Horizontal-Loadings-of-Liquid-concrete-on-skew-arches

7/30/2019 Horizontal-Loadings-of-Liquid-concrete-on-skew-arches

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University College Cork

Coliste na hOllscoile Corcaigh

Dept. of Civil & Environmental Engineering

Horizontal Loadings of Liquid Concrete on Skew

Arches

Paul Anthony Cahi l l

Eoin Thomas Al len

Supervisor : Dr . Vikram Pakrashi

Submit ted as par t of BE (Civi l ) degree


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1. Introduction

Curren t ly the re a re no gu ide l ine s in the f i e ld o f a rch de s ign pe r ta in ing

to l a te ra l fo rce s in l iqu id conc re te a c t ing on p reca s t a rches , wh ich a reto be u sed in road b r idge cons t ruc t ion . The a im o f th i s p ro jec t i s to s e t

up and de s ign work ing mode l s wh ich wi l l have an app l ied ho r izon ta l

load , a s a re su l t o f the l iqu id conc re te . A ca se mode l wh ich i s

re fe renced in ou r inves t iga t ions i s the Leenane Br idge in coun ty

Ga lway . The b r idge i t s e l f , be fo re be ing newly cons t ruc ted , was ove r

100 yea rs o ld bu t in 2008 was de s t royed by s eve re f lood ing and ra in s .

The b r idge was rebu i l t in 2009 and was inc rea sed in s i z e . The b r idge

cons i s t s o f 3 p reca s t a rches wh ich re s t on in s i tu re in fo rced conc re te

abu tmen ts . Dur ing cons t ruc t ion p rob lems re la t ing to the l a te ra l fo rce s

a s soc ia ted wi th l iqu id conc re te a rose , due to no d i rec t gu ide l ine s

pe r ta in ing to a rch de s ign be ing in c i rcu la t ion . Th rough the u se o f

de ta i l ed mode l s u s ing a f in i t e e lemen ts p rog ramme S t rand7 and wi th

the u se o f Exce l p rog rammes , va ry ing pa rame te r s such a s b r idge

d imens ions and cha rac te r i s t i c s a re a ccoun ted fo r and mode l led to

p roduce d i f fe ren t ho r izon ta l load ings . F rom the se ho r izon ta l load ings

Bend ing momen ts in bo th X and Y ax i s wi l l be ob ta ined . Wi th th i s , a

s e r i e s o f da ta cha r t s and g raphs a re c rea ted to d i sp lay the max imum

hor izon ta l load ings to be wi th s tood by the g iven a rch . Wi th a s e t o f

t ab le s o r g raphs the fa i lu re modes can be ach ieved . In pa r t i cu la r these rv iceab i l i ty s t a te o f c rack ing i s a v i t a l mode o f fa i lu re , wh ich fo r

ae s the t i c pu rpose s mus t be avo ided . Us ing the ob ta ined cha r t s ,

gu ide l ine s fo r a rch de s ign can be c rea ted and thus the se modes o f

fa i lu re avo ided o r s een how to be avo ided i n t e rms o f added d imens ions

o r s t reng th cha rac te r i s t i c s . The re wi l l be s even va ry ing pa rame te r s wi th

rega rds to ou r a rch mode l .


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F i g u r e 1 . 1 A p i c t u r e o f a b r i d g e i n t h e w e s t o f I r e l a n d w h i c h w i l l a c t a m o d e l f o r o u r

c a s e s t u d y . 1

F i g u r e 1 . 1 A 3 - D g e n e ra t e d m o d e l o f o n e p r e c a s t a r c h w i t h l a y e r s o f c o n c r e t e b e i n g

p o u r e d a s a t i m e .


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F i g u r e 1 . 2 A r e a l t i m e r e n d e r i n g 3 - D i l l u s t r a t i o n o f t h e a r c h w i t h c o n c r e t e s e t a t

e i t h e r s i d e .

2. Literature Review and Theory

Conc re te a rches have been used to so lve numerous cons t ruc t ion

p rob lems wor ldwide . I t i s the pu rpose o f th i s pape r to f ind the l imi t s a

p reca s t a rch can wi th s tand wi th re spec t to bend ing momen ts and fo rce s

ac t ing a t a po in t on va ry ing d i f fe ren t d imens ions . F i r s t we mus tunde rs tand and the p re s su re s invo lved when the conc re te a t a l iqu id

s tage i s pou red on to the p reca s t s l ab . The re a re va r ious d i f fe ren t

mode l s a s soc ia ted wi th ca lcu la t ing p re s su re s bu t f ew wh ich g ive

re fe rence to a rch c r i t e r ion in pa r t i cu la r .

2.5 Models .

There a re va ry ing d i f fe ren t mode l s wh ich dea l wi th the l a te ra l

p re s su re s c rea ted when l iqu id conc re te i s p laced in fo rmwork and

beg ins to s e t . I t was dec ided fo r the pu rpose s o f th i s r epo r t to focus on

mode ls re su l t ing f rom expe r imen ta l da ta ra the r than theo re t i c a l

mode l l ing o r mode l s made us ing a s sumed va lue s . Th i s dec i s ion was

made to re f l e c t cond i t ions p re sen t on cons t ruc t ion s i t e s and the

re su l t ing conc lus ions wou ld be more re levan t fo r u se in the so lv ing o f

the p rob lem o f l a te ra l fo rce s in conc re te . I t was dec ided to u se s eve ra l

d i f fe ren t mode l s in t he ca lcu la t ion o f the re su l t s . Us ing s eve ra l va ry ing

mode ls and no t ju s t a one g ive s a b roade r de sc r ip t ion and v iew o f the

max imum p re s su re s wh ich t ake p lace . As the mode l s inc luded have a

range o f va ry ing pa rame te r s wh ich wi l l g ive a more accu ra te

rep re sen ta t ion o f the p re s su re s invo lved . The ob ta ined re su l t s w i l l

inc lude many cond i t ions wh ich a re p re sen t in the max imum p re s su re o f


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l a t e ra l fo rce s deve loped on the p reca s t a rch , bu t i f a r ange o f mode l s

were no t to be cons ide red , a l e s s a ccu ra te s e t o f r e su l t s wou ld be

a t t a ined a s a l l va r iab le s ava i l ab le wou ld no t be inc luded in ou r

ana lys i s .

