TAAJISVERSJU... L0/10 DISTRIBUTION ON BRIDGE OEI:I<S -~•---•--=-••~c:-:::u:aaa,aa.llll.CUIIJ-a

by AVRIOO SPN.AATirl' and JESUS MAATINEZM

M orthotropic rectangular plate is e.na:iysed. The plate has been considarad simply supported in two opposite edges and general boundary conditions aloqg the rEIIIaindar edges. Matrix formulation, v::?ry conv~Sttient for prognsnrning in digital computer, is used through the text . This technique is applied to an actual bridge deck and the results are compered Ylith those obtained by means the G.lyo~ssonet-fbwe method.

M alterretive analysis using the folded plate s tructure theory is cons ider­ed. This method represents a more sophisticated analysis, but appsars to be also more approximate and suitable than the previous one in order to math9111!!. tically model actual bridges deck structures. Comparative numerical results are presented.

• Dr. Ingeniero de Caminos. Laboratorio Central

M Or . Ingeniero de Caminos. Ministerio de Obras PUblicas

·1 .- ORTI-OTROPIC PLATE THEORY

1. 1. Introduction.

The analysis of some types of the bridges deck structur es is usual ly divided i n two major steps: 1) Longitudinal stress analysis (beam anal ysis) 2) Transversal load distribution.

This paper i s mainl y concerned to the l ast step. Some conventional analysis replaces during this study the actual bridge deck by an orthotr opic plate. These ideas were first used , by Guyon, and successives improvements in the mathemati cal model were introducted by Massonet and Rowe among others. In -reference (1) a complete analysis· is presented and tables and charts for-­pratica1 applications are also included. Unfortunately, the method presen-­ted there, that will be call G.M .A. method, has several limitations pointed out in references (2) a~d (3).

In t his paper an U!1ified and c::onsistent theory of the orthotropic plate, -suitable for computer analysis is presented. The theory includes all the -­rfnges of variation of the torsional parameter ol. 1 and it is given with some detail i n ap 4,

1.2. Theory

The or thotropic plate governing equation i s

k1jw'iijj • Z(x1,x2) ·( i, j • 1,2)

and the stress-resultants are given by the following formulas :

m11 • - {k11

w, 11 + k1w,22·)

m12 • - d1w' 12

q1 • -{k11w'111 + {k1+d2)w,122)

r , • q1 + m12'2

r1

and r2

a r e the Kirchoff shears .

- 2 -

m21 '" - d2w' 12

q2 • -(k22w' 222 + (k2 + d.1)w, 122)

r 2 • q2 + m21' 1

Einstein ' s convention is used and the notation and sign conventions in fi­gures 1 and 2 . The constant kii can be obtained from the actual properties of the bridge deck by the for~ulas:

k 11 ,. Ei K22 = Ej

d "'Gi 1 0 (2)

k1- ...,1 k11

21<12 a: d1 + d2 + k1 + k2

were i and j are the unit flexural inertias and i and j are the unit torsional inertias along sections direction 1 and

02 resp~ctively .

It is convenient to remember that the equation ( 1). assuming free b. c. along the edges x

2 .. 0, x ... l , is auto-adjoint if and only if k

1 = k -

( Betti theorem) . 2 2

2

In order to obtain this result, the auto-adjoint property is equival~nt, according to E. A. Coddington and N. Levinson "Theory of Ordinary Di ffere!! tial Equations" Me Graw Hill. 1955 to the following conditicn:

~ ~-1(a)MT ~ ~ ~-1(b) ~T (3)

wher e in this case: a • b, b -= 1 2

~(x2) •

M .. 0 m 0

m' 0 - p 0

0 0 0 0 0 0

k an integer number

0 p2 0

' - p 0 - p 2 0

0 PO 0

- p 0

0 0

-p 0

N ..

0

0 0

2 m' "" n' .. k ~

2 k

- 3-

po

0

0

0

0 0 0 0 0 0 0 0 0 n 0 - p

0

n' 0 - p 0 0

.A I< .. k iT 11

~t is easy to see the equa.tion (3) leads to the more simple condition -

k1 - k2.

One of the more important difficulties on the applimhon of the orthotr~ pie plate theory to the bridge design is the traslation of mec~~nical pr2 perties of the actual deck into the parameters of the orthotropic plate -

(kij).

In order to fulfil the Betti conditions sometimes the following assumption is used:

k1 .. k • \] k 2 22

or in another occasions:

k1 - k2- 'lEh3/12

where h is the upper slab thickness and\) :i .. Poisson ratio.

