PREDICTION OF THE ULTIMATE LONGITUDINAL COMPRESSIVE CONCRETE STRAIN AT HOOP FRACTURE USING ENERGY CONSIDERATIONS

The fracture of transverse hoop reinforcement can lead to the collapse of a reinforced concrete column, as has been observed in concrete bridges and buildings attacked by severe earthquakes as well as in laboratory tests. To predict the longitudinal concrete strain at the stage of first hoop fracture a theoretical method based on considerations of strain energy referred to as "Energy Balance Theory" has been proposed by Mander et al. This paper reviews the "Energy Balance Theory" and then proposes several modifications for this theory based on a failure model of a reinforced concrete column subject to axial compression. These modifications take into account significant energy factors neglected in the theory by Mander et al and correct some unrealistic assumptions made in that theory. The predictions of the modified theory are then compared with the results obtained from concentric loading tests on 18 reinforced concrete columns conducted at the University of Canterbury and the validity of the modified theory is assessed.

Hence hoop fracture should be avoided within an expected range of column deformation under any design load condition especially such as the deformations imposed by a severe earthquake.
In order to provide such a restriction to the column deformation, a method for the prediction of the available maximum deformation of the column section limited by the occurrence of hoop fracture is required > The available maximum deformation of a column section could be expressed in terms of an ultimate longitudinal compressive strain of the core concrete at an associated curvature.Note that for the calculation of U g in the last term in Eqs. 3 and 4, de s #as used instead of dt where t = axial compressive strain in the s longitudinal bars.However the axial strain of the concrete core will only be equal to the axial strain in the longitudinal bars before buckling of the longitudinal bars occurs and provided that there is no bond slip of the longitudinal bars.
Once buckling of the longitudinal bars has commenced the strain t c should be expressed as the sum of ^the axial compressive strain of the longitudinal bars t and an apparent strain due to the axial shortening of the longitudinal bars induced by the bar buckling e 1 .
This can be expressed in the form c = t + f. ' (5) ess ' The buckled condition of a longitudinal bar may be modelled as shown in Fig. 3.The total axial displacement AT can be separated into the displacement due to the axial compression £A and the displacement due to the buckling AB. t c , t s and t ?g in Eq. 5 correspond to AT/1, AA/1 and AB/1 respectively, where 1 is the initial length of the bar before loading.
The energy relation between the external work done by the axial force P and the corresponding strain energy in €ne longitudinal bar and the hoop reinforcement can be written in the form is not so important here because the magnitude of the strain energy of that term is found to be considerably smaller than other terms in Eq.6 due to small bending stiffness of the longitudinal bar in the inelastic range and thus that term may be neglected.Now, reconsider the meaning of Eq.4 here.Rearranging Eq.4, The left hand side of the equation is almost equivalent to the external work done on an axially loaded column per unit length (U ).
To be more precise, the external work per unit length of the column U can be expressed in the form ^ The energy absorbed by the cover concrete U cov ^s usual ly only a small fraction of U because cover concrete area is usually mtlch smaller than the core concrete area and the strain at spalling is also smaller than the ultimate concete core strain (see U CQv in Table 2).
Therefore, U can be approximated by the terms on the lift hand side of Eq.7.
The meaning of Eq.8 can be reconfirmed by recalling the typical procedure used to determine the stressstrain relation of confined concrete from a concentric loading test on a reinforced concrete column.
As the first step, a total load versus axial strain relation.Curve (A) in Fig. 4 Eg.9 is identical with Eq. 8 .
Hence, it can be considered that the left hand side in Eq.7 is almost equivalent to the external work done on the column.Now some part of the external work done on the column by the loading machine should be transferred to the longitudinal bars.However, a term with the strain energy directly related to the longitudinal bars can not be found in the right hand side of Eq.7.
Therefore, to ensure that the terms on the right hand side of Eq.7 are adequate, all of the external work done on the longitudinal bars needs to be transferred to the hoop reinforcement.
In this case, in Eq. 6, this requires that A 1 s ,de =0 and si s 0 As a matter of course, this model can only be adequate from a certain stage of loading where the longitudinal bars have been compressed far into the plastic range and their bending stiffness has become negligibly small due to axial yielding.
