Indian Journal of Engineering & Materials Sciences Vol. 8, December 2001, pp. 341-346

Diagonal cracks of structural concrete beams under short-term load with unloading

Terezia Nilrnbergerova, Martin Krizma & Jan Hajek

Institute of Construction and Architecture, Slovak Academy of Sciences, Ddbravska cesta 9, 842 20 Bratislava, Slovak Republic

Received 22 May 2001; accepted 12 September 2001

Characteristics of the process of diagonal cracking have been studied and their evaluation based on the experimental investi ga ti on of reinforced and prestressed concrete beams subj ected to short term load with unloading has been carried out. The relati ons between shear deformations and the characteristics of cracking are explained in the paper. The relationship between the reversible part of diagonal crack opening density and the diagonal crack opening density immediately be fore unloading is also presented.

Considerable attention is paid to the limitation of crack widths in view of their unfavourable effects on the serviceability and durability of concrete structures. This is observed from the fact that R&D efforts have been directed towards reliable estimation of crack widths and subsequently to the design of construction rules limiting their widths. Mostly the development of vertical (bending) cracks has been studied. The issue of diagonal cracks has not been addressed adequately because it has been assumed during the design stage of a structural element, that the width of diagonal crack would be restricted by the reinforcement. Still , in some cases, such as deep or slender beams, the diagonal cracks are not sufficiently arrested by means of reinforcement. Therefore, in such cases, several studies have been concerned with the more detailed investigation of diagonal cracking l

-4

• In cases, when a beam is subjected to moving loadS (e.g. bridges, rails, beams in movable structures like cranes and gantries), intersections of inclined cracks take place even in the middle part of the beam6

. An experimental research on the influence of moving load on the deformational properties of concrete members has been carried out at the Institute of Construction and Architecture in Bratislava. As a logical extension of the above re­search an experimental investigation of the reinforced and partially prestressed beams under gradually in­creasing load with unloading branches has been car­ried out. The tests were mainly concerned with the study of the relation between the elastic deformations and the deformations immediately before unloading. During this study, development of vertical? and di­agonal cracking was observed and studied. This paper

deals with the studies related to the diagonal cracking of reinforced and prestressed concrete members.

Experimental Procedure The diagonal cracks were evaluated from the test

results of three structural concrete (SC) beams with I and T cross-sections. The span/depth ratio (llh) of the I-beams was 7.5, and that of the T-beam was 11 .4. One of the concrete I-beams (I-PCB) was partially prestressed. The degree of prestressing was 0.614 (ref. 8). Details of the cross-sections of the tested beams, reinforcement and side-views are shown in Fi g. 1.

A verage material characteristics of the deformed bars of the non-prestressing reinforcement were: yield strength j,y = 454 MPa, tensile strength Is, = 649 MPa, and modulus of elasticity E,. = 213 GPa. The charac­teristics of prestressing tendons with the nominal di­ameter of Q> 12,5 mm were: it" = 1700 M Pa, E" = 202 GPa. The average mechanical characteristics of the concrete are given in Table I .

Simply supported beams were tested under a con­centrated force F in the middle of the span. The force was induced by means of the two tie rods which con­nected a steel rolled I-beam with the hydraulic jack placed under the tested beam. The test set-up is shown in Fig. 2. The displacement rate control loading was applied on both the loading as well as unloading branches . For the control value, step-like increase of the displacement of the order of 4 mm of the loading of hydraulic jack was taken . The measurements (de­formations of concrete, cracks development) on the all loading and unloading branches of a beam were performed at the loading steps with the same deflec-

342 INDIAN J. ENG. MATER. SCI. , DECEMBER 2001

® T- RCB I -RCB 1- PCB

I

.r~w. STRANOS 5..,12.5

, F J BASE LI NES

11 x 360

Fig. I---Cross-section , reinforcement of tested e lements and posi­ti on of measuring bases

