Calculation method of ultimate moment resistance and moment-rotation curve for steel beam to concrete-filled steel tube column connections with bidirectional bolts

Abstract

The invention discloses a calculation method of ultimate moment resistance and moment-rotation relation for steel beam to concrete-filled steel tube column connections with bidirectional bolts, wherein the calculation method as follows: calculate ultimate moment resistance values of the connection for different failure modes, i.e. yielding of the endplate or T-stub in bending, failure of bolts in tension, failure of column in transverse compression, failure of panel zone in shear, and yielding of the steel beam in bending; the smallest one in the five ultimate moment resistance values is taken as the ultimate moment resistance of the connection; obtain the initial rotational stiffness of the connection by the test, simulation or theoretical calculation; then, the moment-rotation curve of the connection is obtained by substituting the initial rotational stiffness and the ultimate moment resistance into the proposed exponential model for the moment-rotation relation.

Claims

1. A calculation method of ultimate moment resistance for steel beam to concrete-filled steel tube column connections with bidirectional bolts, wherein the calculation method is applicable to an extended unstiffened endplate bidirectional bolted connection, an extended stiffened endplate bidirectional bolted connection, a flush endplate bidirectional bolted connection, and a bidirectional bolted T-stub connection; the calculation method comprising steps of: 1) calculating the ultimate moment resistance of all of the connections for yielding of all of the endplates or T-stub in bending, wherein firstly, yield line parameters for four common types of connections are calculated as follows: for the extended unstiffened endplate bidirectional bolted connection, yield line parameter L.sub.1 is obtained by equation (1): $\begin{array}{cc}{L}_1=\frac{b_p}{2}\left[h_2\left(\frac{1}{p_{\mathrm{fi}}}+\frac{1}{s}\right)+h_1\left(\frac{1}{p_{fe}}\right)-\frac{1}{2}\right]+\frac{2}{g}\left[h_2\left(p_{\mathrm{fi}}+s\right)\right]& \left(1\right)\end{array}$ where b.sub.p is a width of the endplate; h.sub.1 is a distance from centerline of first bolt row to lower surface of compression flange; h.sub.2 is a distance from centerline of second bolt row to lower surface of the compression flange; p.sub.fi is a distance from centerline of the second bolt row to lower surface of tension flange; p.sub.fe is a distance from centerline of the first bolt row to upper surface of tension flange; g is bolt gage; s=0.5√{square root over (b.sub.pg)} is a distance from centerline of second bolt row to edge of the yield line pattern; for the extended stiffened endplate bidirectional bolted connection, if d.sub.e≤s, yield line parameter L.sub.2 is obtained by equation (2): $\begin{array}{cc}{L}_2=\frac{b_p}{2}\left[h_2\left(\frac{1}{p_{\mathrm{fi}}}+\frac{1}{s}\right)+h_1\left(\frac{1}{p_{fe}}+\frac{1}{2s}\right)\right]+\frac{2}{g}\left[h_2\left(p_{\mathrm{fi}}+s\right)+h_1\left(d_e+P_{\mathrm{fe}}\right)\right]& \left(2\right)\end{array}$ where d.sub.e is a distance from the centerline of the first bolt row to the edge of the endplate; if d.sub.e>s, the yield line parameter L.sub.2 is obtained by equation (3): $\begin{array}{cc}{L}_2=\frac{b_p}{2}\left[h_2\left(\frac{1}{p_{\mathrm{fi}}}+\frac{1}{s}\right)+h_1\left(\frac{1}{p_{fe}}+\frac{1}{2s}\right)\right]+\frac{2}{g}\left[h_2\left(p_{\mathrm{fi}}+s\right)+h_1\left(s+P_{\mathrm{fe}}\right)\right]& \left(3\right)\end{array}$ for the flush endplate bidirectional bolted connection, if there is only one row of bolts in tensile zone, the yield line parameter L.sub.3 is obtained by equation (4): $\begin{array}{cc}{L}_3=\frac{b_p{h}_1}{2}\left(\frac{1}{p_{\mathrm{fi}}}+\frac{1}{s}\right)+\frac{2h_1}{g}\left(p_{\mathrm{fi}}+s\right)& \left(4\right)\end{array}$ if there are two rows of bolts in the tensile zone, the yield line parameter L.sub.3 is obtained by equation (5): $\begin{array}{cc}{L}_3=\frac{b_p}{2}\left[h_1\left(\frac{1}{p_{\mathrm{fi}}}\right)+h_2\left(\frac{1}{s}\right)\right]+\frac{2}{g}\left[h_1\left(p_{\mathrm{fi}}+0.75p\right)+h_2\left(s+0.25p\right)\right]+\frac{g}{2}& \left(5\right)\end{array}$ where p is a distance between the two rows of bolts; for the bidirectional bolted T-stub connection, yield line parameter L.sub.4 is obtained by equation (6): $\begin{array}{cc}{L}_4=\frac{b_s}{2}\left[h_2\left(\frac{1}{p_{\mathrm{fi}}}\right)+h_1\left(\frac{1}{p_{\mathrm{fe}}}\right)\right]& \left(6\right)\end{array}$ where b.sub.s is a width of the T-stub flange; then, substituting the above yield line parameter L into equation (7), the ultimate moment resistance M.sub.ep of all of the connections for yielding of the endplate or T-stub in bending is obtained:
M.sub.ep=f.sub.yt.sub.ep.sup.2L  (7) where f.sub.y is yield strength of the endplate, t.sub.ep is the thickness of the endplate or T-shaped flange, and L is the yield line parameter of the connection, i.e. L.sub.1, L.sub.2, L.sub.3 or L.sub.1; 2) calculating the ultimate moment resistance of the connection for failure of bidirectional bolts in tension, wherein for endplate bidirectional bolted connections, the ultimate moment resistance of the connection for failure of bidirectional bolts in tension M.sub.bo1 is obtained by equation (8):
M.sub.bo1=n.sub.t.Math.min(0.9f.sub.ubA.sub.s,0.48πd.sub.mt.sub.epf.sub.up).Math.(h.sub.b−t.sub.bf)  (8) for the bidirectional bolted T-stub connection, the ultimate moment resistance of the connection for failure of bidirectional bolts in tension Mbo2 is obtained by equation (9):
M.sub.bo2=n.sub.t.Math.min(0.9f.sub.ubA.sub.s,0.48πd.sub.mt.sub.epf.sub.up).Math.(h.sub.b+t.sub.bf)  (9) where n.sub.t is the number of bolts in tension zone; f.sub.ub is the ultimate tensile strength of the bolt, A.sub.S is the effective tensile area of the bolt, d.sub.m is a nominal diameter of the bolt head, h.sub.b is the height of the steel beam, t.sub.bf is the thickness of beam flange, and f.sub.up is the ultimate tensile strength of the endplate or T-stub flange; 3) calculating the ultimate moment resistance of the connection for failure of the panel zone in shear, wherein for endplate bidirectional bolted connections, the ultimate moment resistance of the connection for failure of the panel zone in shear M.sub.pz1 is obtained by equation (10): $\begin{array}{cc}{M}_{\mathrm{pz}1}=\left(\frac{1.8f_{y,\mathrm{cw}}{A}_{vc}}{\sqrt{3}}+0.85{\mathrm{vA}}_c{f}_{cd}\sin \alpha \right).Math.\left(h_b-t_{bf}\right)& \left(10\right)\end{array}$ for the bidirectional bolted T-stub connection, the ultimate moment resistance of the connection for failure of a panel zone in shear M.sub.pz2 is obtained by equation (11): $\begin{array}{cc}{M}_{\mathrm{pz}2}=\left(\frac{18f_{y,\mathrm{cw}}{A}_{vc}}{\sqrt{3}}+0.85{\mathrm{vA}}_c{f}_{cd}\sin \alpha \right).Math.