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PERIODICA POL)'TECii!;jCA SER. cn'n EXG. VOL 3o, SO. 3. pp 25}-270 (199::)

CRACKS AS IMPORTANT CONSTITUENTS OF STRUT AND TIE MODELS

.J. AU.!,~SI

Department of Reinforced Concrete Structures Technical l"niversity of Budapest

Received: 00vember 10. 1992

Abstract

Construction of strut and t:e mode! IST\!j at D-regions is based on elastic stress distribu- tions by imagining the force, 2S re,ultants of stresses. Beside 'snlOoth' stress-trajectories there are ·turbulent' places having a decish'e influence on the cracks. Therefore. it is useful to use alternative ST\is. onp con:itructed on the base of smooth trajectories. and another which fits the turbulent stre,ses. Examples are gi\'en for different types of structural elements.

Keyword,,: strut and tie model (ST\\). elastic stress-trajectories. smooth and turbulent stress-t rajectories. altprnative ST\!s for nTH' st met ural element.

of the Strut and Tie Model

Principles of the strut and tie model, and a short review of its application is given according to SCHLAICH (1991) as follows. Internal stresses of RC structures are characterized by bending moments, axial and shear forces that are determined using well-known methods of the structural analysis.

Connections between bending moments and deformations as well as dis- tributions of stresses due to internal forces and moments are given on the base of a slightly modified elementary bending theory of bars which also takes the specific behaviour of the structural concrete into account.

However, sections or regions of an RC structure may behave differently and a linear distribution of axial strains and stresses due to the so-called Bernoulli-Navier hypothesis can only be assumed for limited parts of the structure, referred to as B-regions. (Fig. 1).

Sections or regions where the distribution of strains and stresses sig- nificantly differs from those obtained by the elementary bending theory of bars are called D-regions. At D-regions, e.g. near concentrated loads, frame corners (Fig. 2), etc. assumptions of elementary bending theory of RC beams have to be replaced by those of a strut and tie model. B-regions and D-regions can be separated using Saint,-Venant's principle. The first

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252 J. ALMAsI

0) b)

Fig. 1. Elementary bending theory for prismatic beams

step of developing a strut and tie model is the assignment of D-regions of the structure in question, then, more or less correct distributions of elastic stresses and the principal stress directions are to be determined for the D-regions (for example by FEM analysis) (Fig. 3a). This way more or less correct stress diagrams at the boundaries of the D-region also become known.

Assuming that pencils of trajectories of compressive and tensile prin- cipal stresses in a D-region represent 'load paths' of compressive and tensile forces and replacing these pencils by properly directioned straight struts and ties, respectively, a contiguous and rigid plane truss or space grid can be constructed (Fig. 3b).

This truss or grid is called the replacement strut and tie model of the D-region. Bar forces acting at the struts and ties are the resultants of compressive and tensile stresses. The directions of struts have to be taken in the average direction of the trajectories of compressive stresses and located about the central lines of the pencils. The ties should follow the tensile stresses in the same way.

Strut and tie lllUU.tl111J.Y-, ol)'\ilClW31 nrrn;';ciF<: the structural cL!!a! v 0" \7lith some freedom of choice that can be used to aim either at the safest or at the cheapest or at an otherwise optimised solution.

For practical reasons (e.g. to produce a simpler replacement truss or to simplify the manufacturing of the reinforcement) one usually does not closely follow the principal and tensile stress directions. In this case it is necessary to consider the consequences of these deviations, that to check the equilibrium and to adjust the amOU'1t of reinforcement for

in'-o aCCO-lln-'- j-'-s de--:a-'-l'ons rrom -'-ho D-inri~al di-oct:on<: (ii';a !)

.... 1 t< l_D _1" V.!. L L 1_ i _ t...L ... .!. l ___ ,-_p·_ _l.\.... ! _w \J... " ... --;.

