Lateral Stability of Precast Members During Handling and Placing

Lateral Stability of Precast Members

During Handling and Placing

By Jean Muller* 1.-GENERAL:

The problem of lateral stability of long span precast prestressed members is often of prime impor-tance since such members are usu-ally slender in order to keep their weight to a minimum. The most critical conditions are normally en-countered during lifting and plac-ing and/ or before individual gird-ers are connected together by the deck slab or diaphragms.

The following is based upon the results of a very comprehensive study ( 1) undertaken by Pierre Lebelle who made a definitive step forward into the study of such prob-lems.

Notations used are as follows: lx longitudinal moment of inertia lg transverse moment of inertia of

cross section

J transverse moment of inertia of top or bottom flange

Kt torsion modulus

E and G modulii of elasticity

E

withG=----2(1

+

JL) (JL: Poisson's ratio)

A = Elx principal bending rigidity B

=

Ely transverse bending

rigid-ity

C

=

GKt torsional rigidity

D

=

EJ transverse bending rigid-ity of top or bottom flange 2a = L span length

"'Assistant Chief Engineer Enterprises Campenon Bernard Paris, France

20

z

p

q

moment arm of internal stresses (usually called j.d.)

vertical uniform load horizontal (lateral) uni-form load.

Definition of Lateral Buckling: When a slender beam is subjected to an increasing vertical load ( uni-form or concentrated) there is a critical value of this load (which may be much lower than that pro-ducing flexural or shear collapse) for which rapidly increasing lateral deflections and corresponding rota-tions take place suddenly near the center of the beam: this is lateral buckling, and can result in collapse of the member.

Such a phenomenon may occur even where no lateral loads or di-rect forces are applied.

2.-LATERAL BUCKLING OF A SLENDER SYMMETRICAL BEAM UNDER UNIFORM VERTICAL LOAD:

The load is assumed to be ap-plied at the center of gravity. The critical load Per producing lateral buckling may be written as follows:

myBC P e r =

-m = variable coefficient dependent principally upon the end sec-tion condisec-tions with regard to:

(a) : vertical bending (b) : lateral bending (c) : torsion

Results for several typical conditions are on page 21:

End section conditions _{Value of} for torsion for vertical bending for lateral bending m

built-in simply supported hinged 28.3

built-in cantilever hinged 12.8

built-in built-in both ends hinged 98

built-in at one end

built-in and simply supported hinged 54

at the other end

built-in simply supported built-in

50

built-in built-in built-in 137

3.-INFLUENCE OF THE SHAPE OF THE CROSS SECTION AND POINT OF APPLICATION OF LOAD:

The preceding results are valid for a rectangular section with small transverse dimensions in proportion to the span length; and the load is calculated as applied at the level of the centroid.

(a)-Case of symmetrical I beams:

Top and bottom Ranges introduce additional restraint towards lateral deflections. The following dimen-sionless factor is used:

D z2 _EJ

z2

{ 3 = 2 . . = 2 .

-c

U GKt U For wide Range beams, the mo-ment arm z represents the distance between centers of top and bottom Ranges.

The increase of buckling load k2

due to bending rigidity of Ranges has the following value:

beam hinged for lateral bending k2

=

j

1

+ :

2 . f3

beam built-in for lateral bending k2

=

v

1

+

1T2 f3

The above coefficient of increase is usually small.

(b)-Case of unsymmetrical I beams

(most common case of pre-stressed concrete girders). Let

J

1 and

J

2 be the transverse

February, 1962

moments of inertia of the top and bottom Ranges respectively. The equivalent transverse moment of in-ertia of the cross section is given by

2 1 1

= +

-J

}1 ]2

Above formula may be used with the corresponding value of

J.

(c)-Influence of the point of appli-cation of load:

If the load is not applied at the centroid of the section, but rather at a distance "d" from it, a rotation of the cross section results in an additional transverse load. Buckling load is thereby affected. The cor-recting factor is as follows :

k1

=

1-0.72 8

where 8 is a dimensionless factor

2d

;n

equal to: 8

= -

-L'/C

k1 is smaller than unity if the load is

applied above the center of gravity. For an unsymmetrical cross sec-tion, d represents the distance of the point of application of the load to the center of torsion and not to the centroid.