3.1 Introduct ion

A 3D b r ick mode l was s e t up u s ing S t rand7 f in i t e e lemen ts p rog ramme .

A base mode l was in i t i a l ly de s igned wi th ou r ba se d imens ions s e t a t an

in te rmed ia te va lue s . The re a re many va ry ing pa rame te r s invo lved wi th

the mode l f rom the change o f the ang le o f skew o f the a rch , h

which i s the he igh t o f conc re te pou red e i the r s ide a t a t ime o f the a rch ,

R the ra te o f p lacemen t , T the t empe ra tu re o f the conc re te , Ra t io o f

d imens ions o f ou r mode l , f c u the conc re te s t reng th , T the th icknes s o f

the p reca s t s t ruc tu re . These s even pa rame te r s wi l l have a range o f

va lue s wh ich wi l l be inpu t ted in to ou r ba se mode l and thus re su l t ing inva ry ing ho r izon ta l load ings fo r ou r a rch .

F i g u r e 3 . 1 A s c r e e n s h o t o f t h e a r c h m o d e l w i t h a p p l i e d h o r i z o n t a l l o a d s l o n g t h ea r c h a n d i n t e r n a l s t r e s s e s .

3.2 Models used to create hor izontal loadings

Using fou r d i f fe ren t mode l s to c r ea te va lue s fo r p re s su re s a t inc rea s ing

he igh t s a long the a rch i t w i l l g ive va ry ing re su l t s and th en be compu ted

as ho r izon ta l fo rce s and used in the mode l . The cons ide red mode l s

b e in g us e d : Ad ams , C i r i a , Ga rn er s an d Ro d en s . Th es e fo u r d i f f e r en t

mode ls cons ide r va ry ing a spec t s o f fo rmwork p re s su re s and wi l l then

re su l t in va ry ing ho r izon ta l load ings . F rom ou r re su l t s we have found


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tha t load ings a re cons i s ten t f rom be ing h ighes t to lowes t in s i z e . The

mode ls range f rom la rge s t to sma l l e s t in ho r izon ta l load ings a s be low.

1 - C i r i a

2 Ro d en s

3 Ad ams

4 Gard en er s

The re i s a gene ra l pa t t e rn s een th roughou t the cou rse o f r e su l t s o f the

ho r izon ta l load ings and u l t ima te ly the Bend ing momen ts wh ich adhe re

to the above o rde r . Once ho r izon ta l load ings a re e s tab l i shed fo r e ach o f

the mode l s they a re u sed to ge t va lue s fo r t he bend ing momen ts in bo th

X & Y ax i s .

6. Creation and Analysis of 3-D Models and Graphs.

6.1 Strand 7 f in i te e lements programme.

The compu te r p rog ram used to compu te the re su l t s was a f i n i t e e lemen t

ana lys i s p rog ram ca l l ed S t rand7 . The so f twa re a l lows the u se r to c rea te

a mesh o f the mode l , a s s ign p rope r t i e s and load ings qu ick ly and

e f f i c ien t ly . I t i s qu i t e me thod ica l in i t s app roach to comple t ing the

mode l . I t compu te s the s t re s s , fo rce and d i sp lacemen t a t e ach po in t on

the mesh and the re su l t s func t ions , inc lud ing g raph ica l i l lu s t ra t ion o f

the s t re s s in the mode l , g raph ing op t ion and sp read shee t fo rma t o f the

re su l t s a l low fo r qu ick ana lys i s o f the s t ruc tu re .

I t was dec ided to c rea te 3D mode ls o f the a rches ra the r than 2D mode ls

a s we fe l t the 3D mode ls wou ld be more accu ra te than i t s 2D

coun te rpa r t . A lso , by u s ing 3D mode ls , the p re s su re load ings on the

a rch cou ld be en te red d i rec t ly in to the p rog ram a s a Globa l Face

Pre s su re . Th i s p re s su re ac t s ove r the face o f the b r ick on the X-ax i s ,

mi r ro r ing the rea l l i f e e f fec t s o f the conc re te a c t ing ho r izon ta l ly . S tand7 inc ludes the func t ion fo r en te r ing equa t ions fo r app ly ing p re s su re and

th i s p roved bo th he lp fu l and accu ra te . Fo r example , when app ly ing

p re s su re re l i an t on he igh t , the exac t he igh t o f the b r ick i s u sed in the

p rog ram and p roves h igh ly accu ra te when compared to the p re s su re

va lue s gene ra ted in Exce l .


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F i g u r e 6 . 1 A s c r e e n s h o t o f t h e t h r e e d i m e n s i o n a l m o d e l i l l u s t r a t i n g t h e l o a d s a c t i n g

o n t h e a r c h h o r i z o n t a l l y .

