1. The document discusses the development of a feedback loop algorithm for pseudo-dynamic testing of structures. Pseudo-dynamic testing combines experimental testing with numerical integration techniques to simulate dynamic loading. 2. Various numerical integration schemes are explored for use in the pseudo-dynamic test method. The Newmark explicit method is analyzed and shown to be unconditionally stable for certain time step values. 3. The stability and accuracy of the integration scheme is evaluated by examining the amplification matrix and eigenvalues. The Newmark explicit method is found to have no numerical damping and minimal frequency distortion for optimal time steps.

1. INDIAN INSTITUTE OF TECHNOLOGY ROORKEE
DEVELOPMENT OF FEEDBACK LOOP ALGORITHM
FOR PSEUDO-DYNAMIC TESTING
SHREYANSH AJMERA
M. Tech in Structural Dynamics
Enrolment No: 17526001
Department of Earthquake Engineering

2. 2
• I n tr o du ctio n
• Var io u s Lab o r ato r y Testin g Meth o d s
• Pseu d o d ynamic Test Meth o d
• N u mer ical I n teg r atio n Tech n iq u es
• Er r o r s in Pseu d o d ynamic Testin g
• Meth o d o log y
• Resu lts
• Co n clu sio n
• Fu tu r e w o r k
Overview of Presentation

3. 3
• A lthou gh an alytical meth o ds ar e av ailab le for th e
p red iction of in elastic b eh av io r of th e stru ctu ral
system, th e r esu lts ar e sever ely limited b y th e
u n cer tain ties in tr oduced b y th e simp lif ied math emati cal
id ealisation o f th e stru ctu r e and o f its non lin ear me mb er
p r o p erties .
• Ex p er imental testing r emain s th e mo st reliab le mean s
to ev alu ate th e in elastic b eh av io r of th e stru ctur al
system.
• Th e p seud od yn amic test metho d comb in es well
establish ed an alytical techn iq u es in Str u ctu ral
D yn amics w ith ex p er imen tal testin g .
Introduction

4. 4
• Sh ak in g tab le testin g
- mo st r ealistic an d r eliab le
- limitatio n o f size, w eig h t o f th e str u ctu r e
- r ed u ced scale str u ctu res an d h en ce similitu d e
p r o b lems
• Q u asi - Static testin g
- mo st eco n o mical an d v er satile
- larg e scale str u ctu res can b e tested
- su b jected to p r ed ef ined lo ad in g o r d ef o r matio n h isto r y
- u sed in stu d yin g th e d iff er ent str u ctu ral d etails an d
o th er p r o p er ties
Various Laboratory Testing Methods

5. 5
Pseudo-dynamic test method
• Realism o f sh ak in g tab le test + eco n o my an d co n v en ien ce
o f q u asi - static test.
• Co mb in atio n o f ex p er imen tal setu p an d an alytical
co mp u tatio n .
• Str u ctu re is mo d elled as a d iscr ete p ar ameter mo d el w ith
f in ite n u mb er o f mo d es an d d eg r ees o f f r eed o m.
• Mass matr ix an d d amp in g matr ix ar e an alytically
co mp u ted an d fed in to th e co mp u ter mo d el.
• D u r in g th e test, th e eq u atio n s o f mo tio n ar e so lv ed b y
mean s o f d ir ect step b y step n u mer ical in teg r ation .
• Th e d isp lacemen t resp o n se co mp u ted in each step o f a
test is imp o sed q u asi - statically o n th e test str u ctu re.

6. 6
Pseudo-dynamic test method
• Resto r in g f o r ces ar e r eco r d ed f o r th e str u ctu r al
d ef o r matio n s u sin g lo ad tr an sd u cer s.
• Th is is f ed in to th e n u mer ical alg o r ithm f o r th e
co mp u tatio n o f th e d isp lacemen t r esp o n se.
• H en ce n o u n cer tainties in mo d ellin g o f n o n - linear
mater ial b eh av io u r.
• Testin g is d o n e o n an ex ten d ed time scale w h ich u su ally
g o es u p to 1 0 0 times th e actu al earth q u ake d u ratio n.

