords: Shape memory alloys; Pseudoelasticity; P –D effect; Dynamic analysis; Earthquake engineering applications; Centering skill
1. Introduction As well known [1–3], Shape memory alloys (SMAs) are a class of materials able to undergo reversible micromechanical phase transition processes, changing their crystallographic structure; this capacity results in two major features at the macroscopic level: the shape memory effect (depending on their capability in recovering possible accumulated deformations by heat treatment) and super-elasticity (i.e. the recovery of large deformations in loading-unloading cycles, occurring at sufficiently high temperatures). The shape memory effect derives from a firstorder martensitic phase transformation and gives
*
Corresponding author.
the SMAs a high dissipative capacity with comparison to ordinary metals, achieving large hysteretic loops (similar to those exhibited by conventional steels) without incurring plastic deformation; the hysteresis, here actually due to the growth and re-orientation of the martensite crystals (that can be reduced to their original configuration upon the application of heat), renders particularly interesting the SMAs applications in the field of earthquake engineering for the realization of dissipative devices. Furthermore, the super-elastic behaviour of the SMAs, due to elastic loading of the austenitic parent up to the threshold stress where-upon the transformation from austenite to martensite occurs, is able to provide an energy-absorbing effect combined with a theoretically zero residual strain upon unloading [1,4].
0927-0256/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 2 ) 0 0 2 4 5 - 8
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Nomenclature f horizontal excitation w vertical load a0 horizontal excitation amplitude xf horizontal excitation pulsation tf excitation time duration m beam mass c, c0 , c00 , cs frame, piles and rods damping coefficients k, ks frame and rods stiffness Rtot model total resistance H columns height L beam length ‘0 , ‘0 , ‘00 initial and instantaneous rods length D‘0 , D‘0 , D‘00 initial and instantaneous rods elongation and shortening D‘, D‘_ instantaneous elongation and its rate of the currently stretched rod a0 , b0 initial rods inclination angles a0 , a00 , b0 , b00 instantaneous rods inclination angles a instantaneous inclination angle of the currently stretched rod n, p, q, cw , m, n, c, w, D‘^ SMA parameters t time variable u, u_ , € u beam horizontal displacement, velocity and acceleration /, /_ , /€ columns average rotation and its derivatives
For detailed characterization, procession and metallurgical information of this material a large amount of literature is available. For their particular properties, SMAs lend themselves to be successfully adopted in a broad set of advanced applications, ranging from orthodontic appliances to microstructures used in the treatment of the blood-vessel occlusion, to devices such as robotic arms, artificial hearts, heat engines, force and displacements actuators [5–7]. The SMAs unique mechanical characteristics are widely exploited in high performance seismic engineering applications.
/00 , /000 /0 , /00 /p Tp T T Tp0 , Tp00 T 0 , T 00 T00 , T000 N N Np0 , Np00 Nr0 , Nr00 Nr N 0 , N 00 N00 , N000 Nc0 , Nc00 H
initial rotation yield values instantaneous rotation yield values plastic cumulated rotation frame restoring force strain-dependent shear force in the frame T ð/Þ cos /, equivalent strain-dependent shear force in the frame shear forces in the piles strain-dependent shear forces in the piles yield strain-dependent shear forces in the piles strain-dependent normal force in the currently stretched rod N ð/Þ cos a, equivalent normal force in the stretched normal forces in the piles normal forces in the diagonal rods normal force in the currently stretched rod strain-dependent normal forces in the diagonal rods yield strain-dependent normal forces in the diagonal rods critical yield strain-dependent normal forces in the diagonal rods SMA evolutionary backstress
2. The structural model Let us consider the portal frame of Fig. 1a, subject to an horizontal excitation f acting on its beam and a vertical load w ¼ mg, with m the mass of the beam; denote by u the horizontal displacement of the beam with respect to the base of the columns, by / the columns average rotation and by the superimposed dots their time derivatives. Moreover let the frame be able to move only according to the single degree of freedom mechanism, as shown. The frame itself is assumed to exhibit an elastic–plastic behaviour while the
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Fig. 1. Elastic–perfectly plastic frame endowed with tendons.
tendon reinforcements are shown as cross-bracing in Fig. 1a. By assuming the rotation / as the Lagrangian parameter, positive for counter-clockwise rotations, with reference to the instantaneously formed ACD and ABD triangles under the time-varying horizontal forcing action, one can write the geometric relations 8 H L ‘0 ð/Þ ‘0 ð/Þ ‘0 þ D‘0 ð/Þ > > ¼ ¼ ¼ ¼ > 0 0 ð/Þ > sin a sinðp=2 /Þ cos / cos / sin b ð/Þ > > > ( > > 0 0 > b ð/Þ ¼ p=2 ½a ð/Þ / > > > > < sin b0 ð/Þ ¼ cos½a0 ð/Þ / > H L ‘00 ð/Þ ‘00 ð/Þ ‘0 þ D‘00 ð/Þ > > ¼ ¼ ¼ ¼ > 00 00 ð/Þ > sin a sinðp=2 þ /Þ cos / cos / sin b ð/Þ > > > ( > > 00 00 > ð/Þ ¼ p=2 ½a ð/Þ / b > > > : sin b00 ð/Þ ¼ cos½a00 ð/Þ /;
ð1Þ where H is the height of the columns, L the length of the beam, ‘0 the initial length of both tendons,