6.3 Establ ish ing Models and respect ive pressures and forces

As s ta ted in the l i t e ra ry rev iew we have worked wi th fou r d i f fe ren t

mode l s wh ich t ake d i f fe ren t app roaches in c rea t ing ho r izon ta l p re s su re s

due to l iqu id conc re te . I f wh en we s ta r t ed to u se the se mode l s and the re

fo rmula s to c rea te ou r load ings fo r e ach spec i f i c c a se by hand and

us ing ju s t a c a lcu la to r and pen i t wou ld be in to nex t yea r be fo re any

re levan t da ta wou ld be a t t a ined . Us ing ou r chosen sp read shee t , we can

s imp ly inpu t the g iven equa t ion f rom the chosen mode l s and in se r t the

requ i red pa rame te r s fo r e ach mode l c a se . A l l - ready d i s cus sed in

chap te r 4 we re the va ry ing pa rame te r s to be cons ide red fo r e ach mode l

and spec i f i c c a se . F rom d rawing up a s e r i e s o f equa t ions and a t ab le o f

wh ich inc ludes co lumns o f the pa rame te r head ings and rows o f theva lue s o f the pa rame te r s . Wi th th i s t ab le c rea ted i t i s e a sy to a cce s s the

va lue s o f p re s su re s o r fo rce s fo r e ach spec i f i c c a se and s imp ly inpu t

tha t va lue o r range o f va lue s in to the de s ign mode l in the f in i t e

e lemen ts p rog ramme and u l t ima te ly ou tpu t the fo rce s impac ted on to the

a rch and the re su l t ing momen ts and shee r fo rce s . In to ta l we wi l l c rea te

a s e r i e s o f mode l s in t e rms o f the ang le o f skew o f the a rch f rom the

p r inc ipa l ax i s and va ry one pa rame te r e ach t ime we wi l l do th i s fo r

each mode l fo r 5 to 8 va ry ing pa rame te r s and fo r 7 d i f fe ren t ang le s o f

skew. Th is wi l l r e su l t in 236 d i f fe ren t mode l s cove r ing a l l a spec t s o f a rch s i z e and the va r ia t ions in t e rms o f the ang le o f skew o f the a rch .


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Each mode l wi l l ana lysed and i t s max imum s t re s s and bend ing momen ts

reco rded and g raphed a s d i s cus sed be low. Each ang le o f skew wi l l have

9 va ry ing g raphs wi th each o f the fou r p re s su re mode l s rep re sen ted on

the each o f the 9 g raphs . The 9 g raphs wi l l r ep re sen t e ach change in

the va ry ing pa rame te r .

F i g u r e 6 . 5 A s e c t i o n o f t h e d i s c u s s e d t a b l e f r o m e x c e l , w h e r e s p e c i f i c m o d e l s c a n b e

e a s i l y t r a c k e d a n d p r o d u c e t h e r e s u l t i n g f o r c e s .

A sec t ion o f such a t ab le i s shown above . When the t ab le i s c rea ted i t

c an be s imp ly cop ied aga in and used fo r the o the r mode l s . H igh l igh ted

in ye l low a re the chang ing va lue s fo r tha t pa rame te r . By wr i t ing the

g iven equa t ion fo r the mode l and pa s t ing i t in to the s ame row ac ros sthe bo t tom o f the t ab le and a s s ign ing the coe f f i c ien t s fo r tha t c a se to

the requ i red va lue s ( the boxes in ye l low) and the rema in ing va lue s a re

le f t a t ba se l ine va lue s . The neces sa ry va lue fo r p re s su re i s then

ou tpu t ted in tha t box unde r the re levan t co lumn . Us ing th i s sy s tem

gave us qu ick , a ccu ra te and the requ i red va lue fo r the spec i f i c c a se to

inpu t to ou r 3 -D mode l . Hav ing deve loped th i s sy s tem we found i t e a sy

to t r ans fe r va lue s f rom the fo rmwork p re s su re mode l s wi th any

d imens ion o r pa rame te r and re lay th i s da ta in to the f in i t e e lemen ts

p rog ramme and p roduce a mode l o f an a rch wi th ho r izon ta l load ings

ac t ing a t va ry ing he igh t s a long the mode l led a rch .

6.4 Creat ion of 3-Dimensional Models

The f i r s t s t ep in the deve lopmen t o f the mode l was c rea t ing beam

e lemen ts u s ing the po in t s and l ine s func t ion , c rea t ing 180 beams in the

shape o f the pa rabo la . The beams we re then conve r ted in to Quad8

p la te s , by ex t rud ing the beams by the de s i red th icknes s o f the a rch . By

subd iv id ing the se p la te s in to fou r , squa re p la te s we re c rea ted th rough

the th icknes s . I t i s v i t a l tha t a l l f a ce s o f p l a te s and b r icks a re squa re in

shape , fo r comple t ing the f in i t e e lemen t app roach . These p la te s we rethen conve r ted in to Hexa8 b r icks , by once aga in u s ing the ex t rud ing


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func t ion , and ex t rud ing the b r icks by the wid th o f the a rch .

Subd iv id ing the se l a rge rec tangu la r p r i sm b r icks a long the Z ax i s

re su l t s in the de s i red b r icks u sed in the mode l l ing . The nodes a t the

base o f the a rch we re then re s t ra ined aga ins t t r ans la t ion in the X , Y

and Z d i rec t ion . The p rope r t i e s o f the conc re te we re then app l ied to theb r icks , comple t ing the mode l .

F i g u r e 6 . 7 D i s p l a y i n g h o w t h e b r i c k i s r e l a t i v e l y s q u a r e i n s h a p e a n d i t s b a s i c d a t a .