1.3. Example.

A computer program has been written according to the orthotropic pl~te -theory developped in the Appendix -A. The following example have been analysed, correspqnding to the data (~igu re 4). -

11

• 50 ft . • 18 . 29 m. 12

~ 50 ft. a 15.24 m. 2 \11 -=\1

2 • 0.15 E"' 2300000 t/m

i • 2057.2 in4/in = O.fr33711468 m

4/m

j .. 422.0 " .. 0.00691534 1 " i - 189.0 11

- 0 . 003103?10 " 0

j .. 52.0 0

" .. 0 . 000855405 "

The orthotropic plate parameters corresponding to the equations (2 ) a nd the aditional condition

k 1 .. k .. "'k 2 22

- 4 -

Two loading conditions have been studied 1) Influence lines (Excentricity coefficientes). 2) Abnormal transportation loading.

1) Influence lines.

The acting "knife loading" has·· been aproxirnated by 1 or 5 Fourier serie {S.F.) terms. The results (vertical deflections wand longitudinal bending moments m

11) are represented in tables I and II.

The results A correspond to the G.M.A. method assumi~g the interpolation -formula for the excentricity coefficient:

K- .. KO + K, v;; where K

0 and K

1 are coefficient corresponding to extreme cases r:f..• 0 and -

·Cil• 1 and are given in charts (reference ( 1)).

The results 8 and C are obtained by applying the orthotropic plate theory given here and not further approximation have introduced, except the linear elasticity hypothesis. Results 8 correspond to 1. S. F. term and results G -

to 5 S.F. terms.

The excentricity coefficien~given in the tables have been obtained from­the following definition:

The excentricity coefficient of a result r , k , at a given section is the r

ratio between the result r obtained by means orthotropic plate theory and ~he result r assuming beam theory, ·corresponding to the same loading condi tion.

In the G.M.A. it is· assumed the following additional simplification:

k .. k m w

In some text books (5) and (6) it is recommended the practice: k .. 1.1 k if the G.M.R. theory is applied. m w

In the figures 6 and 7 the results from the tables I and II are represen­ted,

b) Abnormal transportation l oading.

The extraordinary vehicle represented in figure 8 has been considered.

-5-

The following results helve been obtained.

Section: x1

• 0.416 11 • 7.609 m.

x2

• 0.07512

• 1.143 m.

G. M. A. : m11

• 1.1 x 5. 742 P • 6.020 P mt/ml.

Orthotropic plate : m11 •

(The fa~tor 1.1 has been used in the G.M. A.)

Section: x1

• 0,5 11

• 9.145 m.

x2

• 0,5 12

• 7.620 m.

G.M.A. Orthotropic plate

: m22

• 1.148 P mt/ml . m

22 • 1.167 P "

- 5.974 p 11

The position of the vehicle correspond in this case to the figure 9.

1. 5. Comparison between G.M.R. and the orthotropic plate.

G.M.R. Method presents the following features:

a) The mathematical formulation is not suitable for programming .

b) In the free boundary conditions the Poisson ratio is assumed to be zero .

c) The validity of the empirical rules (only first S.F. term, factor 1.1 ) should be recoris1der8d.

d) Only results helve been produced (tables, charts and interpolation formulas ) for the range 0 < o(. ~1.

Contrarily the orthotropic plate theory needs in order to be applied a dig! tal computer, but not presents the above limitations.

From the numerical results of the example it is observed.

a) The influence lines for w and m 11

are not identical and they can not been obtained oner from another by a !)imple scale factor.

- 6-

b) The convergence of w is greater than the convergence of m11

•

c) In general, in the applications,. both method would lead to the same r_!!! sults.

2. FOLDED PLATE STRUCTURES

2.1. Introduction

The orthotropic plate theory represents a consistent mathematical model -but in its application to the actual bridge deck analysis after difficul­ties arises :

-First, about the translation to the geometric and elastic properties of­the deck into the orthotropir, values k . . Second, about the interpretation of the results obtained fro~ the orthot~opic plate theory, where not dis­tintion have been made among the ciiferent parts of the transversal section of the bridge, like webs, upper slab etc . The last difficulty is to obtai~ t he in- plane stress distribution, because the orthotropic plate theory CO!! sider only f l exural and torsional stress-resultants; that means, for exam­ple,is not possible to know the value of the shear:stress existing at the joint section between a ·web and a s l ab .

2 . 2 . Folded plate structures theory,

The folded plate structures approach avoid most of the above mentioned -difficulties , but increasing the computation cost . Prismatic folded plate structures theory has been very well stablished, in the elastic range . In the references (7) and (9) the more important features of the theory are shown .

Usually, for sake the analysis convenience 1some simplifications into this theory are introducted (long prismatic folded plates), for example refe­rences {10) and (4),

- 7 ....

2 . 3 . Example

The bridge deck shown in figure 4, has been analysed alternatively by means of the orthotropic plate (O . P.) and tha long prismatic folded plate (L . P. ­F. P.) theories.

The parameters used in the O. P. tt'!eory were:

11

.. 18. 19 m.