‘0 ð/Þ and ‘00 ð/Þ the instantaneous lengths of the tendons, D‘0 ð/Þ and D‘00 ð/Þ the instantaneous elongation and shortening of the tendons, and a0 ð/Þ, a00 ð/Þ, b0 ð/Þ, b00 ð/Þ the angles given in Fig. 1a. The instantaneous expression of the angles a0 ð/Þ, a00 ð/Þ and of the current lengths of the rods is given as cos / a0 ð/Þ ¼ tan1 ; L=H sin / cos / ‘0 ð/Þ ¼ ‘0 þ D‘0 ð/Þ ¼ H sin a0 ð/Þ ð2Þ cos / ; a00 ð/Þ ¼ tan1 L=H þ sin / cos / : ‘00 ð/Þ ¼ ‘0 þ D‘00 ð/Þ ¼ H sin a00 ð/Þ From the vertical equilibrium condition of the beam, neglecting the vertical inertia forces, one gets
8 N 0 ð/; /_ Þ ¼ cs D‘_0 ð/; /_ Þ þ N 0 ð/Þ > > 0 _ Þ sin a0 ð/Þ þ N 00 ð/; /_ Þ sin a00 ð/Þ > r00 w þ N ð/; / < 00 r r _ _ _00 0 00 _ _ _ N Np ð/; /Þ þ Np ð/; /Þ ¼ TP ð/; /Þ tan / r ð/; /Þ ¼ cs D‘ ð/; /Þ þ N ð/Þ cos / 0 _ Þ ¼ c0 H /_ þ T 0 ð/Þ ð/; / T > p > Tp ð/; /_ Þ þ Tp0 ð/; /_ Þ þ Tp00 ð/; /_ Þ ¼ cH /_ þ T ð/Þ; c ¼ c0 þ c00 ; T ð/Þ ¼ T 0 ð/Þ þ T 00 ð/Þ > : 00 Tp ð/; /_ Þ ¼ c00 H /_ þ T 00 ð/Þ
ð3Þ
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A. Baratta, O. Corbi / Computational Materials Science 25 (2002) 1–13
where Tp0 ð/; /_ Þ and Tp00 ð/; /_ Þ, Np0 ð/; /_ Þ and Np00 ð/; /_ Þ are respectively the shear and normal forces in the piles, Nr0 ð/; /_ Þ and Nr00 ð/; /_ Þ are the forces in the diagonal stay-rods, c, c0 , c00 , cs are the damping coefficients of the elastic–plastic frame, of the piles and of the tendons. Furthermore Tp ð/; /_ Þ is the restoring force of the frame, T 0 ð/Þ, T 00 ð/Þ, T ð/Þ, N 0 ð/Þ and N 00 ð/Þ are the strain-dependent forces respectively in the beam-columns, in the frame and in the diagonal rods, that are discussed in Section 3. In Fig. 1b the T –/ relationships in the static range for an elastic–plastic frame are plotted both for w ¼ 0 and in the presence of the P –D effect. On the other side, the horizontal equilibrium equation m€ uð/; /_ ; /€Þ ¼ Nr0 ð/; /_ Þ cos a0 ð/Þ Nr00 ð/; /_ Þ cos a00 ð/Þ þ ½N 0 ð/; /_ Þ þ N 00 ð/; /_ Þ sin / p
p
Tp ð/; /_ Þ cos / þ f
ð4Þ
combined with Eq. (3) yields m€uð/; /_ ; /€Þ ¼ Nr0 ð/; /_ Þ cos a0 ð/Þ Nr00 ð/; /_ Þ cos a00 ð/Þ þ N 0 ð/; /_ Þ tan / sin a0 ð/Þ r
ð5Þ
After some algebraic operations one gets 00
cos½a ð/Þ þ / m€uð/; /_ ; /€Þ ¼ Nr0 ð/; /_ Þ cos / Nr00 ð/; /_ Þ
cos½a00 ð/Þ þ / cos /
Tp ð/; /_ Þ þ f þ w tan /: cos /
ð6Þ
Considering the case when the tendons are made of slender rods, unable to sustain compressive forces, only positive values of Nr00 ð/; /_ Þ and negative values of Nr0 ð/; /_ Þ are admissible. Moreover, taking into account that, if the two diagonal rods are equal and the frame is symmetric (i.e. Tp0 ð/; /_ Þ ¼ Tp00 ð/; /_ Þ ¼ Tp ð/; /_ Þ=2), the system behaves symmetrically for motion both in the leftward and in the rightward direction and N 0 ð/Þ ¼ N 00 ð/Þ, a0 ð/Þ ¼ a00 ð/Þ; then it is possible to write Eq. (6) in the simpler form
ð7Þ
where Nr ð/; /_ Þ and að/Þ are respectively the normal force and the inclination angle of the currently stretched tendon. As the displacement, with its time derivatives, is related to the Lagrangian parameter by uð/Þ ¼ H sin /; u_ ð/; /_ Þ ¼ H cos / /_ ; €uð/; /_ ; /€Þ ¼ H ðcos / /€ sin / /_ 2 Þ:
ð8Þ
After substitution of Eq. (8) into Eq. (7) and some further algebraic operations one gets mH cos / /€ mH sin / /_ 2 þ cs cos að/ÞD‘_ð/; /_ Þ þ cos að/ÞN ð/Þ tan /bsin að/ÞNr ð/; /_ Þ sin /Tp0 ð/; /_ Þc 1 þ cos2 / þ ½c0 H /_ þ T 0 ð/Þ cos / ¼ f þ w tan /;
þ Nr00 ð/; /_ Þ tan / sin a00 ð/Þ Tp ð/; /_ Þ sin / tan / Tp ð/; /_ Þ cos / þ f þ w tan /:
cos½að/Þ þ / cos / 0 _ 2Tp ð/; /Þ þ f þ w tan u; cos /
m€uð/; /_ ; /€Þ ¼ Nr ð/; /_ Þ
ð9Þ
where N ð/Þ is the strain-dependent normal force in the currently stretched rod, D‘ð/Þ and D‘_ð/; /_ Þ are the active tendon elongation and its rate, and can be deduced, with að/Þ, by Eq. (2), yielding D‘ð/Þ ¼ ‘ð/Þ ‘0 ; d cos / /_ D‘_ð/; /_ Þ ¼ H d/ sin að/Þ sin / 1 L sin / þ H _ ¼ H /; sin að/Þ sin að/Þ L þ H sin / cos / að/Þ ¼ tan1 : ð10Þ L=H þ sin /
3. The mechanical models 3.1. Elastic–perfectly plastic frame The restoring force portion T ð/Þ of the simple elastic–perfectly plastic oscillator, included in Tp ð/; /_ Þ in the motion equation Eq. (6), can be expressed as a function of the piles rotation /
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T ð/Þ ¼ kHRð/Þ; 8 / /p > > < Rð/Þ ¼ /00 > > : 00 /0
if /00 6 / 6 /0 ; if /_ P 0; / P /0 ;
ð11Þ
if /_ 6 0; / 6 /00 ;
where k is the piles stiffness, /00 , /000 and /0 , /00 are respectively the initial and instantaneous positive and negative yielding values of the Lagrangian parameter, and /p is the plastic rotation cumulated throughout the loading history. The straindependent shear stress yield limits T00 , T000 in the piles are then given as T00 ¼ kH /00 > 0;
T000 ¼ kH /000 < 0:
ð12Þ
3.2. Shape memory alloys tendons As is well known [1], under dynamic action, a stress-induced micromechanical phase transition occurs in SMAs that causes inelastic deformation. The super-elasticity is one of the main SMAs characteristics, and it consists of their capability in providing hysteresis loops with a consequent energy-absorption effect, and a theoretically zero residual strain upon unloading. This natural damping capacity and centering effect, due to the reversible micromechanical phase transition process, renders the SMAs particularly attractive for applications in the fields of vibration control and earthquake resistant engineering [8–10]. As a consequence of the increasing scientific and commercial interest attracted by SMAs, many constitutive models able to reproduce the main macroscopic effects have been proposed in literature. In the following, a simplified one-dimensional SMA constitutive model is considered, deduced by a general material characterization, developed as an extension of the Bouc–Wen hysteretic model [11] and on the basis of some experimental studies [12–14]. When applying the isothermal material stress– strain rate relation to the stretched SMA tie-rod inserted in the considered model, one can get the rate relation between the N ð/Þ portion of the SMA rod restoring force and its elongation