The load ings we re app l ied in d i f fe ren t load ca se s . Each load ca se

rep re sen t s a d i f fe ren t pou r o f conc re te , and range f rom the a rch be ing

empty to the comple t ion o f the pou r ing o f conc re te . Fo r e ach load ca se

g rav i ty was a s s igned a s -9 81m/s2. The p re s su re was a s s igned to the

b r icks a s a g loba l p re s su re in the X d i r ec t ion , on the face s wh ich wou ld

be in con tac t w i th the conc re te in a work ing env i ronmen t . As

p rev ious ly s t a ted the loads wi l l be a s s igned a t he igh t s r i s ing up the y

ax i s o f the a rch acco rd ing to the i r va lue s in the Exce l sp read shee t .

They wi l l by de fau l t a c t ho r izon ta l ly to the su r face o f the s t ruc tu re .


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F i g u r e 6 . 8 A s c r e e n s h o t d i s p l a y i n g t h e v a r i o u s p o u r o r l o a d i n g c o n d i t i o n s

a p p l i c a b l e t o t h a t m o d e l .

Once the mode l ha s been co r rec t ly deve loped and ha s the a s s igned

loads . The use r wi l l chose an op t ion f rom the so lve r and run the l inea r

s ta t i c so lve r . In the so lve r th e re a re va r ious op t ions to cons ide r . As we

c rea ted ou r mode l u s ing a g loba l sy s tem whe re po in t s a re a l l r e la ted to

an o r ig in po in t and when the re su l t f i l e i s comple te the u se r c an s e lec t

and x , y o r z co -o rd ina te and ge t r e su l t s a t tha t po in t , l ine ac ros s the

s t ruc tu re and a t a p lane ac ros s the a rch u s ing the co -o rd ina te s . Fo rm

se t t ing the co r rec t s e t t ings fo r the so lve r and s e lec t ing the ok bu t ton

the p rog ramme wi l l so lve and p roduce the re su l t s fo r the s e r i e s o f loads

a s s igned to the s t ruc tu re . As spoken abou t p rev ious ly the s i z e o f a

th ree d imens iona l Hex-8 mode l the re i s a cons ide rab le run t ime

invo lved wi th such a mode l . When howeve r the mode l i s r an , I cou ld

open the re su l t s f i l e and the p rog ramme wi l l au toma t ica l ly run the

re su l t s f i l e on the s t ruc tu re . When the p rog ramme has ran the re su l t s

f i l e , the mode l ha s been comple ted f rom in i t i a l de s ign a l l the ways

th rough to re su l t ana lys i s and app l ic a t ion . I t i s now down to the u se r

be i t myse l f , my pa r tne r o r any de s ign consu l tan t to a cce s s re su l t s anduse them in any way they de s i re . The re su l t s c an be t aken a t a g lance

f rom the s c reen by co r re spond ing the co lou rs on the s t ruc tu re to the

cha r t on the top r igh t hand o f the s c reen .

6.5 Accuracy of Models and Data

Using a f in i t e e lemen ts p rog ramme such a s S t rand7 fo r a f i r s t t ime use r

can be d i f f i cu l t and ha rd to mas te r . F rom ou r r e sea rch and cons tan t u se

o f th i s so f twa re package we go t to g r ip s wi th the ava i l ab le too l s and

func t ions and l e a rned how fu l ly exp lo i t the p rog ramme to deve lop and

c rea te u sab le and s ign i f i c an t da ta . Hav ing spen t weeks work ing a t bo th


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the mesh ing o f ou r mode l and the a s s ign ing o f loads to the mode l on a

t r i a l and e r ro r ba s i s we f ina l ly c rea ted a work ing and t e chn ica l ly

co r rec t mode l .

F i g u r e 6 . 9 A m o d e l s u m m a r y f r o m s t r a n d d i s p l a y i n g t h e v o l u m e a n d n u m b e r o f b r i c k

e l e m e n t s u s e d i n t h i s s p e c i f i c m o d e l .

Having o r ig ina l ly dec ided to u se two-d imens iona l mode l l ing , the semode ls u sed p lane s t ra in p la te s a s a mode l type . Th i s app roach had

advan tages in the fo rm o f qu ick run t ime , e a s i ly app l ic ab le load ings

and sma l le r mode l to work on thus mak ing i t e a s ie r to a s s ign loads o r

compu te any changes . Whi le u s ing th i s type o f mode l we encoun te red

p rob lems whe re the th icknes s o f the p la te s e f fec ted the dens i t i e s o f the

p la te s them se l f and gave a subs tan t i a l ly l a rge r re s i s t ance than tha t o f

a p reca s t conc re te a rch and thus re su l t ing in f l awed re su l t s . Hav ing

t r i ed rec t i fy ing the se fau l t s w i th ou r mode l and even t r i ed chang ing to

a d i f fe ren t p la te sy s tem in the fo rm o f p la in s t ra in p la te s . These

changes re su l t ed in fu r the r i l l eg i t imac ie s in the mode l p roduc ing

re su l t s fo r s t re s s in the range o f 60Kpa . These va lue s we re obv ious ly

we l l ou t s ide any p laus ib le range o f re sul t s . Th i s u l t ima te ly ca l l ed fo r

a new mode l type and de sign . As can be s een f rom the mode l summary

the re a re ove r 129 ,600 d i f fe ren t b r i ck e lemen ts , th i s shows the shea r

amoun t o f e lemen ts i t t ake s to c rea te such a mode l and u l t ima te ly tha t

the mesh ing needs to be co r rec t o r e l s e one mis take wi l l th row ou t the

re su l t s o r the mode l ana lys i s when loads a re app l i ed .