On the other hand, if the axial force.P ^ reduces to zero due to serious buckling tne elastic energy released from the longitudinal bars U gb (which is represented by the shaded area in Fig. 6) might be transferred to the hoop reinforcement.However, this elastic energy U , may be negligibly small compared with the total strain energy absorbed in the Longitudinal bars in the case where the hoop spacing and the amount of hoops are adequate to achieve sufficient ductility for a column during severe earthquake loading.
For example, such serious buckling can be expected only at very high axial compressive strain in the case of a column designed by the New Zealand concrete code [5] which is based on ductile design philosophy.
Hence, it might be unnecessary to take into account this factor in Eq.6 for a ductile column.
From the considerations described above, it can be predicted that Eq. be used to determine U h .This value of Eq.10 was based on several tensile tests on reinforcing steel bars [1,2].Note that the integration range in Eq.10 is from zero to the hoop fracture strain t Usually f the measurement of fracture strain of a steel bar e f is significantly affected by the gauge length used in the tensile test.The strain distribution along the bar axis at the bar fracture can be represented as in Fig. 7.
It is evident that large strains concentrate over a small length of necked bar at the fracture.
The fracture strain calculated from the elongation between points A and B, which includes the necking part of the bar, is the average strain between those points.
The corresonding stress-strain curves can be expressed as shown in Fig. 8  = strain at the ultimate tensile strength of steel (see Fig. 8), instead of Eq.10.
The details of these tests are described in the following section.Hence, the gauge length used in the measurement of strain needs to be properly chosen for the estimation of the strain energy capacity of the hoop reinforcement U . .For example, if circular spirals with Sb mm bar diameter are provided in a column with a concrete core diameter of 1 m, the total length of the spiral in one turn of the spiral is about 200 times the spiral bar diameter.In this case, for the strain energy capacity of the spiral at first spiral fracture, 70 to 100 MJ/m calculated from Eq.11 will give a better estimation than 110 to 150 MJ/m calculated from Eq.10.
Moreover, the strain distribution of the spirals or hoops along the column axis also needs to be taken into account for the estimation of the U This is because the corresponding strain energy stored in the core concrete has to be calculated from a theoretical or a measured stress-strain relation of confined concrete over a defined gauge length.That is, the longitudinal concrete strain is an average strain within the gauge length used for the concrete strain definition or measurement.Hence, the calculated strain energy of the concrete core does not usually just correspond to the strain energy stored in the damaged region concentrated within one hoop spacing at which the first hoop fracture occurs.The longitudinal concrete strain used in the stress-strain relation might mean an average strain within a plastic hinge region of the column over a length of one-third to a full section depth of the column.
Therefore, if the buckled shape of the longitudinal bar is represented as in Fig. 9, the average strain of the hoops within the above mentioned length of the column needs to be used for the estimation of the strain energy stored in the hoops.
As a result, the value of U sh needs to be modified again based on the buckled shape expected.
If buckling over more than two hoop spacing is not expected prior to the hoop fracture, the value of U , obtained from Eq.11 can be used without further modificaiton because the hoop strains distributed along the coulmn axis will be almost constant.However, if buckling is limited within one hoop spacing, the external work done on the longitudinal bars may hardly be transferred to the hoop reinforcement because such buckling condition corresponds to the case without a tie bar representing hoops in Fig. 5 and thus the last term in Eq.6 must be eliminated.
In this case, the proposed equations (Eqs.(2) Core concrete: In the initial stage, works as a compression strut (spring k ) in the same manner as the cover cc concrete, until the complete formation of the shear sliding surfaces, and some percentage of the external work is transferred to hoops by Poisson's effect as represented by a pantograph in Fig. 13(a).
The angle H in the pantograph corresponds to the Poisson's ratio of the concrete treated as a unit solid material.It is assumed that the second stage commences with the complete formation of the shear sliding surfaces which is simultaneous with the complete spalling of cover concrete.After formation, the shear sliding surfaces act as a transformer which changes the axial load into lateral load by the wedging effect of the top and bottom cones in the case of type A, or wedge shaped lumps in the case of type B, shear sliding surfaces of Fig. 10.