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" N ð/Þ Hð/Þ n1 _ _ N ð/; /Þ ¼ ks D‘_ð/; /_ Þ jD‘_ð/; /_ Þj Nc # N ð/Þ Hð/Þ ; ð13Þ Nc
N ð/Þ w Hð/Þ ¼ ks c D‘^ þ 2 jD‘ð/Þjp ½tan1 ðxÞ ; ks p x ¼ x3 jxjq ;
x ¼ nD‘ð/Þ;
q < 1;
ð14Þ
where Hð/Þ is the evolutionary one-dimensional backstress that accounts for the metastable forms of SMA behaviour, Nc is the critical yield normal force and all other symbols represent material constants. One can notice that, for specified rotations (and thereafter tie length variations), the force-elongation relationship for the stretched SMA tie-rod can be obtained by numerical integration of Eq. (13) while N ð/Þ can be deduced by simultaneous integration of Eq. (13) and the motion equation. Furthermore, by assuming that the mechanical model shows an isoresistant behaviour, the following relations hold ( 0 N0 if N ð/Þ > 0 N0 ¼ ; N000 if N ð/Þ < 0 ( 0 ð15Þ Nc ¼ mN00 if N ð/Þ > 0 Nc ¼ ; 00 00 Nc ¼ mN0 if N ð/Þ < 0 c¼
N0 Nc ; ks w N0
w ¼ cw
N0 ks
being N0 , Nc respectively the limit and critical normal forces related to the relevant yield values N00 , N000 , m a given prefixed percentage and cw a given constant. 3.3. Slender elastic–plastic tendons In Fig. 2a a possible loading sequence is depicted, referred to the rod status in Fig. 2b. One possible load cycle may be carried on by following the path ABCDE; hence, under a zero force, the rod can be driven to any of the positions E, F, A, G, H, since, being devoid of flexural stiffness, it deflects freely. The subsequent cycle starts, for
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Fig. 2. (a) Possible loading sequence for the unilateral tendon; (b) rod status. The cycle path is followed by the sequence of capital letters.
instance, from the point H, but the rod cannot resist the displacement before it has recovered both the deflection and the previously cumulated plastic elongation; so it cannot go into tension before the point E is attained, whence it follows firstly the elastic branch EC, subsequently it undergoes the further plastic elongation CI and, thereafter, it is unloaded down to the zero value of the force (point L) and it assumes any position (E, F, A, G, . . .) on the left of E. Subsequent cycles follow analogous paths, after updating the length (the original one plus plastic elongation) of the rod.
4. Numerical investigation 4.1. Structural model characteristics The structural model of Fig. 1a, with a total resistance of Rtot ¼ T0 þ N0 cos a0 ¼ 40 000 cos a0 kg, with T0 ¼ T00 ¼ T000 , subject to a periodic timedependent forcing action f ¼ ma0 sinðxf tÞ, with pulsation xf , amplitude ma0 and duration tf , is considered for numerical investigation. The frame characteristics are m ¼ 100 kg s2 cm1 ; 1
k ¼ 140 000 kg cm ;
H ¼ 300 cm;
L ¼ 500 cm;
c ¼ 187 kg s cm1 ;
ð16Þ
while, for the no-hardening SMA tendons, the following parameters are assumed
ks ¼ 140 000 kg cm1 ; cs ¼ 69 kg s cm1 ; D‘^ ¼ 0 m; n ¼ 120; p ¼ 0; cw ¼ 200; m ¼ 0:200; n ¼ 20:5; q ¼ 3=4:
ð17Þ 4.2. Numerical results In the following, in order to better analyse the usefulness of adopting some SMA devices, three cases are considered: • elastic–plastic frame with SMA tendons (Fig. 3a) • elastic–plastic frame with elastic–plastic tendons (Fig. 3b) • elastic–plastic frame with unilateral elastic– plastic tendons, unable to sustain compression forces (e.g. steel wires, Fig. 3c) In Figs. 4 and 5 the results relevant to the case of the frame endowed with SMA tendons, which absorb a 20% of the total resistance, are reported for two different amplitudes, a0 ¼ 800 cm s2 and a0 ¼ 1600 cm s2 , of the sine forcing function acting at xf ¼ x0 , with x0 ¼ 37:42 s1 . In Fig. 4 the equivalent shear stress in the piles T ¼ T ð/Þ cos / and normal stress in the stretched SMA tendon N ¼ N ð/Þ cos a is depicted versus the frame response in terms of piles rotation, while, in Fig. 5, the phase diagrams /–/_ for the two forcing amplitude cases are shown.
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Fig. 3. (a) Frame with a couple of SMA tendons; (b) frame with a couple of elastic–plastic tendons; (c) frame with a couple of unilateral elastic–plastic tendons. The cycle path followed is given by the sequence of capital letters.
Fig. 4. Hysteretic cycles for the elastic–plastic frame endowed with SMA tendons, for xf ¼ x0 and (a) a0 ¼ 800 cm s2 , (b) a0 ¼ 1600 cm s2 .
The same results are produced for the two cases of the frame equipped with elastic–plastic tendons (Figs. 6 and 7) or with unilateral elastic–plastic tendons (Figs. 8 and 9), always accounting, in both cases, for the 20% of the total model resistance. In details, the equivalent stresses in the frame piles and in the stretched elastic–plastic or unilateral elastic–plastic tendons are respectively illustrated in Figs. 6 and 8 while Figs. 7 and 9 show the response phase diagrams for the two mentioned cases. From the comparison of these latter with the one relevant to the model provided with SMA
tendons (Fig. 5) one can immediately figure out that, in this case, the phase closing loop stabilizes on a residual response value much closer to the zero than in both the other cases, with a high SMA centering effect; moreover, while the response of the structural models equipped with (standard or unilateral) elastic–plastic tendons monotonically shifts along the / axis in the phase plane, in the SMA case it tends to a stationary response, emphasizing the lower sensitivity of this system to the P –D effects induced by the applied vertical loads. This result is clear by looking at Fig. 10 where the
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Fig. 5. Phase /–/_ diagrams for the elastic–plastic frame endowed with SMA tendons, for xf ¼ x0 and (a) a0 ¼ 800 cm s2 , (b) a0 ¼ 1600 cm s2 .
