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| body | --uriencoded--$$ \normalsize \text %7B(a) When%7D \quad \frac%7BP_r%7D%7BP_c%7D \geq 0.2 \\ & \frac%7BP_r%7D%7BP_c%7D + \frac%7B8%7D%7B9%7D \left ( \frac%7BM_%7Brx%7D%7D%7BM_%7Bcx%7D%7D + \frac%7BM_%7Bry%7D%7D%7BM_%7Bcy%7D%7D \right) \leq 1.0 & \qquad \qquad \qquad (H1.1a) \\ \text %7B(b) When%7D \quad \frac%7BP_r%7D%7BP_c%7D < 0.2 \\ & \frac%7BP_r%7D%7B2P_c%7D + \left ( \frac%7BM_%7Brx%7D%7D%7BM_%7Bcx%7D%7D + \frac%7BM_%7Bry%7D%7D%7BM_%7Bcy%7D%7D \right) \leq 1.0 & \qquad \qquad \qquad (H1.1b) $$ |
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Where
Pr : = required axial strength using LRFD or ASD load combinations, kips(N)
Pc : = available axial strength, kips(N)
Mr : = required flexural strength using LRFD or ASD load combinations, kips-in.
Mc : = available flexural strength, kips-in. (N-mm)
x : = subscript relating symbol to strong axis bending
y : = subscript relating symbol to weak axis bending
For design according to Section B3.3 (LRFD):
Pr = required axial strength using LRFD load combinations, kips (N)
Pc = ɸcPn = design axial strength, determined in accordance with Chapter E, kips (N)
Mr = required flexural strength using LRFD load combinations, kip-in. (N-mm)
Mc = ɸbMn = design flexural strength, determined in accordance with Chapter F, kip-in. (N-mm)
ɸc = resistance factor for compression = 0.90
ɸb = resistance factor for flexure = 0.90
For design according to Section B3.4 (ASD):
Pr = required axial strength using ASD load combinations, kips (N)
Pc = Pn/Ωc = allowable axial strength, determined in accordance with Chapter E, kips (N)
Mr = required flexural strength using ASDload combinations, kip-in. (N-mm)
Mc = Mn/Ωb = allowable flexural strength, determined in accordance with Chapter F, kip-in (N-mm)
Ωc = safety factor for compression = 1.67
Ωb = safety factor for flexure = 1.67
TS EN 1991-1-3
6.2.9 Bending and axial force
(4) Members which are subjected to combined bending and axial compression should satisfy:
| Mathinline |
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| body | --uriencoded--$$ \normalsize \frac %7BN_%7Bed%7D%7D%7B \frac %7B\chi_y N_%7BRk%7D%7D%7B\gamma_%7BM1%7D%7D%7D + k_%7Byy%7D \frac %7BM_%7By,Ed%7D+\Delta M_%7By,Ed%7D%7D%7B \chi_%7BLT%7D \frac %7B M_%7By, Rk%7D%7D%7B\gamma_%7BM1%7D%7D%7D + k_%7Byz%7D \frac %7BM_%7Bz,Ed%7D+\Delta M_%7Bz,Ed%7D%7D%7B \frac %7B M_%7Bz, Rk%7D%7D%7B\gamma_%7BM1%7D%7D%7D \leq 1 & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (6.61) \\ \frac %7BN_%7Bed%7D%7D%7B \frac %7B\chi_z N_%7BRk%7D%7D%7B\gamma_%7BM1%7D%7D%7D + k_%7Bzy%7D \frac %7BM_%7By,Ed%7D+\Delta M_%7By,Ed%7D%7D%7B \chi_%7BLT%7D \frac %7B M_%7By, Rk%7D%7D%7B\gamma_%7BM1%7D%7D%7D + k_%7Bzz%7D \frac %7BM_%7Bz,Ed%7D+\Delta M_%7Bz,Ed%7D%7D%7B \frac %7B M_%7Bz, Rk%7D%7D%7B\gamma_%7BM1%7D%7D%7D \leq 1 & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (6.62) $$ |
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where NEd, My,Ed and Mz,Ed are the design values of the compression force and the maximum moments
about the y-y and z-z axis along the member, respectively
ΔMy,Ed, ΔMz,Ed are the moments due to the shift of the centroidal axis according to 6.2.9.3 for
class 4 sections, see Table 6.7,
χy and χz are the reduction factors due to flexural buckling from 6.3.1
χLT is the reduction factors due to lateral torsional buckling from 6.3.2
kyy, kyz, kzy, kzz are the interaction factors
Örnek ÇYTHYE 2016
Aşağıdaki şekilde verilen sistemde yükleme durumu verilen HE 300 B enkesitli kolon elemanın sabit ve hareketli yükleri verilmiştir. Sistemin birleşik etkiler altında elemanının tasarımını GKT ve YDKT dayanım kontrollerini kullanarak tasarım kuvvetlerini bulunuz.