'V .. y .. 0.15 1 2 4

i .. 0 . 033?11468 m /m j - 0 . 000281250 11

i - 0 .003103?10 11

0

j - 0 . 000562500 0

11

12

• 15. 24 m.

E • 2300000 t/m2

and the 8etti condition was assumed : k1

.. k2

.. ~ 1 k22

The ea~ameters of the L. P. F.P. theory are ste)ghforward obtained,assuming -the idealised transversal section of the figure 10.

Similary to the example 1.3 . the same two loading conditions have been con­sidered 1) Influence lines 2) Abnormal transportation loading.

1) Influence lines.

In the tables V, VI, VII the results obtained from the two approach are -­shown. These c6mpared results have been deflections (w), longitudinal normal stress (S) and transversal bending moments (M) . In the table VIII are repre sented the values of the shear results (T) existing at the jolnt section of the upper slabs and the vertical webs. These results can be obtained only -from the L. P.F. P. theory.

In the figures 11 , 12, 13, and 14, the above results have been drawn.

- 8-

i2) Abnormal transports tion loading.

Assuming the vehicle position given in .the fig . 8 the following results are obtained:

Section x1

• 0.416 11

• ? . 609 m.

x2

• 0,075 12

• 0.762 m.

O.P. m11

• 4.471 mt/ml. and the .corresponding stress at the

flange level is: 2

S - 8.842? x 4.471 • 40 t/m 2

L.P.F.P. Directly form the analysis S a 53 t/m

Considering now the vehicles position of the figure 9, the results are:

Section x1

• 0.5 ~1 • 9,145 m.

x2

• 0.512

· ·?.620 m.

O.P. m22

• M • 0.280 mt/ml

L.P.F.P. M .. 0 •. 240 . .,

2.4. Comparative study between O. P. end L.P.F.P. methods.

From the results of the above example same discrepancies between. results -obtained from the two methods are observed. Pn explanation for this diffe­rence can be the relatively small span/width of the deck, particulary im­portant in the L.P.F.P. theory.

By another hand, the L.P.F,P. mathematical model describes more accurately the actual behavior of the bridge· deck than the O.P. model. In this last­theory only mean values of the stress and deflection are obtained as re-­sults and it is not distinghished between different parts of the transver­sal section. At end, the last disadvantage of the O.P. theory is its inab! lity to give any idea about in- plane s t ress distribution.

-9-

3 . Conclusions.

Although a more exhaustive study is needed in order to obta in a more defini tive conclusions, particulary in reference to the imporU!nC8 of the <..pan/ width ratio, number and distance be tween longi t udirol bE-:cl.ms , s l a b thl.ckness, type of transversal section etc, the following provisional titatonentc can -be presented:

1) The orthotropic plate theory appears to be a more cons istent muthr.ona ti­cal model that the previous one Guyon-Massonet-Rowe.

2)8oth methods, orthotropic plate and Q.Jyon-Massonet-Aowo , present in its -applications, the difficults to model correctly the most relevant pr·opcrties of an actual bridge deck structure.

3) The folded plate structure represents a more a dvance theory in ordt:r to model this type of bridge structu~es than the t wo a bove mentioned t heories . Particulary the in-plane stress are given from t his approach and ir'l'(-:gular types of transversal section can be handled wit hout additional computa tio­nal effort.

4) If the number of longitudinal b~ms is eleva tsd and thP dist~nce between them ~s small the orthotropic plate theory may represent a ccurately enough the actual bridge deck behaviour . In other case this statement can not be -valid.

- 10 -

l'AOL.E I

DEFLECTION EXCENTniCITY CO:::r-ICIENTS k SECnON x :: 0.5 11 VI 1

z 3. 45 2. 65 1. 92- 1. 26 0.71 0 .31 -0. 02 -0. 26 f0 .51 A J ~ l 0.125 3.61 2.81 2.00 1.28 0 .71 o. 28 -o.os -o·.·22 · to. 57 n 1 ~ j. ' · 59 2.79 1.99 1. 28 0.71 o . ~! -0.01 -<l .JZ ~-57 C I

2 .01 , 1.92 1.77 1. 43 1.02 o . 0. 30 . -0.02 . 30 A I ~ 0.250 2.17 2. 00 1.79 1.43 1.01 0 .62 .0. 27 -0. 05 I . 35 lJ ~ 2.18 1.99 1.78 1.42 1.01 o .62 o . 27 -o.os l:.0 . 3S_,JL.j la 1.02 1. 26 1. 44 1.48 1. 32 1.01 0 .66 0. 31 ki.04 A u o.:rJs 1.12 1.28 1. 43 1.47 1.28 0.96 o .58 o. 2~ .os n 0~ ~ I 1.12 1.28 1. 42 1.45 .1. 27 0.96 0 .62 0. 28 . .05 c

o . r; 0.71 1.02 1. 32! 1. 45 1. 32 1.02 0 .71 0. 37 A 0. $00 0 . 41 0.71 1.01 1. 28 1.40 1.28 1.01 0.71 0. 41 D

0. 41 0 .71 1.01 1. 27 1. 39 1.7.7 1.01 0 .71 0 . 41 c

TABLE li

BEI'OING MOMENT EXCENTRICITY Cet.TICIENTS km SECTION x 1 .. 0 . 5 1 1

- --·>!-----------------~-·~---------..------~-------r-""' 0,'~~- LOPDED ~CTION 0,-fJ l ~.:.:...::...t---r----r---.....,-~--r---,----• .....-_...,_"!' ... -""'1.,_ ..... -~--M---r"",_"""''-''-t" __ _,.,..• ,. 0. r-1 -!