Table I- Mean values of strength s of concrete and moduli of elasticity

Designa- Compressive strength Direct Modulus tion [MPa] tens ile of elasticity

of beams strength [GPa] [MPa]

on cubes on pri sms j ;.", 1;)", l ei Ec

I-RCB 55.32 44.80 2.42 40.83 I-PCB 52.34 38.25 2.35 37.81

T-RCB 4 1.43 33.11 2.09 39.74

tion. Fig. 3 shows the load deformation characteristics of beam I-PCB . The symbols (filled circles to repre­sent the loading branches, empty circles for the un­loading branches) represent the measured loading steps. In the case of the T -beam, the last three un­loading cycles were made after the yield stress of the tensile reinforcement was reached. Thi s corresponds to the value of the loading force F = 245 kN . The tests of I-bea ms were limited by the capacity of loading equipment, which was, FlII{{x = 400 kN. The tests on J­beams were therefore interrupted at the loading levels A = 0.74 and A = 0.68 for prestressed concrete and

450 I-PCB

400

350

300

Z 250 ::. I.L.

200

150

100

50

Fig. 2-Test set- up

5 10

alol (mm )

.. . ;;.o--!

........ Q., 0 '

~Ioading

. unloading

15 20

Fig. ~Loading force versus deflec tion relationshi p of beam 1-PCB

reinforced concrete specimens (I-PCB and I-RCB), respective ly.

At every control displacement level, the strains of continuously linked-up base-lines at the compressed and tensioned edges as well as at the crossing diago­nals were regi stered . The metal plugs embedded into the surface layer of the beam served to measurements of the distance changes between two adjacent ones by means of the mechanical strain gauges. The base-lines at the top and bottom edges were of the nominal length s = 360 mm and their nominal vertical di stance was v = 460 mm. The base-lines formed an imaginary truss consisting of "struts" and "ties" enabling the cal­cu lation not only of the strains but also of the deflec­tions. The network of base-lines (Fig . I) was decom­posed into two truss systems. The metal plugs were located in the vertices of trusses. The measurements of elongations between two adjacent meta l plugs were then evaluated by us ing a method based on Williot-

NURNBERGEROV A el at.: DIAGONAL CRACKS OF STRUCTURAL CONCRETE BEAMS 343

Mohr translocation polygons (in numerical form). The input data consisted of co-ordinates of vertices of a truss, elongations of struts, and of an incident matrix formed by the number of vertices adjacent to a given vertex. From the input data, the vertical and horizon­tal displacement of the vertices (deflections) were calculated. The widths of cracks were registered at the two (or four) loading steps before unloading and after unloading to the zero level at each loading cycle. The widths of vertical and diagonal cracks were measured at the position of tensile reinforcement and in the middle of the web, respectively . The following char­acteristics of cracking were evaluated, namely : the crack density , the crack opening density , and the crack widths. The detailed loading procedure has been reported earlier7

•

Results and Discussion In accordance with the theory of elasticity, the dif­

ferential equation of the deflection line due to shear forces is

... (1)

where }',,, is the shear deformation resulting from the relative displacement of two adjacent cross-sections at the distance dx. According to the known relations the shear deformations may be calculated as follows:

KV YII/= GA .. . (2)

where V is the shear force, C is the shear modulus of elasticity, A is the cross-sectional area and K is the non-dimensional coefficient depending on the shape of the cross-section. The quotient of the shear force and shear deformation VI}',,, , i.e., the expression CAlK represents the shear stiffness Bq . However, the calcu­lation of the value of shear stiffness after cracking is more complicated.

Following the method of measuring the deforma­tions as described, the deflections of the beams at specified locations are computed by separating the effects of bending moments and those of shear forces. In a similar manner, vertical displacements (deflec­tions) Cl" due to shear force V for the each node of the bottom chord of the imaginary truss , and subsequently the shear deformations }',/1 as a difference between two

adjacent deflections a".i - a".i.j divided by the length of the base line Iv are obtained. The experimental values of shear stiffness B" of the beam I-RCB versus the

Z ~

E "..

>

Z ~

>

350,-------------------------------,

I-RCB 300 . .. . ....... : .. _. _ ...... .; . .. .. - - .. .

Experimental

Trilinear (proposal)

Trilinear (experiment) x: : . . . 250

200 •• ;; '~ ,) •••• .•••• ' .... ...... ! •••• ... .....•.. .. ..... . ,. .

150 . .. .. .. .... ~ . .. .. .. .. -: .. . . .. ..... : ......... . . ~ . ... ... ... . .", . .

100

50

i 'X . : .

