Composite mechanical performance of prefabricated concrete based on hysteresis curve equation

At present, the retrofitting of existing houses has become one of the hot issues in urban construction in my country. To ensure the normal use of existing houses in the process of building and adding floors, some scholars have put forward the idea of ‘self-supporting floors in the construction phase.’ In the construction phase, the prestressed steel truss bears the frame beam's weight and construction load. This solves the problem that the original roof cannot bear the new construction load, ensures the normal use of the original building during the construction process, saves investment, and shortens the construction period due to the integration of structural force and construction measures. However, the angle steel of the built-in truss chord of the frame beam occupies a larger space when passing through the built-in column. This affects the arrangement of the frame column longitudinal reinforcement, the frame beam prestressed reinforcement, and the horn tube, and at the same time, it is not convenient for the prestressed reinforcement to be anchored.

The angle steel-concrete column used in the new frame structure means that the rigid angle steel is placed in the column instead of the longitudinal steel bars. The steel plate hoop welded to the longitudinal angle steel replaces the stirrup to form a space steel skeleton and concrete component. Because there is no additional longitudinal reinforcement on the outside of the angle steel, the thickness of the concrete protective layer of the angle steel skeleton can be relatively reduced, roughly the same as that of ordinary reinforced concrete columns [1]. This increases the moment arm of the angle steel, enables the column to play a greater role, and facilitates embedded parts. At the same time, the angle steel frame can be prefabricated in the factory to simplify the construction procedure and speed up the construction progress. Good economic benefits will be achieved in actual projects.

There are not many reports on the study of the force performance of the normal section of the angle steel-concrete column. The corresponding skeleton curve is extracted from the hysteresis curve of the specimen [2]. The specimens all suffered a large eccentric bending failure. Based on experimental research, the force performance of the normal section of the angle steel-concrete column was discussed. The experiment in this article provides a reference for the elasticity analysis of the structure.

The analysis of the previous test process shows that the damage of the test piece occurs at the end of the test piece, and the ultimate flexural bearing capacity of the damaged section can be obtained by Eq. (1): (1) Mut=Pu(L/2)+NΔm/2 M_u^t = {P_u}(L/2) + N{\Delta _m}/2 As shown in Table 2, L is the clear height of the test column and N is the axial force exerted on the test piece [11]. The Mut M_u^t obtained from the test is shown in Table 2. The assumption of flat section is still valid, and the ultimate compressive strain of concrete when reaching the limit state can still be taken as 0.0033. The concrete stress pattern in the compression zone is equivalent to a rectangular stress pattern. The limit height x_b of the compression zone can be calculated according to formula (2): (2) xb=0.8(h−as)1+fv0.0033Ess {x_b} = {{0.8\left( {h - {a_s}} \right)} \over {1 + {{{f_v}} \over {0.0033{E_{ss}}}}}} a_s is the distance from the inner edge of the tension angle to the tension edge. f_y is the yield strength of the tension angle steel; E_ss is the elastic modulus of the angle steel. Figure 6 shows the cross-sectional stress when the full cross-section of the large-eccentric column under compression and tension angle steel yields. In the figure b is the width of the column section.