>. ::>' o.125 o. 25o o.:r1s o.soo o. C>25 o.150 o.37s 1.ooo .- g ~

~--+---~r-~~+-----4-----4------+-----~-------~----4---~·~ 4.80 3.46 2. 01 1.02 0 . 37 -0. 04 -0. 30 . - 0 . 52 -0.69 ~ :j 5.118 3.69 z.z3 1.1s o. 42 -o.o5 -0. 36 -o.sa -o.781? ·l

--~~->5'-'-· • ..;;;.1<?_ 3. 57 2. 24 1.11 o. 11.2 -o.os -o . ~o -o. 52, - 0. 78 c ! 3. 45 2.65 1.92 1. 26 o. 11 o. 31 -o.oz -o. z6 -u~s~ A l

0

0

~ 3. 61 2.91 2. 07 1. 32 0.72 0 . 28 -O. t15 -o. 33 - U. SC) u I

---+-:hlol 2.80 2. ro 1. 31 0.73 0. 28 - o .os -q_.J3_t.:D·59 C _!

E o.12s m 0

~ 0. 250 2.13 2.03 1. 90 1. so 1. 05 o. 63 o. z.u -v.cq - o. 3S r. l hJ 0

lil

2. 01 1.92 1.77 1.4-3 1. 02 0. 66 0 . 30 1-0. 02 1-0.30 A I

---+-?.~··14 1.~J •• 8.L 1.4.-t 1. 0.1 .'".Q.._Q!'~ ~-r--·o . :')_j-o . m_ :-o. a§ .. r£--.1 l.Oa1.26 _l . f:-4 l. lj3 1.32 1.01 0 . 66 0 . 31 i - 0. 04 A I 1.05 1.27 1.49 1.59 I 1. 35 L OO 0 . 6~ 0. 25 1-0.10 H !

---+ l.CI? ! . 26 l:..·.!!-2 • ..,..._h:W ~_!. 29 . - . .!!~9.2__~!.!.('3 ,...Q...L·~--:~~J-.9_ ~!-.! 0 . 37 0. 71 1. 02 1. 32 I 1. t;5 1. 32 J.. D~- 0.71 () . 3i A I

?.:

8 0.375

o.soo 0 . 3,~ n.69 1.03 1. 35 I 1.53 1. 35 1.<>3 0. 69 0 . 34 n I .____~ _ _ _....._0..:..· =3.)J o. 69 __ j_) . ti2 1_.=~9-~1- __ 1.~9 , L02 o.6~) . o. ~5 c _l

TABLE V

VERTICAL DEFLECTIONS w (mm)

' -'""-" , .. __,_ 'f I · , .. )-". / .'.· · •• • (,' I",.·' .'· ;

{<~::··/ /. CONSIDERED SECTION t·1 / ·:·: ~ :: ........... , : 1.· t ,. . .·. I --~-------· --·-... ·-~---.... ··-----4---.··"- ------·--~-~-·-~-·-..... · .. .. ; !· ·~··>:>J ~ 1 2 6 10 14 18 22 26 30 34 38 42 f.:·>> l l i ' ' l I 1 11. 15 8 . 22 3. 60 1.05 0;01 -0.26 -0.23 -0. 13 -0. 06 -0.01 - 0. 01 0. 02 i ' l ; 2 8. 22 6.8o 3.81 1. 6o o.~,3 -0. 03 -0-.. 13 -0. 11 -o. o6 -o.oz -a.oo o. o1 i ! l 1 2-6 s.G5 5. 27 3-97 2. 17 0.89 0. 2_2 - 0. 03 - 0. 08 - 0.06 -0.04 -0. 01 -o. oo I r i \ 6 I 3. 60 3.81 3.91 2.76 1. 41 o·.s3 o. ll - o. o4 - o. o6 - o.os - o.o2 - o. 01 I w \ I ' 6 I 8 6 . , l""t 1 1 - 10 2.0 2.5 3.44 3. 25 2.00 0. 92 0. 30 0. 03 -0.05 -c.oo -0. 04 -0.03 5 1 i i 10 1.05 1.59 2.76· 3. 4.4 2. 62 1.41 0 ~ 56 o.13 -o.o3 -a.o6 -a.o(.l -a.o6 1 a. 1 1 7 10-14 I 0. 39 0.90 2.04 3. 17 3. 14 1.9S 0.93 0. 31 o. o3 -0.06 -o. o8 -0.~:9 j ~ f l 1 14. 0.01 0. 4.3 1.41 2.62 3. 67 2. 59 1 •• ~0 0. 57 0. 13 -0.04 -0. 11 -0. 13 : ~ i ) z ! 14.18 - 0. 19 0. 13 0.91 1.99 3. 12 3. 11 1.97 0.93 0. 30 0.01 - 0. 13 -0. 18 § \