(d)-Approximate formula for usual problems:

For the usual dimensions of a simple supported prestressed

crete girder, the correcting factors relating to bending of the flange and location of load may be taken as follows:

kl. k2

=

1

+

1.25 f3 - 0.72 8

D Z2 2d

JB

where f3

=

2 - - and 8

= -

-C U L C

4.-BEAMS WITH ELASTIC RE-STRAINT AT THE ENDS: End conditions of a beam may be such as to provide for an elastic re-straint with regard to torsion. Let Rx be the spring constant of such restraint. For a rotation of of the end section, the induced tor-sional moment is then equal to Mx = -Rx. of

Assume moreover the following conditions:

-Negligible effect of bending of the flanges.

-Uniform load applied at center of gravity.

-Beam hinged at both ends with regard to vertical and lateral bend-ing.

The critical buckling load may be

yBC

still expressed as: Per= m -La Values of m versus the spring factor Rx are plotted on fig. 1. It may be seen that for large values of R, (beam fixed at the ends with re-gard to torsion) K approaches the value of 28.3 previously mentioned. 5.-RECTANGULAR BEAM

SUS-PENDED AT BOTH ENDS: Assume a beam carrying a load p (its own unit weight for example) suspended at both ends at points located at a distance e above the center of gravity (see fig. 2a). A small deviation of from the vertical position, induces a torsional

mo-22

pL

ment equal to - - . e of in the beam. 2

Thus, the spring constant has the pLe

following value: Rx

= - - .

2

Usually, a beam is lifted by cables as shown in fig. 2b and because of the elastic elongation of cables, the instantaneous center of rotation is not located at the height of the pick up point, but rather at a distance e

H

given by: e

=

d

+ - -

with l+K V COS E

K =

-2 Es As sin3 _{a sin}2 E

where V reaction at the end section pL

2

Es modulus of elasticity of cables

As area of cables

a and e angles shown on fig-ure 2b.

d distance between centroid and points where lifting cables are fastened to the girder

Knowing the e value, Rx is easily computed and the critical load may be obtained from the chart given in the preceding paragraph.

6.-RE CT ANGULAR BEAM LIFTED AT INTERMEDIATE POINTS:

When a beam is lifted at points

L

located at a distance p. - on either 2

side of the centerline section, the critical load depends upon the value of p. Derivation of the equations has been made by Lebelle and corres-ponding charts are given in his origi-nal paper. The presence of the can-tilevered section causes the critical

100

50

00 c c '-It) a. '-"C c a.

....

0 c

.2

W) c 0 u

10

5

4

3

2,5

2

1,5

1,0

0,9

0,8

0,7

0,6

0,5

0,4

0,3

m 0,

c

2

·c::

a.

cl)

0,1

0

0

I I I

I

I

I

I

I I :/ j I

7

I

I

I

I

I

I

II

I

I

I

pcr:m

VBC

_L~

v

)

/

___.-/

v

_{Valuers. oP m}

5

10

15

20

25

28,3

Fig. !-Buckling Load of Girder with Elastic End Restraint.

pL 2 -E.

(a)

Girder Elevation Fig. 2 - - - -H Side view

(b)

load to increase very rapidly. As an

example, fig. 3 gives the variation of m versus the ratio p between dis-tance of pick up points and total length of the beam, with the assump-tion that the beam is fixed-in with regard to torsion at A and B.

Thus, the most efficient way to

in-crease the safety factor of a given beam against lateral buckling is to

reduce the distance between pick up points. The optimum location is obtained for p = 0.50 (pick up points

L 3

located at - and - L). Such an

4 4

arrangement may seldom be used for a prestressed concrete girder be-cause of the internal stresses in-duced.

Such as:

-at pick up points where exces-sive tensile stresses develop at top fiber.

-at the center line where exces-sive compression stresses develop at bottom fiber and tensile stresses at top fiber.

Final location of pick up points should be chosen after a careful check of actual stresses at all critical sections.

7.-E F FE C T 0 F LATER A L LOADS ON SLENDER BEAMS. INTERRELATION WITH LAT-ERAL BUCKLING.

Assume a beam subjected to a vertical uniform load p per unit length (its own weight for example) and a lateral uniform load W (wind load for example). The beam is also assumed to be hinged at both ends with regard to vertical and lateral

bending.

In the case where p is negligible the maximum lateral moment would

wu

be equal to - - . Because of the

8

tendancy for lateral buckling, the

February, 1962

effect of a vertical load p is to

in-crease the lateral moment above the previous value while torsional mo-ments take place along the beam.

(a)-Beam fixed-in with regard to torsion:

The actual moment may be ex-pressed as

wu

My=fLl·--8

where fL1 is a factor greater than unity depending upon the magnitude of the vertical load.

Variation of fL1 versus the ratio

of actual vertical load p and critical load Per is given on fig. 4 for a beam of rectangular cross section fixed-in at both ends with regard to torsion.