7. 7
Figure 1: Schematic of the Pseudodynamic test method [2]
Pseudo-dynamic test method

8. 8
Discrete Parameter Model
• Stru ctur al system can b e con sid er ed as a d iscrete
p ar ameter mo d el .
• Eq u ation of mo tio n o f a d iscrete p ar ameter mod el is
rep resen ted as:
𝑚𝑥2𝑖+1 + 𝑐𝑥1𝑖+1 + 𝑓𝑠𝑖+1 = 𝑝𝑖+1 (1)
• Sin ce mo de p ar ticip ation factor is less for h ig h er mo d e
respo nses, this d iscretization is v alid an d p rov ides
su ff iciently accu r ate r esu lts .

9. 9
• Th e n u mer ical meth o d s ar e classif ied in to tw o typ es:
1 . I m p l i c i t S c h e m e s – M a n y a r e u n c o n d i t i o n a l l y s t a b l e
2 . E x p l i c i t S c h e m e s – C o n d i t i o n a l l y s t a b l e
• Stab ility an d accu r acy ar e th e tw o imp o r tan t cr iter ia f o r
d eter min in g th e r eliab ility o f th e n u mer ical I n teg r ation
sch eme.
Numerical Integration Schemes
E x p l i c i t S c h e m e s :
1 . C e n t r a l d i ff e r e n c e m e t h o d
2 . N e w m a r k ’s e x p l i c i t m e t h o d
3 . M o d i f i e d N e w m a r k ’s
m e t h o d
I m p l i c i t S c h e m e s :
1 . N e w m a r k ’s m e t h o d
2 . α - O p e r a t o r S p l i t t i n g
m e t h o d
3 . G e n e r a l i z e d a l p h a m e t h o d

10. 10
Numerical Integration Schemes
• G en er alized in teg r ation meth o d is N ew mar k ’s alg o r ithm
w h ich is g iv en as:
𝑥1𝑖+1 = 𝑥1𝑖 + 1 − γ ∆𝑡 𝑥2𝑖+1 + γ∆𝑡 𝑥2𝑖+1 (2)
𝑥𝑖+1 = 𝑥𝑖 + ∆𝑡 𝑥1𝑖 + 0.5 − 𝛽 ∆𝑡 2 + 𝛽 ∆𝑡 2 𝑥2𝑖+1 (3)
• Here, γ an d β d efin es th e v ariatio n o f acceleratio n in
g iv en time step .
• Fo r γ = 1 2 , th er e is n o n u mer ical d amp in g o b ser v ed in
th e so lu tio n.
• By p u ttin g 𝛽 = 0, g iv es N ew mar k ’s ex p licit meth o d .

11. 11
Numerical Stability and Accuracy
• Th e d ir ect in teg r ation sch eme can b e r ep r esented in a
r ecu r sive matr ix f o r m as g iv en :
𝑥𝑖+1 = 𝐴𝑥𝑖 + 𝐿𝑓𝑖+𝑣 (4)
• Th e n u mer ical an alysis is d o n e f o r f r ee v ib r ation
r esp o n se, w e h av e
𝑥𝑛 = 𝐴𝑥𝑛−1 = 𝐴𝑛
𝑥0 (5)
• Fo r stab le in teg r atio n sch eme, th e ab o v e mu st yield a
b o u n d ed so lu tio n f o r an y ar b itr ar y in itial so lu tio n
v ecto r.