F i r s t ly bo th the beam and a rch we re c rea ted in the s ame p roces s ,inc lud ing a l l the s ame c r i t e r i a fo r mesh ing , u s ing the s ame b r ick

e lemen ts (hex -8 ) and keep ing a l l cond i t ions the s ame wi th in the mesh

and b r ick s e t t ings . Wi th bo th mode l s c rea ted we s e t abou t to s e t the

end cond i t ions o f bo th mode l s . The a rch wou ld be fu l ly f ixed a t one

end and f ixed in ro ta t ions a t the o the r . A t the end o f the a rch wi th on ly

f ixed ro ta t ions i t wou ld b e a l lowed to move in bo th the x and y ax i s . In

keep ing wi th th i s s t a t i c cha rac te r i s t i c , the beam was f ree a t one end

and fu l ly f ixed a t the o the r end . Us ing the se now re levan t and work ing

mode ls we s e t the s t rand7 p rog ramme to ana lys ing bo th beam and a rch


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with on ly s e l f -we igh t be ing app l ied ba s ica l ly an U .D.L . (Un i fo rmly

Dis t r ibu ted Load) ac t ing ac ros s the l eng th and b read th o f the s t ruc tu re .

insert length 20 m

insert breadth 10 m

insert height 0.4 m

insert y 0.2 m

Theory for a simply supported beam

SW rho*h 9.6 KN/m^2

for 10m width 96 KN/m width

I = bd^3 /12 0.053333333 m^4

M = (w(l^2))/8 4800 kNm

Stress 18000 KPa

Results

For Simply Supported

Percentage

Correct

Brick Middle Stress (KPa) Arch 19067.35 105.9297222 %

Beam -17850.7 -99.1705556 %

Moment

(KNm) Arch 5084.626667 105.9297222 %Beam -4760.186667 -99.1705556 %

As seen above , u s ing an Exce l sp read shee t we cou ld ca lcu la te the

momen ts f rom the re su l t ing s t re s se s ou tpu t ted by the re su l t s f i l e o f ou r

mode l in the f in i t e e lemen ts p rog ramme . The s econd momen ts a rea was

ca lcu la ted wh ich wou ld be the s ame a t the c rown o f the a rch a s

anywhere a long the beam. We took va lue s f rom the a rch a t th i s po in t .

Keep ing in l ine wi th ou r re su l t s and the accu racy the a rch shou ld c rea te

s l igh t ly l a rge r s t re s s e s and momen ts a s a l l tough i t a c t s ove r the s ame

span i t ha s a l a rge r vo lume due to i t s shape and thus re su l t ing in a


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l a rge r UDL due to the inc rea se in s e l f -we igh t . F rom the re su l t s

ob ta ined fo r bo th the a rch and beam they we re inpu t ted to the sp read

shee t a s s een above and then compared back aga ins t e ach o th e r and the

hand ca lcu la t ions o f wha t the s t ruc tu re shou ld p roduce in t e rms o f

momen ts and shee r . F rom ca lcu la t ing the s t re s s and momen ts fo r abeam 20 me t re s long , 0 .4 me t re s in he igh t and 10 me t re s in b rea th . The

Second momen t o f a rea was ca lcu la ted g iv ing 0 .0533m4. S imp ly by

us ing a fundamen ta l ru le o f s t a t i c s fo r s t re s s e s the momen ts c an be

ca lcu la ted .

( E q u a t i o n 6 . 1 ) . We can now ca lcu la te the momen t

c rea ted in the a rch a s S t rand& on ly ou tpu t s s t re s s e s .

3-D Mode l Arch St re s s % co r rec t : 105 .9% 3-D Mode l Br ick S t re s s % co r rec t : 99 .17% 3-D Mode l Arch Momen t % co r rec t : 105 .9% 3-D Mode l Br ick Momen t % co r rec t : 99 .17%

With th i s s imp le t e s t a ch ieved and pa s sed we cou ld move on wi th the

c rea t ion o f the o the r mode l s fo r va ry ing pa rame te r s and load ca se s ,

happy in the knowledge tha t ou r mode l wou ld be co r rec t and p roduce

p rope r and t rue re su l t s and da ta a s the new th ree d imens iona l mode l

was work ing acco rd ing ly .

6.6 Use and calculat ion of establ ished data

Having succes s fu l ly c rea ted work ing mode l s wi th re levan t and usab le

da ta i t i s now t ime to t ake th i s da ta and use i t to c a lcu la te momen ts and

thus u l t ima te ly to a s ce r ta in whe the r o r no t the s e rv iceab i l i ty l imi t

s t a te s wi l l be pa s sed o r fa i l due to exces s momen ts expe r ienced wi th in

the a rch . S t rand7 p roduces an ex tens ive s e r i e s o f r e su l t s , when us ing

the g loba l s e t t ing in t e rms o f the mode l s pos i t ions re su l t s c an be

a t t a ined f rom any pos i t ion o r po in t on the g raph . When the re su l t s f i l e

ha s f in i shed runn ing i t w i l l d i sp lay g raph ica l ly . The s t re s se s

th roughou t the a rch wi l l be shown in co lou r wh ich can be re fe renced to

the cha r t on the top r igh t o f the s c reen when the re su l t s f i l e ha s been

opened ( re fe r to f igu re 3 .1 page 21 ) . S t rand7 ana lys i s a l l the load ca se sac t ing on the s t ruc tu re , then t akes in to cons ide ra t ion the a s s igned

cha rac te r i s t i c s o f the a rch and then a t t r ibu te s the s t re s s e s a cco rd ing ly

ac ros s and th roughou t the a rch . As now reac t ions , de f le c t ions and

s t re s se s a re now a l l p re sen t on the mode l , a cqu i r ing the se re su l t s in

re levan t pos i t ions and us ing the re su l t s to p roduce momen ts and

u l t ima te ly re la t ionsh ip g raphs be tween a l l the 4 p re s su re mode l s ( re fe r

to chap te r 2 .5 page 16 ) .