If modelling does not closely follow the stress-nows, it can cause in- compatibilities in the corresponding strains, that means, cracks a,nd plastic deformations have to develop. It is well known that concrete has a low tensile st:i'ength and permits limited plastic compreSSlve deformations. To

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CRAG.f:S AS U.fPORTANT CO."·~·STITUENTS OF STRUT AND TIE .MODELS 253

h b) 1~ ~

"

+

d)

~D ~~~r",·" '~Er

/region-jj, ," , '. reg ion //;/ ~"" "

Fig, 2, Regions wherp there are non-linear strain distributions, separation of sections

avoid developing of wide cracks and exceeding plastic limit. an addit.ive reinforcement. of t.wo directions has to be used,

The Problems of Strut and Tie Model

The use of strut. and t.ie models is st.rongly hampered by problems, not.

perfectly clarified so far, as follows:

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254 J. ALMASJ

0)

et )

b)

Fig. 3. Principal stresses and the imagr of forces

First Group of Questions

In order to determine whether principles used heretofore form a sufficient base to develop strut and tie models, which can properly model the real behaviour of structures, it is necessary to improve the adequateness of the modelling by refining the fundamental assumptions.

Questions connecting to this are as follows:

- What dense should a replacement truss be?

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CRACKS,AS IMPORTANT CONSTITUENTS OF STRUT AND TIE MODELS

a)

b)

T (As}

{

PRINCIPAL TENS!ON CRACK

-4, Correction of reinforcement

What are the physical limits for constructing the truss?

How does the reinforcement influence the truss?

255

When using two strut and tie models for the same structures how can the load be split into parts born by each model?

How can the effect of the prestressing be taken into account in strut and tie modelling?

How have statically indeterminate strut and tie models to be correctly solved?

How does a complicated cross-section influence the modelling?

Second Group of Questions

These are related to the accuracy of calculation. Connected to this one can ask:

How does the deformation of strut and ties influence the action- effect of the truss elements?

How can be compensated the neglection of the compabitility condition for the changes in length of the fictitious bars?

What is the minimal amount of reinforcement for assuring the 'suffi- cient ductility'?

What kind of safety measures have to be used to avoid erroneous dimensionings?

What kind of results do give the comparison between the calculated and the measured values?

How does the bond change in nodes?

How does the deviation of the strength of the concrete influence the results obtained by STMs?

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256 J. AL.H.4S1

The Influence of Cracks on the Strut and Tie Model

Cracks on a well-designed structure or structural element gradually appear as the intensity of the load increases, they are uniformly distributed and not concentrated in narrow strips of the structure, their widths remain moderate in the service state of the structure.

It is impossible to avoid cracks at the level of load of service state, even in case of optimal design, however, crack widths and the crack pattern can be influenced in many ways. Factors influencing the crack pattern are as follows (Fig. 5):

the geometry and cross-section of the structure, the loads and their characteristics,

the variance of strength of the concrete, the concrete covering the reinforcing bars, the temperature and the free motion hinder, the amount of reinforcement,

the diameter of reinforcing bars, the distances of reinforcement,

the direction of reinforcement (how it follows the direction of principal tensile stresses),

the types of reinforcement (normal, prestressed or mixed), the bond and the anchorages.

At abrupt changes of the cross-section, so-called stress peaks develop (Fig. 6). Here the maximum stresses can multiply exceed the average values caiculated by usual methods. Steep changes in stresses especially in tensile stresses are hardly born by materials having a iimited ductility like concrete .. Around stress peaks, tensile stresses quickly increase and exceed the tensile strength of the concrete. Cracks arise, large deformation of the tensile reinforcement starts and plastic zones develop while other parts of the structure are in elastic range. The appearance of cracks that the equalizing of stresses has begun.

Cracks at stress peaks have a decisive influence on the load-bearing capacity of structures. These stress peaks form only small parts of the Vllhole system of stresses and they can hardly be fit by any strut and tie model. However, disregarding them would be a bad mistake. Researches prove that unlike the other structural parts, where tensile forces can be conveniently covered by simple webs of orthogonal reinforcement, at stress peaks this method only gives a reduced-value solution both in ULS and SLS.

Researches have also shown that at places where stress peaks can develop the load bearing capacity can be increased by 20-30 % if reinforcing bars are put at right angles to the cracks. The solution can be improved

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CRACKS AS IMPORTANT CONSTITUBNTS OF STRUT AND TIB MODBLS

i) al

cl

o

di

f)

"

,.