12. 12
• Fo r N ew mar k ex p licit meth o d , th e in itial so lu tio n
v ecto r is g iv en as:
𝑥0 =
𝑥0
𝑥10
𝑥20
(6)
• Th e amp lif icatio n matr ix is g iv en as:
𝐴 =
1 ∆𝑡
∆𝑡2
2
−𝜔2∆𝑡
2
1 −
𝜔2∆𝑡
2
∆𝑡
2
−
𝜔2∆𝑡3
4
−𝜔2 −𝜔2∆𝑡
−𝜔2∆𝑡2
2
(7)
Numerical Stability and Accuracy

13. 13
• Th e matr ix A h as 3 d istin ct eig en v alu es, λ1 , λ2 , an d λ3
an d th er e a d iag o n al matr ix J su ch th at
𝐽𝑛 = Φ−1𝐴𝑛Φ (8)
w h er e, Φ = 𝜙1 , 𝜙2 , 𝜙3 an d J = 𝑑𝑖𝑎𝑔 𝜆1, 𝜆2, 𝜆3 , an d th e
v ecto r s 𝜙𝑖 ar e th e eig en v ecto r s o f A co r r esp on ding to th e
eig en v alu e 𝜆𝑖.
• U sin g Eq u atio n 5 an d 8 , w e g et
𝑥𝑛 = 𝑐1 𝜆1
𝑛
+ 𝑐2 𝜆2
𝑛
+ 𝑐3 𝜆3
𝑛
( 9 )
Numerical Stability and Accuracy

14. 14
• Th ese 𝑐1 , 𝑐2 , an d 𝑐3 ar e th e co n stan ts b ased o n in itial
co n d itio ns.
• Ou t o f th e th ree eig en so lu tio ns, λ1 ,2 sh o u ld b e co mp lex
co n j u gate an d λ3 < λ1 , 2 ≤ 1 .
• Th e meth o d w ill b e stab le if th e ab o v e co n d itio ns ar e
satisfied .
• By so lv in g 𝐴 − 𝜆𝐼 = 0 , w e g et
𝜆1 ,2 = 𝐴 ± 𝑖 𝐵 = 𝑒 − 𝜁 +𝑖 Ω ( 1 0 )
Numerical Stability and Accuracy

15. 15
w h er e,
𝐴 = 1 −
𝜔 2 Δ𝑡 2
2
( 11 )
an d ,
𝐵 =
4 − 𝜔 2 Δ 𝑡 2 −2 2
2
( 1 2 )
• Fo r th e r esp o nse to b e stab le, th e co n d itio n 𝐴2
+ 𝐵 2
≤ 1
sh o u ld b e satisf ied an d B sh o u ld b e r eal. Th e v alu e
𝐴2 + 𝐵 2 w ill alw ays b e eq u al to 1 an d th e B sh o u ld b e
r eal, h en ce th e stab ility co n d itio n,
𝜔 2 Δ𝑡 2 − 2 2 ≤ 4 ( 1 3 )
Numerical Stability and Accuracy

16. 16
lead s to ,
0 ≤ 𝜔Δ𝑡 ≤ 2 ( 1 4 )
• For 𝜔Δ𝑡 = 2 , th e respo nse will no t b e accur ate b ut stab le.
• Th e eig en so lu tio n s can b e r ep r esented as g iv en in th e
Eq u atio n 1 0 , wh ere 𝜁 an d Ω is d efin ed as
𝜁 = −
𝑙𝑛 𝐴2 +𝐵 2
2 Ω
( 1 5 )
an d ,
Ω = 𝑎𝑟𝑐𝑡𝑎𝑛
𝐵
𝐴
( 1 6 )
Numerical Stability and Accuracy

17. 17
su b stitu ting Eq u atio n 1 0 in Eq u atio n 9 , w e g et
𝑥𝑛 = 𝑒 −𝜍 𝜔 ∆𝑡 𝑛 𝑐1 𝑐𝑜𝑠 𝜔 ∆𝑡𝑛 + 𝑐2 𝑠𝑖𝑛 𝜔 ∆𝑡𝑛 + 𝑐3 𝜆3
𝑛
( 1 7 )
w h er e, 𝜔 = Ω Δ𝑡 . Co mp ar in g th e n u mer ical r esp o n se w ith
th e u n d er damp ed f r ee v ib r ation r esp o n se o f a SD O F system
w h ich is g iv en as u n d er :
𝑥 𝑡 = 𝑒−𝜍𝜔𝑡(𝑐1cos𝜔𝐷𝑡 + 𝑐2𝑠𝑖𝑛𝜔𝐷𝑡) ( 1 8 )
w h er e, 𝜁 an d 𝜔𝐷 ar e th e v isco u s d amp in g r atio an d th e
d amp ed n atu r al f r eq u ency o f th e SD O F. Co mp ar in g
Eq u atio n s 1 7 an d 1 8 , we can say th at 𝜁 an d 𝜔 are th e
n u mer ical d amp in g an d th e n u mer ical f r eq u ency
co r r esp ond in g to th e system r esp ectiv ely.
Numerical Stability and Accuracy

