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Mathematical model of RC banded column behaviour
Construction and Building Materials 15 Ž2001. 351᎐359
Mathematical model of RC banded column behaviour L. CirtekU Brno Uni¨ ersity of Technology, Faculty of Ci¨ il Engineering, Udolni 53, Brno 662-42, Czech Republic Received 21 February 2000; received in revised form 15 November 2000; accepted 15 February 2001
Abstract This paper presents a mathematical model of banded columns and theoretical analysis of factors influencing their behaviour. The equations for the real load᎐carrying capacity are given. The results of the theoretical behaviour are compared with experimental results. 䊚 2001 Elsevier Science Ltd. All rights reserved. Keywords: Mathematical model; Reinforced concrete; Columns; Steel banding
1. Introduction Steel bandage is a structure comprising longitudinal steel angles and transversal strips fixed to a column of a squarerrectangular section. The steel angles are embedded into cement mortar at the column corners. After cement mortar hard-setting, pre-heated strips are welded to the steel angles. Bandage reduces the deformations of concrete in transversal direction of the columns loaded by pressure, thus increasing the compression strength of concrete. The testing program consisted of 34 banded Žfullyrpartially . columns and five columns without bandage cast from concrete C12r15 w1x. Identical bandage was applied on tworthree columns. The dimensions of the specimens were 300 = 300 = 1500 mm, the diameter of longitudinal reinforcement was 14 mm, stirrups of 5.5 mm in diameter were spaced 200 mm apart. Partial bandage considers using non-continuous steel angles Žin lengths of 90 mm for the given dimension of
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the columns., while full bandage considers the continuous ones. The influence of axial spacing astr and crosssectional area A str of the strips on the load᎐carrying capacity of the columns was tested on the partially banded columns ŽPBCs.. The influence of the angle cross-section on the load᎐carrying capacity was tested on the columns fully banded ŽFBCs.. The columns were loaded by the central pressure up to the onset of the visible faults of concrete. The longitudinal and transversal deformations of concrete and the deformation of strips and angles were measured in the course of the loading. The onset of the faults was examined by acoustic emission and visually. Different qualities of concrete of the columns tested made it difficult to exactly formulate the examined effects. Behaviour and load᎐carrying capacity relations of the FBCsrPBCs to the RC columns can be influenced by further factors: side ratio of a rectangular column; cross-sectional area; largest nominal maximum aggregates size in concrete mixture; number and arrangement of reinforcement; and by the load actions before and in the course of the bandage application. Since it was impossible to implement further experimental tests for economical reasons, mathematical modelling of the banded columns was performed.
0950-0618r01r$ - see front matter 䊚 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 5 0 - 0 6 1 8 Ž 0 1 . 0 0 0 1 4 - 9
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2. Deformation and strength of concrete at the triaxial state of stress The strength of concrete f c3 at the triaxial state of stress depends mainly on the intensity and directions of the principal stresses Ž 1 G 2 G 3 ., on the quality of concrete and on the level and history of loading, respectively. From the tests results is derived the strength area w2᎐5x. Some properties of concrete influencing behaviour of the banded columns are described further later w5x. 䢇
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Particularly at the triaxial pressure, the strength and deformability of concrete is considerably higher than at the uniaxial one. The stress᎐strain diagrams of concrete for the clamping effects 1 s 2 are possible to determine in accordance with w5x. It is evident that variation of the values 1 s 2 results in changes of the stress᎐strain diagrams. Deformability of concrete at the ultimate strength f c3 goes down with the increasing quality of concrete and arises with the increasing clamping effects Ž 1 s 2 . ŽFig. 1.. The relative strength of concrete f c3 rfc grows with the clamping effects Ž 1 s 2 . and is independent on the quality of concrete. Ratio of the unit strains Ž 1 s 2 .r 3 at the ultimate strength of concrete is independent on the quality of concrete but depends on the clamping effects Ž 1 s 2 ..
mathematical modelling of the above mentioned methods, rheological properties of concrete are neglected and, in most cases, it is considered constant transversal deformation along the full height of the columns. For that reason, recommendations presented in these models were not taken into account for the banded columns modelling. At the banded columns loading, different deformations develop themselves in each point of concrete with reference to the actual state of stress. Since the clamping effects 1 s 2 of bandage differ in each point of concrete, the stress᎐strain diagram cannot be applied as a constant one. In the case of banded columns it was found that the problem could be eliminated by introduction of the multilinear stress᎐strain diagram at the ultimate state of concrete. The strain᎐stress diagram was constructed by interconnecting n q 1 points X i w 3i , f c3 x for the various clamping effects 1i s 2 , i i i s 1, . . . n Žsee Fig. 2.. For the transversal strain of concrete was derived ratio Ž cx rcz . s 0.280, Ž cx s cy ., as the mean value for the entire interval of the loading. The ratio value agreed with the values obtained from measuring and with the ones derived from the code w5x. A RC column with bandage was modelled as the three-dimensional structure with physically non-linear behaviour of all materials. The column was loaded by the concentric normal force. Regarding the steel angles behaviour in the course of the tests w1x and their potential behaviour in practice, three types of models were considered: 䢇
3. Mathematical model In practice, steel tubes filled with concrete and RC columns confined by circular spirals w7,8x are mostly used to increase the strength of concrete. The effects of stirrups for RC columns are described in w9᎐12x. At
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PBC model (P). Slipping of the angles was left out of consideration. The slip is understood as a fault of static bond between the steel angles and mortar along height of a column. FBC model considered partial slip of the angles (F). Moreover, the model took into account different axial strain of the four angles. Prediction of the
Fig. 1. Influence of the relative clamping effects Ž 1 s 2 .rfcm on the relative unit deformation 3 r c1 for concrete C12r15 to C50r60 reaching the ultimate strength f c3 . c1 s y0.0022 is a compressive strain Žuniaxial . in concrete on the peak stress f cm , f ctm is mean value of the axial tensile strength, Ec is mean value of secant modulus of elasticity w6x.
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Table 1 The test results and prediction for the middle area of the PBCs ŽP. and FBCs ŽF. a
Fig. 2. Derivation of the multilinear stress᎐strain diagram at the ultimate strength of concrete C12r15.
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load᎐carrying capacity of concrete was made for the areas at the top, middle and bottom of the columns and it differed in the values of slip. The equations, therefore, will be used for comparison of the tested columns carrying capacities. FBC model considered total slip of the angles (CD). From the structural safety design standpoint the effect of normal stress of the angles is ignored.
The relation between the state of stress of concrete and the strength of concrete w5x was calculated with the concrete strength coefficient cf . Algorithm of the calculation was designed to express cf either a multiple of the unaxial compressive strengths f c or a multiple of the unaxial tensile strengths f ct for the case when the tensor of stress develops the state corresponding to the strength area of concrete. This coefficient has analogous expressing ability as the relation crfc Ž ctrfct . for the case of the uniaxial pressure Žtension.. The stress᎐strain diagrams for steels were considered according to Fig. 2 w1x. Software ANSYS w13x was used for mathematical modelling. The scale space grid was 25 = 25 = 20 mm. Reinforced concrete was modelled by three-dimensional element Slolid65. Steel bandage and mortar between the angles and concrete of the PBCs was modelled by three-dimensional ‘bricks’. Mortar of the FBCs was modelled by element Contact52. Heating of the steel strips at bandage application on temperature T was modelled with the strip cooling on temperature y2r3⌬T Ž ⌬T s T y 20⬚C.. Since the results of the mathematical modelling proved that the shrinking and creeping effects of concrete influence the load᎐bearing capacity of the banded columns insignificantly, the model of these phenomena is not presented.