In conjunction with this moment torsional moments take place along the beam. The maximum value is obtained at the end section and has the following value:

2C W

Mt= T l . .

-L p

p

Variation of r1 versus is also

Per given on fig. 4.

(b)-Beam with partial restraint at end sections with regard to tor-sion:

The lateral moment depends upon

p

the ratio - - and the spring con-Per

stant Rx at the end sections. The maximum lateral moment un-der a uniform load W is given by the equation:

wu

/-tl M1

= .

-8 /Ll and r1 are in fig. 4. 2C 1 - - - . T l LRx

given by the chart 25

300

200

100

E

(&... 0

•

tt ::>

->

28,3

t--1,0

26

'

A

B

J

I

I

J

l

~

PL

l

l

J

L

_"'

I

"f

,

YBc

pcr:m

7

L~

I

I

I

J

I

v

/

----Va.\u«E»

of

P

0,8

0,6

Fig. 3-Buckling Load Versus Location of Pick-Up Points.

~.0

3,0

c 0 u (II

...

_.,

"'0 c.. 0

-

c

_2,0

(II

E

0

E

0 c..

.,

c..

-0 0 .-u

....

0 0

...

•

c II)

1,0

0 0 "ii (II c.. c.. 0 u ... c

~~

=i:.

0,5

I

!J

J

VJ

lJY

v

/

---

'I'

7

7

/

p

-

-per

0,2

o.~

0,6

qa

1,0

Fig. 4-Lateral Stability of Girders. Influence of Vertical Load on Transverse Bending.

Rectangular Beam Subjected to an Uniform Load and A Concentrated Load:

It has been previously described how safety against lateral buckling can be greatly increased by reduc-ing the distance between pick up points. The same result may be achieved by lifting the beam at the ends but applying an upward con-centrated load at the middle. If a crane is used for handling the mem-ber, a third cable creating an ad-justable load may be used for this purpose.

The chart of fig. 5 gives the varia-tion of m versus the magnitude of the center load Q. Care must be taken that no excessive tensile or compressive stresses appear at criti-cal sections. This condition usually gives an upper limit for the applied load Q.

8.-BEAM SUBJECTED TO VER-TICAL LOAD AND DIRECT

FORCE.-Collapse of the beam in such a case may be due to:

-Lateral buckling under vertical load (or vertical bending) ;

-Direct buckling under the nor-mal force;

-Torsional buckling under the normal force.

Notations:

28

N actual normal force (a com-pressive force is positive) NE Euler buckling load

M.,

r2B

L2

Torsional buckling load S . C ( S cross sectional area )

= - - (

lv

=

lx

+

ly polar )

lv ( moment of inertia) Critical moment with regard to lateral buckling for a beam subjected to a circular vertical bending and no direct force

1ry!BC L

Mer Actual critical moment with regard to lateral buckling for a beam subjected to vertical bending and normal force. Interaction formula between the above factors is:

(

Mer) 2 N N

- = ( 1 - - ) ( 1 - - )

Mo NE NT

This formula is valid only when a beam is subjected to a constant moment throughout the length.

However, the same formula may be used for a beam subjected to a uniform load when N is small com-pared to the buckling forces NE and NT· Such is the usual case where handling makes use of inclined cables inducing a direct force in the beam of moderate magnitude.

Typical Example:

An investigation of a precast gird-er for the Tancarville (Dec. 1960

PC! Journal) bridge with regard to

lateral stability is presented here.

(a)-Girder properties:

Section properties at mid span are as follows:

Location of centroid: 50 in. from the top fiber

Transverse moments of inertia: -Ton flange: J1

=

17,100 in4

-Bottom flange:

J

2

=

89,000 in4

-Equivalent moment of flanges 2

J

= - - - =

28,700 in4 1 1

+

-h

Jz

-Overall section: ly = 108,000 in4 Torsional modulus: Kt

=

31,200 in4

Moment arm of internal stresses: z

=

104in. Location of center of torsion: 23 in. from top fiber

150

E

~ 0

"'

~

lrrr

r

1

_{~r ~h}

J

I

l_

l

L

l

1 :::J

1

1

pc.r

:.m

ve:c

~

100 L~

1

v

I

L

50

L't"

L

/

~

v

...-_g_

~\.. 0,1 0,2 0,3 0,-4

o,s

0,6

Fig. 5-Buckling of Beam with Uniform and Concentrated Loads.

Average girder weight is 1620 lbs per ft.