Fig. 6. Hysteretic cycles for the elastic–plastic frame endowed with elastic–plastic tendons, for xf ¼ x0 and (a) a0 ¼ 800 cm s2 , (b) a0 ¼ 1600 cm s2 .
time histories to the same forcing actions for the three cases are compared. Coming back to the original frame endowed with SMA tendons, some interesting considerations can be carried out if one considers the effects induced in the response of the model, given with the prefixed total resistance and subject to a defined excitation, by varying the distribution of such resistance between the SMA tendons and the
elastic–plastic frame itself. Actually, by increasing the percentage of contribution of the SMA tendons to the total resistance, one can observe that, for many amplitudes a0 of the excitation and two different forcing pulsations, respectively x0 ¼ 37:42 s1 and xc ¼ 52:91 s1 , the response shows the residual and maximum values given in Figs. 11 and 12. One can notice that the higher the intensity of the forcing action gets, the more the
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Fig. 7. Phase /–/_ diagrams for the elastic–plastic frame endowed with elastic–plastic tendons, for xf ¼ x0 and (a) a0 ¼ 800 cm s2 , (b) a0 ¼ 1600 cm s2 .
Fig. 8. Hysteretic cycles for the elastic–plastic frame endowed with unilateral elastic–plastic tendons, for xf ¼ x0 and (a) a0 ¼ 800 cm s2 , (b) a0 ¼ 1600 cm s2 .
response is reduced, both in terms of peak and residual values, when increasing the amount of resistance absorbed by the SMA members; also an improvement (whose entity always rises for higher forcing amplitudes) is introduced in the system behaviour if operating a comparison with the simple elastic–plastic frame without tendons and with resistance equal to the total one, corre-
sponding to the initial values (zero SMA percentage, dashed lines). Finally a comparison is given (Figs. 13 and 14), for varying percentages of contribution of the tendons resistance to the total model resistance, between the SMA case and the introduced elastic– plastic (unilateral or not) tendons cases, for particular forcing intensities and frequencies. From
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Fig. 9. Phase /–/_ diagrams for the elastic–plastic frame endowed with unilateral elastic–plastic tendons, for xf ¼ x0 and (a) a0 ¼ 800 cm s2 , (b) a0 ¼ 1600 cm s2 .
Fig. 10. Structural models time histories for xf ¼ x0 and (a) a0 ¼ 800 cm s2 , (b) a0 ¼ 1600 cm s2 .
here one can again emphasize the high centering skill of SMA in the considered application: for peak values substantially equal (Figs. 13a and 14a) or lower (Figs. 13b and 14b) than the ones corresponding to the frame with unilateral elastic– plastic tendons (and much lower than the ones relevant to standard elastic–plastic tendons) a very consistent reduction of the residual response values can be pursued even for low SMA resistance percentages. One can, then, conclude that, in the considered application, the adoption of such alloys results
desirable, due to their good performance either in terms of dissipative capacity (always higher than the one relevant to standard materials), or, especially, in terms of centering skill, which was the objective of the present study.
5. Conclusions In the paper, the influence of SMA elements collaborating to the overall strength of a simple structural model undergoing horizontal shaking
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Fig. 11. (a) Residual and (b) maximum response for varying percentages of SMA tendons resistance for xf ¼ x0 .
Fig. 12. (a) Residual and (b) maximum response for varying percentages of SMA tendons resistance for xf ¼ xc .
has been investigated in order to show them preferable to other ordinary or even special materials in terms of increasing of the structural system dissipative capacity and high re-centering skill.
The basic structure is assumed to exhibit a elastic– perfectly plastic material behaviour, while the tendons are supposed to behave according to a pseudoelastic model; the performance of such
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Fig. 13. Residual and maximum response for varying percentages of the tendons resistance for xf ¼ x0 , (a) a0 ¼ 800 cm s2 , (b) a0 ¼ 1600 cm s2 .
Fig. 14. Residual and maximum response for varying percentages of the tendons resistance for xf ¼ xc , (a) a0 ¼ 800 cm s2 , (b) a0 ¼ 1600 cm s2 .
system is compared with the response of a similar structure, where the tendons are supposed to be
fully elastic–plastic, as the main structure, or, alternatively, unilaterally plastic, i.e. unable to sup-
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port compression forces, as is the case if the tendons are very slender. The results prove that pseudoelastic tendons decisively improve the dynamic response capacity of the structure with respect to the case in which the tendons, although supporting the same fraction of static horizontal strength, are made by thin elastic–plastic wires, and even more with respect to classically elastic–plastic rods. In both cases, in fact, pseudoelastic tendons produce smaller maximum amplitude of the response and much smaller residual drift, as can be verified by inspection of the diagrams in Figs. 11–14. This result can be attributed to at least two main phenomena that are peculiar to the pseudoelastic property: the ability to dissipate energy while recovering the free-flow deformation, that is probably the reason for the first drop of the structure’s response in the above plots for small SMA contribution (see, i.e. Fig. 12), and the ability to recover the initial status after the stress is released, that is the reason for the second drop when the SMA percentage is very high (say larger than 60%). A second feature, that is very attractive for aseismic design, is that such positive influence becomes more significant the higher is the intensity of the shaking, as can be confirmed, for instance, from inspection of Fig. 11.
Acknowledgement Paper supported by grants of the Italian National Council of Research (CNR).
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References [1] C.M. Wayman, T.W. Duerig, An introduction to martensite and shape memory, in: T.W. Duerig, K.N. Melton, D. Stolen, C.M. Wayman (Eds.), Engineering Aspects of Shape Memory Alloys, 1990, pp. 3–20. [2] C.M. Wayman, Shape memory and related phenomena, J. Prog. Mater. Sci. 36 (1992) 203–224. [3] C.M. Wayman, Introduction to the Crystallographic of Martensitic Transformations, Macmillan, New York, 1964. [4] W. Buehler, R. Wiley, Nickel-based alloys, Technical Report, Naval Ordinance Lab, US-Patent 3174851, 1965. [5] T.W. Duerig, K.N. Melton, D. Stolen, C.M. Wayman (Eds.), Engineering Aspects of Shape Memory Alloys, Butterworth-Heinemann, London, 1990. [6] K. Funakubo, Shape Memory Alloy, Gordon and Breach Science Publishers, New York, 1984. [7] D.E. Hodgson, Using Shape Memory Alloys, Shape Memory Applications, Inc., 1988. [8] W.D. Armstrong, A one-dimensional model of a shape memory alloy fiber reinforced aluminium metal matrix composite, J. Intell. Mater. Syst. Struct. 7 (1996) 448–454. [9] A. Baz, K. Iman, J. McCoy, Active vibration control of flexible beams using shape memory actuatuors, J. Sound Vibrat. 140 (1990) 437–456. [10] B.J. Maclean, G.J. Patterson, M.S. Misra, Modeling of a shape memory integrated actuator for vibration control of large space structures, J. Intell. Mater. Syst. Struct. 2 (1991) 71–94. [11] Y.K. Wen, Method for random vibration of hysteretic systems, J. Eng. Mech. Div., ASCE 102 (2) (1976) 249–263. [12] E.J. Graesser, F.A. Cozzarelli, Shape memory alloys as new materials for aseismic isolation, J. Eng. Mech., ASCE 117 (1991) 2590–2608. [13] E.J. Graesser, F.A. Cozzarelli, A proposed three-dimensional constitutive model for shape memory alloys, J. Intell. Mater. Syst. Struct. 5 (1994) 78–89. [14] Z.C. Feng, D.Z. Li, Dynamics of a mechanical system with a shape memory alloy bar, J. Intell. Mater. Syst. Struct. 7 (1996) 399–410.