Bandage
Nf ŽkN.
⌽c
NR ŽkN.
RC1 P1 P2 P3 P4 P5 P6 P7 F1 F2 F3 F4
1500 2050 2400 1850 1750 2100 2050 1800 2500 2550 2750 3050
1 1.543 1.643 1.431 1.287 1.612 1.485 1.326 1.670 1.628 1.610 1.685
᎐ 1946 2428 1742 1730 1930 1992 1809 2581 2430 2849 3130
a
Nf is the normal force produced with the hydraulic press on the boundary of the visible fault onset, ⌽c is the ultimate capacity factor of concrete at the values Nf and NR is the prediction of the normal force at the failure limit of a column at the volumes of concrete Vc2 s 50 = 50 = 40 mm
columns P1 to P7 and F1 to F4 was simulated on the mathematical model described above. For the characteristics of concretes, there were considered mean values of two identically banded columns. The columns were loaded by the increasing normal force up to the faults of concrete in the middle area. The deformations on the materials between the points shown in the Fig. 1 w1x, the stress of longitudinal reinforcement, stirrups, strips, angles and the strength coefficient of concrete cf of all elements of concrete was calculated. Consequently, cf was calculated for the volumes of concretes from Vc2 s 50 = 50 = 40 to Vc4 s 100 = 100 = 100 mm ŽFig. 3.. From the coefficient values it was possible to calculate the side length a v of the cubes for the case when the tensor of the stress Žat loading of the column by the force Nf . generates the state of stress corresponding to the concrete strength area Ž cf s 1..
3.1. Verification of the mathematical model General information about the tests and the results are presented in Table 1 w1x. The behaviour of the
Fig. 3. Relation between the normal force N and the concrete strength coefficient cf for P2 ŽActivation of the bandage at N s 0 by heating of the strips, T s 150⬚C..
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The clamping effect of bandage on the concrete compressive strength f c3 can be expressed by the ultimate capacity factor ⌽c s Fc3rFc . The resultant of the normal stresses of concrete at the failure limit for the RC columns is possible to express approximately Fc s 0.85 A c f c and the resultant Fc3 q4 of the normal stress of concrete at the failure limit for the banded columns can be expressed as follows: Fc3 s Nf q A s s y Fan , where 䢇
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A c is total cross-sectional area of a concrete section; f c compressive strength q4 of concrete; A s Ž s . area Žcompressive stress y4. of longitudinal reinforcement; and Fan resultant of axial normal stresses of the angles at the failure limit of the FBCs that was determined from the measured values of ⌬ an and the strain᎐stress diagrams of the angles. For the PBCs it is Fan s 0.
The difference between the unit strain values obtained from the tests w1x and modelling were found Žat the normal force Nf , that corresponds to the ultimate capacity in the middle area of the columns. in range: 䢇
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from y11.0 to 10.2 Ž%. for the longitudinal strains of concrete ⌬ cz ; from y23.2 to 19.60 Ž%. for the transversal strains of concrete ⌬ cz ; and from y34.5 to 23.8 Ž%. for the strain of strips ⌬ str .
The ultimate capacity factor of concrete ⌽c Žat Nf . and the normal force at the failure limit of the columns NR Žat the concrete volumes Vc2 s 50 = 50 = 40 mm. were calculated Žsee Table 1.. Fig. 3 shows the extremes of the maximum values of the concrete strength coefficient cf for the column P2 at the volumes Vc2 to Vc4 . The analysis of the modelling and the test results showed that for the determination of the boundary of the visible faults onset, it is possible to consider as a ‘smeared’ state of strain in the concrete blocks of the volume approximately Vc2 s 50 = 50 = 40 mm and it corresponds to the volume of the cubes: V ( Ž 3dg .
3
of FBC concrete express the values of ultimate capacity factor of concrete ⌽c s 1.287 to 1.685 ŽTable 1.. 4. Factors effecting the ultimate capacity The effects of various factor parameters of bandage on the column ultimate capacity are presented. The ultimate capacity will be expressed either by the objective ultimate capacity factor of concrete ⌽ i or by the relative factor ⌿I s ⌽ ir⌽ i,ref , where ⌽ i,ref is the ultimate capacity factor for the bandage with the mean values of its parameters. Eqs. Ž2P., Ž2F. and Ž2CD. ᎐Eq. Ž9P,F,CD. were obtained by data approximation calculated on the models P, F, CD. The characteristics of materials used for the mathematical modelling of the column behaviour were considered to be of the mean values of the measured ones. The bandage is always activated by heating Žprestressing. of the strips on temperature T s 150⬚C on the columns that are identical in shape, reinforcement and in quality of concrete with the columns tested. 4.1. Spacing of steel strips The influence of axial spacing of the steel strips astr in proportion to the side length a of the square section of a column on the ultimate capacity of FBC and PBC concrete is expressed by the strip spacing factor Žsee Fig. 4.: ⌽ar s 1 q 0.283
ž aa / str
y1
,
Ž 2P .
⌽ar s 1 q 0.90y 0.45
ž
astr a
/
⌽ar s 1 q 1.02y 0.51
astr a
/
ž
1.3
,
Ž 2F.
.
Ž 2CD.
1.3
The equations are valid for the interval 0.333F Ž astrra. F 1.333.
Ž1.
where dg is the largest nominal maximum aggregate size. For the columns tested, dg s 16 mm. The favourable effects of bandage on the ultimate capacity
Fig. 4. The influence of the strip spacing astr on the carrying capacity of concrete expressed by factor of the strips spacing ⌽ar .
L. Cirtek r Construction and Building Materials 15 (2001) 351᎐359
355
4.2. Cross-sectional area of steel strips The influence of the cross-sectional area A str of the steel strips in proportion to the cross-section area A of a column on the ultimate capacity of FBC and PBC concrete is expressed by the relative factor of strip sectional area: ⌿Ar s 1 q
Ž A strrA . y 0.00333 9.42 Ž A strrA .
0.0022F Ž A strrA . F 0.0061
0.8
, Ž 3P,F,CD.
According to this equation the increase of the bandage efficiency for Ž A str rA. ) 0.004 is very low. 4.3. Steel angles This prediction relates only to the influence of restriction of the concrete lateral strain due to the effects of the steel angle stiffness on FBC concrete ultimate capacity. J is the moment of inertia of the steel angles to the central axis of inertia perpendicular to the axis of symmetry at the cross-section of the angles. The influence of the angles on the column 300 = 300 mm of size is expressed by the steel angle factor: ⌿an s 0.671q
0.363 J J q 1 ⭈ 10y8
.
Fig. 6. The influence of square section size on concrete loading capacity of the banded columns expressed by factor ⌿ag Ž AstrrAs 0.0033, dg s 16 mm.. Extreme of maximum and minimum impacts. The three data in the legend indicates type of prediction, astrra, quality of concrete.
creasing quality of concrete ŽFig. 6.. This property of concrete is accompanied by the increasing efficiency of the bandage. The effect is expressed by the quality factor of concrete: ⌿c s 0.882y
0.596 f c y 4.97 f q 1.187 ct , f c q 77 fc
Ž 5P .
⌿c s 0.880y
0.621 f c y 5.186 f q 1.397 ct , f c q 48.65 fc
Ž 5F.
⌿c s 0.992y
0.585 f c y 4.888 f q 1.171 ct . f c q 32.65 fc
Ž 5CD.
