DYNAMIC EFFECTS OF ECCENTRIC LOADS IN COMPOSITE
CONCRETE AND STEEL BRIDGE DECKS MADE OF A PAIR OF
MAIN BEAMS*
F r a n c e s c o M a r t i n e z * * and G i o v a n n i M e n d i t t o * *
S O M M A R I O : Gli Autori estendono in campo dinamico la c) the pulsating loads are always in reciprocal contact
teoria statica della torsione troll uniforme delle sezioni misle in with the extrados surface of the bridge deck;
parete sottile. d) in dynamic range the deflection curve of the struc-
In particolare vieue ricercato 1o stato effettivo di soltecitazione ture has an affinity to that due to static loads, with affinity
in impalcati da ponte costitu#i da una coppia di travi principali ratio given by two temporal functions f ( t ) and g(t);
a sezione mista sotto l'azione di carichi pu/santi.
e) the damping effects of the bridge deck are not taken
into consideration;
S U M M A R Y : The Authors extend in dynamic range the static
theory of non-uniform torsion of thin-walled composite concrete g) the mass of the pulsating loads with respect to that
and steel sections. of the bridge deck are negligible;
ParticularO,, the actual range of stresses in bridge decks made g) the deflection curve of the bridge deck is such that
by a pair of main composite concrete and steel beams under pul- the stresses remain in an elastic stage;
rating loads is investigaled.
the study is carried out in a general way by the solution
of the system of Lagrange's equation.
1. Introduction.
Technical evolution of the past few years is going to-
wards, in the case of composite concrete and steel wide I , , ~ "i r¸ /~
span bridges, the reduction, with equal width of the
bridge deck, of the number of beams up to a limit of two.
The structural features of such arrangements has imposed,
with particular reference to the problems connected with " /: r(,~
the eccentricity of loads, a more careful analysis in which
has found its proper place the study of static range in
non-uniform torsion.
The static theory of non-uniform torsion of thin walled / 7
composite concrete and steel sections has been presented //
in a uniform manner by Heilig and Steinbach [5], [10].
Here the Authors extend the studies of the abovemen-
tinned in dynamic range in consideration of the great
Fig. 1
importance of research into the actual range of stresses
in such composite concrete and steel bridge decks under
pulsating loads.
3. L a g r a n g i a n function.
The Lagrangian function
2. Hypotheses of the problem.
~=T--V
In the hypotheses that:
implies the determination of potential energy (V) and
a) the section of the bridge deck is rigid transversally
kinetic energy (T) of the following system.
so that its deflection curve is known by the displacement
(vc) of the centroid and the rotation (tp);
3.1 Potential energy.
b) the distribution of the mass is uniform;
3.1.1. Bending rate.
The general expression of the potential bending energy is:
* The Authors state that they have contributed in equal
measure to the drafting of this paper.
** Istituto di Scienza e Tecnica delle Costruzioni, Poli-
tecnico, di Milano.
-~- T t
Lif, e, dV,
f
JUNE 1972 ll[
where (x) or, being
M~ IV* v~ ----vo + x, 9
e* = B x B u - - B*v "q C*
with vG refered to the centroid G and resulting as
Buy, + B.vx~ ~o*
with the index i relative to the concrete sections and the
O~va and 0 ~
rolled profile sections.(~)
0e 0e
terms proportional to the curvature (and therefore con-
T stant in the section)
.... • .... 7 - ' ........ --~
1 B, t. / 02vo \~e~
2 v,
being
Fig. 2 A* : 8"
*u + ~v2D __ 2 n
~-3'T~v ~ t r~**
- - Lj T~.v
The section of Fig. 2 is symmetrical with respect to the with
vertical axis, so that
a*~, = z, e, £,,x,'y,'aA,
B. v= r~ Et [ . xo'~ dA = O (s)
D= Z~N, fa dA,
furthermore: t
IV* = C* 0!,~
L'
OZ= f
W'g= Vo+ x~r = f bds-- x, ~ 3.1.2. Torsional rate.