On the dynamic behaviour of elastic–plastic structures equipped with pseudoelastic SMA reinforcements A. Baratta *, O. Corbi Department of Scienza delle Costruzioni, University of Naples Federico II, Via Claudio 21, 80125 Naples, Italy
Abstract In the paper the dynamics of a structural elastic–plastic frame, endowed with pseudoelastic shape memory alloy (SMA) tendons and subject to some vertical and horizontal loading conditions, are analysed. A wide numerical investigation is carried out in order to show how SMAs are able, more than ordinary or other special materials, to give to the structure good dissipative skill and high re-centering capacity. At this aim, some comparisons are presented with the cases of a simple oscillator, of a frame with elastic–plastic tendons and of a frame with unilateral elastic–plastic tendons unable to sustain compression. 2002 Elsevier Science B.V. All rights reserved. PACS: 81.05.Bx; 43.40.At; 43.40.Tm Keywords: Shape memory alloys; Pseudoelasticity; P –D effect; Dynamic analysis; Earthquake engineering applications; Centering skill
1. Introduction As well known [1–3], Shape memory alloys (SMAs) are a class of materials able to undergo reversible micromechanical phase transition processes, changing their crystallographic structure; this capacity results in two major features at the macroscopic level: the shape memory effect (depending on their capability in recovering possible accumulated deformations by heat treatment) and super-elasticity (i.e. the recovery of large deformations in loading-unloading cycles, occurring at sufficiently high temperatures). The shape memory effect derives from a firstorder martensitic phase transformation and gives
*
Corresponding author.
the SMAs a high dissipative capacity with comparison to ordinary metals, achieving large hysteretic loops (similar to those exhibited by conventional steels) without incurring plastic deformation; the hysteresis, here actually due to the growth and re-orientation of the martensite crystals (that can be reduced to their original configuration upon the application of heat), renders particularly interesting the SMAs applications in the field of earthquake engineering for the realization of dissipative devices. Furthermore, the super-elastic behaviour of the SMAs, due to elastic loading of the austenitic parent up to the threshold stress where-upon the transformation from austenite to martensite occurs, is able to provide an energy-absorbing effect combined with a theoretically zero residual strain upon unloading [1,4].
0927-0256/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 2 ) 0 0 2 4 5 - 8
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Nomenclature f horizontal excitation w vertical load a0 horizontal excitation amplitude xf horizontal excitation pulsation tf excitation time duration m beam mass c, c0 , c00 , cs frame, piles and rods damping coefficients k, ks frame and rods stiffness Rtot model total resistance H columns height L beam length ‘0 , ‘0 , ‘00 initial and instantaneous rods length D‘0 , D‘0 , D‘00 initial and instantaneous rods elongation and shortening D‘, D‘_ instantaneous elongation and its rate of the currently stretched rod a0 , b0 initial rods inclination angles a0 , a00 , b0 , b00 instantaneous rods inclination angles a instantaneous inclination angle of the currently stretched rod n, p, q, cw , m, n, c, w, D‘^ SMA parameters t time variable u, u_ , € u beam horizontal displacement, velocity and acceleration /, /_ , /€ columns average rotation and its derivatives
For detailed characterization, procession and metallurgical information of this material a large amount of literature is available. For their particular properties, SMAs lend themselves to be successfully adopted in a broad set of advanced applications, ranging from orthodontic appliances to microstructures used in the treatment of the blood-vessel occlusion, to devices such as robotic arms, artificial hearts, heat engines, force and displacements actuators [5–7]. The SMAs unique mechanical characteristics are widely exploited in high performance seismic engineering applications.
/00 , /000 /0 , /00 /p Tp T T Tp0 , Tp00 T 0 , T 00 T00 , T000 N N Np0 , Np00 Nr0 , Nr00 Nr N 0 , N 00 N00 , N000 Nc0 , Nc00 H
initial rotation yield values instantaneous rotation yield values plastic cumulated rotation frame restoring force strain-dependent shear force in the frame T ð/Þ cos /, equivalent strain-dependent shear force in the frame shear forces in the piles strain-dependent shear forces in the piles yield strain-dependent shear forces in the piles strain-dependent normal force in the currently stretched rod N ð/Þ cos a, equivalent normal force in the stretched normal forces in the piles normal forces in the diagonal rods normal force in the currently stretched rod strain-dependent normal forces in the diagonal rods yield strain-dependent normal forces in the diagonal rods critical yield strain-dependent normal forces in the diagonal rods SMA evolutionary backstress
2. The structural model Let us consider the portal frame of Fig. 1a, subject to an horizontal excitation f acting on its beam and a vertical load w ¼ mg, with m the mass of the beam; denote by u the horizontal displacement of the beam with respect to the base of the columns, by / the columns average rotation and by the superimposed dots their time derivatives. Moreover let the frame be able to move only according to the single degree of freedom mechanism, as shown. The frame itself is assumed to exhibit an elastic–plastic behaviour while the
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Fig. 1. Elastic–perfectly plastic frame endowed with tendons.