18. 18
• I f 𝜁 ≠ 0 , w e g et n u mer ically in d u ced d amp in g . A lso th e
d iff er ence b etw een th e n u mer ical f r eq u ency 𝜔 an d 𝜔
g iv es th e d isto r tio n in th e f r eq u en cy in th e n u mer ical
tech n iq u e.
• Pu ttin g th e v alu e o f A an d B f r o m th e Eq u atio n 11 an d 1 2
in th e Eq u atio n 1 5 , w e g et 𝜁 = 0 w h ich sh o w s th at th e
N ew mar k Ex p licit meth o d d o es n o t h av e n u mer ical
d amp in g . Also , su b stitu ting Eq u atio n 11 an d 1 2 in th e
Eq u atio n 1 6 , w e g et
𝜔 =
1
∆𝑡
𝑎𝑟𝑐𝑡𝑎𝑛
4−(𝜔2∆𝑡2−2)2
2−𝜔2∆𝑡2 ( 1 9 )
Numerical Stability and Accuracy

19. 19
• Th e d iff er ence in th e n atu r al f r eq u en cy an d th e
n u mer ical f r eq u en cy w ill g iv e th e amo u n t o f f r eq u en cy
d isto r tion in th e system. Th e p lo t o f p er cen tag e o f p eriod
d isto rtion 𝑇 − 𝑇 𝑇 an d ∆t/T sh o ws th at th e we g et a
r easo n ab ly accu r ate so lu tio n w h en ∆t/T is less th an 0 .0 3
an d th e p er io d d isto r tion v an ish es w h en ∆t g o es to zer o .
Numerical Stability and Accuracy

20. 20
Numerical Stability and Accuracy
Fig u re 2 : Perio d S h rin k a g e b y N ewma rk Ex p licit met h o d

21. 21
Numerical Stability and Accuracy
• H en ce, th e stab ility r eq u ir emen ts f o r th e N ew mar k
Ex p licit meth o d is 𝜔Δ𝑡 ≤ 2 .
• Pu ttin g 𝛽 = 0 an d 𝛾 = 1 2 , in th e Eq u atio n 2 an d 3 , we
g et
𝑥1𝑖+1 = 𝑥1𝑖 + (𝑥2𝑖 + 𝑥2𝑖+1)
∆𝑡
2
( 2 0 )
𝑥𝑖+1 = 𝑥𝑖 + ∆𝑡 𝑥1𝑖 +
∆𝑡2
2
𝑥2𝑖 (21)
a nd ,
𝑥2𝑖+1 = 𝑚 +
∆𝑡
2
𝑐
−1
𝑝𝑖+1 − 𝑓𝑠𝑖+1
− 𝑐𝑥1𝑖 −
∆𝑡
2
𝑐𝑥2𝑖 (22)

22. 22
Fig ure 3 : Newma rk’s Ex plicit Alg o rithm
Newmark’s Explicit Algorithm

23. 23
Er r o r s in
• Stru ctu ral id ealisatio n
• N u mer ical in teg r ation
• Ex p er imen tatio n
Errors in Pseudodynamic testing

24. 24
• Lu mp ed mass mo d el
• D amp in g ch ar acter istics
• Str ain r ate eff ects
Error in Structural Idealisation

25. 25
• S tab ility an d accu r acy
• F r eq u en cy d isto r tion
• G r o w th o f sp u r io us r o o ts
Errors in Numerical Integration