Ž 4F.
The efficiency of full bandage formed by the corner angles 60 = 60 = 6 mm of size is considerably high and applying angles of a large size has no corresponding effects on the ultimate capacity of concrete ŽFig. 5.. Side length aan and thickness tan of the 60 = 60 = 6 mm angles correspond to the equations: aan s 0.2 a, tan s 0.1aan . 4.4. Quality of concrete The deformability of concrete arises with the de-
The equations are derived for concrete of the compression strength f c s 8᎐25 Nrmm2 and for the tensile strength f ct corresponding to f ctrfc s 0.08᎐0.14. 4.5. Rectangular section of a column The banded column can be of a rectangular section with the sides a, b Ž a- b .. The influence of the side ratio on the ultimate capacity of concrete respects the section shape relative factor: ⌿ab s 0.2
a q 0.8, 0.5F Ž arb . F 1. b
Ž 6P,F,CD.
4.6. Square section of columns, maximum size of aggregate in concrete mixture Fig. 5. The influence of moment of inertia of the angles J on FBC concrete ultimate capacity expressed by the steel angle factor ⌽an s 1489 ⌿an .
The influence of size of the square section Žlength side a. and the maximum size of aggregate dg on FBC and PBC concrete ultimate capacity has a common
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‘denominator’ ᎏ the strength effect of the scale space grid of the model ᎏ see Eq. Ž1.. The section size and the maximum size of aggregate are expressed by the following factor:
⌿ag s 1 q
ž
ag y 0.3 8.6 ag
exp
/
,
where: ag s a
0.016 , a,d g w mx , dg
Ž 7P . Fig. 7. The influence of aggregate maximum size dg of concrete mixture to concrete loading capacity of the banded columns by factor ⌿ag .
exp s y2.143 q 2.464, ) 1.150, f y2 s ⌽ar⌿Ar ⌿C 1.287y 2.117 ct , fc
ž
ag y 0.3 ⌿ag s 1 q 14.5ag
ž
/
exp
/
,
where: exp s y3.333 q 3.963, ) 1.190, f y2 s ⌽ar⌿Ar ⌿C 1.107y 1.325 ct , fc
ž
ag y 0.3 ⌿ag s 1 q 7.1ag
ž
4.7. Banding of loaded columns
Ž 7F.
/
exp
/
,
where: exp s y1.972 q 2.132, ) 1.081, f s ⌽ar⌿C 1.150y 1.842 ct . fc
ž
Ž 7CD.
/
Eqs. Ž7P., Ž7F. and Ž7CD. are derived for the banded columns of the length side from as 0.3 to 1.2 m and the maximum aggregates size from dg s 8 to 32 mm. With the increasing carrying capacity of concrete of the banded columns Ž . of the section 300 = 300 mm the values of the factor ⌿ag decrease Žat the constant value dg . and thus the carrying capacity of concrete decreases for a) 300 mm. Dependence of the maximum aggregate size on the load᎐carrying capacity of concrete is presented in Fig. 7. Eqs. Ž7P., Ž7F. and Ž7CD. were verified by tests only for dg s 16 mm. For design of the FBC columns in accordance of the code w5x is, therefore, suitable to consider negative effect of this influence in the interval 0.008F dg F 0.016 wmx only, given by factor ⌿dg maximum aggregate size according to the equation: ⌿dg s 0.892q 6.8dg w mx .
Ž 8CD.
In case a column is loaded before the bandage application, it can influence its carrying capacity, especially when the compressive stress of concrete < c < G 0.6 f c . Development of microscopic faults occurs at that stress. For that reason, the mathematical model considers the lower tensile strength of concrete. The negative effects of shrinkage and the positive ones of creeping concrete on the carrying capacity of the banded columns were also calculated within the mathematical modelling. The analyses showed very low influence of these effects on the carrying capacity. For prediction of the FBC and PBC faults, the columns that were identical with the shape and reinforcement of the tested ones were considered. Prior to the bandage application the columns that were loaded by the normal force N1 relating to the ratio N crfc Ns 0, 0.50, 0.75 and 1.00. The relative state of stress of concrete crfc of the columns before banding can be approximately expressed by the ratio of the normal forces: crfc f N1rNfo . The equation for the normal force Nfo at the column failure limit on the instant before banding is: Nfo s 0.85 A c f c q A s f y , where A s is area of longitudinal reinforcement and f y is the yield stress of reinforcement. The effect of loading in the course of the bandage application can be expressed by the factor Žsee Fig. 8.: ⌿N s 1 y 0.2
N1 Nfo
ž /
3
.
Ž 9P,F,CD.
4.8. Pre-heating (pre-stressing) of strips Heating of the strips on the temperature 0 F T F 215⬚C showed that the carrying capacity of concrete
L. Cirtek r Construction and Building Materials 15 (2001) 351᎐359
partial bandage fulfils the conditions aan s 0.2  , tan s 0.1a an , l an ( 0.3  wmx; the length side aan , the thickness tan of the angle cross-section and the length l an of the angle for full bandage fulfils the conditions 0.133  F aan F 0.233  , 0.100 aan F tan F 0.125 aan , l an ( l col y 0.05 wmx; prior to welding, the strips are pre-heated to the temperature T ( 150 Ž120.⬚C when the mean compressive strength of concrete fulfils Ždoes not fulfil. the condition f c - 19 Nrmm2 ; the yield stress of reinforcement is equal 288 Nrmm2 at least; and the strength of all materials are considered to be the mean values determining by the tests.
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Fig. 8. The relation of the stress᎐state of concrete crfc to the banded columns ultimate capacity on the instant before banding is expressed by the factor ⌽ N s ⌽ar ⌿Ar ⌿an ⌿C ⌿ab ⌿ag ⌿N .
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enlarges with the increasing heating temperature. However, extremely high temperatures can cause the onset of microscopic cracks. To eliminate this mode of failure, it is necessary to pre-heat the strips to the temperature T ( 150 Ž120.⬚C when the mean compressive strength of concrete fulfils Ždoes not fulfil. the condition f c - 19 Nrmm2 . 4.9. Longitudinal reinforcement The influence of positioning, number and yield stress of reinforcement on the bandage efficiency was examined. The effects of these parameters on the ultimate capacity were insignificant.