\left(h_b+t_{\mathrm{bf}}\right)& \left(11\right)\end{array}$ where f.sub.y,cw is yield strength of a steel tube web of the concrete-filled steel tube column; A.sub.vc is the cross-sectional area of the steel tube web in shear; v is a reduction factor to allow for the effect of longitudinal compression of steel tube in the concrete-filled steel tube column on resistance in shear; A.sub.c is the cross-sectional area of the encased concrete; f.sub.cd is a design value of the cylindrical compressive strength of an encased concrete of the concrete-filled steel tube column; α is an angle in a panel zone; the reduction factor to allow for the effect of longitudinal compression of the concrete-filled steel tube column on resistance in shear, i.e. v, is calculated according to equation (12): $\begin{array}{cc}v=0.55\left(1+\frac{2N}{N_u}\right)& \left(12\right)\end{array}$ where N is a design compressive normal force in the concrete-filled steel tube column, and N.sub.u is an ultimate compressive bearing capacity of the concrete-filled steel tube column; the angle α in the panel zone is calculated according to equation (13):
α=arctan[(w.sub.c−2t.sub.cf)/z]  (13) where w.sub.c is a width of the concrete-filled steel tube column; t.sub.cf is the thickness of the steel tube flange of the concrete-filled steel tube column; z is a length of the lever arm for the connection, and z is (h.sub.b−t.sub.bf) for endplate bidirectional bolted connections, while z is (h.sub.b+t.sub.bf) for the bidirectional bolted T-stub connection; 4) calculating the ultimate moment resistance of the connection for failure of column in transverse compression, wherein for endplate bidirectional bolted connections, the ultimate moment resistance of the connection for failure of column in transverse compression Mc1 is obtained by equation (14):
M.sub.c1=[2ωρk.sub.cwb.sub.eff,cwt.sub.cwf.sub.y,cw+0.85k.sub.cb.sub.eff,c(d.sub.c−2t.sub.cw)f.sub.cd].Math.(h.sub.b−t.sub.bf)  (14) for the bidirectional bolted T-stub connection, the ultimate moment resistance of the connection for failure of column in transverse compression M.sub.c2 is obtained by equation (15):
M.sub.c2=[2ωρk.sub.cwb.sub.eff,cwt.sub.cwf.sub.y,cw+0.85k.sub.cb.sub.eff,c(d.sub.c−2t.sub.cw)f.sub.cd].Math.(h.sub.b+t.sub.bf)  (15) where ω is the reduction factor to allow for the effects of panel zone in shear on a transverse compression resistance of the steel tube web of the concrete-filled steel tube column; ρ is the reduction factor to allow for the steel tube web buckling in the concrete-filled steel tube column; k.sub.cw and k.sub.c are two factors to allow for the effect of longitudinal compressive stress on the transverse compression resistance of the steel tube and the encased concrete, respectively; b.sub.eff,cw and b.sub.eff,c are effective lengths of steel tube web and encased concrete in compression respectively; d.sub.c is section depth of the concrete-filled steel tube column; t.sub.cw is the thickness of the steel tube web; the reduction factor to allow for the effects of panel zone in shear on the transverse compression resistance of the steel tube web of the concrete-filled steel tube column, i.e. ω, is obtained by equation (16): $\begin{array}{cc}\omega =\{\begin{array}{cc}\frac{1}{\sqrt{1+1.3{\left(b_{\mathrm{eff},\mathrm{cw}}{t}_{cw}/A_{vc}\right)}^2}}& \left(\mathrm{for}\mathrm{exterior}\mathrm{connections}\right)\\ 1& \left(\mathrm{for}\mathrm{interior}\mathrm{connections}\right)\end{array}& \left(16\right)\end{array}$ the reduction factor to allow for the steel tube web buckling in the concrete-filled steel tube column ρ, is obtained by equations (17): $\begin{array}{cc}\rho =\{\begin{array}{cc}\left(\lambda -0.2\right)/{\lambda }^2& \left(\lambda >0.72\right)\\ 1& \left(\lambda \le 0.72\right)\end{array}& \left(17\right)\end{array}$ $\begin{array}{cc}\lambda =0.932\sqrt{\frac{b_{\mathrm{eff},\mathrm{cw}}\left(w_c-2t_{\mathrm{cf}}-2t\right)f_{y,\mathrm{cw}}}{E{t}_{\mathrm{cw}}^2}}& \left(18\right)\end{array}$ where λ is slenderness ratio of the steel tube web in the concrete-filled steel tube column; for rolled section of the steel tube, t is the inner radius of corners in the steel tube with rolled section; for welded section of the steel tube, t is the height of the steel tube weld leg; E is elastic modulus of the steel tube; the factors to allow for effect of longitudinal compressive stress on the transverse compression resistance of the steel tube and encased concrete or k.sub.cw and k.sub.c, are obtained by equations (19) and (20): $\begin{array}{cc}{k}_{\mathrm{cw}}=\{\begin{array}{cc}1& ({\sigma }_{\mathrm{col}}\le 0.7f_{y,\mathrm{cw}})\\ 1.7-\frac{{\sigma }_{\mathrm{col}}}{f_{y,\mathrm{cw}}}& ({\sigma }_{\mathrm{col}}>0.7f_{y,\mathrm{cw}})\end{array}& \left(19\right)\end{array}$ $\begin{array}{cc}{k}_c=1.3+3.3\frac{{\sigma }_c}{f_{\mathrm{cd}}},\mathrm{and}{k}_c=2\mathrm{if}\mathrm{it}\mathrm{is}\mathrm{greater}\mathrm{than}2& \left(20\right)\end{array}$ where σ.sub.col is the longitudinal compressive stress in the steel tube of the concrete-filled steel tube column, σ.sub.c is longitudinal compressive stress in encased concrete; the effective lengths of steel tube web and encased concrete in compression b.sub.eff,cw and b.sub.eff,c, are obtained by equation (21):
b.sub.eff,cw=b.sub.eff,c=t.sub.bf+2√{square root over (2)}h.sub.e+5(t.sub.cf+t)+s.sub.p  (21) where h.sub.e is an effective height of a weld seam between a steel beam flange and the endplate; s.sub.p is length obtained by dispersion through the endplate, and its minimum is the thickness of the endplate, and its maximum is double thickness of the endplate; 5) calculating the ultimate moment resistance of the connection for yielding of the steel beam in bending, wherein the ultimate moment resistance of the connection for yielding of the steel beam in bending M.sub.b is obtained by equation (22):
M.sub.b=f.sub.ybW.sub.p  (22) where f.sub.yb is yield strength of the steel beam; W.sub.p is a plastic section modulus of the steel beam; 6) calculating the final ultimate moment resistance and failure mode of the connection, wherein the smallest value of ultimate moment resistances for the above five failure modes is the final ultimate moment resistance of the connection, and the corresponding failure mode is an actual failure mode; for endplate bidirectional bolted connections, the ultimate moment resistance M.sub.u1 can be obtained:
M.sub.u1=min(M.sub.ep,M.sub.bo1,M.sub.pz1,M.sub.c1,M.sub.b)  (23) for the bidirectional bolted T-stub connections, the ultimate moment resistance M.sub.u2 can be obtained:
M.sub.u2=min(M.sub.ep,M.sub.bo2,M.sub.pz2,M.sub.c2,M.sub.b)  (24).