*"wini"c.r

~ -'300 t1

~ R n ~

~o=====~>

b I

h 1

\~ o

E:ZS2

j)l

n I

Fig. 5. Factors influencing cracks

257

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258 J. ALMASI

0)

b)

( :) I

I

c. )

d}

e) f)

1

I

I

-

-

1_

Fig. 6. Stress peaks, decisive effect on cracks

by application of wedgings, roundings up, that is, gradual and not abrupt changes of the cross-sections. These geometrical refinements also permit good possibilities to refine the reinforcement as well.

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CRACKS AS IMPORTANT CONSTITUENTS OF STRUT AND TIE MODELS 259

As previously mentioned, the reinforcement has a decisive effect on cracks. The correct direction of tensioned bars of strut and tie model around the stress peaks has a great importance because this gives a great influence on behaviour at S1S.

In the subsequent sections typical cracks on different structures are shown that has to be taken in consideration in constructing strut and tie models.

0.)

b) c. )

3

d) e.) f)

L[J

Fig. 7. Principal stresses and what influences the cracks

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260 1. ALMASI

Beam with Dapped End

Tests on beams with dapped end show that the first crack appears at relatively low load level, (at about 25 % of the ultimate load) and it starts with nearly 45° angle at the corner (Fig. 1). If the beam has a correct reinforcement, the number of cracks increases to 2-3, while the length and the width of the first crack are growing. The place of the starting points and the direction of cracks are influenced by the geometrical proportion of the dapped end (such as wedging, rounding, up, etc.). Two types of the strut and tie models are recommended to use (Fig. 8). The load has to be split into two parts according to the proportion of stiffness of the two models.

a) bl c.l

1"10d8\ 1 t-i\ odC?\ TI

8. Alternative models for the same structure

Tests on corbels show four characteristic failures: bending, shear, dl;agon;al splitting and under the load 9). In the first three cases

charaC"tel~lS1G1C cracks develop feature the manner and the cause of the failure. If the bracket is connected to a column then the load has an infiuence on the direction of the bending cracks (Fig. 9).

When ratio aj d is higher than 1 then vertical stirrups are more ef- fective than horizontal ones (Fig. 10). Cracks on the bracket show the importance of using oblique reinforcement. Doing so, the basic strut and tie model has to be completed with ties crossing the cracks (Fig. 11). Al- ternative solution for the struts is also given on Fig. 11. Again, the load has to be divided between the two models.

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CRACKS AS IMPORTANT CONSTITUENTS OF STRUT AND TIE MODELS

a)

bl

T--_H

__ r

d)

---W

1/(1

Lr

f)

C.l

e.)

-::.t-__

0

- r

"-..ft ~

r-~j

r

___ .l

-~ k

Fig. 9. Princlpal stresses and failure modes

261

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262 J. ALMASI

a ) a

b) -, - - - a JL 11

1111111 }

a Id D 1 a Id >1

- -

Fig. 10. Main reinforcements for corbel and cantilever

a)

Ir~,- / b)

tl"

Modell

Model 11

cl

d) e)

Fig. 11. Alternative models of corbel alld refinements of struts

Frame Corners

The action effects arising on frame corners are very different, depending on stiffness ratio and loads. There are two characteristic cases: frame corners loaded by: 'opening' moment and those loaded by 'closing' ones (Fig. 12).

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CRACKS-AS IMPORTANT CONSTITUENTS OF STRUT AND TIE MODELS

bl cl

0)

~ )

-no,

... _-"""" I

_L

i I

I I

I I

I

0 -l. _ _ _ ..L

d)

I

I

-~

e) 'f

Fig. 12. Characteristic cracks and stresses at frame, corners

263

)

The cracks and their forms depend on the stresses of the used reinforcement type.

The diagonal cracks are deep on the frame corner if they are loaded by a 'closing' bending moment, therefore, taking off the tensile forces we need more ties than one. Example is given in Fig. 13.

At frame corners loaded by 'opening' bending moment, the crack starts from the inside corner and after a short way, it splits into two direc- tions perpendicular to the original direction (Fig. 12). This again makes necessary to use two strut and tie models. The basic model and the alter- native version is shown in Fig. 14.

