Theory Grouted Elements

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USFOS

Grouted Members

Theory


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where the plastic capacities in bending and axial tension/compression are

( ) ( )3 23 2 2

,6 4

Ps ys Pg ys

D D t D tM f N f

= =

(1.3)

Wherefys is the yield stress for steel andDand trepresents the diameter and thickness ofthe tube, respectively

For the grouted section there is obtained:

3 1 1sin , sin 22Pg Pg

M N

M N

= =

(1.4)

where the plastic capacities in bending and axial compression are

( ) ( )3 2

2 2

,12 4Pg yg Pg yc

D t D t

M f N f

= =

(1.5)

where fygs is the yield stress for grout. The grouted section contributes only on the

compression side of the interaction.

For the composite section the two interaction equation can be added. The angle to the

neutral axis, , constitutes the interrelation. The resulting equation is plotted in Figure1-2, for an example where all plastic capacities are set equal to unity. Compression isdefined positive.

It is observed that the maximum bending moment occurs when the plastic neutral axis isat midsection (i.e.= /2) when D Ps PsM M= + andND= 0.5NPg. The points B and Ccorrespond to bending momentsMB= MCandNB= 0,ND= NPg. The bending moments

can be determined solving for from the condition Nsteel= - Ngrout, see Section 1.1.1.

The axial force at points A and F areNA= NPs+ NPg andNF= - NPs, respectively

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-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5

Moment

Axialforc

e

Steel

Grout

Total

D

F

C

B

A

Figure 1-2 Example of plastic interaction function for grouted steel tube

Interaction equations that can be calculated for tubes with grout Durocrit D4 and S5 are

compared with interaction equation and FE analysis results given in Safetec Report OD-2000-0065, 2001. The diagrams are presented in Appendix B. It is observed that the

agreement with Eurocode 4 curves is excellent, as expected.

1.1.1 Simplified interaction equation

Haukaas, M. and Yang, Q. (2000) proposed to use 1/cos type function as an

approximation to the interaction curve. Two curves are defined; one above the maximum

bending moment and the other below maximum bending moment. The equations aregiven as

11/

1

ln cos2

cos ,2

ln

C D

A DD

D A D C

D

N N

N NN NM

M N N M

M

= =

(1.6)

above the maximum bending moment and



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21/

2

ln cos2

cos ,2

ln

B D

F DD

D F D B

D

N N

N NN NM

M N N M

M

= =

(1.7)

below the maximum bending moment. It is seen that the factor is determined so that theinteraction curve goes through the points (NC,MC) and (NB,MB) in addition to (ND,MD),

(NA,0) and (NF,0). Confer Figure 1-3 for definition of the various points on the

interaction curve.

Figure 1-3 Points on yield surface ( Haukaas, M. and Yang, Q. (2000))

The various points are, as shown before, defined by:

D Ps PgM M= +

1

2D PgN N=

c PN N= g

Pg

s

= + A PsN N CompFac N

= - F PN TensFac N

(1.8)



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The factor TensFac takes into account additional tension capacity due to the grout. If

there is assumed that the grout prevents the steel from contracting when subjected to

plastic tension, a tensile stress equal to Y is produced in the circumferential direction.The effective yield stress then becomes 1.12Y. Hence, the default value assumed inUSFOSis TensFac= 1.12.

The factor CompFac takes into account any reduction in the compression capacity of the

grout. By default CompFac = 1.0

The moments MBand MCare determined from the requirement NB= 0. This yields thefollowing equations:

( )sin2

22

BB Pg Ps Pg PN N N Ns

+ = + (1.9)

3sin sinB Ps B PgM M M B = + (1.10)

The angle B(0,/2)has to be solved by an iteration procedure. The Interval bi-section

method, which is a stable method, is used.

Figure 1-4 shows an example of the simplified interaction equation along with the exact

relationship. The agreement is good. Note: The tension capacity is augmented by 12% on

the tension side only for the approximate curve.

The exponent = 1.14below the maximum moment and = 1.25above.