2. The calculation method of ultimate moment resistance for steel beam to concrete-filled steel tube column connections with bidirectional bolts according to claim 1, for endplate bidirectional bolted connections, the relation between moment M.sub.1 and rotation θ.sub.1 can be obtained by equation (25):
M.sub.1=M.sub.u1(1−e.sup.−s.sup.j.sup.θ.sup.1.sup./M.sup.u1)  (25) for bidirectional bolted T-stub connections, the relation between moment M.sub.2 and rotation θ.sub.2 can be obtained by equation (26):
M.sub.2=M.sub.u2(1−e.sup.−s.sup.j.sup.θ.sup.2.sup./M.sup.u2)  (26) where S.sub.j is initial rotational stiffness of the connection, which can be obtained by experiment, simulation or analytical model; the calculation method of moment-rotation curve is applicable to four common types of connections: the extended unstiffened endplate bidirectional bolted connection, the extended stiffened endplate bidirectional bolted connection, the flush endplate bidirectional bolted connection, and the bidirectional bolted T-stub connection.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is the three-dimensional schematic diagram for two of the four connection types, for which the present invention is applicable.

(2) FIG. 2 is the schematic diagram of the extended unstiffened endplate bidirectional bolted connection, for which the present invention is applicable.

(3) FIG. 3 is the schematic diagram of the extended stiffened endplate bidirectional bolted connection, for which the present invention is applicable.

(4) FIG. 4 is the schematic diagram of the flush endplate bidirectional bolted connection with only one row of bolts in the tensile zone, for which the present invention is applicable.

(5) FIG. 5 is the schematic diagram of the flush endplate bidirectional bolted connection with two rows of bolts in the tensile zone, for which the present invention is applicable.

(6) FIG. 6 is a schematic diagram of the bidirectional bolted T-stub connection, for which the present invention is applicable.

(7) FIG. 7 is a detailed drawing of the extended stiffened endplate bidirectional bolted connection.

(8) FIG. 8 is the moment-rotation curve of the extended stiffened endplate bidirectional bolted connection.

DETAILED DESCRIPTION

(9) In order to make the above features of the invention more understandable, the calculation method of the invention is described in detail below with the drawings and the exemplary embodiment.

(10) FIG. 1 shows the three-dimensional schematic diagram for two of the four connection types for which the present invention is applicable. FIGS. 2 to 6 are schematic diagrams for four types of connections, in which the physical meanings of symbols in formulas for yield line parameters of such connections are indicated. FIG. 7 is a detailed drawing of the present exemplary embodiment for the extended stiffened endplate bidirectional bolted connection, and its relevant parameters shown in Table 1 can be obtained from an existing reference:

(11) TABLE-US-00001 TABLE 1 Relevant parameters of the connection Axial f.sub.yb f.sub.y, cw f.sub.y f.sub.up f.sub.ub h.sub.e compression (Mpa) (Mpa) (Mpa) (Mpa) (Mpa) (mm) ratio 379.3 311.8 273.6 406.0 892 5.66 0.4

(12) The calculation method of ultimate moment resistance and moment-rotation curve for the present exemplary embodiment is conducted according to the following steps: 1) Calculate the ultimate moment resistance of the connection for yielding of the endplate in bending:

(13) The connection type of the exemplary embodiment is the extended stiffened endplate bidirectional bolted connection, so its yield line parameter can be obtained by Eq. (2):

(14) $L_2=\frac{b_p}{2}\left[h_2\left(\frac{1}{p_{\mathrm{fi}}}+\frac{1}{s}\right)+h_1\left(\frac{1}{p_{\mathrm{fe}}}+\frac{1}{2s}\right)\right]+\frac{2}{g}\left[h_2\left(p_{\mathrm{fi}}+s\right)+h_1\left(d_e+p_{\mathrm{fe}}\right)\right]=3015.8$

(15) The ultimate moment resistance of the connection for yielding of the endplate in bending, i.e. M.sub.ep, can be obtained by Eq. (7):
M.sub.ep=f.sub.yt.sub.ep.sup.2L=118.82 kN.Math.m 2) The ultimate moment resistance of the connection for failure of bidirectional bolts in tension can be calculated by Eq. (8):
M.sub.bo=n.sub.t.Math.min(0.9f.sub.ubA.sub.s,0.48πd.sub.mt.sub.epf.sub.up).Math.(h.sub.b−t.sub.bf)=146.9 kN.Math.m 3) Calculate the ultimate moment resistance of the connection for failure of panel zone in shear:

(16) The reduction factor to allow for the effect of longitudinal compression of the concrete-filled steel tube column on resistance in shear, i.e. v, can be calculated by Eq. (12):