C----- Z, E~ fA v?{ dA~
1 x,
-g- fv,,,y,dV, l x, ~ l
= -T" f *q ~ dVt
g
f
and therefore: where:
O~v~ Oz9
e~ = - - OZ~ y, + (~o + xo,r ) OZ~
O* f * OSv~ , o . 0 3 9 1
with which: + + xo' )
z
I : - - E' I-- s' - W - °' WJ
being
2 v, v,
=± z, e , f L L '/ 02Of \ 2 2 st = 3'fdsf ;
2
( o2~ ,~ +
+ (~o + xo,r) ~ \ OZ~ / St = - - f, xo'~ ds~ + ~o d,~ + yT xt ds~ ;
f , f
a~v~ o~o
2 &---r- o~ (~ + xo'~)y* 1 d A d z
1 1 E~ ( * Osva
--21 X, ~ f v, r~gV, = -~- X,--~-fv, -- s, ---~-~Z3+
(q See [5].
(~) The effect of the reinforcement of the bridge deck is
here considered negligible. In the case that there is also present + S* Oz31 d V = =
prestressed reinforcement, and that the reinforcement is not
negligible, it is sufficient to extend the sum to them also.
= 21 [A,,f0 L/{,~]~'vG"d.z ~-' B** "OrL(.~_Z8)0a~0
'., +
(a) Being f x,y~ dA~ = 0 because x and y are principal
axes of inertia.
_2c**f~o o3vo
OZ~
o,9
OZ3
dz]
112 MECGANICA
being : 3.2. Calculation of the kinetic energy (T).
Following the hypothesis already formulated, that it is
~2 f A s~*2 dA~
A** = Z,-----~ possible to neglect the mass of the pulsating load with
respect to that of the whole system and remembering
E~ "2 the general expression of kinetic energy
B** = El ~ fa S~ dA~
1 . 1
T= T f #v2 d V + T (AP2 + Bq2 + Cr2)
E~ * *
C** = Xi ~ L t s~ St dA~ where p, q, r are the conponents of the angular velocity
and A , B and C the principal .moments of inertia, one
obtains with /~, the density of the material forming the
3.1.3. Potential energy of external loads. structure
a) Rate of dead load. din, = #tdVt
Let q be the dead load of the structure for unit of length,
and v~(z , t) the component a b o u t y of the displacement 1 f [Ov,(z, t)]2
T = - - f - Z~ Vf #l Ot dV~ +
of the centroid G. The potential energy is:
1 t) 2 2
+ ~ [ o~(~,ot ]z,f~, m(xi +fi) dV,
-- e J: v~(z, t)d~.
or, with reference to the centroid
b) Rate of live loads (Fig. 3)
l fv#,[°vo&'t) x, O~(Z"t) ]2 d V +
T=--~- x~ Ot - - + Pt J
+ + D* [09(z' 012
L---3T-J
P
r
I
" i -¢.
",'Z2/s.
$o
Fig. 4
Fig. 3 with
D* = E, f tz,(x~ + y~) dV, .
Vi
Potential energy is (4)
- P [va(z, t) + e ~(~, t)] Let f(t) and g(t) be two coefficients of proportionality,
functions of time but not of place. Holding therefore that
in the case of a single load; the deflection curve in the dynamic range has an affinity
to that obtained by effect o f the same loads in a static
range, it can be said of fixed loads,
- - f : P(~O [vG(~, t) + e ~(Z, t)l&
re(z, t) = f ( t ) v a ( z )
in the case of a load distributed along Z. ~(z, t) =g(t)~(~9
it will be, remembering the results achieved in paragraph 3,
(~) It is to be pointed out that, strictly speaking, the rotation
takes place around the shear centre T so that the rotational
contribution of the work with reference to a general point is T = - y1 x, f v m [ vG(~O--;i-
df(t)+x,9(Z)_~t]2dV,+
(Fig. 4)" i
Ps~ = P(s cos a) =
= P(~0dT) cos a ----P e ~0. dt J ;
JUNE 1972 113
a ~[ a2v~(z) ]2dz + is obtained:
v = - ~ - B. f£ f(t) &,
a d2f(t) , d~g(t)
1 L r ... d29(Z) 12 n~-t- aa2----~F + axsf(t) + a a 4 g ( t ) + a l s = O
+ -T A* f£ [~t,J----~--z~] dZ+
1 A** L d~f(t) d2g(t)
&+ a~,d--Tv-+a~---iV-+.~ag(t)+ a24f ( t ) = 0
&+
5. The solution of t h e s y s t e m of differential
-- C**f~f(t)g(t) d3va(z) equations.