26. 26
• Co mp u ted d isp lacemen ts cann ot b e ex actly imp o sed on
th e str u ctu re .
• Er r o r s in measu r emen t o f r esto r in g f o r ces .
• Th ese error s can gener ate hug e cu mu lative error s and
can b e tak en car e of using nu mer ical damp ing
p rop er ty of th e algor ith m u sed f or th e disp lacemen t
co mp u tatio n s .
Errors in Experimentation

27. 27
• A p seu d o dyn amic testin g simu latio n co d e is d ev elo p ed
in MATLA B u sin g N ew mar k Ex p licit sch eme.
• Th e co d e tak es th e static p u sh o ver cu r v e d ata f o r th e
u p d atio n o f r esto r ing f o r ce v alu e in th e co d e.
• W h en th e v elo city ch an g es its sig n , th e co d e u ses
in itial stiff n ess till th e u n lo ad ing p h ase ch an g es to
lo ad in g p h ase.
• Th e co d e is v er if ied u sin g th e so lv ed ex amp le g iv en
in th e N PTEL co u r se o n “ I n tro d uction to Ea rth q u ake
En g in eerin g” f o r n o n - linear seismic r esp o n se o f th e
str u ctu res w ith b i - lin ear f o r ce d ef o r matio n cu r v e.
Methodology

28. 28
Methodology
• Fo r elasto - p lastic SD O F system,
• Elastic stiff n ess = 3 9 .4 7 8 N /m,
• Mass = 1 k g ,
• D amp in g co n stan t = 0 .2 5 1 N sec /m,
• El- Cen tr o ear th q u ake
• U sin g av er ag e acceler atio n meth o d
Th e d isp lacemen t r esp o n se o f th e system g iv en in th e
f o llo w in g ex amp le is co mp ar ed w ith th e r esu lts o b tain ed
u sin g th e simu latio n co d e. Th e simu latio n co d e g iv es
co r r ect r esu lts an d h en ce is v er if ied.

29. 29
Methodology
Fig u re 4 : C o mp a rison o f d isp la cemen t resu lt s o f a S D OF
sy st em wit h N ewma rk Ex p licit sch eme a n d a v era g e
a ccelera t io n met h o d

30. 30
Results
• Th e p seu d o dyn amic simu latio n is car r ied o u t f o r a
f ix ed b ase steel can tilev er steel co lu mn as a SD O F
system.
• A f o r ce d ef o r matio n r elatio n is g en er ated u sin g
O p en Sees an d th e p u sh o v er cu r v e is p lo tted .
• Th e simu latio n is d o n e f o r 3 d iff er ent ear th q u ak es an d
th e r esp o n ses ar e co mp ar ed w ith th e r esp o n ses u sin g
Lin ear A cceler atio n meth o d .

31. 31
• Pu sh o v er cu r v e is mad e u sin g O p en Sees f o r a steel
can tilev er co lu mn h av in g f o llo w in g p r o p erties:
Leng t h: 2 0 0 0 mm
Yield St reng t h: 2 5 0 MPa
M o dulus o f Ela st icit y : 2 1 0 G Pa
U nit Weig ht : 7 8 5 0 k g /m3
Po st - Yield St reng t h: 1 % o f I n itial Str en g th
Results

32. 32
• Str ess - Strain cu r v e f o r th e Steel0 1 mater ial is sh o w n
b elo w :
Fig ure 5 : Stress - Strain curv e fo r Steel0 1 Ma teria l
Results

33. 33
(a) Steel cantilever column
(b) Section (a-a)
Results
Fig ure 6 : SD OF sy stem fo r pseudo dy namic simula tion

34. 34
Th e in itial in p u t p ar ameter s f o r th e an alysis is g iv en as
u n d er:
• mass 𝑚 = 254 3 .4 𝑘𝑔
• in itial stiff n ess 𝑘𝑖𝑛𝑖 = 312 .559 𝑘𝑁 𝑚
• in itial d isp lacemen t 𝑥 0 = 0
• in itial v elo city 𝑥 1 0 = 0
• d amp in g r atio 𝑧 = 0 . 05
• n atu r al f r eq u ency 𝜔 = 𝑘
𝑚 = 11 .0 856 𝑟𝑎𝑑 𝑠
Results