tendon reinforcements are shown as cross-bracing in Fig. 1a. By assuming the rotation / as the Lagrangian parameter, positive for counter-clockwise rotations, with reference to the instantaneously formed ACD and ABD triangles under the time-varying horizontal forcing action, one can write the geometric relations 8 H L ‘0 ð/Þ ‘0 ð/Þ ‘0 þ D‘0 ð/Þ > > ¼ ¼ ¼ ¼ > 0 0 ð/Þ > sin a sinðp=2 /Þ cos / cos / sin b ð/Þ > > > ( > > 0 0 > b ð/Þ ¼ p=2 ½a ð/Þ / > > > > < sin b0 ð/Þ ¼ cos½a0 ð/Þ / > H L ‘00 ð/Þ ‘00 ð/Þ ‘0 þ D‘00 ð/Þ > > ¼ ¼ ¼ ¼ > 00 00 ð/Þ > sin a sinðp=2 þ /Þ cos / cos / sin b ð/Þ > > > ( > > 00 00 > ð/Þ ¼ p=2 ½a ð/Þ / b > > > : sin b00 ð/Þ ¼ cos½a00 ð/Þ /;
ð1Þ where H is the height of the columns, L the length of the beam, ‘0 the initial length of both tendons,
‘0 ð/Þ and ‘00 ð/Þ the instantaneous lengths of the tendons, D‘0 ð/Þ and D‘00 ð/Þ the instantaneous elongation and shortening of the tendons, and a0 ð/Þ, a00 ð/Þ, b0 ð/Þ, b00 ð/Þ the angles given in Fig. 1a. The instantaneous expression of the angles a0 ð/Þ, a00 ð/Þ and of the current lengths of the rods is given as cos / a0 ð/Þ ¼ tan1 ; L=H sin / cos / ‘0 ð/Þ ¼ ‘0 þ D‘0 ð/Þ ¼ H sin a0 ð/Þ ð2Þ cos / ; a00 ð/Þ ¼ tan1 L=H þ sin / cos / : ‘00 ð/Þ ¼ ‘0 þ D‘00 ð/Þ ¼ H sin a00 ð/Þ From the vertical equilibrium condition of the beam, neglecting the vertical inertia forces, one gets
8 N 0 ð/; /_ Þ ¼ cs D‘_0 ð/; /_ Þ þ N 0 ð/Þ > > 0 _ Þ sin a0 ð/Þ þ N 00 ð/; /_ Þ sin a00 ð/Þ > r00 w þ N ð/; / < 00 r r _ _ _00 0 00 _ _ _ N Np ð/; /Þ þ Np ð/; /Þ ¼ TP ð/; /Þ tan / r ð/; /Þ ¼ cs D‘ ð/; /Þ þ N ð/Þ cos / 0 _ Þ ¼ c0 H /_ þ T 0 ð/Þ ð/; / T > p > Tp ð/; /_ Þ þ Tp0 ð/; /_ Þ þ Tp00 ð/; /_ Þ ¼ cH /_ þ T ð/Þ; c ¼ c0 þ c00 ; T ð/Þ ¼ T 0 ð/Þ þ T 00 ð/Þ > : 00 Tp ð/; /_ Þ ¼ c00 H /_ þ T 00 ð/Þ
ð3Þ
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where Tp0 ð/; /_ Þ and Tp00 ð/; /_ Þ, Np0 ð/; /_ Þ and Np00 ð/; /_ Þ are respectively the shear and normal forces in the piles, Nr0 ð/; /_ Þ and Nr00 ð/; /_ Þ are the forces in the diagonal stay-rods, c, c0 , c00 , cs are the damping coefficients of the elastic–plastic frame, of the piles and of the tendons. Furthermore Tp ð/; /_ Þ is the restoring force of the frame, T 0 ð/Þ, T 00 ð/Þ, T ð/Þ, N 0 ð/Þ and N 00 ð/Þ are the strain-dependent forces respectively in the beam-columns, in the frame and in the diagonal rods, that are discussed in Section 3. In Fig. 1b the T –/ relationships in the static range for an elastic–plastic frame are plotted both for w ¼ 0 and in the presence of the P –D effect. On the other side, the horizontal equilibrium equation m€ uð/; /_ ; /€Þ ¼ Nr0 ð/; /_ Þ cos a0 ð/Þ Nr00 ð/; /_ Þ cos a00 ð/Þ þ ½N 0 ð/; /_ Þ þ N 00 ð/; /_ Þ sin / p
p
Tp ð/; /_ Þ cos / þ f
ð4Þ
combined with Eq. (3) yields m€uð/; /_ ; /€Þ ¼ Nr0 ð/; /_ Þ cos a0 ð/Þ Nr00 ð/; /_ Þ cos a00 ð/Þ þ N 0 ð/; /_ Þ tan / sin a0 ð/Þ r
ð5Þ
After some algebraic operations one gets 00
cos½a ð/Þ þ / m€uð/; /_ ; /€Þ ¼ Nr0 ð/; /_ Þ cos / Nr00 ð/; /_ Þ
cos½a00 ð/Þ þ / cos /
Tp ð/; /_ Þ þ f þ w tan /: cos /
ð6Þ
Considering the case when the tendons are made of slender rods, unable to sustain compressive forces, only positive values of Nr00 ð/; /_ Þ and negative values of Nr0 ð/; /_ Þ are admissible. Moreover, taking into account that, if the two diagonal rods are equal and the frame is symmetric (i.e. Tp0 ð/; /_ Þ ¼ Tp00 ð/; /_ Þ ¼ Tp ð/; /_ Þ=2), the system behaves symmetrically for motion both in the leftward and in the rightward direction and N 0 ð/Þ ¼ N 00 ð/Þ, a0 ð/Þ ¼ a00 ð/Þ; then it is possible to write Eq. (6) in the simpler form
ð7Þ
where Nr ð/; /_ Þ and að/Þ are respectively the normal force and the inclination angle of the currently stretched tendon. As the displacement, with its time derivatives, is related to the Lagrangian parameter by uð/Þ ¼ H sin /; u_ ð/; /_ Þ ¼ H cos / /_ ; €uð/; /_ ; /€Þ ¼ H ðcos / /€ sin / /_ 2 Þ:
ð8Þ
After substitution of Eq. (8) into Eq. (7) and some further algebraic operations one gets mH cos / /€ mH sin / /_ 2 þ cs cos að/ÞD‘_ð/; /_ Þ þ cos að/ÞN ð/Þ tan /bsin að/ÞNr ð/; /_ Þ sin /Tp0 ð/; /_ Þc 1 þ cos2 / þ ½c0 H /_ þ T 0 ð/Þ cos / ¼ f þ w tan /;
þ Nr00 ð/; /_ Þ tan / sin a00 ð/Þ Tp ð/; /_ Þ sin / tan / Tp ð/; /_ Þ cos / þ f þ w tan /:
cos½að/Þ þ / cos / 0 _ 2Tp ð/; /Þ þ f þ w tan u; cos /
m€uð/; /_ ; /€Þ ¼ Nr ð/; /_ Þ
ð9Þ
where N ð/Þ is the strain-dependent normal force in the currently stretched rod, D‘ð/Þ and D‘_ð/; /_ Þ are the active tendon elongation and its rate, and can be deduced, with að/Þ, by Eq. (2), yielding D‘ð/Þ ¼ ‘ð/Þ ‘0 ; d cos / /_ D‘_ð/; /_ Þ ¼ H d/ sin að/Þ sin / 1 L sin / þ H _ ¼ H /; sin að/Þ sin að/Þ L þ H sin / cos / að/Þ ¼ tan1 : ð10Þ L=H þ sin /
3. The mechanical models 3.1. Elastic–perfectly plastic frame The restoring force portion T ð/Þ of the simple elastic–perfectly plastic oscillator, included in Tp ð/; /_ Þ in the motion equation Eq. (6), can be expressed as a function of the piles rotation /
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T ð/Þ ¼ kHRð/Þ; 8 / /p > > < Rð/Þ ¼ /00 > > : 00 /0
if /00 6 / 6 /0 ; if /_ P 0; / P /0 ;
ð11Þ
if /_ 6 0; / 6 /00 ;
where k is the piles stiffness, /00 , /000 and /0 , /00 are respectively the initial and instantaneous positive and negative yielding values of the Lagrangian parameter, and /p is the plastic rotation cumulated throughout the loading history. The straindependent shear stress yield limits T00 , T000 in the piles are then given as T00 ¼ kH /00 > 0;
T000 ¼ kH /000 < 0:
ð12Þ