Us ing the g raph func t ion in S t rand 7 a l lowed us to ge t s t r e s s e s a t any

po in t s we de s i red and hence inc rea se the accu racy o f ou r ana lys i s and


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repo r t . Hav ing read th rough ou r re su l t ing mode l s we found on eve ry

mode l tha t the max imum s t re s se s occu r red mos t ly on the penu l t ima te

pour . Look ing a t the mode l and i t s s t r e s s e s h igh l igh ted in va r ious

co lou rs we cou ld s ee tha t s t r e s s e s occu r red in ma in ly two pos i t ions .

The re was howeve r 3 pos i t ions bu t f rom consu l t ing a pub l ic a t ion on t hedynamic o f a rches tha t , the load a t the top o f the c rown makes each

e lemen t a long the a rch o f the b r idge p re s s on the one nex t to i t . Th i s

happens un t i l the push i s app l ied to the end suppor t s o r abu tmen ts ,

wh ich a re f ixed in the g round o r to ano the r suppor t2. F rom th i s we

dec ided to t ake re su l t s o f s t re s s e s f rom po in t s s t a r t ing , in the midd le

and a t the end o f the po r t ions o f h igh s t re s s reg ions . Th i s we fe l t gave

a more accu ra te range o f re su l t s to show the max imum fo rce s gene ra ted

in the g raph . The po in t s o f s e lec t ion wi l l be shown on the f i gu re be low.

F i g u r e 6 . 1 2 A s c r e e n s h o t i l l u s t r a t i n g w h e r e t h e r e s u l t s w i l l b e t a k e n i n r e d a n d t h e

e x c l u d e d z o n e i n p i n k .

As i l lu s t ra ted in the f igu re above the ma in reg ions o f s t re s s a c t ing

wi th in the a rch . The above d iag ram has been t runca ted a t i t s c en t re

a long the b read th o f the a rch (cu t t aken pa ra l l e l to the x ax i s ) . Th i s

then shows the s t re s s e s a t the cen t re o f the s t ruc tu re . When ob ta in ing

re su l t s o r de f le c t ion in the a rch u s ing the g loba l coo rd ina te s sys tem the

p rog ramme a l lows the u se r to ge t r e su l t s a c ros s a p lane o r a long a l ine

a t any po in t a long the s t ruc tu re u s ing x , y and z co -o rd ina te s . As s een

in the aga in in the d iag ram 6 .12 the po in t s whe re we p laced a l ine

ac ros s the th icknes s o f the a rch ( in pu rp le ) and a t t a ined re su l t s fo r tha t

s ec t ion o f the a rch , wh ich was p laced aga in a t i t s c en t re , whe re the

max s t re s se s wou ld occu r , and no t a t the edges o r the a rch . The reg ion

whe re we wou ld be exc lud ing fo r ana lys i s i s ou t l ined in p ink a s the se
http://www.amazon.com/Dynamics-Arches-Frames-Developments-engineering/dp/044499792X/ref=sr_1_11?s=books&ie=UTF8&qid=1299249690&sr=1-11http://www.amazon.com/Dynamics-Arches-Frames-Developments-engineering/dp/044499792X/ref=sr_1_11?s=books&ie=UTF8&qid=1299249690&sr=1-11


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fo rce s wi l l be t r ans fe r red to the b r idge abu tmen ts .

F i g u r e 6 . 1 3 A g r a p h c r e a t e d b y s t r a n d 7 a c r o s s a l i n e s e c t i o n o f t h e a r c h s h o w i n g t h es t r e s s e s c r e a t e d .

The p roduced g raph i s shown above o f a l ine s ec t ion o f s t re s s e s t aken

ac ros s a s ec t ion o f the a rch . S t re s se s on the ou t s ide and in s ide o f the

a rch a re shown e i the r s ide o f the g raph . The s e r i e s o f load ca se s a re

d i sp layed and a re d i s t ingu i shab le v ia the l egend . As can be s een load

case 8 ha s the c rea ted the max imum s t re s s on the ou t s ide face and i s

d i sp layed in the t ex t box to the top r igh t o f the g raph . The max imum

s t re s s on the in s ide face ha s been c rea ted by the fu l l load ca se and i s

s een by the co ra l r ed l ine in te r s ec t ion the l e f t y ax i s by the -1200KPamark .

6.7 Result s and graphed data

As above , s t a ted i s how whe re and why we c rea te the da ta and s e t o f

r e su l t s needed fo r the ca lcu la t ion and de te rmina t ion o f the

se rv iceab i l i ty l imi t s t a te check fo r c rack ing and whe the r o r no t i t i s

a t t a inab le fo r c e r ta in a rches o f spec i f i c d imens ions . F rom the c rea ted

re su l t s wh ich a re exp la ined above , to whe re we ge t those re su l t s . He re

I wi l l show the f ind ings o f one s e t o f r e su l t s fo r one spec i f i c c a se . The

case chosen he re i s the ba se l ine range o f pa rame te r s fo r a l l the fou r

mode ls . The en t i re s e t o f r e su l t s w i l l be pub l i shed in an append ix . Fo r


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the c la r i ty o f th i s r epo r t I w i l l show the s e t o f r e su l t s and exp la in how

they we re a t t a ined . As s a id shown he re wi l l be the s e t o f r e su l t s fo r the

fou r mode l s a l l s e t a t Base l ine va lue s . The ba se l ine va lue s a re the

de fau l t va lue s wh ich a re a l l s e t a t in te rmed ia te va lue s fo r e ach va ry ing

pa rame te r s d i s cus sed in chap te r 4 . The en t i re p roces s a s de sc r ibed f romth i s chap te r wi l l imp lemen ted f rom c rea t ing the mode l , a s s ign ing a l l

the cha rac te r i s t i c s , check ing the mode l tha t i t works co r rec t ly ,

app ly ing the load ca se s , runn ing the so lve r , ana lys ing the fo rce s and

graph ing the da ta . These a re the s t eps to be t aken to end up a t comple te

s tage whe re the mode l s c an be compared to each o the r an d then used to

a t t a in s e rv iceab i l i ty checks fo r c rack ing .