5. Real carrying capacity Determination of the normal force at the failure limit NR is described further. The tests of the experimental columns and the mathematical modelling were based on the following pre-conditions: 䢇
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the largest nominal maximum aggregates size dg meets the condition 8 F dg F 32 wmmx; the column is loaded by the concentric compressive force; the sectional area is either of a square or a rectangular shape of sides a, b, where the longer side b is maximally 1.2 m and side ratio is arbG 0.5; the axial distance of the strip astr fulfils the condition, 0.333F Ž astrr . F 1.333, where  s 0.5 Ža q b.; Ž10. the sectional area A str of the strips corresponds to the interval 0.0022F Ž A strrab. F 0.0061; the compression strength f c and the tensile strength f ct of concrete fulfil the conditions 8 F f c F 25 wNrmm2 x, 0.08 F f ctrfc . F 0.14; the length side aan , the thickness tan of the angle cross-section and the length l an of the angle for
357
The carrying capacity factor of concrete is expressed by ⌽c s ⌽ar ⌿Ar ⌿an ⌿C ⌿ab ⌿ag ⌿N , where the equations for the factors are given in parentheses: ⌽ar Eq. Ž2P,2F.; ⌿Ar Eq. Ž3P,F,CD.; ⌿an Eq. Ž4F. for the full bandage; and ⌿an s 1 for the partial bandage; ⌿C Eq. Ž5P,5F.; ⌿ab Eq. Ž6P,F,CD.; ⌿ag Eq. Ž7P,7F.; and ⌿N Eq. Ž9P,F,CD.. The side length a of the column section have to be substituted by the mean length  Eq. Ž10. in the case, where the parameter is included in the equations. The equations marked with P have to be used for the partial bandage. The equation marked with F have to be used for the faults at the top, middle and bottom of the fully banded columns. Table 2 Test results and prediction. Nf is the normal force induced by the press at the visible failure limit of concrete, NR -prediction Žaccording to the chapter 3. of the normal force at the failure limit of the column concrete Bandage
Nf ŽkN.a
NR ŽkN.a
P1 P2 P3 P4 P5 P6 P7 F1 F2 F3 F4 F5 F6 P8 F7
2050 2400 1850 1750 2100 2050 1800 25004 Ž2100. 25504 Ž2300. 27504 Ž2550. 30504 Ž2650. 2870 2575 2533 2917
1930 2392 1769 1716 1992 2056 1863 24934 Ž2379. 24454 Ž2208. 26284 Ž2474. 31284 Ž2860. 2795 2663 2477 2916
a
The fault along the depth of the columns are located: wherever corresponding values Nf , NR ; in the middle Nf , NR 4; and at the toprbottom Ž Nf , NR ..
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The resultant of the normal axial stresses of the angles is expressed by:
tom of the columns, for the bandages P8 ŽF7. it was an s 0.15 Ž1.0.. The values NR are presented in Table 2.
Fan s y4an A an , where the axial stress of the angles an y4 is taken from the stress᎐strain diagram. The unit strain an of the angles is: an s an ⌬ cz
Ž 11 .
Where an is the angle slip factor considered by value: 䢇
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for the fault of concrete at the toprbottom of the columns from an s 0.15 to 0.40; and for the fault of concrete in the mid of the columns...... an s 0.5. ⌬ cz is the guaranteed relative unit strain of concrete considered by value:
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for the columns unloaded before the banding ⌬ cz s 0.00674y 0.00595 ⌽c for the interval 1 - ⌽c - 2.054:
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for the columns loaded before banding ⌬ cz s 0 for the interval 1 - ⌽c - 1.179 and ⌬ cz s 0.00875⌽c y 0.01032 for the interval 1.179 - ⌽c - 2.054.
For the angles of partial bandage it is Fan s 0. Prediction of the real normal force NR {q} of the banded columns at the failure limit is expressed as:
Appendix A. Notation CWB FBC PBC a Ž b. A T aan Ž tan . astr dg fc Ž fct . fctm fc3 fy lan lcol tstr Aan Ac As Astr Fan Ec Fc
NR s Fc q Fan q Fs , where Fc is resultant of the normal stress at the ultimate strength of concrete Fc s 0.85⌽c A c f c and Fs is the compressive force in longitudinal reinforcement at the ultimate strength Fs s A s fs . 5.1. Verification of equations for real carrying capacity determination
The normal forces NR at the concrete failure limit for the bandages presented in the Table 1 w1x were calculated as described in the chapter 5. The angle slip factor an in the Eq. Ž11. was design by the mean value an s 0.275 for the faults of concrete at the toprbot-
Ž Fc3 . Fs J Nf Nfo NR N1
 1 , 2 , 3 an
Columns without bandage Fully banded column Partially banded columns Short Žlong. side of the rectangular section of the column Cross-sectional area of the column Pre-heating temperature Žprestressing. of the strips Side length Ždepth. of the equilateral angle Axial distance of the strips Maximum aggregate size Compression Žtensile . strength of the concrete Mean value of axial tensile strength Concrete strength at the triaxial state of stress Yield stress of reinforcement Total length of the angle Length of the column Depth of the strips Area of the steel angles Cross-sectional area of the column concrete part Area of longitudinal reinforcement Cross-sectional area of the strip Resultant of axial normal stresses of the angles at the failure limit of the banded columns Modulus of elasticity in compression of the concrete Resultant of the normal stresses of concrete at the failure limit for the RC Žbanded. column Compressive force in longitudinal reinforcement at the ultimate strength Moment of inertia of an angle Normal force at the visible failure limit of concrete induced by the hydraulic press Normal force at the column failure limit immediately before banding Prediction of the normal force at the failure limit of the banded columns Normal force acting on the column in the course of banding Mean side length of the section Principal stresses Stress of the steel angles
L. Cirtek r Construction and Building Materials 15 (2001) 351᎐359
c Ž ct . s 1 , 2 ,
3 c1 cx , cy , cz ⌬ an ⌬ cz , ⌬ cx , ⌬ str an cf ⌽ar ⌽c ⌿ab ⌿an , ⌽an ⌿ag ⌿Ar ⌿C ⌿N
359
Compression Žtensile . stresses of the concrete
References
Stress of reinforcement Unit strain corresponding principal stresses 1, 2 , 3
w1x Cirtek L. RC columns strengthened with bandage ᎏ experimental programme and design recommendations. w2x Cedolin L-, Crutzen RJ, Dei P. Triaxial stress᎐strain relationship for concrete. J Eng Mech Div ASCE Proc. Paper 12360 1977;103:423᎐439. w3x Ahmad SH, Shah SP. Complete triaxial stress-strain curves for concrete. J Struct Div ASCE Proc. Paper 1982;108:728᎐741. w4x William KJ, Warnke ED. Constitutive model for triaxial behaviour of concrete. Proceedings, International Association for Bridge and Structural Engineering, vol 19. Bergamo, Italy: ISMES, 1975. w5x CEB-FIP Model Code 90, 1991, Lausanne, pp. 34᎐51. w6x ENV 1992-1-1 ŽEurocode 2., Design of concrete structures. Part1: General rules and rules for buildings, 1991. w7x Sheikh SA, Toklucu M. Reinforced concrete columns confined by circular spirals and hoops. ACI Struct J 1993;5:542᎐553. w8x Pessiki S, Pieroni A. Axial load behaviour of large-scale spirally-reinforced high-strength concrete columns. ACI Struct J 1997;3:304᎐314. w9x Abdel-Halim MAH, Abu-Lebdch TA. Analytical study for concrete confinement in tied columns. J Struct Eng 1989; 115Ž11.:2810᎐2828. w10x Sheikh SA, Shah DV, Khoury SS. Confinement of high-strength concrete columns. ACI Struct J 1994;1:100᎐111. w11x Sheikh SA, Uzumeri SM. Strength and ductility of tied concrete columns. J Struct Div 1980;106ŽST5.:1079᎐1102. w12x Fafitis A, Shah SP. Predictions of ultimate behaviour of confined columns subjected to large deformation. ACI J 1986:423᎐433. w13x ANSYS software, Swanson Analysis System, Inc, Houston.