3.2. Shape memory alloys tendons As is well known [1], under dynamic action, a stress-induced micromechanical phase transition occurs in SMAs that causes inelastic deformation. The super-elasticity is one of the main SMAs characteristics, and it consists of their capability in providing hysteresis loops with a consequent energy-absorption effect, and a theoretically zero residual strain upon unloading. This natural damping capacity and centering effect, due to the reversible micromechanical phase transition process, renders the SMAs particularly attractive for applications in the fields of vibration control and earthquake resistant engineering [8–10]. As a consequence of the increasing scientific and commercial interest attracted by SMAs, many constitutive models able to reproduce the main macroscopic effects have been proposed in literature. In the following, a simplified one-dimensional SMA constitutive model is considered, deduced by a general material characterization, developed as an extension of the Bouc–Wen hysteretic model [11] and on the basis of some experimental studies [12–14]. When applying the isothermal material stress– strain rate relation to the stretched SMA tie-rod inserted in the considered model, one can get the rate relation between the N ð/Þ portion of the SMA rod restoring force and its elongation
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" N ð/Þ Hð/Þ n1 _ _ N ð/; /Þ ¼ ks D‘_ð/; /_ Þ jD‘_ð/; /_ Þj Nc # N ð/Þ Hð/Þ ; ð13Þ Nc
N ð/Þ w Hð/Þ ¼ ks c D‘^ þ 2 jD‘ð/Þjp ½tan1 ðxÞ ; ks p x ¼ x3 jxjq ;
x ¼ nD‘ð/Þ;
q < 1;
ð14Þ
where Hð/Þ is the evolutionary one-dimensional backstress that accounts for the metastable forms of SMA behaviour, Nc is the critical yield normal force and all other symbols represent material constants. One can notice that, for specified rotations (and thereafter tie length variations), the force-elongation relationship for the stretched SMA tie-rod can be obtained by numerical integration of Eq. (13) while N ð/Þ can be deduced by simultaneous integration of Eq. (13) and the motion equation. Furthermore, by assuming that the mechanical model shows an isoresistant behaviour, the following relations hold ( 0 N0 if N ð/Þ > 0 N0 ¼ ; N000 if N ð/Þ < 0 ( 0 ð15Þ Nc ¼ mN00 if N ð/Þ > 0 Nc ¼ ; 00 00 Nc ¼ mN0 if N ð/Þ < 0 c¼
N0 Nc ; ks w N0
w ¼ cw
N0 ks
being N0 , Nc respectively the limit and critical normal forces related to the relevant yield values N00 , N000 , m a given prefixed percentage and cw a given constant. 3.3. Slender elastic–plastic tendons In Fig. 2a a possible loading sequence is depicted, referred to the rod status in Fig. 2b. One possible load cycle may be carried on by following the path ABCDE; hence, under a zero force, the rod can be driven to any of the positions E, F, A, G, H, since, being devoid of flexural stiffness, it deflects freely. The subsequent cycle starts, for
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Fig. 2. (a) Possible loading sequence for the unilateral tendon; (b) rod status. The cycle path is followed by the sequence of capital letters.
instance, from the point H, but the rod cannot resist the displacement before it has recovered both the deflection and the previously cumulated plastic elongation; so it cannot go into tension before the point E is attained, whence it follows firstly the elastic branch EC, subsequently it undergoes the further plastic elongation CI and, thereafter, it is unloaded down to the zero value of the force (point L) and it assumes any position (E, F, A, G, . . .) on the left of E. Subsequent cycles follow analogous paths, after updating the length (the original one plus plastic elongation) of the rod.
4. Numerical investigation 4.1. Structural model characteristics The structural model of Fig. 1a, with a total resistance of Rtot ¼ T0 þ N0 cos a0 ¼ 40 000 cos a0 kg, with T0 ¼ T00 ¼ T000 , subject to a periodic timedependent forcing action f ¼ ma0 sinðxf tÞ, with pulsation xf , amplitude ma0 and duration tf , is considered for numerical investigation. The frame characteristics are m ¼ 100 kg s2 cm1 ; 1
k ¼ 140 000 kg cm ;
H ¼ 300 cm;
L ¼ 500 cm;
c ¼ 187 kg s cm1 ;
ð16Þ
while, for the no-hardening SMA tendons, the following parameters are assumed
ks ¼ 140 000 kg cm1 ; cs ¼ 69 kg s cm1 ; D‘^ ¼ 0 m; n ¼ 120; p ¼ 0; cw ¼ 200; m ¼ 0:200; n ¼ 20:5; q ¼ 3=4:
ð17Þ 4.2. Numerical results In the following, in order to better analyse the usefulness of adopting some SMA devices, three cases are considered: • elastic–plastic frame with SMA tendons (Fig. 3a) • elastic–plastic frame with elastic–plastic tendons (Fig. 3b) • elastic–plastic frame with unilateral elastic– plastic tendons, unable to sustain compression forces (e.g. steel wires, Fig. 3c) In Figs. 4 and 5 the results relevant to the case of the frame endowed with SMA tendons, which absorb a 20% of the total resistance, are reported for two different amplitudes, a0 ¼ 800 cm s2 and a0 ¼ 1600 cm s2 , of the sine forcing function acting at xf ¼ x0 , with x0 ¼ 37:42 s1 . In Fig. 4 the equivalent shear stress in the piles T ¼ T ð/Þ cos / and normal stress in the stretched SMA tendon N ¼ N ð/Þ cos a is depicted versus the frame response in terms of piles rotation, while, in Fig. 5, the phase diagrams /–/_ for the two forcing amplitude cases are shown.
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Fig. 3. (a) Frame with a couple of SMA tendons; (b) frame with a couple of elastic–plastic tendons; (c) frame with a couple of unilateral elastic–plastic tendons. The cycle path followed is given by the sequence of capital letters.