I. g \ 18 - o. z6 - 0. 03 0. 53 1.41 2.59 3. 43 2. 58 1.40 o. s6 o.11 -o.13 - 0. 23 FE ~

! ~ l. 18 .22 -0. 26 -0. 11 0.27 0.93 1.97 3.io 3. 10 1.97 0.93 0 . 21 -0. 11 -0. 26 1 g; f 0 - . )

• a. : I ! 1 ! Q i 1 r 13.ti,O 9.50 3-59 0.78 -0.07 - 0. 16 - 0.10 - 0.11 - 0. 19 - 0. 34 -o.sl - O. oO I I I; o :: 2 9.50 7. 52 3.84 1.49 0.43 o.o8 -o.o2 -0. 07 -0.16 -0.29 - 0. 4.4 -0. 51 j ' -' . ~

f ~ 2 - 6 6. 15 5. 58 4. 06 2. 19 0.94 0.32 o. o? -e.o4 -0.13 -o.24 - o. 36 -o.43 i : ! 6 3.59 3.84 4.01 2.84 1.47 o. 6o o. us -o.oo -0.10 -0. 19 -0 . 29 -0. 34 w ! l 6-1o 1.ss 2 . ~$ 3.ss 3. 37 2. o6 o.9s o. 33 o.o6 -o.o6 -o.14 - o. 22 -o. z6 5 t i 10 0.7S 1. 49 2.84 3.56 2. 66 1.39 0. 56 0. 15 -0.01 -0.01 -0. 16 - 0. 19 c. ~ ll ! 10-14 0.20 0.84 2. 11 3. 26 3.16 1.94 0.88 0. 30 o.os -0.05 -0. 11 -0.14 @ fl

I • Cl \ 14 -0.07 0.43 1. 47 2. 66 3.37 2.52 1. 32 0. 53 0 .15 o. oo -0.07 -o.ll d .

! { 14.- 18 - 0. 16 .0.20 0. 97 1.99 3.09 3. 05 1. 88 0.87 0.31 0.07 -0.04 - O. o<) u. ! , I ___ 1s l -0.16 o. oa o.6o 1.39 2.sz 3. 27 2. 48 1. 32 o.56 o. 1s .-e.o2 - o.1o 'r

L_t_=:~~:_.! .. .2 .. 13 . -0.-02. :0:..135 0. 9.1 a.90 s.o2 3. 02 1.:90 0.91 0. 35 -0 . 02-~_J __ J

TABLE VI

L ONGITLOINAL NORMAL STRESSES S ( t/m2

)

! ·:-' . i CONSIDERED SECTION - .! <.· _.. l I . . . ...------- .. --.J . \ i .. . ·. ! f i ·. ) 1 :;- . 1 1 2 6 10 111. 18 22 26 . 30 34 3s 42 i ·

~

}

f ~ : i i ; I )

I i j s i ~

l !

' I I !

! ' I I

i ! I ' I I I

7. 0 1:-{ 1-'-' '"' '· . a c.

s .. a -

1 211 160 73 21 -0 -5 -.5 - 3 - 1 -0 -0 - 1 2 160 132 76 32 8 - 1 - 3 - 2 -1 -0 -0 - 1 ~

2- 6 113 103 77 44 18 4 - 1 - 2 - 1 - 1 - 1 - 1 i ~ 6 73 76 76 55 2.8 11 2 - 1 - 1 - 1 - 1 - 1 ! ~

6-10 42 52 67 63 4.1 18 6 0 - 1 - 1 - 1 - 1 l u ! 10 21 32 55 67 52 2<'3 11 3 -0 - 1 -2 - 2 j g; !

10- 14 8 18 41 61 61 40 19 6 0 - 1 - 2 - 3 l ~ i 14 0 9 29 52 65 51 28 11 3 - 1 - 3 - 4 l § !'