Compressive strain Žuniaxial . in concrete on the peak stress fcm w5x Unit strain of concrete
Axial strain of the steel angles Longitudinal unit strain of concrete, transversal unit strain of concrete and strips Angle slip factor Concrete strength coefficient Factor of the strips spacing Ultimate capacity factor for concrete of the banded columns Factor of the section shape Factor of the steel angle Factor of the section size and the maximum size of aggregate Factor of strip sectional area Factor of concrete quality Factor for the load on RC column prior the bandage application
Mathematical model of RC banded column behaviour L. CirtekU Brno Uni¨ ersity of Technology, Faculty of Ci¨ il Engineering, Udolni 53, Brno 662-42, Czech Republic Received 21 February 2000; received in revised form 15 November 2000; accepted 15 February 2001
Abstract This paper presents a mathematical model of banded columns and theoretical analysis of factors influencing their behaviour. The equations for the real load᎐carrying capacity are given. The results of the theoretical behaviour are compared with experimental results. 䊚 2001 Elsevier Science Ltd. All rights reserved. Keywords: Mathematical model; Reinforced concrete; Columns; Steel banding
1. Introduction Steel bandage is a structure comprising longitudinal steel angles and transversal strips fixed to a column of a squarerrectangular section. The steel angles are embedded into cement mortar at the column corners. After cement mortar hard-setting, pre-heated strips are welded to the steel angles. Bandage reduces the deformations of concrete in transversal direction of the columns loaded by pressure, thus increasing the compression strength of concrete. The testing program consisted of 34 banded Žfullyrpartially . columns and five columns without bandage cast from concrete C12r15 w1x. Identical bandage was applied on tworthree columns. The dimensions of the specimens were 300 = 300 = 1500 mm, the diameter of longitudinal reinforcement was 14 mm, stirrups of 5.5 mm in diameter were spaced 200 mm apart. Partial bandage considers using non-continuous steel angles Žin lengths of 90 mm for the given dimension of
U
Tel.: q420-5-4114-6229; fax: q420-5-4321-2106. E-mail address: [email protected] ŽL. Cirtek..
the columns., while full bandage considers the continuous ones. The influence of axial spacing astr and crosssectional area A str of the strips on the load᎐carrying capacity of the columns was tested on the partially banded columns ŽPBCs.. The influence of the angle cross-section on the load᎐carrying capacity was tested on the columns fully banded ŽFBCs.. The columns were loaded by the central pressure up to the onset of the visible faults of concrete. The longitudinal and transversal deformations of concrete and the deformation of strips and angles were measured in the course of the loading. The onset of the faults was examined by acoustic emission and visually. Different qualities of concrete of the columns tested made it difficult to exactly formulate the examined effects. Behaviour and load᎐carrying capacity relations of the FBCsrPBCs to the RC columns can be influenced by further factors: side ratio of a rectangular column; cross-sectional area; largest nominal maximum aggregates size in concrete mixture; number and arrangement of reinforcement; and by the load actions before and in the course of the bandage application. Since it was impossible to implement further experimental tests for economical reasons, mathematical modelling of the banded columns was performed.
0950-0618r01r$ - see front matter 䊚 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 5 0 - 0 6 1 8 Ž 0 1 . 0 0 0 1 4 - 9
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2. Deformation and strength of concrete at the triaxial state of stress The strength of concrete f c3 at the triaxial state of stress depends mainly on the intensity and directions of the principal stresses Ž 1 G 2 G 3 ., on the quality of concrete and on the level and history of loading, respectively. From the tests results is derived the strength area w2᎐5x. Some properties of concrete influencing behaviour of the banded columns are described further later w5x. 䢇
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Particularly at the triaxial pressure, the strength and deformability of concrete is considerably higher than at the uniaxial one. The stress᎐strain diagrams of concrete for the clamping effects 1 s 2 are possible to determine in accordance with w5x. It is evident that variation of the values 1 s 2 results in changes of the stress᎐strain diagrams. Deformability of concrete at the ultimate strength f c3 goes down with the increasing quality of concrete and arises with the increasing clamping effects Ž 1 s 2 . ŽFig. 1.. The relative strength of concrete f c3 rfc grows with the clamping effects Ž 1 s 2 . and is independent on the quality of concrete. Ratio of the unit strains Ž 1 s 2 .r 3 at the ultimate strength of concrete is independent on the quality of concrete but depends on the clamping effects Ž 1 s 2 ..
mathematical modelling of the above mentioned methods, rheological properties of concrete are neglected and, in most cases, it is considered constant transversal deformation along the full height of the columns. For that reason, recommendations presented in these models were not taken into account for the banded columns modelling. At the banded columns loading, different deformations develop themselves in each point of concrete with reference to the actual state of stress. Since the clamping effects 1 s 2 of bandage differ in each point of concrete, the stress᎐strain diagram cannot be applied as a constant one. In the case of banded columns it was found that the problem could be eliminated by introduction of the multilinear stress᎐strain diagram at the ultimate state of concrete. The strain᎐stress diagram was constructed by interconnecting n q 1 points X i w 3i , f c3 x for the various clamping effects 1i s 2 , i i i s 1, . . . n Žsee Fig. 2.. For the transversal strain of concrete was derived ratio Ž cx rcz . s 0.280, Ž cx s cy ., as the mean value for the entire interval of the loading. The ratio value agreed with the values obtained from measuring and with the ones derived from the code w5x. A RC column with bandage was modelled as the three-dimensional structure with physically non-linear behaviour of all materials. The column was loaded by the concentric normal force. Regarding the steel angles behaviour in the course of the tests w1x and their potential behaviour in practice, three types of models were considered: 䢇
3. Mathematical model In practice, steel tubes filled with concrete and RC columns confined by circular spirals w7,8x are mostly used to increase the strength of concrete. The effects of stirrups for RC columns are described in w9᎐12x. At
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PBC model (P). Slipping of the angles was left out of consideration. The slip is understood as a fault of static bond between the steel angles and mortar along height of a column. FBC model considered partial slip of the angles (F). Moreover, the model took into account different axial strain of the four angles. Prediction of the
Fig. 1. Influence of the relative clamping effects Ž 1 s 2 .rfcm on the relative unit deformation 3 r c1 for concrete C12r15 to C50r60 reaching the ultimate strength f c3 . c1 s y0.0022 is a compressive strain Žuniaxial . in concrete on the peak stress f cm , f ctm is mean value of the axial tensile strength, Ec is mean value of secant modulus of elasticity w6x.
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Table 1 The test results and prediction for the middle area of the PBCs ŽP. and FBCs ŽF. a
Fig. 2. Derivation of the multilinear stress᎐strain diagram at the ultimate strength of concrete C12r15.
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load᎐carrying capacity of concrete was made for the areas at the top, middle and bottom of the columns and it differed in the values of slip. The equations, therefore, will be used for comparison of the tested columns carrying capacities. FBC model considered total slip of the angles (CD). From the structural safety design standpoint the effect of normal stress of the angles is ignored.
The relation between the state of stress of concrete and the strength of concrete w5x was calculated with the concrete strength coefficient cf . Algorithm of the calculation was designed to express cf either a multiple of the unaxial compressive strengths f c or a multiple of the unaxial tensile strengths f ct for the case when the tensor of stress develops the state corresponding to the strength area of concrete. This coefficient has analogous expressing ability as the relation crfc Ž ctrfct . for the case of the uniaxial pressure Žtension.. The stress᎐strain diagrams for steels were considered according to Fig. 2 w1x. Software ANSYS w13x was used for mathematical modelling. The scale space grid was 25 = 25 = 20 mm. Reinforced concrete was modelled by three-dimensional element Slolid65. Steel bandage and mortar between the angles and concrete of the PBCs was modelled by three-dimensional ‘bricks’. Mortar of the FBCs was modelled by element Contact52. Heating of the steel strips at bandage application on temperature T was modelled with the strip cooling on temperature y2r3⌬T Ž ⌬T s T y 20⬚C.. Since the results of the mathematical modelling proved that the shrinking and creeping effects of concrete influence the load᎐bearing capacity of the banded columns insignificantly, the model of these phenomena is not presented.