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264 J. ALMA.SI

)

Mode! Model

II

Fig. 13. Alternative models for corners with 'closing' bending moment

0.) b)

=-r--1~ ~

! . ' "

1)

Q " I

Model 1 Model II

14. A..lternative rnodels for corners v-iith ~opening~ bending mOIllent

on Beams

One of the most uncomfortable tasks for a Ue:t51,lSl],t:l is to make openings on structural parts great statical importance. Nevertheless, the special difficulties in these cases have to be solved in a safe vvay.

Openings in homogeneous stress fields cause singularities in the stress- es and make rise special cracks in well determinated places (Fig. 16).

Possible strut and tie models for these cases are given in Fig. 17. In these cases it is also useful to use the combinations of tWD strut ana tie models or to use a refined model.

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CRACKS AS IMPORTANT CONSTITUENTS OP STRUT AND TIE MODELS 265

a) b)

15. Influence of geon1etry proportion on the model

Beams

If a structure bears large loads, beams have to be used. A survey of the literature shows that a lot of different recommendations have been worked out to control the stresses and to design the reinforcement in deep beams.

Tests show four characteristic failure modes (Fig. 18). Up to now methods mainly concentrated on avoiding bending failure but did not prop- erly discuss the other cases, e.g. the large local stress peaks caused by large concentrated forces. In the case of normal beams, second, third, etc. cracks often rise in slightly different directions, while in the case of deep beams this divergence cannot be observed.

To approach correct force flows we again need two combinations of strut and tie models (Fig. 9).

Tests also show that the cracks can be essentially influenced by the proportion of horizontal and vertical reinforcement. As a rule, a larger load bearing capacity can be achieved by an enlarged vertical proportion of reinforcement. This also supports the usefulness of the application of a second model as shown in Fig. 19.

Conclusions

Construction of strut and tie models is based on elastic stress distributions, by imagining the strut and tie forces as resultants of stresses.

Abrupt changes in geometry at D-regions of structures cause singular stresses or turbulent places which disturb the generally smooth trajectories of principal stresses. Unlike the places where the trajectories can be easily

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266 J. ALMAsI

cl bl

l

e)

Fig. 16. Cracks around openings

described by a strut and tie model, forces which arise at the places of singularities cannot be fit in a simple way by commonly used strut and tie models. If we force to fit these forces by the model, we easily lose the simplicity of the calculation.

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CRACKS AS IMPORTANT CONSTITUENTS OF STRUT AND TIE MODELS 267

Model n

-'1 Model I If-

Fig. 17. Alternative models for a deep beam with opening

(18)

268

a)

c)

-- -

.n .. ":'): n.<r -

----

3'

J. ALMAsI

b}

d)

~'

L~3 I~.\~

Fig. i 8. Characteristic cracks and failures of deep beams

~\

3

However, these turbulent places have decisive influence on the crack at the and their existence must not be disregarded in the calculations. Taking into account tensile forces at these places needs a special treatment. It is necessary to construct alternative models to control the first cracks appearing at singular places. This can be done by the following way: beside of the strut and tie model, constructed on the base of smooth stress trajectories, also a second model has to be constructed the ties (or tensile forces) of which are passing through these places and make the stress peaks to distribute. The main part of this article shows such models for different structures.

The load distribution between two replacing models can be divided - in lack of a better solution - according to the proportion of stiffness of the two models. It also has to be taken into consideration that codes require more than one load case or load combination.

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CRACKS AS IMPORTANT CONSTITUENTS OF STRUT AND TIE MODELS

d 1

I

1/

i,

/ /

,

I ; , L'-L_J

]

1

i77

11 .-

,."

e)

\\

\ \

\ \

\ \

\

- ' ! > -

\

\

\

ill

Modal IT

1

\ ,I, /,h

\ / I i

\ ,,/ I I

\ f-!---}

\ \ I,"

\ \ I 1 ; "

\, , / /

-1-1+--<0- ---.Ii

l

tI

Mode.l 11.

Pig. 19. Alternative models for deep beams

269

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270 J. ALMi.SI

References

SCHLAICH, J. - SCHAEFFER, K.: Design and Detailing of Structural Concrete Using Strut and Tie Models. The Structuml Engineer, No. 6, March, 1991.

Address:

Dr. J6zsef ALMASI

Department of Reinforced Concrete Structures Technical University H-1521 Budapest, Hungary

Hivatkozások

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