Hence, the angle 6 1 of the pantograph in Fig. 13(b) can be determined from the inclination of the shear sliding surfaces.
The friction between the shear sliding surfaces (spring k ^) also provides resistance against axial load.
(3) Longitudinal reinforcement: In the initial stage, works as a compression strut (spring k ) against the axial load.In the final stage, after the onset of buckling, works as a pantograph which can transfer the axial load work to the hoop reinforcement by changing the load direction from axial direction to lateral direction.The angle B" in Fig. 13(c) can be determined from the buckling mode of the longitudinal bars.
(4) Hoop reinforcement: In the initial stage, works as lateral confinement which provides quasi-fluid pressure to the core concrete.
In the second stage, after the formation of the shear sliding surfaces in the core concrete, works as hoops to prevent sliding of the top and bottom cones or the wedge shaped lumps until buckling of the longitudinal bars commences.
In the final stage, works as anti-buckling support until the hoop fracture.
It is of interest that the angles 6,9' and B n in Fig. 13 need not be determined in order to calculate the longitudinal concrete strain at first hoop fracture.This is because only the strain energy capacity of the hoop reinforcement up to its fracture is required to be determined for this case.Only when the stress-strain relation of the confined concrete needs to be computed step by step, it is necessary to determine the angles 6 , $ ' and 0 " .In Fig. 13 On the other hand, in the high strain region in the second stage after the formation of the shear sliding surfaces, the improvement of the load carrying capacity is attributed to the restraint against the shear sliding along the surfaces between the core concrete parts separated by cracks shown in Fig. 10.For reinforced concrete column?subjected to concentric axial compression, a value for ?s of 0.004 may be assumed because of abseftce of the strain gradient in the column section.
For the reinforced concrete columns subjected to combined axial load and bending moment, a value for t of 0.006 or more may be adequate if the transverse reinforcement provided in the columns is not extremely congested, as has been found appropriate in a previous study [15] .
The value of a varies with the expected buckling mode of the longitudinal bars.The buckling mode can be expressed as a combination of the modes of a gross buckling and a local buckling as shown in Fig. 14.
If the expected local buckling is limited to within one hoop spacing without occurrence of the gross buckling a = 1 may be adopted because the hoop strain distribution along the column length will be approximately constant.
In the case of a column with the gross buckling of longitudinal reinforcement as shown in Fig. 9, a can be calculated as follows.If the core diameter of column is d , the c' total height of column is 3d_ and the gauge length of the concrete"" core strain measurement is d , the gross buckled shape of the longitudinal bars may be approximated by a sinusoidal curve expressed by the following equation.In Eq.19 it is assumed that the lateral expansion of the column at each end is negligible because even spalling of cover concrete cannot usually be observed in the end portions of the column, due to the effects of friction between the loading plate and the contact surface of the concrete at each end of the column.The average lateral displacement within the gauge length 1 of d c in the middle third of the column San be expressed in the form  Hence, in this case a = 0,9 might be an adequate value.
If the local buckling occurs over several hoop spacings in addition to the above gross buckling, the value of a needs to be reduced to a value which is less than 0.9.
If the local buckling in addition to the above gross buckling does not occur or is limited within one hoop spacing, the value of a need not be reduced from 0.9.The value of (i which express the ratio of strain energy absorbed in the compression reinforcement by axial yielding to external work done on the compression reinforcement is not so simple to calculate theoretically, because it depends on several factors such as the hoop spacing, the stiffness of the hoop reinforcment against the longitudinal bar buckling, the bending stiffness of the longitudinal bar in plastic buckling, and so on.However, the value of $ could be approximated using Fig. 15

Fig. 3
Fig. 3 Buckling Model for a Longitudinal Bar satisfy the above conditions, it would be necessary to assume that the buckling of the

Fig. 6
Fig. 7 Strain Distribution with Necking in a Steel Bar

Fig. 11
Fig. 11 Typical Failure Mode of a Circular Column the midheight of the column the height of the column (see Fig.14(a))