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3. STEEL BAR STRAIN AND STEEL BAR FRACTUREThe fracture of steel bar under cyclic load is closely related to the cyclic plastic strain which cannot be measured directly by the wire strain gage because it is too large. In order to study the steel bar fracture in relation with the plastic strain, the plastic strain equation of steel bar is derived here. The plastic strain equation of steel bar is obtained by the RC column model (Figure 8) which satisfies following conditions.(i) Steel bar stresses in the tension side distribute linearly in the regions of CA and CB respectively and the steel bar stress in the fixed-end section is equal to the tensile strength (σu).(ii) Plastic strains of steel bar in the tension side distribute linearly in the regions of CA and CB respectively. (iii) The position of neutral axis is the same in every section and the plane strain distribution in the section is assumed.(iv) Elastic strain and elastic deformation are neglected.According to the condition (i), the plastic ranges (Lp, Lpb) of steel bar in the upper region and the lower region of the fixed-end are given by (1).in which Lc: column length, σy: yield stress, τb: bond strength, ψ: perimeter of steel bar. The plastic deformation (δp) of RC column includes the two components of δp1 and δp2 which are corresponding to the plastic strains in the lower region (CB) and in (1) ēķēĪøσŀσļôôô ô –⎝⎠⎜⎟⎛⎞= ēķĩ ψσļ σŀ –ùπτĩôôôôôôôôôôôô ô = , the upper region (CA) respectively. The components of δp1 and δp2 are obtained from the condition (ii) and expressed by the curvature (κp) in the fixed-end section. Substituting in the relation of δp =δp1 +δp2, the curvature (κp) of the plastic column deformation is expressed only by δp.According to the condition (iii) the plastic strain (εp) of steel bar is decided by (4).in which X, Xn are the coordinates of the steel bar and the neutral axis in the section.The plastic strains (ε1, ε2) of the left side steel bar (X2) and the right side steel bar (X1) can be expressed by the plastic column deformation (δp) which can be measured strictly in the RC column test.In Figure 10 the calculated steel bar strains by (5) are compared with the test results δķøκķēķĩēĪùôôôôôôôôôôôôôô ô = (2) δķùκķēķēĪùôôôôôôô ôēķùýôôôôô ô –⎝⎠⎜⎟⎜⎟⎛⎞= ,(3) κķδķλôôô ô = λēķúēĪēķ– ()ýôôôôôôôôôôôôôôôôôôôôô ôēķĩēĪùôôôôôôôôô ô + = in which(4)εķğğĵ–λôôôôôôôôô ôδķ–=(5) εøğøğĵ–λôôôôôôôôôôô ôδķ–= εùğùğĵ–λôôôôôôôôôôô ôδķ–= , which are measured directly by the wire strain gage (WSG-0). The measured plastic strain of steel bar may include the effect of bending deformation which is not included in the calculation by (5). We can see that the peak-to-peak amplitude and the number of cycle, which effect on the very low cycle fatigue, are approximated well by (5).
The cyclic plastic strain calculated by (5) are shown in Figure 11. The white circles in Figure 11 are the peak points in every cycle of plastic strain. The cycles whose ampli-tudes are smaller than the critical strain (εcr) are neglected because the damage (Dcr) by the cycles are small enough (Dcr<0.007). The critical strain (εcr) is corresponding to the maximum amplitude of cycle between the skeleton curves and explained in Figure 9. By the use of (5), the plastic strain cycles (Nf)Tes t until the steel bar fracture of all specimens are obtained and shown in Table 1. 4. ANALYSIS OF STEEL BAR FRACTUREIn this study the steel bar fracture under strong cyclic load is assumed to be the very low cycle fatigue and the analysis method of it is derived on the basis of the Coffin-Manson expression and the Palmgren-Miner rule. According to the Coffin-Manson expression the relation between the number of cycles (Nf) until the steel bar fractures and the plastic strain amplitude (εpa) is given by (6). in which εf : fracture strain, c: constant. The number of cycle to fracture (Nf) of the Coffin-Manson expression is defined under the cyclic load with constant amplitude. In this study it is assumed that the Coffin-Manson expression is applicable to the steel bar fracture under the cyclic load with varying amplitude and the relation between the plas-tic strain amplitude (εpaj) and the cycles to fracture (Nfj) in the j-cycle is expressed by (7).The damage ratio of steel bar fracture (Dcr) under cyclic load is given by the sum-mation of the damage of every cycle and expressed by (8).The fracture condition of steel bar is given by Dcr=1 and it can be calculated by the plastic deformation (δpaj) of RC column because εpaj is given by δpaj as shown in (5).The modulus (c) in the Coffin-Manson expression is calculated under the condition εķĨεĭùĕĭ()Ī = (6)εķĨıεĭùĕĭı()Ī = (7)(8) ċcrøĕĭıôôôôô ôı∑ = when the calculation coincides with test result with the test results