6.7 .1 Tables of created data .

Full Pours Full PoursInside Face

MomentsOutside Face

MomentsHeights

(m) Stress (Mpa)Moments

(kNm) Heights (m) Stress (Mpa)Moments

(kNm)

0.7400 0.5432 14485.3333 0.7800 0.6246 16656.0000

1.1800 0.7861 20962.1333 1.5800 0.9558 25488.0000

1.4800 0.8293 22114.6667 1.8800 1.0482 27952.0000

1.8460 0.7868 20981.3333 2.2460 1.0449 27864.0000

2.0300 0.5537 14765.3333 2.4300 0.8627 23005.3333

2.3600 0.6778 18074.6667 2.7600 0.4993 13314.66672.5500 1.0150 27066.6667 2.9500 0.3580 9546.6667

2.6000 1.2306 32816.0000 3.0000 0.6207 16552.0000

The above t ab le shows the range o f momen ts expe r ienced in the a rch on

bo th the ou t s ide and in s ide face . The he igh t co lumn re fe r s to the he igh t

a long up the a rch in the y ax i s the whe re the s t re s s i s t aken . The s t re s s

co lumn i s s e l f -exp lana to ry whe re tha t i s the s t re s s found a t tha t he igh t

up the a rch a t the cen t re o f the a rch . The momen ts co lumn i s then thes t re s s conve r ted by us ing fo rmula 6 .1 u s ing the s econd momen t o f a rea

to ge t the momen t a t tha t po in t . The above t ab le i s the t ab le fo r the

Base l ine va lue s fo r the Rod in mode l . F rom the se va lue s wh ich wi l l be

c rea ted 3 more t imes fo r e ach o the r mode l and thus the re wi l l be fou r

s imi la r t ab le s fo r e ach mode l a t the se s e t o f va lue s . Wi th t ab le s c rea ted

fo r a l l the fou r re su l t s we then , u s ing the g raph func t ion in Exce l

combine each mode l s s e t o f r e su l t s fo r momen ts wh ich wi l l be a s s igned

to the y ax i s . The he igh t wi l l be a s s igned to the x ax i s , bu t wi l l be the

same fo r a l l the mode l s and thus c rea t ing an equa l ba s i s o f ana lys i s to

compare each mode l to the nex t . The t ab le i s d iv ided in to two pa r t s the


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moments a t the inne r fa ce and th e momen ts a t the ou te r f a ce wh ich wi l l

p rov ide two d i f fe ren t g raphs . No t ice tha t the he igh t s inc rea se by the

th icknes s o f a rch and thus d i sp lay ing wh ich a re the ou te r f a ce

momen ts . The two g raphs p roduced f rom combin ing the fou r ranges o f

momen ts a s shown be low aga in fo r the Base l ine s e t o f va lue s fo r thefou r chosen mode l s . The re a re c e r ta in compar i sons tha t c an be t aken

f rom the se g raphs and thus l e ad ing to fu r the r ana lys i s .

F i g u r e 6 . 1 4 S h o w s t h e g r a p h o f t h e f o u r m o d e l s g r a p h e d w i t h M o m e n t s V H e i g h t s f o r

t h e i n s i d e f a c e m o m e n t s .


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F i g u r e 6 . 1 5 S h o w s t h e g r a p h o f t h e f o u r m o d e l s g r a p h e d w i t h M o m e n t s V H e i g h t s f o r

t h e o u t s i d e f a c e m o m e n t s .

Fro m lo o k in g a t b o th g rap h s i t can b e seen th a t Ro d in s an d Ad ams

mode ls fo r ho r izon ta l p re s su re s a re s imi la r in the re range o f va lue s fo rmomen ts . F rom look ing a t the i r app roaches fo r how p re s su re i s c rea ted

one wou ld expec t th i s they sugges t tha t the p re s su re enve lope i s c lo se

to be ing hyd ros ta t i c in i t s de s ign and then once the p re s su re max imum

is reached i t adhe re s to i t s own se t o f gu ide l ine s wh ich a re re la t ive ly

s imi la r . On the o the r c a se C i r i a and Ga rdne rs mode l s a re no t in keep ing

wi th th i s r ange o f va lue s . The concep t C i r i a ha s adop ted i s to ma in ta in

a cons tan t un i fo rm p re s su re a long the s t ruc tu ra l fo rmwork o r wha teve r

the l iqu id conc re te i s in con tac t w i th in th i s c a se the p reca s t a rch . Th i s

des ign p rocedure can be s een in f igu re 6 .16 whe re the p re s su re

enve lope i s d ic ta ted by a s t ra igh t l ine a t the p re s su re face . Ga rdne r

howeve r p roces se s a s imi la r app roach un t i l the he igh t o f conc re te

poured reaches the min imum immers ion dep th fo r an immers ion rod o r

v ib ra to r . Th i s he igh t i s s e t a cco rd ing to Ga rdne r a t 1 me t re . Af te r th i s

the p re s su re g raphs e s tab l i she s a new p re s su re va lue and inc rea se s

l i n ea r ly as th e h e ig h t o f co n cre te r i ses u n t i l , l i k e Ro d in s an d Ad ams

mode ls when i t r e aches the p re s su re max imum (P , max) i t no longe r

inc rea se s and becomes cons tan t .