The following data need to be stored for the grouted cross-section:

MD= maximum plastic bending moment

NA= Maximum compressive capacity

NF= Maximum tension capacity

1= Exponent for on compression side

2= Exponent on tension side

Alternatively the three first properties may be normalized against the yield stress for steel

to obtain:

DPys

Wf

= = equivalent plastic section modulus

A

cys

N

A f= = equivalent plastic compressive area

Ftys

NA

f= = equivalent plastic tension area

In addition the equivalent elastic area needs to be stored as discussed below



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-2

0

2

4

6

8

0 1 2 3

Moment

Axialforce

Steel

Concrete

Total

cos-beta

Figure 1-4 Exact (Total) and approximate (cos1

) interaction curve

1.1.2 Reduced bending capacity on compression side

It is possible to specify a reduced bending capacity of the grout, so that the nose on the

compression side is not taken fully into account. This is done such that a new point D*is

defined

( )' 0 1= + - D B Red D B RedM M M M M M (1.11)

where the user can define the bending reduction factorMRed. Default is 1.0.

The corresponding axial force, 'DN has to be determined from the interaction relationship

below theND:

21/

cos 2

D D

D F

D

D

N N

N N

= (1.12)

The yield surface above ( ,D DN ) is then described by

11/cos2

D

D A

N NM

M N

= DN (1.13)



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For simplicity the 1-factor will not be changed, i.e. the shape of the yield surface isassumed to remain unchanged.

Below the point D( ,D DN ) the non-modified interaction equation will be used.

The modified interaction function is illustrated by the dotted line in Figure 1-5

If MRed= 1 the surface is not modified, if MRed= 0, the maximum bending moment is at

N = 0 and is equal to MB.

-2

0

2

4

6

8

0 1 2 3Moment

Axialforce

SteelConcrete

Totalcos-beta

Reducedcapacity

MD`,ND`

Figure 1-5 Possible modification of yield surface (dotted line)

1.1.3 Yield surface convexity

The yield surface should preferably be convex. Otherwise, spurious elastic unloading

from the yield surface could result. A convex surface is achieved as long as the betafactor is larger than unity. For potential combinations of plastic section moduli and

plastic axial capacities of the grout and the steel, the yield surface could become non-

convex. This is evident in the example given in Figure 1. The plastic capacities used toderive Figure 1 are, however, probably not realistic for grouted steel sections. Another

factor that could trigger non-convex surface is if the tension capacity is considerably

larger than 1.12 of the steel plastic tension force.



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A warning be given if < 1 and is put equal to 1.This will relax the requirement thatM = MBfor N = 0. (M will be slightly larger than MB). Another option is to reduce the

plastic axial capacity until > 1.

For the various Durocrit sections studied it seems that > 1 for all the cases.

1.1.4 Local buckling

Finite element analysis shows that the effect of local buckling is significant for members

with D/t exceeding 40. Haukaas, M. and Yang, Q. (2000) propose to take this into

account by the reducing the compressive strength of the grout by the factor

max 1, 0.003 0.88ygD

t

= +

(1.14)

so that

/yg yg ygf f (1.15)

1.1.5 Shear force interaction

For short members the shear force may become significant, notably because the bending

moment capacity may increase significantly due to the grout. In order to limit the shear

force it must be implemented in the interaction equation. This is accomplished bymodifying the bending moment term in the interaction equation. In the three-dimensional

case the following equivalent non-dimensional bending moment is calculated:

2 2 4 4

y z x ym m m m q q + + + +4

z (1.16)

where , , , ,y yxz z

y z x y z

D D T P

M QM

P

Qm m m q q

M M Q= = = = =

Q

The influence from shear (and torsion) is moderate until the shear force becomessignificant.

The shear force and torsional moment is assumed carried primarily by thesteel pipe. The

shear capacity may be augmented beyond the steel shear capacity by a user-definedfactor, ShearFac 1, such that

t= P Y sQ ShearFac A (1.17)

where Asis shear area of the steel tube.



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1.2 Elastic properties

The effective bending stiffness is taken as the sum of the steel and grout bending

stiffness:

( ) ( ) ( )= + Reds gEI EI I EI (1.18)

Red is a factor that accounts for a reduced contribution from the grout. According toEurocode 4, Red= 0.8, while Haukaas, M. and Yang, Q. (2000) recommend Red= 1.0. It

is possible to specify value in the input. The default value is Red= 1.0

The elastic axial stiffness is taken as the sum of the steel and grout axial stiffness

( ) ( ) ( )s g

EA EA EA= + (1.19)

The same stiffness is used for axial tension and compression for the following reasons:

It is practical to have the same elastic properties, because the sign of the axialforce is not known a priori

It is physically plausible that the grout will take moderate tension The bending stiffness used assumes implicitly that the grout acts in tension

It is better to limit the tension taken by the grout by means of the plasticinteraction function, as shown in the previous paragraph

It is beneficial to modify the elastic bending stiffness such that the steel properties can be

used, i.e. equivalent moment of inertia and equivalent area are defined

= + g

eq s Red g

s

EI I I I

E

g

eq s g

s

EA A A

E= + (1.20)

1.3 Mass calculation

An equivalent density may be defined such that the total mass of the grouted pipe is

correct:

r r r= +g

eq s g

s

A

A (1.21)

where sis density of steel and gis density of grout.