(17) $v=0.55\left(1+\frac{2N}{N_u}\right)=0.99$

(18) The angle α in the panel zone can be calculated by Eq. (13):
α=arctan[(w.sub.c−2t.sub.cf)/z]=33.02°

(19) The ultimate moment resistance of the connection for failure of panel zone in shear, i.e. M.sub.pz, can be calculated by Eq. (10):

(20) $M_{pz}=\left(\frac{1.8f_{y,\mathrm{cw}}{A}_{vc}}{\sqrt{3}}+0.85\nu A_c{f}_{cd}\sin \alpha \right).Math.\left(h_b-t_{\mathrm{bf}}\right)=385.17\mathrm{KN}.Math.m$ 4) Calculate the ultimate moment resistance of the connection for failure of column in transverse compression:

(21) The reduction factor to allow for the effects of panel zone in shear on the transverse compression resistance of the steel tube web of the concrete-filled steel tube column, i.e. ω, is calculated by Eq. (16):

(22) $\omega =\frac{1}{\sqrt{1+1.3{\left(b_{\mathrm{eff},\mathrm{cw}}{t}_{\mathrm{cw}}/A_{\mathrm{vc}}\right)}^2}}=0.98$

(23) The reduction factor to allow for the concrete-filled steel tube column web buckling, i.e. ρ, is calculated by Eq. (17) and Eq. (18):

(24) $\lambda =0.932\sqrt{\frac{b_{\mathrm{eff},\mathrm{cw}}\left(w_c-2t_{\mathrm{cf}}-2t\right)f_{y,\mathrm{cw}}}{E{t}_{cw}^2}}=0.87\rho =\left(\lambda -0.2\right)/{\lambda }^2=0.89$

(25) The factors to allow for the effect of longitudinal compressive stress on the transverse compression resistance of the steel tube and the encased concrete, i.e. k.sub.cw and k.sub.c, are calculated by Eq. (19) and Eq. (20):
k.sub.cw=1
k.sub.c=2

(26) The effective lengths of steel tube web and encased concrete in compression, i.e. b.sub.eff,cw and b.sub.eff,c, can be calculated by Eq. (21):
b.sub.eff,cw=b.sub.eff,c=t.sub.bf+2√{square root over (2)}h.sub.e+5(t.sub.cf+t)+s.sub.p=72.84 mm

(27) The ultimate moment resistance of the connection for failure of column in transverse compression, i.e. M.sub.c, can be calculated by Eq. (14):
M.sub.c=[2ωρk.sub.cwb.sub.eff,cwt.sub.cwf.sub.y,cw+0.85k.sub.cb.sub.eff,c(d.sub.c−2t.sub.cw)f.sub.cd].Math.(h.sub.b−t.sub.bf)=336.11 kN.Math.m 5) The ultimate moment resistance of the connection for yielding of the steel beam in bending can be calculated by Eq. (22):
M.sub.b=f.sub.ybW.sub.p=194.97 kN.Math.m 6) The final ultimate moment resistance and failure mode of connection can be calculated by Eq. (23):
M.sub.u=min{M.sub.ep;M.sub.bo;M.sub.pz;M.sub.c;M.sub.b}=118.82 kN.Math.m

(28) The actual failure mode of the connection is the failure mode when the ultimate moment resistance equals to 118.82 kN.Math.m in step 1), i.e. yielding of the endplate. 7) Calculate the moment-rotation curve of the connection:

(29) According to an existing reference, the initial rotational stiffness of the connection is 33405 kN.Math.m/rad, and then the moment-rotation curve can be obtained by substituting the ultimate moment resistance and initial rotational stiffness into Eq. (24):
M=118.82.Math.(1−e.sup.−281.14θ)

(30) FIG. 8 is a comparison of moment-rotation curves for the exemplary embodiment between results based on the calculation method in this invention and experiments in the existing reference; it can be seen that the calculation method in the invention can well predict the ultimate moment resistance and moment-rotation curve of the exemplary embodiment.