35. 35
Fig ure 7 : Pusho v er curv e o f Steel C a ntilev er co lumn
Results

36. 36
Results
• Th e mo d el is su b j ected to th e f o llo w ing 3 d iff er ent
earth q u akes:
El- Cen tr o ear th q u ake, PG A : 0 .3 2 g
N o r th r idge ear th q u ake, PG A : 0 .5 7 g
K o b e ear th q u ake, PG A :0 .3 4 g
• Th e r esp o n se o f th e mo d el w ith El - Cen tr o ear th q u ake
is sh o w n :

37. 37
Results
Fig ure 8 : C o mpa rison o f A ccelera t io n respo nse o f a SD OF
sy stem with Newma rk Ex plicit a nd Linea r Accelera tio n
met ho d subj ect ed t o El - C ent ro ea rt hqua ke

38. 38
Results
Fig ure 9 : C o mpa rison o f Velo cit y respo nse o f a SD OF sy st em
with N ewma rk Ex plicit a nd Linea r A ccelera tio n metho d
subj ect ed t o El - C ent ro ea rt hqua ke

39. 39
Results
Fig ure 1 0 : C o mpa riso n o f D ispla cement respo nse o f a SD OF
sy stem with N ewma rk Ex plicit a nd Linea r A ccelera tio n
met ho d subj ect ed t o El - C ent ro ea rt hqua ke

40. 40
Results
Fig ure 11 : C o mpa rison o f Fo rce - D ef o rma tion respo nse o f a
SDOF sy stem with Newma rk Ex plicit a nd Linea r Accelera tio n
met ho d subj ect ed t o El - C ent ro ea rt hqua ke

41. 41
Results
• Th e ab o v e co mp ar iso n sh o w s th at th e r esu lt o f th e
p seu d o dyn amic simu latio n f o r a S D O F system u sin g
N ew mark Ex p licit meth o d match es w ell w ith th e
n o n lin ear time h isto r y an alysis o f th e same system
u sin g Lin ear A cceler atio n meth o d .

42. 42
Conclusion
• P seu d o d ynamic testin g meth o d is a p o w er fu l tech n iq u e
f o r testin g larg e scale str u ctu res.
• Mo st r eliab le as w e g et th e r esto r ing f o r ces d ir ectly
f r o m th e ex p er imen t h en ce n o sco p e f o r er r o r in
mo d ellin g o f stiff n ess matr ix .
• N ew mark Ex p licit meth o d is an ex p licit sch eme an d can
b e u sed f o r th e system h av in g lo w er d eg r ees o f
f r eed o m.
• D u e to ex p er imen tal er r o r s, th er e is a sp u r ious g r o w th
o f h ig h freq u ency resp o n se in MD O F an d to su p ress th at
Mo d if ied N ew mar k alg o r ith m can b e u sed in w h ich
f r eq u ency p r o p o rtional n u mer ical d amp in g is p r esen t.

43. 43
Future work
• Th e simu latio n is carried o u t fo r SDOF system an d can
b e ex ten d ed to MD O F systems.
• Th e p r o g ram n eed s to b e w r itten f o r th e co n tr o ller o f
th e actu ato r f o r th e p seu d o dyn amic testin g p r esen t in
th e Pseu d o dyn amic Testin g Lab o rato ry, Dep artmen t o f
Ear th q u ake En g in eer ing , I I T Ro o r k ee an d th e
ex p er imen tal r esu lts n eed to b e co mp ar ed w ith th e
an alytical r esu lts.
• Su b structuring can red u ce th e co st o f th e testin g to a
g r eat ex ten t an d larg er str u ctu r es in w h ich n o n lin earity
is co n cen tr ated can b e tested u sin g th is co n cep t.

44. 44
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Thank you