: .. :.::::. I: ~,I ~~t~1:1:-';-i. :t :.:.: ~~.: . : ..... :.: 0

0 50 100 150 200 250

V (kN)

Fig. 4-Shear stiffness versus shear force re lati onship

2S0

200

1S0

I-RCB ......... ;, ...... ': .. .............. -~ .. £.i.~ .... .... . ... ...... ~ .... ..... :. ... . . . .. . .... .... :.It:AA

• • ••• : • • •• •• •••

100

SO

0

:::::::::r:::::::::.::::::::: ~r~A: :: .. ::::: :::::::: : : ~ : '

.. . .. .. -so

-100

-1S0

-200

····· ····;·· ········:··2..:·· ·······,·········;·········. .. ....... ~ ......... ': ~i-•... _ .... ... -.. ~ ........ ",' ....... .

. c~ .

: ::::::~~~~~~i~: :::::: ::: ::::::: :::::: : ~ :: o 4 .0 10

·2S0 '-----------------'---------------'

-6,0 -4 ,0 -2,0 0,0 2.0 4,0 6,0

Ym (mm.m·1)

Fig. 5--Shear force versus shear deformations rel ationship

shear force V are plotted in Fig. 4. The course of this relationship may be quite satisfactoril y approximated by the trilinear regression (dashed lines). The first part represents the stage up to cracking, the second part corresponds to the formation of diagonal cracks, and the third part approximates the stabilised stage where only the crack widths increase. As a matter of fact, the shear stiffness has to be constant up to cracking but the scatter of results in this part is large as the meas­ured values are relatively small. It seems to be suit­able to replace also the third part of the relationship representing the stabili sed cracking with a constant. The value of this constant may represent the mini­mum shear stiffness related to the service load . Simi­lar approach of calculation of shear stiffness has been used as reported earlier 9. The shear deformations af­ter cracking may be also determined as specified elsewhere'o. More detai led description of the CEB method is reported earlier". The relationship between

344 INDIAN 1. ENG. MATER. SCI. , DECEMBER 200 1

the shear deformation and shear force for beam 1-RCB is shown in Fig. 5.

The next step is to find the relation between the shear deformations and the crack widths. In order to obtain such a relation, the diagonal crack opening densi ty l2 a cr,d was evaluated, which is the sum of widths of diagonal cracks occurring between two di­agonals of the imaginary truss divided by the length of the diagonal. Even if the cracks were, as a rule, not perpendicular to the direction of the diagonals, the difference of inclinations were small and these could be neglected. To mitigate the effect of the choice of the base lines, the moving average method of second degree was applied in the evaluation of both the shear deformations as well as the diagonal crack opening densities. The diagonal crack opening densities as a function of shear deformations of the beam ]-RCB are plotted in Fig. 6. For approximation of the experi­mental values the regression straight line of the shape y = ax+b was applied. Similar relationships were ob­tained for the beams J-PCB and T-RCB. The parame­ters of the regression straight lines are given in Table 2 together with the correlation coefficients for the three beams.

As in the case of vertical cracks, the diagonal crack opening density is related to the average crack width. But the crack width limit is usually applied to the maximum crack width . Therefore, the relationship between the diagonal crack opening density and the maximum diagonal crack width was established from the tests. Unlike the linear dependence between the vertical crack opening density and maximum vertical crack width6

, the relationship between the diagonal crack opening density and the maximum diagonal crack width cannot be approximated by a straight line. The third degree polynomial of the shape y=ax3 +bx2

3500 ,-------- -----------,

3000

~- 2500 'E E 2000

.3 '! 1500 U

1:l 1000

500

I-RCB

. . .... . . .. : . . .. . .. . ... ; . . . . . ~. ~ . . . . . . . KX .......... -- .. ... -. . . . . . .... . . . .. ~. ~ . ~ .. ~: ; ~~

Re9r. straight line

O~~--T---~--~---r---~

0.0 1.0 2.0 3,0

1m (mm,m·l )

4,0 5,0

Fig. &- Diagonal crack opening density versus shear defor­mations

+ ex was applied, As earlier, the span/depth ratio of the tested beams or their reinforcement were different and accordingly also the ratio of shear force induced deflection to the total deflection differed considerably (Table 3), Despite these differences, it was possible to apply one regression curve for all the three beams. Its parameters of regression were as follows :