-daqo(Z)
-dz+
The system of two differential equations, linear and
--qf(t) f~o~(~0 & - ao
f~p(z) [f(t)va(Z) + e g(t) 9(Z)] dz. having constant coefficients, is reproposed in the form
4. Differential equation of motion. a21 amJ
r;y l= [ - - a . 4 f ( t ) - - a 2 s g ( l )
l g (t) ]
Lagrange's equations (in the so-called second form)
from which, being
d 00o~ OoLP (s)
it ook Oq~ (q. = f , g) A = [a~, al2]
ta,,l a~2J
noting that the potential energy V" is independent from 1
a, = W [a,~o~-- ~ , ~ ]
3~ and g it becomes:
1
h = - 2 - [a,~a~-- o,~a~]
d OT OT OV
4' o/ -~f +-Wf=° cl = - -
1
A a15a22
d OT OT OV
1
W+w=o a2 = -2- [a,,o~, - - o,,o..,]
1
b.,. = - ~ - [a2,a,4 - - aria23]
from which, posing
1
C'2 = " - ~ alsa21
au = Zt f I~lv~ d V l
vf
it results that:
al o. = 5"1 f ! IAIXIVG9 dV,
rf
l f " ( t ) = aaf(t) + b~g(t) + cl
al, = B , Ii \ ~ ] dz + A** fl ~ , ~ 1 g"(t) = ao.f(t) + b2g(t) + co..
aia = - - C** f ~ dav° d*9 d z
The above can be reduced to a differential system of
four equations of the first order. Giving in fact:
a~s = - - q j'o~ [ , , ~ - / , ( ~ g M &
I f'(t) = a = A ( t )
a21 = Z, f t*tcpva d V i
v1 i f ( t ) = a ' = al f ( t ) + big(t) + cx = B(t, f , g)
2 2 ,'(t) = 3 = c(t)
0=2 = x, fv, I~lxtq~ d V i + D*rp2(Z)
g (t) = fl' = a2f(t) + b2g(t) + c, = D ( t , f , g )
L / d29,2 z ( dsg_]2
and changing the name of the variables in the following
way:
-- f; P(Z) " &
" dava damp f(t) =yi A(t) = F~
as, = - C** dz g(t) = y ~ C(t) = F~
a =ya B(t,f,g) = F~(t,f,g)
(5) Dot over letters denotes derivative with respect to time. fl =y4 D(t,f,g) = F,(t,f,g)
114 MECCANICA
it results that 4
yt = y , + Zj e j j , (i = 1, 2, 3, 4)
l
y'== F,(t) where )'~t in respect of
y's = Fs(t, y l , 3'~)
.yll ,,. ynl [
,)"4 = F4(t,yt ,A'~) v(t) . . . . . . . . . . ]# o
lyln • • • ynn
or, in a general form
form a fundamental system of the associated homogeneous
j , q = F,(t, y l , y=) (i = 1, 2, 3,4) differential system. Furthermore
_ t 4
A fundamental system of solution of the associated f Y'j Mij(t, ~)O(~) d~ (i = 1, 2, 3, 4)
,.yl ~ t0 1
homogeneous system, can be obtained by the characteristic
equation
where (10 is a point which has been arbitrarily chosen in
the range in which t is defined) and the 4 ~ functions
I al
--2 01 0
bx - - 2
=0 y**(~) ... Y,,,(¢)
a~ b2 0 -- U?..
1
which leads to v(t) yl,,(t) ... A,,,(t)t.,_ . (i, j = 1 .... 4
yi,j+t(t) ... An,J+1( )
24--(al+b~) 2~+(aib2--a2bi)= 0
[ yln(1) ,...ynn ([)
Indicated then with ej arbitrary constants, the solution
are the resolutory kernels.
of the n o n h o m o g e n e o u s system can be expressed in the
form Received 19 June 1972.
LIST OF SYMBOLS By = ZfEt L x t£d A ~ ;
I
= f&ds ; Bxy = ZI E~ fat x~y, dAI ;
= - - f , [(x - - xa~) sin a + ( y --.),~,) cos a] ds ;
Swv = Z~ E( fa~ t#~.x'l dAc ;
C* = Y'¢ E( f 'ql:2 d A t ; 02P(
aAf M~ = Bx-A--z-. : bending-moment about axis x;
oZ ~
IV* L bridge span;
G centroid of the complete section: steel and homogenized
concrete;
B~ = Z~Et
L (
ytdAf; T shear center of the complete section: steel and homogenized
concrete.
JUNE 1972 115
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l ]6 MECCANICA