1 14-18 -4 3 18 /;0 00 60 40 19 0 0 - 3 - 5 ! § 1

! 18 1 -5 - 1 11 23 51 64 51 28 11 2 - 2 - 5 ! l ~ 13-22 I - 5 - 1 6 19 t;.O 60 6o 4.0 19 6 - 1 - 5 l J , I l 1 ' 127 121 84 t~7 22 8 o - 5 - 9 -15 -2~~ - :;o • l . ' ; 2 ,. 112 106 77 ~.s 26 12 4 - 2 - 7 - 13 - 20 - 26 ~ i 2- 6 96 91 71 50 31 16 7 1 - 5 - 10 - li - 22 ·~· 6. I 80 76 M 51 35 21 11 3 -2 -7 -13 - lG

! 6- 10 65 63 58 50 39 26 14 6 0 -5 - 10 -14

10 52 50 so 4S 42 30 19 10 3 - 2 - 7 - 10 I 1o-1t-. c.o 39 t.3 a~ 4-3 ~s u 14 6 -·t. _,: - 6 ! . ; . . . '"' . " : lt, ,. 30 30 35 11,1 4-3 3.S 29 18 10 J.!. - 1 - 3 I , . . -~ ? ., 2 1 zc "6 A 1 '1 3 A 2 3 1 ( 7 .... • , •. t:,.-; •...,. ~ --~ .u · 0 " '+ 'v L} ~ LS. ..) A

/ 18 : :.3 16 21 30 33 41 33 7.9 19 11 6 5 ', ,~-·~·, 1 1" 1 0 16 ? • ~· ~o t"~ -:' ?I. ,,; , " h' - - J... - ; - """ ... ..,,;. .....-I+ L;. ....... v ·r --: -v -""' _..,

w

5 L::..

P. (';

3 <.:...

! .__! ___ (_ -- -----~-··-------.. --·-··--··~-----------· -·~--·-···--

TABLE VII

TRANSVERSAL BENDING MOoi.C:NTS M ( mt/ml )

t<~~:/::: ~ .· !PLATE 1 PLATE 2 - ~-PLA~ 3 PLATE 4 PLATE 5 PLATE 6 PLATE 7 PLATE 8 PLATE 9 :.:.ei.JS.IE·. lo ""'rPLATE11r ·.,/ .... · - - -- - _...,...._ ------------·--r ........ .... , ... ~· ··· -:. , .. . . . · . I ! I • I I . : I ' . ! ... · ·, • . 1 " " . 6 6 j 10 10 1 14 14 ! 18 18 ?2 22 ! 26 26 : 30 30 ; "A 3A ; "0 "~ I!" .· !··· . . ":-' · •·. ' "'1 ''z I 1 2 ; 1 2 ! 1 2 r 1 2 l - 1 2 : 1 2 : 1 2 1 "'~1 'f-2 • -""1 1 ""2 ,_ ·

! ,. 1 ~- o . 0-0. 39j-o . 39~o. 6o -0.6~.42 -o.~o. 21 -o.21~ . oa 1 -e.os~o.01 -o. 01: o.o1 0.01: o. o1 o .o~ o. o1l c . ot o.co ' o . o~~ o. o1 J , 2 !v.o o. 26j~0 .26~ . 24 -o. z~~o. 3o -0 .3~.20 -0.2~.09 -o.o9~ .03 -o. o3,-o.o3 -o.o1' o. oo · o. oQ o.oo o.oQ o.co c.co o.ci f ! 2-6

1.o.o 0. 14.1 0 . 1~-0. 14 0.1~. 16 -0. 16-0. 18 -0. 18~. 11 -o.11Lo. os -o.os-o.o2. -o.o2!..o. c.o -o.oo o.uo o.od o.oo o. oo o.d

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lf" .O.BLE VIII

SiEAA- REsULTS T (t) SEGTION x1 "' 0 , 1 1

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Pppendix A

General Theory of the Orthotropic Pl~e~

Governing equation

Stress-resultants

m11 • - (k11w'11+k1w' 22)

m22 • - d1w'12

q1 • -(k11w'111+(k1+d2)~ '122) r

1 D ~1+fl112'2

Complementary solution.

(i , j .. 1 , 2) (I )

~22 • - (k22w' 22+k2w ' 11)

m21 .. - d2w, 12

q2 • -(k22w,~+(k2+d1)w,122)

r 2 • q2+m21'1

By definition the complementary solution is the solution of the homogerilous equation (1) ·i.e . ~

kij •c,iijj .. 0 (II)

Assuming simply support along x1

.. 0 af1d x1

.. 11

the complementary solu-­tion can be written as:

Substituyj--q into (n) and considering the· n-th term it is obtained

k11Jh w~ (x2) + 2k1~~ w~,22(x2) + k22w~~2222 (x2 ) • 0 (III)

or for convenience the subscript n can be dropped i n all the following -formulas and the equation (III) becomes:

(IV)

are The algebraic roots t (j a 1,2 , 3 ,4 ) of the characteristic equation (IV)

j

t - + ~ k,,±i[K j -

k22

(v)

The following posible three cases are considered .