_ 3 2 Wcr ,d,ma x - 2.83£ - 08acr ,d - 1.993£ - 04acr ,d

+ O.5115acr,d

. . . (3)

where a cr,d is diagonal crack opening density and W cr,d, IIIQX is maximum width of diagonal cracks. The

Table 2--Paramcters of regress ion straight lines of re lati onship between the di agonal crack opening densities and shear deforma­

tions

Designation Parameters o f regress ion Corre lation of beams strai ght lines coeffi cielll

y = ax +b a b r

I-RCB 655 .85 -121.l 7 0 .925 1 I-PCB 532.53 - 99 .52 0.8097

T-RCB 400.94 -1 63.75 0 .5841

Table }--Ratio of the shear force induced d_flections to the IOta I deflection

Designation of beams

I-RCB I-PCB

T-RCB

Service load

0.30 0.19 0.08

Max imum load

0.342 0.280 0. 11 8

600 ,----------- --------,

500 . ... .. . . . . . .. . . . ..

E 400

2-~ 300 E .q

.; 200

100 .. x I·RCB • I·PCB + T·RCB

- Poly. regression O ~----~-----_r----~

o 1000 2000 3000

Ucr ,d (j.1m.m·l )

Fig. 7-Maximum diagonal crack width versus di agonal crac k opening density

NURNBERGEROV A et al.: DIAGONAL CRACKS OF STRUCTURAL CONCRETE BEAMS 345

correlation coefficient was r = 0.9655 . The experi­mental values of maximum diagonal crack width against the diagonal crack opening density and regres­sion polynomial are shown in Fig. 7.

As the beams were tested for unloading it was pos­sible to establish the relationship between the diago­nal crack widths immediately before unloading and those after unloading to zero level. The loading steps with measured crack widths are illustrated in Fig. 8, where the relationship between the loading force and the sum of diagonal crack width of the beam I-ReB is plotted. The diagonal crack widths measured at the loading steps immediately before unloading and after unloading to zero level and consequently the diagonal crack opening density known for each base line pro­vided sufficient amount of data to investigate the re­lationship between the total and reversible diagonal crack opening densities. This relationship shows rela­tively small scatter and could be approximated by a straight line with sufficient accuracy . The parameters of regression straight lines are given in Table 4 both for reversible and irreversible part of diagonal crack opening density. As the values of parameters arr,v

(airrev) differed only slightly, it was possible to ap­proximate the results of all three beams with a straight line as follows:

aer.d .rev = 0.7 6acr .d .be/ore··IIIl/oadillg (4)

The correlation coefficient r was 0.9887 . In the case of the irreversible part of diagonal crack opening density the scatter of results is bigger (r = 0.8773) because the measured values are small. In Fig. 9, the reversible part of diagonal crack opening density is plotted against the diagonal crack opening density immediately before unloading for the three beams. In Fig. 10 the relationship of the irreversible diagonal crack opening density versus the diagonal crack opening density immediately before unloading for the three beams is shown. The linear relationship between the reversible/irreversible diagonal crack opening density and the diagonal crack opening density before unloading are in a good agreement l

) with the results.

Conclusions On the basi s of experimental results , it may be con­

cluded that there is a relationship between the shear

Table 4--Parameters of regression stra ight lines of re lationship between the reversible/irrevers ible part of the density of cracks opening and the density of cracks opening immediately before unloading

450

4()0

350

300

Z 250 :;s u.. 200

150

100

50

0

I-RCB

0 2000

Designation of beams

I-RCB I-PCB

T-RCB

Reversible part

Parameter of regression Correlation straight lines coefficient

an'"

0.7741 0.7459 0.7280

r rel'

0.9908 0.9833 0.9866

'E E

Irreversible part

Parameter of regression straight lines

Correlation coefficient

Qirre,'

0.2259 0.2541 0.2720

0.8511 0.8659 0.9167

4000.-------------------------------~

, . . 3000 ...... . -. -;- . . . .. . . - . ; . -. _. -. _. -:- - .... . . ~ -. ~:' . . x ,, ' .

'.'

2: 2000 ... .. . x . .:

"' "6;'

. . ·r,//".,,,,,-/·· - .. -.: ....... : - ...... : ... .

. . .