Bandage
Nf ŽkN.
⌽c
NR ŽkN.
RC1 P1 P2 P3 P4 P5 P6 P7 F1 F2 F3 F4
1500 2050 2400 1850 1750 2100 2050 1800 2500 2550 2750 3050
1 1.543 1.643 1.431 1.287 1.612 1.485 1.326 1.670 1.628 1.610 1.685
᎐ 1946 2428 1742 1730 1930 1992 1809 2581 2430 2849 3130
a
Nf is the normal force produced with the hydraulic press on the boundary of the visible fault onset, ⌽c is the ultimate capacity factor of concrete at the values Nf and NR is the prediction of the normal force at the failure limit of a column at the volumes of concrete Vc2 s 50 = 50 = 40 mm
columns P1 to P7 and F1 to F4 was simulated on the mathematical model described above. For the characteristics of concretes, there were considered mean values of two identically banded columns. The columns were loaded by the increasing normal force up to the faults of concrete in the middle area. The deformations on the materials between the points shown in the Fig. 1 w1x, the stress of longitudinal reinforcement, stirrups, strips, angles and the strength coefficient of concrete cf of all elements of concrete was calculated. Consequently, cf was calculated for the volumes of concretes from Vc2 s 50 = 50 = 40 to Vc4 s 100 = 100 = 100 mm ŽFig. 3.. From the coefficient values it was possible to calculate the side length a v of the cubes for the case when the tensor of the stress Žat loading of the column by the force Nf . generates the state of stress corresponding to the concrete strength area Ž cf s 1..
3.1. Verification of the mathematical model General information about the tests and the results are presented in Table 1 w1x. The behaviour of the
Fig. 3. Relation between the normal force N and the concrete strength coefficient cf for P2 ŽActivation of the bandage at N s 0 by heating of the strips, T s 150⬚C..
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The clamping effect of bandage on the concrete compressive strength f c3 can be expressed by the ultimate capacity factor ⌽c s Fc3rFc . The resultant of the normal stresses of concrete at the failure limit for the RC columns is possible to express approximately Fc s 0.85 A c f c and the resultant Fc3 q4 of the normal stress of concrete at the failure limit for the banded columns can be expressed as follows: Fc3 s Nf q A s s y Fan , where 䢇
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A c is total cross-sectional area of a concrete section; f c compressive strength q4 of concrete; A s Ž s . area Žcompressive stress y4. of longitudinal reinforcement; and Fan resultant of axial normal stresses of the angles at the failure limit of the FBCs that was determined from the measured values of ⌬ an and the strain᎐stress diagrams of the angles. For the PBCs it is Fan s 0.
The difference between the unit strain values obtained from the tests w1x and modelling were found Žat the normal force Nf , that corresponds to the ultimate capacity in the middle area of the columns. in range: 䢇
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from y11.0 to 10.2 Ž%. for the longitudinal strains of concrete ⌬ cz ; from y23.2 to 19.60 Ž%. for the transversal strains of concrete ⌬ cz ; and from y34.5 to 23.8 Ž%. for the strain of strips ⌬ str .
The ultimate capacity factor of concrete ⌽c Žat Nf . and the normal force at the failure limit of the columns NR Žat the concrete volumes Vc2 s 50 = 50 = 40 mm. were calculated Žsee Table 1.. Fig. 3 shows the extremes of the maximum values of the concrete strength coefficient cf for the column P2 at the volumes Vc2 to Vc4 . The analysis of the modelling and the test results showed that for the determination of the boundary of the visible faults onset, it is possible to consider as a ‘smeared’ state of strain in the concrete blocks of the volume approximately Vc2 s 50 = 50 = 40 mm and it corresponds to the volume of the cubes: V ( Ž 3dg .
3
of FBC concrete express the values of ultimate capacity factor of concrete ⌽c s 1.287 to 1.685 ŽTable 1.. 4. Factors effecting the ultimate capacity The effects of various factor parameters of bandage on the column ultimate capacity are presented. The ultimate capacity will be expressed either by the objective ultimate capacity factor of concrete ⌽ i or by the relative factor ⌿I s ⌽ ir⌽ i,ref , where ⌽ i,ref is the ultimate capacity factor for the bandage with the mean values of its parameters. Eqs. Ž2P., Ž2F. and Ž2CD. ᎐Eq. Ž9P,F,CD. were obtained by data approximation calculated on the models P, F, CD. The characteristics of materials used for the mathematical modelling of the column behaviour were considered to be of the mean values of the measured ones. The bandage is always activated by heating Žprestressing. of the strips on temperature T s 150⬚C on the columns that are identical in shape, reinforcement and in quality of concrete with the columns tested. 4.1. Spacing of steel strips The influence of axial spacing of the steel strips astr in proportion to the side length a of the square section of a column on the ultimate capacity of FBC and PBC concrete is expressed by the strip spacing factor Žsee Fig. 4.: ⌽ar s 1 q 0.283
ž aa / str
y1
,
Ž 2P .
⌽ar s 1 q 0.90y 0.45
ž
astr a
/
⌽ar s 1 q 1.02y 0.51
astr a
/
ž
1.3
,
Ž 2F.
.
Ž 2CD.
1.3
The equations are valid for the interval 0.333F Ž astrra. F 1.333.
Ž1.
where dg is the largest nominal maximum aggregate size. For the columns tested, dg s 16 mm. The favourable effects of bandage on the ultimate capacity
Fig. 4. The influence of the strip spacing astr on the carrying capacity of concrete expressed by factor of the strips spacing ⌽ar .
L. Cirtek r Construction and Building Materials 15 (2001) 351᎐359
355
4.2. Cross-sectional area of steel strips The influence of the cross-sectional area A str of the steel strips in proportion to the cross-section area A of a column on the ultimate capacity of FBC and PBC concrete is expressed by the relative factor of strip sectional area: ⌿Ar s 1 q
Ž A strrA . y 0.00333 9.42 Ž A strrA .
0.0022F Ž A strrA . F 0.0061
0.8
, Ž 3P,F,CD.
According to this equation the increase of the bandage efficiency for Ž A str rA. ) 0.004 is very low. 4.3. Steel angles This prediction relates only to the influence of restriction of the concrete lateral strain due to the effects of the steel angle stiffness on FBC concrete ultimate capacity. J is the moment of inertia of the steel angles to the central axis of inertia perpendicular to the axis of symmetry at the cross-section of the angles. The influence of the angles on the column 300 = 300 mm of size is expressed by the steel angle factor: ⌿an s 0.671q
0.363 J J q 1 ⭈ 10y8
.
Fig. 6. The influence of square section size on concrete loading capacity of the banded columns expressed by factor ⌿ag Ž AstrrAs 0.0033, dg s 16 mm.. Extreme of maximum and minimum impacts. The three data in the legend indicates type of prediction, astrra, quality of concrete.
creasing quality of concrete ŽFig. 6.. This property of concrete is accompanied by the increasing efficiency of the bandage. The effect is expressed by the quality factor of concrete: ⌿c s 0.882y
0.596 f c y 4.97 f q 1.187 ct , f c q 77 fc
Ž 5P .
⌿c s 0.880y
0.621 f c y 5.186 f q 1.397 ct , f c q 48.65 fc
Ž 5F.
⌿c s 0.992y
0.585 f c y 4.888 f q 1.171 ct . f c q 32.65 fc
Ž 5CD.