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Befo re the fo rce s a re app l ied in the f in i t e e lemen ts p rog ramme mode l

we g raphed the gene ra ted fo rce s in exce l wh ich mi r ro red the re su l t s

a t t a ined in the i r r e spec t ive pub l ica t ions .

F i g u r e 6 . 1 6 T h e G r a p h e d p r e s s u r e s a c c o r d i n g t o t h e f o u r m o d e l s i n E x c e l ( B a s e l i n e

V a l u e s ) .

From s imp ly look ing a t how the mode l s compare to a s imp ly fu l l

hyd ros ta t i c app roach i t c an be s een tha t i t wou ld l e ad to exces s ive

vo lumes o f conc re te and re in fo rcemen t to c rea te a c rack ing o r bea r ing

capac i ty h igh enough to p reven t fa i lu re due to the se modes . Hav ing

gene ra ted a subs tan t i a l amoun t o f da ta pe r ta in ing to the fou r chosen

pre s su re mode l s , co r rec t ly inpu t ted the se range o f p re l imina ry re su l t s

in to work ing mode l s , we have in - tu rn c rea ted an accu ra te sys tem to

quan t i fy the ho r izon ta l loads impa r ted on to a p reca s t a rch due to the

pour ing o f l iqu id conc re te on to tha t s t ruc tu re . D iscus sed in 6 .7 and

6 .7 .1 a re the g raphed re su l t s and da ta fo r one range o f va lue s wi th s e tpa rame te r s wh ich we re chosen to be the ba se l ine va lue s . A fu l l s e t o f

r e su l t s a re pub l i shed in the append ix .

7. Conclusion

The ana lys i s o f ho r izon ta l load ings o f l iqu id conc r e te on conc re te skew

arches was ca r r i ed ou t u s ing the u se o f exce l sp read shee t s and the

f in i t e e lemen ts p rog ramme S t rand7 . The a ims o f th i s r epo r t we re

ach ieved th rough ana lys i s o f the p roduced s t re s se s and momen ts re su l t

da ta , c rea ted in the a rch when ho r izon ta l p re s su re s we re app l ied v ia


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va ry ing fo rmwork p re s su re mode l s . The gene ra t ion o f re su l t s gave a

wide range o f va lue s fo r va ry ing s iz e a rches and thus c rea ted a

ca ta logue o f g raphs o f the mode l s and the fo rce s c rea ted when inpu t ted

to the 3 -Dimens iona l mode l . These ob jec t ive s inc luded e s tab l i sh ing the

l imi t s and c rack ing capac i t i e s o f skew a rches due to the impa r ted loaddue to the l iqu id conc re te , the bene f i t s o f such a sys tem a s a means o f

s t ruc tu ra l a s s e s smen t , i s the ea se in iden t i fy ing s te e l r e in fo rcemen t

capac i t i e s , the impac t o f ove r s t re s s ing o r ove r load ing the p reca s t a rch

and l e ad ing to damag ing the conc re te s t ruc tu re a t s e rv iceab i l i ty l imi t

s t a te s such a s c rack ing , and the succes s o f c rea t ing such a sys tem to

a l low an end use r to iden t i fy the se l imi t s and de s igna te d imens ions to

such p reca s t a rch de s igns a t the de s ign s tage .

Asse s smen t o f the re la t ionsh ips be tween the mode l s and the i r r e su l t an t

da ta gave an ind ica t ion to the max imum c rack ing re s i s t ance o f an a rcho f a p rede te rmined s iz e and d imens ions . Us ing S t rand7 wh ich i s a

re la t ive ly u se r f r i end ly f in i t e e lemen ts p rog ramme a l lows fo r

app l ic a t ion o f mu l t ip le load ca se s a t one o r va ry ing t imes a l lowing the

rep l i c a t ion o f re a l cons t ruc t ion p rac t i c e s p roduc ing p rec i s e and usab le

re su l t s . The u se o f such a t e chn ica l and spec i f i c p rog ramme ou tpu t ted

ve ry manageab le and func t iona l da ta wh ich gave th i s r epo r t such

succes s fu l r e su l t s . F rom acqu i r ing the a l a rge s e t o f ana lysed mode l s i t

gave u s the oppor tun i ty to compi le a pape r to d i sp lay the sugges ted

ranges o f c apac i t i e s o f p reca s t skew a rches wi th fou r va ry ing p re s su re

mode ls compared and used in the deve lopmen t o f t he da ta .

9. Reference

1 Heneghan , K . 2010 F igu re 1 .1 Wes t Ga lway Renowned S tone Br idges .

[on l ine ] .Ava i l ib le a t : h t tp : / /ke i thheneghanpho to .pho toshe l t e r . Com

/ i mage / I00 00qKcvQ CC HNn0 . [acce s sed on 17 November 2010]

2 Hen rych , J . 1992 The Dynamic o f Arches and f rames (deve lopmen t o f

C iv i l Eng inee r ing ) . V i rg ina Tech , USA.
http://www.amazon.com/Josef-Henrych/e/B001KDXPSK/ref=sr_ntt_srch_lnk_11?qid=1299249690&sr=1-11http://www.amazon.com/Dynamics-Arches-Frames-Developments-engineering/dp/044499792X/ref=sr_1_11?s=books&ie=UTF8&qid=1299249690&sr=1-11http://www.amazon.com/Dynamics-Arches-Frames-Developments-engineering/dp/044499792X/ref=sr_1_11?s=books&ie=UTF8&qid=1299249690&sr=1-11http://www.amazon.com/Josef-Henrych/e/B001KDXPSK/ref=sr_ntt_srch_lnk_11?qid=1299249690&sr=1-11

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