- 11 -

Case I . ~<0 Torsionall y soYt and ~lexurally stiff orthotropic plates. (Typical case in bridge analysis) .

Case II. Ll:: 0 Isotropic plate .

Case III.l:\>0 Torsionally sti~f and flexurally soft orthotrapic plate.

Introducing the definition of the . flexural parameter e-~ The equation (v) becomes:

I t~i Casei. tj"";tA9e = ;,t(r;,t si)

k where ri• arc cos 12

k 11 k22

Case II. t .. + ,\e (multiple roots) j - .c:~.

Case III. t j .. ± Ae e--r.;± (_r :t s) ' k

where C(c arg Ch 12

k11 k22

i = v-:; r .. A9cos ~ s .. A9sin ~

2 . 2 ~

s • .J.esh ~ 2

'the complementary solution is obtained from the following formulas~

Ec (x2) .. § ( ~ .E (x2), £ f(l2- x2J) ~1234 where: Be; (x2) .. (w;w, 1 ;w,2 ;m11 ;m22;m12 ;m21 ;q1 ;q2;r 1 ;~ 2)t

Each term of ~ Jx2

) is a function o~ x2

and varies along the direction x1

as sin~x1 except w, 1 ;m 12 ;m21 ;~ 1 ;r 1 wich vary as cos~x 1 •

§ - 1 0 -o 0

A 0 0 - 0 0 1 0 0

k11

A2 0 -k~ 0

k A2 0 -k 0 2

- d A 22

0 0 0 12

0 -d A 0 0

k11~3 2 1

-ck,+d21 JA 0 0

tk2+d12

JA 0 0 -k22

k )? 0 - c2k,2-k2JA 0 11

(2k12-k1Jf 0 0 -k22

- 1 ? -

~1234 = A1

~ A3

A4

A. (1 = 1, 2 ,3,4) are arbitrary constants. l.

The matrices ~and .E (x2

) depend on the orthotropic plate case considered :

Case I .

8 =

Case II.

p =

Case III.

8 =

1 0

- r B

2 2 r - s -2rs 3 2 3 2

-r + 3rs - s +3sr

1

-r

2 r

3 -r

1

- r

2 2 r +s

3 2 - r - 3rs

0

s

- 2rs

~ 2 s :+3sr

and for all the cases C = 0 B

where Q =~~ij} (i , j = 1, 2 , 3 ,4)

Particule1r solution;

P(x ) - 2

and~ . . l.J

- rx = e 2

sin

[

cos

- sin cos

By definition , it is the solution of equation (I) not satisfiyir<g al.l the boundary conditions . It is assumed here the orthotropic plate simply SLIP-

portted at x1

= 0 end x1

= 11

and infinite width .

- 13 -

Two loading conditions are considered.

a) Knife load along x2

. ol2

i.e.

Z {x1

,x2

) -= Z{x2

) . ~ (x2-0(

2)

In this case it can be shown that the particular solution is: '

p (d ) 0 if x2 ~a2

A (x2) E £ - 2-x2 ~12 ..

-o 0

E ~ f (x2-~) ~34 if x2

) ()(2

where Be (x2

) is a similar matrix as ~ (x2

) .

f (x2

) nave been already defined The matrices E• ~. £,

~~2" ~ .[:~: j. H Cases I y III.

0 0

~1210 ~34 [1! l

.. 1r J. H Case II. a

n where H = fv -j( 4 k11 12

and a .. ~ • ; 011 I Z (x) sin l\x.dx n 1

1

in the n-th S.F. term of Z (x1).

The boundary conditions to be considered nre:

l x21~ oo

x2.....,Glz :!: 0

X _.,o{ + 0 2 2 -

w ~ 0

YJ,2 ~ 0 · a

w,222 • ...!2,_

2k11

b) Uniforms distributed load, i.e. Z(x1

,x2

) .. Z(x1

)

The particular solution is:

A (x ) = G 8 an -o 2 - -o k )\ 4

11 Final solution .

where 8 = 8 {assuming r=s=O in every element of B) -o

Is the solution of equation (I). plus the actual boundary conditions, and -its expresion is:

(VI}

- 14 -

The matri.x A of A (x ) is obtained from the boundary conditions, -along x

2=0 ;~§~2=12:Crwo

2cases of the boundary conditions are conside­red :

a) Homogeneous boundary conditions ,

Tre general matri~ form of this conditions is given for each edge i (i=1 , 2: i . e . x

2=0 or x2~12 respectively) .

(VII)

The diagonal matrix ~i end ~i

~1 + ~i = ~ (unit matrix)

satisfy the condition

The unknowns ~1234 are obtained from the following equation:

~1234 = - ~1 k r~.13J..... [~15 + -p 1

/)11 p1 .11

r~23J [~25l ~.... + k2 ~f21 'j) ~2 1!J

where: ~1i is the i -row matrix of (g g, ~£ ~ (12~ ~i" "" " "(gQ~{l2) , E_f]

~1i is the i-element of , ,

~ . " J ... 2i

~~ !.! " .. u "

depending on the loading case considered .