Ž 4F.
The efficiency of full bandage formed by the corner angles 60 = 60 = 6 mm of size is considerably high and applying angles of a large size has no corresponding effects on the ultimate capacity of concrete ŽFig. 5.. Side length aan and thickness tan of the 60 = 60 = 6 mm angles correspond to the equations: aan s 0.2 a, tan s 0.1aan . 4.4. Quality of concrete The deformability of concrete arises with the de-
The equations are derived for concrete of the compression strength f c s 8᎐25 Nrmm2 and for the tensile strength f ct corresponding to f ctrfc s 0.08᎐0.14. 4.5. Rectangular section of a column The banded column can be of a rectangular section with the sides a, b Ž a- b .. The influence of the side ratio on the ultimate capacity of concrete respects the section shape relative factor: ⌿ab s 0.2
a q 0.8, 0.5F Ž arb . F 1. b
Ž 6P,F,CD.
4.6. Square section of columns, maximum size of aggregate in concrete mixture Fig. 5. The influence of moment of inertia of the angles J on FBC concrete ultimate capacity expressed by the steel angle factor ⌽an s 1489 ⌿an .
The influence of size of the square section Žlength side a. and the maximum size of aggregate dg on FBC and PBC concrete ultimate capacity has a common
L. Cirtek r Construction and Building Materials 15 (2001) 351᎐359
356
‘denominator’ ᎏ the strength effect of the scale space grid of the model ᎏ see Eq. Ž1.. The section size and the maximum size of aggregate are expressed by the following factor:
⌿ag s 1 q
ž
ag y 0.3 8.6 ag
exp
/
,
where: ag s a
0.016 , a,d g w mx , dg
Ž 7P . Fig. 7. The influence of aggregate maximum size dg of concrete mixture to concrete loading capacity of the banded columns by factor ⌿ag .
exp s y2.143 q 2.464, ) 1.150, f y2 s ⌽ar⌿Ar ⌿C 1.287y 2.117 ct , fc
ž
ag y 0.3 ⌿ag s 1 q 14.5ag
ž
/
exp
/
,
where: exp s y3.333 q 3.963, ) 1.190, f y2 s ⌽ar⌿Ar ⌿C 1.107y 1.325 ct , fc
ž
ag y 0.3 ⌿ag s 1 q 7.1ag
ž
4.7. Banding of loaded columns
Ž 7F.
/
exp
/
,
where: exp s y1.972 q 2.132, ) 1.081, f s ⌽ar⌿C 1.150y 1.842 ct . fc
ž
Ž 7CD.
/
Eqs. Ž7P., Ž7F. and Ž7CD. are derived for the banded columns of the length side from as 0.3 to 1.2 m and the maximum aggregates size from dg s 8 to 32 mm. With the increasing carrying capacity of concrete of the banded columns Ž . of the section 300 = 300 mm the values of the factor ⌿ag decrease Žat the constant value dg . and thus the carrying capacity of concrete decreases for a) 300 mm. Dependence of the maximum aggregate size on the load᎐carrying capacity of concrete is presented in Fig. 7. Eqs. Ž7P., Ž7F. and Ž7CD. were verified by tests only for dg s 16 mm. For design of the FBC columns in accordance of the code w5x is, therefore, suitable to consider negative effect of this influence in the interval 0.008F dg F 0.016 wmx only, given by factor ⌿dg maximum aggregate size according to the equation: ⌿dg s 0.892q 6.8dg w mx .
Ž 8CD.
In case a column is loaded before the bandage application, it can influence its carrying capacity, especially when the compressive stress of concrete < c < G 0.6 f c . Development of microscopic faults occurs at that stress. For that reason, the mathematical model considers the lower tensile strength of concrete. The negative effects of shrinkage and the positive ones of creeping concrete on the carrying capacity of the banded columns were also calculated within the mathematical modelling. The analyses showed very low influence of these effects on the carrying capacity. For prediction of the FBC and PBC faults, the columns that were identical with the shape and reinforcement of the tested ones were considered. Prior to the bandage application the columns that were loaded by the normal force N1 relating to the ratio N crfc Ns 0, 0.50, 0.75 and 1.00. The relative state of stress of concrete crfc of the columns before banding can be approximately expressed by the ratio of the normal forces: crfc f N1rNfo . The equation for the normal force Nfo at the column failure limit on the instant before banding is: Nfo s 0.85 A c f c q A s f y , where A s is area of longitudinal reinforcement and f y is the yield stress of reinforcement. The effect of loading in the course of the bandage application can be expressed by the factor Žsee Fig. 8.: ⌿N s 1 y 0.2
N1 Nfo
ž /
3
.
Ž 9P,F,CD.
4.8. Pre-heating (pre-stressing) of strips Heating of the strips on the temperature 0 F T F 215⬚C showed that the carrying capacity of concrete
L. Cirtek r Construction and Building Materials 15 (2001) 351᎐359
partial bandage fulfils the conditions aan s 0.2  , tan s 0.1a an , l an ( 0.3  wmx; the length side aan , the thickness tan of the angle cross-section and the length l an of the angle for full bandage fulfils the conditions 0.133  F aan F 0.233  , 0.100 aan F tan F 0.125 aan , l an ( l col y 0.05 wmx; prior to welding, the strips are pre-heated to the temperature T ( 150 Ž120.⬚C when the mean compressive strength of concrete fulfils Ždoes not fulfil. the condition f c - 19 Nrmm2 ; the yield stress of reinforcement is equal 288 Nrmm2 at least; and the strength of all materials are considered to be the mean values determining by the tests.
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Fig. 8. The relation of the stress᎐state of concrete crfc to the banded columns ultimate capacity on the instant before banding is expressed by the factor ⌽ N s ⌽ar ⌿Ar ⌿an ⌿C ⌿ab ⌿ag ⌿N .
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enlarges with the increasing heating temperature. However, extremely high temperatures can cause the onset of microscopic cracks. To eliminate this mode of failure, it is necessary to pre-heat the strips to the temperature T ( 150 Ž120.⬚C when the mean compressive strength of concrete fulfils Ždoes not fulfil. the condition f c - 19 Nrmm2 . 4.9. Longitudinal reinforcement The influence of positioning, number and yield stress of reinforcement on the bandage efficiency was examined. The effects of these parameters on the ultimate capacity were insignificant.