Fig. 6

We did not consider the effect of section strain gradient on improving the concrete compressive strength of eccentrically compressed members. The equivalent rectangular concrete compressive strength is still taken as f_c, and the effect of different eccentricity e_a is not considered in the bearing capacity formula. The bearing capacity of the section shown in Figure 6 can be calculated according to the following formula: (3) {N=fcbx+fy'Ass′−fyAssN(ηe0+h2−ass)=fcbx(h−ass−x2)+fs′Ass′(h−ass−ass′) \left\{ {\matrix{ {N = {f_c}bx + f_y^\prime A_{ss}^\prime - {f_y}{A_{ss}}} \hfill \cr {N\left( {\eta {e_0} + {h \over 2} - {a_{ss}}} \right) = {f_c}bx\left( {h - {a_{ss}} - {x \over 2}} \right) + f_s^\prime A_{ss}^\prime (h - {a_{ss}} - a_{ss}^\prime )} \hfill \cr } } \right. The x calculation by Eq. (3) must satisfy the calculation x < x_b, x_b according to Eq. (2). At the same time: (4) x≥0.8as′1−fv0.0033Ess x \ge {{0.8a_s^\prime } \over {1 - {{{f_v}} \over {0.0033{E_{ss}}}}}} When the condition of formula (4) cannot be satisfied, the compression angle steel cannot reach the full-section yield in the limit state. There may be two situations where the section is yielded, or the whole section is not [12]. When the outer edge of the compression angle steel yields, the concrete height in the compression zone is (5) x=0.8c1−fv0.0033Ess x = {{0.8c} \over {1 - {{{f_v}} \over {0.0033{E_{ss}}}}}} c is the thickness of the angle steel protective layer. (6) 0.0033εy′=x/0.8(x/0.8−c−d) {{0.0033} \over {\varepsilon _y^\prime }} = {{x/0.8} \over {(x/0.8 - c - d)}} The stress of the inner edge of the compressed angle steel is σss′ \sigma _{ss}^\prime : (7) fy′σss′=x/0.8−c−dx/0.8−as′ {{f_y^\prime } \over {\sigma _{ss}^\prime }} = {{x/0.8 - c - d} \over {x/0.8 - a_s^\prime }} At this time, the section bearing capacity can be calculated according to formula (8): (8) {N=fcbx+fy′Assy′+σss′(Ass′−Assy′)+12(fy′−σss′)(Ass′−Assy′)−fyAssN(ηe0+h2−ass)=fcbx(h−ass−x2)+fy′Assy′(h−ass−c−g)+σss′(Ass′−Assy′)⋅[h−ass−c−d−12(as′−c−d)]+12(fy′−σss′)(Ass′−Assy′)⋅[h−ass−c−d−13)(as′−c−d)] \left\{ {\matrix{ {N = {f_c}bx + f_y^\prime A_{ss}^{{y^\prime }} + \sigma _{ss}^\prime \left( {A_{ss}^\prime - A_{ss}^{{y^\prime }}} \right) + {1 \over 2}\left( {f_y^\prime - \sigma _{ss}^\prime } \right)\left( {A_{ss}^\prime - A_{ss}^{{y^\prime }}} \right) - {f_y}{A_{ss}}} \hfill \cr {N\left( {\eta {e_0} + {h \over 2} - {a_{ss}}} \right) = {f_c}bx\left( {h - {a_{ss}} - {x \over 2}} \right) + f_y^\prime A_{ss}^{{y^\prime }}\left( {h - {a_{ss}} - c - g} \right) + \sigma _{ss}^\prime \left( {A_{ss}^\prime - A_{ss}^{{y^\prime }}} \right) \cdot } \hfill \cr {\left[ {h - {a_{ss}} - c - d - {1 \over 2}\left( {a_s^\prime - c - d} \right)} \right] + {1 \over 2}\left( {f_y^\prime - \sigma _{ss}^\prime } \right)\left( {A_{ss}^\prime - A_{ss}^{{y^\prime }}} \right) \cdot \left[ {h - {a_{ss}} - c - d - {1 \over 3})(a_s^\prime - c - d)} \right]} \hfill \cr } } \right. The formula Assy′ A_{ss}^{{y^\prime }} is the area of the pressure angle steel reaching the yield part. g is the distance from the area's center of gravity where the pressure angle steel reaches the yield part to the outer edge of the pressure angle steel. From Eq. (8) x needs to satisfy the following equation: (9) 0.8c1−fv0.0033Ess<x<0.8as′1−fv0.0033Ess {{0.8c} \over {1 - {{{f_v}} \over {0.0033{E_{ss}}}}}} < x < {{0.8a_s^\prime } \over {1 - {{{f_v}} \over {0.0033{E_{ss}}}}}} When x does not satisfy the formula (9), it means that the full section of the compression angle steel has not yielded. This situation in which all the tight angles have not yielded does not fully play the role of the tight angles. This situation should be avoided when designing.

The flexural bearing capacity of the section of the 6 test columns obtained according to the above calculation formula is shown in Table 2. Compared with the test result Mut M_u^t , the average value of Muc M_u^c is the mean square error, and the coefficient of variation is. The formula calculation result is conservative.