Design span length L = 160 ft. (b)-Safety factor of a girder placed

on top of piers:

The girder is assumed to rest on top of the piers without intermedi-ate connection with adjacent gird-ers. Adequate bracing near the ends may be considered as preventing any rotation of the end sections. Be-cause such a loading stage may last for several days, the modulus of elasticity may be as low as:

February, 1962

E = 3 x 106 _{psi and}

G

=

0.4 E

=

1.2 x 106 _psi.

Critical buckling load is:

yBC

per = 28.3 k1 k2 -L3

[from section 3( d)-Afproxi-mate formula for usua prob-lems]

Dimensionless factors f3 and 8 are computed as follows:

D Z2 _{EJ Z}2 { 3 = 2 - . - = 2 x - - - - =

C U GK U

28,700 104 ) 2 2x x---~ 0.4 X 31,200 ( 160 X 12) = 0.013 2d

(B

2d

I

Ely

a=L~./c=L~--cK=

2 23 - 50

J

~~~-108,000 = -0.082 160 X 12 0.4 X 31,200

The correcting factor k1 k2 is thus

equal to: k1 k2 = 1

+

1.25 X 0.013 -0.72 ( -0.082) = 1.08 1.08 X 28.3 X 3 X 106• and Per= X 1603 _{X 1,728}

V

108,000 X 0.4 X 31,200 = 465 lb per in = 5580 lb per ft. 5580 Safety factor:--= 3.4 1.620

(c) -Stability during lifting opera-tions:

Because of the short duration of such an operation the modulus of elasticity may be taken equal to E = 4 x 10H psi.

The beam is lifted with a dis-tance between pick up points equal to L' = 165 ft. Owing to the lift-ing devices used, the beam acts as though it would be free to rotate at the ends around an axis located at a distance e = 10 ft. above the centroid. The spring constant of the pL. e

end restraint is equal to Rx = --~

2 (section 5) and depends upon the magnitude of the load p. (multiply

L

both sides by - ) and get 2C

Dimensionless factor: L Rx pL2. e

2C 4C

Let k be the safety factor against buckling. 30 Thus Per= (k X 1.62) k/ft LRx and - - = 0.32 k. 2C

On the other hand:

(1) k1k2V BC Per= m Per andk =-~ 1.62 With numeral values used above:

k=0.153m (2)

Eliminating k between ( 1) and ( 2), we obtain

LRx

--=0.049m. 2C

Another relationship between LRx

-·· --- and m is given by the chart 2C

of fig. 1. The intersecting point of both curves gives m = 15.5. Safety factor k

=

0.153 x 15.5 = 2.4

The tendancy for lateral buckling could also be investigated by calcu-lating the lowest value of the modu-lus of elasticity inducing buckling for a given load.

The load Per is then assumed to get its actual value of p = 1.62 k/ft~

LRx

thus - - = 0.43. From fig. 1 2C

we obtain m = 13.

Entering the equation Per = m k1 k2y'BC

- - - with m = 13 and Per = L3

1.62, we get E = 2 x 106 _psi.

The safety factor calculated in this way would be only:

3x 106

k = - - = 1.5 2x 106

(d)-Stability of the girder for han-dling with a crane:

Assume that lifting of the same girder be undertaken with a crane according to the procedure outlined on fig. 2 with the following

- - - -

~---~--ical values:

a = 45° H = 80ft d = 4.2 ft Es = 22 x 106 _psi

Area of cables As= 1.5 in.2

For a flange width of 4ft, 2.0

tan E

= - =

0.025

80

sin E = 0.025 cos E = 1.

With the previous notations V COS E K= =8.9 2 Es As sin3 _{a sin}2_e 80 LRx e=4.2+-= 12.3ft --=0.39 9.9 2C

From the preceding chart, the cor-responding value of m is equal to 12.5 instead of 13 as found pre-viously. The operation would thus present approximately the same mar-gin of safety. However, the direct force induced in the girder by the lifting cables reduces somewhat this factor of safety. February, 1962 N = 130k 7T2B NE=--= L2 9.9 X 3 X 106 X 108,000

- - - =

870 k. 1.60 X 1.44 X 1000

The tendency for torsional buck-ling has a negligible effect. The cor-recting factor due to the tendency of direct buckling is thus equal to:

J

1- 870 = 0.92 130

Therefore the safety factor is only reduced by 8%.

Reference:

(1)-"Stabilite elastique des poutres en beton precontraint

a

l' egard du deversement lateral" by Pierre Lebelle. Institute Tech-nique du Batiment et des Tra-vaux Publics. September 1959.