. . . - ... , .. - ... . , ... -... . . -

--- ..... ! . .. .. .. !. - --- .. : .... . .. : .. -- .. . · , . . . · . . . . · . . . . · . . . .

4000 6000

1:Wcr,d

8000

().lm) 10000 12000 14000

x.'

1000 I-RCB

I-PCB

+ T-RCB

... Regr. straight line

O~----~-----+----~r-----+-----~ o 1000 2000 3000 4000 5000

Ucr,d,before unloading (J-lm.m,1)

Fig. &-Loadi ng force versus diagonal crack widths Fig. 9--Reversible part of the diagonal crack opening density versus diagonal crack opening density before unloading

346 INDIAN 1. ENG. MATER. SCI., DECEMBER 2001

'7

E E ..3

>

~ ~ U

CS

1200.-----------------------------------~

1000

800

600

400

200

a a

. . . ..... . ...... .... ... _- .. __ ...... . , ..... . .. . ... ..... .. . .. . . . , .. + . • . - ...... ....... . .. - ... ....... : . ..... ... . . .)(.. .. .

+ : II x :

···· · -_···x':···················· · · · ······ : 6 + ~ .. . ' ,. x : d.~)( x

- - -~ ~ ~ »-X-· ': -:< . ..... - . xj.~:-4x . ... I·RCB

I· PCB

+ T·RCB : ..... Reg r. straight line

1000 2000 3000 4000 5000

acr,d,before unloading (J..lm.m-1

)

Fig. IO--Irreversible part of the di agonal crack opening density versus diagonal crack open ing density before unloading

deformation and the diagonal crack opening density which may be expressed by a straight line, From the diagonal crack opening density the maximum width of diagonal cracks may be obtained by means of the Eq. (3). However, to generalise the parameters of re­lationships between the above-mentioned quantities would require more experiments .

The reversible part of the diagonal crack opening density can be determined in accordance with the Eq, (4) as a multiple of the diagonal crack opening density immediately before unloading. The coefficient a =

0.76 of the Eq. (4) corresponds with the one derived for the relationship between the elastic and total de­flections.

Acknowledgement Authors are grateful to the Slovak Grant Agency

VEGA (Grant No. 217034/2001) for partial support of this work.

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(1997) 572. 4 Shin S W, Lee K S, Moon 1 I & Ghosh S K, ACI Struct J, 96

(1999) 549.

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7 Nlirnbergerova T, Krizma M & Hajek 1, Indian J Eng Mater Sci, 8 (200 1)71.

8 Naaman A & Siriaksorn A, PCI J, 24 (1979) 64. 9 STN 73 120 I, Desigll of COliC rete strucllIres, UNM Praha,

1987 (in Czech). 10 CEB Manual on Cracking and Deformations. Bull

d'illfor/l/atioll n O 158. EPFL Lausanne. 1985. II Leonhardt F, Vorlesungen liber Massivbau (Lecture on solid

structures). Part 4. Berlin-Heidelberg- ew York . Springer­Verlag, 1977.

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13 Nlirnbergerova T, Krizma M, Hajek 1 & Hajek F, Betoll a Zdivo , (2000) II.

Nomenclature A = area of the cross section a

c dx

F Filla.\"

fOil I., /pm 1;" Isy

Is, G II I I, . I'

v W cr.c/, j

W cr .(/,I/UU

aa,d

),;" K

= parameter of regression functi ons = denection due to shear forces = total denection in the middle of the span = shear stiffness = parameter of regression fu nctions = parameter of regression fu ncti ons = relative displacement of two adjacent cross

sections = modulus of elasticity of concrete = modulus of elasticity of prestress ing reinforcemen t = modulus of elasticity of non-prestress ing rein-

forcement = loading force = capac ity of loading equipment = cube strength of concrete = tensil e st rength of concrete = prism strength of concrete = tensile strength of prestressing reinforcement = yield stress of non-prestressing reinforcement = tensile strength of non-prestressing reinforcement = modulus of elasticity in shear = depth of beams = span of beams = length of the base line = correlation coefficient = shear force = width of the i-th diagonal cracI.. = maximum diagonal cracks width in a base line = diagonal crack opening density = shear deformation = non-dimensional coefficient depending on the

shape of the cross-section = loading level