When x > x_b calculated by the formula (3) shows that the full section of the tension angle steel fails to reach the yield strength in the ultimate state. The yield strength should be calculated based on small eccentric compression members. Tensile angle steel may show partial yielding or not yielding at all. When the outer edge of the tension angle steel yields, the concrete height x of the compression zone is: (10) x=0.8(h−c)1+fy0.0033Ess x = {{0.8(h - c)} \over {1 + {{{f_y}} \over {0.0033{E_{ss}}}}}} d₁ is the length of the tension angle steel to yield. The strain indication shows that: (11) 0.0033εy=x/0.8(x/0.8−c−d1) {{0.0033} \over {{\varepsilon _y}}} = {{x/0.8} \over {\left( {x/0.8 - c - {d_1}} \right)}} The stress of the inner edge of the tension angle steel is σ_ss: (12) fyσss=x/0.8−c−d1x/0.8−as {{{f_y}} \over {{\sigma _{ss}}}} = {{x/0.8 - c - {d_1}} \over {x/0.8 - {a_s}}} At this time, the section bearing capacity can be calculated according to formula (13): (13) {N=fcbx+fy′Ass′+σss′(Ass′−Assy)−12(fy−σss)(Ass−Assy)−fyAssyN(ηe0+h2−ass′)=fcbx(ass′−x2)+fyAssy(h−ass′−c−g1)+σss(Ass−Assy)⋅[h−ass′−c−d−12(as−c−d)]+12(fy−σss)(Ass−Assy)[h−ass′−c−d−13)(as−c−d)] \left\{ {\matrix{ {N = {f_c}bx + f_y^\prime A_{ss}^\prime + \sigma _{ss}^\prime \left( {A_{ss}^\prime - A_{ss}^y} \right) - {1 \over 2}\left( {{f_y} - {\sigma _{ss}}} \right)\left( {{A_{ss}} - A_{ss}^y} \right) - {f_y}A_{ss}^y} \hfill \cr {N\left( {\eta {e_0} + {h \over 2} - a_{ss}^\prime } \right) = {f_c}bx\left( {a_{ss}^\prime - {x \over 2}} \right) + {f_y}A_{ss}^y\left( {h - a_{ss}^\prime - c - {g_1}} \right) + {\sigma _{ss}}\left( {{A_{ss}} - A_{ss}^y} \right) \cdot } \hfill \cr {\left[ {h - a_{ss}^\prime - c - d - {1 \over 2}\left( {{a_s} - c - d} \right)} \right] + {1 \over 2}\left( {{f_y} - {\sigma _{ss}}} \right)\left( {{A_{ss}} - A_{ss}^y} \right)} \hfill \cr {\left[ {h - a_{ss}^\prime - c - d - {1 \over 3})\left( {{a_s} - c - d} \right)} \right]} \hfill \cr } } \right. Assy A_{ss}^y is the area of the tensioned angle that reaches the yielding part. Equation (13) x needs to satisfy the following equation: (14) 0.8(h−as)1+fy0.0033Ess<x<0.8(h−c)1+fy0.0033Ess {{0.8(h - {a_s})} \over {1 + {{{f_y}} \over {0.0033{E_{ss}}}}}} < x < {{0.8(h - c)} \over {1 + {{{f_y}} \over {0.0033{E_{ss}}}}}} When x does not satisfy the formula (14), it means that the full section of the tension angle steel has not yielded. This situation in which all the tension angles have not yielded also does not fully play the role of the tension angles. This situation should also be avoided when designing.

We have completed the test of six-angle steel concrete columns with steel plate hoops under the horizontal low-cycle reciprocating load and obtained the test column's failure form and hysteresis curve. The hysteretic curve extracts the skeleton curve of six-angle steel-concrete columns. The specimens all suffered a large eccentric bending failure. The force performance of the normal section of the angle steel-concrete column is discussed based on experimental research. At the same time, the formula for calculating the bearing capacity of the normal section of a large eccentrically loaded angle steel-concrete column is given. The paper introduces the calculation formula for the bearing capacity of the normal section of a small eccentric load angle steel-concrete column.