Fig. 4. Hysteretic cycles for the elastic–plastic frame endowed with SMA tendons, for xf ¼ x0 and (a) a0 ¼ 800 cm s2 , (b) a0 ¼ 1600 cm s2 .
The same results are produced for the two cases of the frame equipped with elastic–plastic tendons (Figs. 6 and 7) or with unilateral elastic–plastic tendons (Figs. 8 and 9), always accounting, in both cases, for the 20% of the total model resistance. In details, the equivalent stresses in the frame piles and in the stretched elastic–plastic or unilateral elastic–plastic tendons are respectively illustrated in Figs. 6 and 8 while Figs. 7 and 9 show the response phase diagrams for the two mentioned cases. From the comparison of these latter with the one relevant to the model provided with SMA
tendons (Fig. 5) one can immediately figure out that, in this case, the phase closing loop stabilizes on a residual response value much closer to the zero than in both the other cases, with a high SMA centering effect; moreover, while the response of the structural models equipped with (standard or unilateral) elastic–plastic tendons monotonically shifts along the / axis in the phase plane, in the SMA case it tends to a stationary response, emphasizing the lower sensitivity of this system to the P –D effects induced by the applied vertical loads. This result is clear by looking at Fig. 10 where the
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Fig. 5. Phase /–/_ diagrams for the elastic–plastic frame endowed with SMA tendons, for xf ¼ x0 and (a) a0 ¼ 800 cm s2 , (b) a0 ¼ 1600 cm s2 .
Fig. 6. Hysteretic cycles for the elastic–plastic frame endowed with elastic–plastic tendons, for xf ¼ x0 and (a) a0 ¼ 800 cm s2 , (b) a0 ¼ 1600 cm s2 .
time histories to the same forcing actions for the three cases are compared. Coming back to the original frame endowed with SMA tendons, some interesting considerations can be carried out if one considers the effects induced in the response of the model, given with the prefixed total resistance and subject to a defined excitation, by varying the distribution of such resistance between the SMA tendons and the
elastic–plastic frame itself. Actually, by increasing the percentage of contribution of the SMA tendons to the total resistance, one can observe that, for many amplitudes a0 of the excitation and two different forcing pulsations, respectively x0 ¼ 37:42 s1 and xc ¼ 52:91 s1 , the response shows the residual and maximum values given in Figs. 11 and 12. One can notice that the higher the intensity of the forcing action gets, the more the
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Fig. 7. Phase /–/_ diagrams for the elastic–plastic frame endowed with elastic–plastic tendons, for xf ¼ x0 and (a) a0 ¼ 800 cm s2 , (b) a0 ¼ 1600 cm s2 .
Fig. 8. Hysteretic cycles for the elastic–plastic frame endowed with unilateral elastic–plastic tendons, for xf ¼ x0 and (a) a0 ¼ 800 cm s2 , (b) a0 ¼ 1600 cm s2 .
response is reduced, both in terms of peak and residual values, when increasing the amount of resistance absorbed by the SMA members; also an improvement (whose entity always rises for higher forcing amplitudes) is introduced in the system behaviour if operating a comparison with the simple elastic–plastic frame without tendons and with resistance equal to the total one, corre-
sponding to the initial values (zero SMA percentage, dashed lines). Finally a comparison is given (Figs. 13 and 14), for varying percentages of contribution of the tendons resistance to the total model resistance, between the SMA case and the introduced elastic– plastic (unilateral or not) tendons cases, for particular forcing intensities and frequencies. From
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Fig. 9. Phase /–/_ diagrams for the elastic–plastic frame endowed with unilateral elastic–plastic tendons, for xf ¼ x0 and (a) a0 ¼ 800 cm s2 , (b) a0 ¼ 1600 cm s2 .
Fig. 10. Structural models time histories for xf ¼ x0 and (a) a0 ¼ 800 cm s2 , (b) a0 ¼ 1600 cm s2 .
here one can again emphasize the high centering skill of SMA in the considered application: for peak values substantially equal (Figs. 13a and 14a) or lower (Figs. 13b and 14b) than the ones corresponding to the frame with unilateral elastic– plastic tendons (and much lower than the ones relevant to standard elastic–plastic tendons) a very consistent reduction of the residual response values can be pursued even for low SMA resistance percentages. One can, then, conclude that, in the considered application, the adoption of such alloys results
desirable, due to their good performance either in terms of dissipative capacity (always higher than the one relevant to standard materials), or, especially, in terms of centering skill, which was the objective of the present study.
5. Conclusions In the paper, the influence of SMA elements collaborating to the overall strength of a simple structural model undergoing horizontal shaking
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Fig. 11. (a) Residual and (b) maximum response for varying percentages of SMA tendons resistance for xf ¼ x0 .
Fig. 12. (a) Residual and (b) maximum response for varying percentages of SMA tendons resistance for xf ¼ xc .
has been investigated in order to show them preferable to other ordinary or even special materials in terms of increasing of the structural system dissipative capacity and high re-centering skill.
The basic structure is assumed to exhibit a elastic– perfectly plastic material behaviour, while the tendons are supposed to behave according to a pseudoelastic model; the performance of such
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Fig. 13. Residual and maximum response for varying percentages of the tendons resistance for xf ¼ x0 , (a) a0 ¼ 800 cm s2 , (b) a0 ¼ 1600 cm s2 .
Fig. 14. Residual and maximum response for varying percentages of the tendons resistance for xf ¼ xc , (a) a0 ¼ 800 cm s2 , (b) a0 ¼ 1600 cm s2 .
system is compared with the response of a similar structure, where the tendons are supposed to be
fully elastic–plastic, as the main structure, or, alternatively, unilaterally plastic, i.e. unable to sup-
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port compression forces, as is the case if the tendons are very slender. The results prove that pseudoelastic tendons decisively improve the dynamic response capacity of the structure with respect to the case in which the tendons, although supporting the same fraction of static horizontal strength, are made by thin elastic–plastic wires, and even more with respect to classically elastic–plastic rods. In both cases, in fact, pseudoelastic tendons produce smaller maximum amplitude of the response and much smaller residual drift, as can be verified by inspection of the diagrams in Figs. 11–14. This result can be attributed to at least two main phenomena that are peculiar to the pseudoelastic property: the ability to dissipate energy while recovering the free-flow deformation, that is probably the reason for the first drop of the structure’s response in the above plots for small SMA contribution (see, i.e. Fig. 12), and the ability to recover the initial status after the stress is released, that is the reason for the second drop when the SMA percentage is very high (say larger than 60%). A second feature, that is very attractive for aseismic design, is that such positive influence becomes more significant the higher is the intensity of the shaking, as can be confirmed, for instance, from inspection of Fig. 11.
Acknowledgement Paper supported by grants of the Italian National Council of Research (CNR).
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