5. Real carrying capacity Determination of the normal force at the failure limit NR is described further. The tests of the experimental columns and the mathematical modelling were based on the following pre-conditions: 䢇
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the largest nominal maximum aggregates size dg meets the condition 8 F dg F 32 wmmx; the column is loaded by the concentric compressive force; the sectional area is either of a square or a rectangular shape of sides a, b, where the longer side b is maximally 1.2 m and side ratio is arbG 0.5; the axial distance of the strip astr fulfils the condition, 0.333F Ž astrr . F 1.333, where  s 0.5 Ža q b.; Ž10. the sectional area A str of the strips corresponds to the interval 0.0022F Ž A strrab. F 0.0061; the compression strength f c and the tensile strength f ct of concrete fulfil the conditions 8 F f c F 25 wNrmm2 x, 0.08 F f ctrfc . F 0.14; the length side aan , the thickness tan of the angle cross-section and the length l an of the angle for
357
The carrying capacity factor of concrete is expressed by ⌽c s ⌽ar ⌿Ar ⌿an ⌿C ⌿ab ⌿ag ⌿N , where the equations for the factors are given in parentheses: ⌽ar Eq. Ž2P,2F.; ⌿Ar Eq. Ž3P,F,CD.; ⌿an Eq. Ž4F. for the full bandage; and ⌿an s 1 for the partial bandage; ⌿C Eq. Ž5P,5F.; ⌿ab Eq. Ž6P,F,CD.; ⌿ag Eq. Ž7P,7F.; and ⌿N Eq. Ž9P,F,CD.. The side length a of the column section have to be substituted by the mean length  Eq. Ž10. in the case, where the parameter is included in the equations. The equations marked with P have to be used for the partial bandage. The equation marked with F have to be used for the faults at the top, middle and bottom of the fully banded columns. Table 2 Test results and prediction. Nf is the normal force induced by the press at the visible failure limit of concrete, NR -prediction Žaccording to the chapter 3. of the normal force at the failure limit of the column concrete Bandage
Nf ŽkN.a
NR ŽkN.a
P1 P2 P3 P4 P5 P6 P7 F1 F2 F3 F4 F5 F6 P8 F7
2050 2400 1850 1750 2100 2050 1800 25004 Ž2100. 25504 Ž2300. 27504 Ž2550. 30504 Ž2650. 2870 2575 2533 2917
1930 2392 1769 1716 1992 2056 1863 24934 Ž2379. 24454 Ž2208. 26284 Ž2474. 31284 Ž2860. 2795 2663 2477 2916
a
The fault along the depth of the columns are located: wherever corresponding values Nf , NR ; in the middle Nf , NR 4; and at the toprbottom Ž Nf , NR ..
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The resultant of the normal axial stresses of the angles is expressed by:
tom of the columns, for the bandages P8 ŽF7. it was an s 0.15 Ž1.0.. The values NR are presented in Table 2.
Fan s y4an A an , where the axial stress of the angles an y4 is taken from the stress᎐strain diagram. The unit strain an of the angles is: an s an ⌬ cz
Ž 11 .
Where an is the angle slip factor considered by value: 䢇
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for the fault of concrete at the toprbottom of the columns from an s 0.15 to 0.40; and for the fault of concrete in the mid of the columns...... an s 0.5. ⌬ cz is the guaranteed relative unit strain of concrete considered by value:
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for the columns unloaded before the banding ⌬ cz s 0.00674y 0.00595 ⌽c for the interval 1 - ⌽c - 2.054:
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for the columns loaded before banding ⌬ cz s 0 for the interval 1 - ⌽c - 1.179 and ⌬ cz s 0.00875⌽c y 0.01032 for the interval 1.179 - ⌽c - 2.054.
For the angles of partial bandage it is Fan s 0. Prediction of the real normal force NR {q} of the banded columns at the failure limit is expressed as:
Appendix A. Notation CWB FBC PBC a Ž b. A T aan Ž tan . astr dg fc Ž fct . fctm fc3 fy lan lcol tstr Aan Ac As Astr Fan Ec Fc
NR s Fc q Fan q Fs , where Fc is resultant of the normal stress at the ultimate strength of concrete Fc s 0.85⌽c A c f c and Fs is the compressive force in longitudinal reinforcement at the ultimate strength Fs s A s fs . 5.1. Verification of equations for real carrying capacity determination
The normal forces NR at the concrete failure limit for the bandages presented in the Table 1 w1x were calculated as described in the chapter 5. The angle slip factor an in the Eq. Ž11. was design by the mean value an s 0.275 for the faults of concrete at the toprbot-
Ž Fc3 . Fs J Nf Nfo NR N1
 1 , 2 , 3 an
Columns without bandage Fully banded column Partially banded columns Short Žlong. side of the rectangular section of the column Cross-sectional area of the column Pre-heating temperature Žprestressing. of the strips Side length Ždepth. of the equilateral angle Axial distance of the strips Maximum aggregate size Compression Žtensile . strength of the concrete Mean value of axial tensile strength Concrete strength at the triaxial state of stress Yield stress of reinforcement Total length of the angle Length of the column Depth of the strips Area of the steel angles Cross-sectional area of the column concrete part Area of longitudinal reinforcement Cross-sectional area of the strip Resultant of axial normal stresses of the angles at the failure limit of the banded columns Modulus of elasticity in compression of the concrete Resultant of the normal stresses of concrete at the failure limit for the RC Žbanded. column Compressive force in longitudinal reinforcement at the ultimate strength Moment of inertia of an angle Normal force at the visible failure limit of concrete induced by the hydraulic press Normal force at the column failure limit immediately before banding Prediction of the normal force at the failure limit of the banded columns Normal force acting on the column in the course of banding Mean side length of the section Principal stresses Stress of the steel angles
L. Cirtek r Construction and Building Materials 15 (2001) 351᎐359
c Ž ct . s 1 , 2 ,
3 c1 cx , cy , cz ⌬ an ⌬ cz , ⌬ cx , ⌬ str an cf ⌽ar ⌽c ⌿ab ⌿an , ⌽an ⌿ag ⌿Ar ⌿C ⌿N
359
Compression Žtensile . stresses of the concrete
References
Stress of reinforcement Unit strain corresponding principal stresses 1, 2 , 3
w1x Cirtek L. RC columns strengthened with bandage ᎏ experimental programme and design recommendations. w2x Cedolin L-, Crutzen RJ, Dei P. Triaxial stress᎐strain relationship for concrete. J Eng Mech Div ASCE Proc. Paper 12360 1977;103:423᎐439. w3x Ahmad SH, Shah SP. Complete triaxial stress-strain curves for concrete. J Struct Div ASCE Proc. Paper 1982;108:728᎐741. w4x William KJ, Warnke ED. Constitutive model for triaxial behaviour of concrete. Proceedings, International Association for Bridge and Structural Engineering, vol 19. Bergamo, Italy: ISMES, 1975. w5x CEB-FIP Model Code 90, 1991, Lausanne, pp. 34᎐51. w6x ENV 1992-1-1 ŽEurocode 2., Design of concrete structures. Part1: General rules and rules for buildings, 1991. w7x Sheikh SA, Toklucu M. Reinforced concrete columns confined by circular spirals and hoops. ACI Struct J 1993;5:542᎐553. w8x Pessiki S, Pieroni A. Axial load behaviour of large-scale spirally-reinforced high-strength concrete columns. ACI Struct J 1997;3:304᎐314. w9x Abdel-Halim MAH, Abu-Lebdch TA. Analytical study for concrete confinement in tied columns. J Struct Eng 1989; 115Ž11.:2810᎐2828. w10x Sheikh SA, Shah DV, Khoury SS. Confinement of high-strength concrete columns. ACI Struct J 1994;1:100᎐111. w11x Sheikh SA, Uzumeri SM. Strength and ductility of tied concrete columns. J Struct Div 1980;106ŽST5.:1079᎐1102. w12x Fafitis A, Shah SP. Predictions of ultimate behaviour of confined columns subjected to large deformation. ACI J 1986:423᎐433. w13x ANSYS software, Swanson Analysis System, Inc, Houston.
Compressive strain Žuniaxial . in concrete on the peak stress fcm w5x Unit strain of concrete
Axial strain of the steel angles Longitudinal unit strain of concrete, transversal unit strain of concrete and strips Angle slip factor Concrete strength coefficient Factor of the strips spacing Ultimate capacity factor for concrete of the banded columns Factor of the section shape Factor of the steel angle Factor of the section size and the maximum size of aggregate Factor of strip sectional area Factor of concrete quality Factor for the load on RC column prior the bandage application