At present the user needs to specify the equivalent density of the grouted members in to

obtain correct weight. In most cases the weight contribution may probably be neglected.

1.4 Buckling

USFOS calculates the exact buckling load for members subjected to end forces and end

moments for a given imperfection. According to Eurocode 4 buckling shall be calculatedaccording toEurocode 3, curve afor concrete-filled hollow sections. USFOScontains an



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automatic module for generation of shape imperfections according to curve a, so this

requirement is easily complied with.

For simultaneous bending and compression, the resistance will be governed by the

combined effect of elastic buckling and the plastic interaction function for the grouted

section.

1.5 Mid-hinge

The grouted section model is easily implemented for beam ends. The conventional beam

element used in USFOSallows automatic subdivision of a beam into two new beams once

the yield surface is exceeded at beam mid-span. This is, however, not straightforward toaccomplish when it comes to grouted beams. The reason for this is that the plastic surface

for a grouted section due to its asymmetry (about N = 0 axis) can be considered a surface

with an off-set corresponding to N = ND= 0.5NPg. This is analogous to the surface offset

produced by kinematic hardening of conventional non-grouted pipe sections.

In order to introduce artificial hardening caused by the asymmetry, the code needs

significant careful rewriting. At present it is therefore suggested that this problem is

circumvented by modelling grouted members with two beams, thus avoiding the need forintroduction of mid-span hinges.

The code seems to work reasonably well, even if mid-hinges are introduced, but therewill be small jumps in the bending moment-axial force histories

1.6 Concluding remarks

The grouted section model is based upon the plastic hinge concept assuming grout to

contribute to the resistance only when subjected to compression. A plastic interactionrelationship for pipes with internal grout is established, mainly according to the

propositions given y Haukaas and Yang (2002)

The module developed is quite flexible, allowing user defined efficiency of the grout

properties in bending and axial compression and tension.

The elastic axial stiffness and moment of inertia are modified to account for thecontribution of the grout. It is found very inconvenient to use different axial stiffness in

compression and tension, so that equal stiffness is assumed. It is believed, however, that

this has a negligible impact on the results in practical cases.

It is verified that the model works correctly by comparing axial force-bending moment

interaction relationships obtained in simulations of column buckling with analyticalrelationships (see Verification manual).



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Grouted column tests carried out at Chalmers University of Technology (1999) have beenanalysed, see Verification manual. Very good agreement between tests results and

numerical predictions are obtained when Durocrit D4 and Durocrit S5 are used as grout

material. When Densiphalt is used the accuracy is fairly poor, but this is due to poor

behaviour of the Densiphalt as such. A significantly better agreement may be obtained byusing reduced yield stress of Densiphalt.

Best correspondence between buckling testsand simulations are obtained when the yieldsurface size is set to 0.6 of the fully plastic surface. This is smaller than the default value

0.79 for ungrouted pipes.

When design or reassessment analysis is performed it is recommended to introduce

imperfections calibrated to column curves, e.g. to Eurocode 3, curve a. Then the default

value 0.79 for the yield surface size may be used, i.e. no modification is necessary

(GBOUNDcommand)



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APPENDIX-B Interaction relationships for grouted sections

-2

0

2

4

0 20 40 60 80

Moment

Axialfor

ce

Steel

Grout

Total

Interaction diagram for Durocrit D4, D/T=40.



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-2

0

2

4

0 20 40 60 80

Moment

Axialforce

Steel

Grout

Total

Interaction diagram for Durocrit D4, D/T=60.



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-2

0

2

4

0 20 40 60 80Moment

Axialforce

SteelGroutTotal

Interaction diagram for Durocrit S5, D/T=40



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-2

0

2

4

0 20 40 60 80Moment

Axialforce

Steel

Concrete

Total

Interaction diagram for Durocrit S5, D/T=80

Grouted Members 2005 07 01

Theory Grouted Elements

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