Then , the final solution A (x ) is obtained from (VI) . - 2

b) Edge beam.

The stiffnes matrix , in local axis, of the edge beam 1 (i=1 , 2 )

- 15 -

; e: • ..... .

where 0

(VIII)

EI i ' GJ i are the flexural and torsional stiffness of the beam.

Using the standard axis transformation technique (rotation and traslution shown i n figure 3) the equation (VIII) becomes:

= T,R.T. -:1.-:1.-:1.

t[-w' 2

plate w plate

-a. cos o(i + l. cos C{i

(IX)

is the transformat ion m~trix .

The . equation (IX) have the sal]le mathematical sb·ucture as (VII) Hhere :

~ . = ,;,_, and ~~ = -iT R. T. t and then, the sol ution steps will be similar to -,.l., G. ~ -:1.-:1.

the. pr.•evious case.

- 16 -

V

w

r. l.

E

k w

k m

S.F.

G.M.R.

L. P.F. P.

O.P.

NOTATION

Orthogonal cartesian coordinates

Vertical surface load on (x1

, x2

) per unit area

Derivative of v respect xi

V variable is a matrix

vertical displacement of the plate middle surface

Force acting on the face j in i direction . (bending or tor sor moment) (i,j • 1.2)

shear force actin~ on the face i

Kirchoff shear

Young' s modulus

Poisson ' s ratio, in i direction

Lengths of the plate (figur e 1) (span and width respectively)

Dirac delta dbtribution, pointc( and x2

variable

Distribution coeficlents of the G,M.R. method corresPO£ ding to orthotropic plate where d= 0 , 1,~ respectively

Defl ection excentricity coeficient

Bending moment excintricity coeficient

Fourier serie

Guyon-Massonet- Aowe

Long prisw~tic folded plate

Orthotropic plate

BIBLIOGRAPHY

(1) "Concrete Bridge Design". R.E. Rowe John Willey and Sons. · Inc. 1962

( 2) "Placa ort6tropa rectangular". A. Samartin Revista de Dbras PUblices ... lm'io 196?

~3) "Distribution of concentrated loads on orthotropic oridge decks". A.R. CUsens and R.P. Poma. The Structural Engineers. Sap. 1969

(4)" A Survey on Folded Plate Structures". A. Samartin and J. Martinez. I.A.s.s. Colloquium of Madrid~ Sept. Oct. 1969 .

(5) "Analysis of right bridge decks subjected to abnormal loadning" p.e. Morice and G. Little. Cement and Concrete Association. London 1956

(6J "Le Calcul des Grillages de Poutres et IA!lles Orthotropes". A. Barl§s et c. Massonet. Ed. Dunod.

(7) "Una apliceci6n de los metodos matriciales al calculo de puontes" A. Samartin. l.aboratorio Central de Ensayos de Materiales de ConsW."'U£ ci6n. Publicaci6n oR 197. 1.970.

(B) "Analysis of Orthotropic Folded Plates with Eccentric Stiffness". K.J. William a nd A.C. Scordelis. Report SESM 7D-2 University of Cali­fornia, Berkeley. Feb 1970.

(9) "Computer Program for prismatic Folded Plates with Plate and Beam Ele­ments". c. Meyer and A. C. Scordelis • . Report SESM 70-3 University of California, Berkeley. Feb. 1970.

(10) "Multiple Folded Plate Structures". K.H. Chu and S.G. Purjarkar. Jburn. of Struc. Div. PSCE nl! ST 2. April 1965.

·-« 0 ll. (1. :;)

~'l

>­.., o. ~

--E

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a f" E "--·· ::!

E

•N ><

VI ~ < w CD

w C)

0 w hJ :t: 1-

~ VI

>< < ..J '( u g ' ...

w 0:: ::> C)

ii:

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' .. _,; I'

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1 I I ··- I ---r I

a) TRANVERSAL SECTION

.... . '& .. 2es. __ .. _

I -

1-·' .:..2.!,.._ I - --·. ------~---I r-'·""~ -+-1 '\ / ___ ..j

-~ I • ( ~ 1->'·l''Lj .~... :~ ·-·- -·· ... / •.7<'!'.1. ..

=n 11 11 u ··- · ~- ..... ~.i:Z.---L- _fo.51.2 1 __ .4Ji72 , i ! l_

j_ --~ b) lONGITUDINA L SECTION

L. ,., ~L_ __ , I

FI GURE 4 .• DEFINI TION OF TftE BRIDGE DECK

0 -'!.

<3 \.I \!;: 7-S'-lj.) ·c ..... ~ ?! g \--• {\.

~ u ,.n bl 0

I ~cl bl 0: :;.:) 0 {;:

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