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Çelik sınıfı
S275 Fy= 275 N/mm2 Fu= 430 N/mm2 (Yönetmelik Tablo 2.1A)
Enkesit Sınıflandırılması
Eksenel Basınç için başlık parçası (Tablo 5.1A, Durum 1)
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| body | --uriencoded--$$ \normalsize \lambda = \frac %7Bb%7D%7Bt%7D = \frac %7Bb_f%7D%7B2t_f%7D = \frac %7B300%7D%7B2 \times 19%7D =7.89 \leq \lambda_r = 0.56 \sqrt \frac %7BE%7D%7BF_y%7D = 0.56 \sqrt \frac %7B200000%7D%7B275%7D = 15.1 $$ |
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Eksenel Basınç için gövde parçası (Tablo 5.1A, Durum 5)
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| body | --uriencoded--$$ \normalsize \lambda = \frac %7Bh%7D%7Bt_w%7D = \frac %7B208%7D%7B11%7D =18.9 \leq \lambda_r = 1.49 \sqrt \frac %7BE%7D%7BF_y%7D = 1.49 \sqrt \frac %7B200000%7D%7B275%7D = 40.2 $$ |
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Eğilme momenti için başlık parçası (Tablo 5.1B, Durum 10)
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| body | --uriencoded--$$ \normalsize \lambda = \frac %7Bb%7D%7Bt%7D = \frac %7Bb_f%7D%7B2t_f%7D = \frac %7B300%7D%7B2 \times 19%7D =7.89 \leq \lambda_p = 0.38 \sqrt \frac %7BE%7D%7BF_y%7D = 0.38 \sqrt \frac %7B200000%7D%7B275%7D = 10.2 $$ |
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Eğilme momenti için gövde parçası (Tablo 5.1B, Durum 15)
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| body | --uriencoded--$$ \normalsize \lambda = \frac %7Bh%7D%7Bt_w%7D = \frac %7B208%7D%7B11%7D =18.9 \leq \lambda_p = 3.76 \sqrt \frac %7BE%7D%7BF_y%7D = 3.76 \sqrt \frac %7B200000%7D%7B275%7D = 101.4 $$ |
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Karakteristik Eksenel Basınç Kuvveti Dayanımı
Enkesitin asal eksenlerine dik eleman burkulma boyları
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| body | --uriencoded--$$ \normalsize L_%7Bcx%7D = K_x L_x = 1.0 \times 6000 = 6000 mm \\ L_%7Bcy%7D = K_y L_y = 1.0 \times 6000 = 6000 mm $$ |
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Yönetmelik 8.1.1 uyarınca, narinlik oranları
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| body | --uriencoded--$$ \normalsize \lambda_x = \frac %7BL_%7Bcx%7D%7D%7Bi_x%7D = \frac %7B6000%7D%7B129.9%7D = 46.2 \\ \lambda_y = \frac %7BL_%7Bcy%7D%7D%7Bi_y%7D = \frac %7B6000%7D%7B75.8%7D = 79.2 \\ \lambda_%7Bmax%7D = ( \lambda_x; \lambda_y ) = (46.2; 79.2) = 79.2 $$ |
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Elastik burkulma gerilmesi ve kritik burkulma gerilmesi
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| body | --uriencoded--$$ \normalsize \lambda_%7Bmax%7D = 79.2 \leq 4.71 \sqrt \frac%7B200000%7D%7B275%7D = 127.0 \qquad F_e = \frac %7B \pi%5e2 E%7D%7B\left ( \frac %7BL_%7Bcy%7D%7D%7Bi_y%7D \right )%5e2%7D = \frac %7B \pi%5e2 200000%7D%7B ( 79.2 )%5e2%7D = 314.69 N/mm%5e2 \\ F_%7Bcr%7D =\left ( 0.658%5e%7B\frac%7BF_y%7D%7BF_e%7D%7D\right ) F_y = \left ( 0.658%5e%7B\frac%7B275%7D%7B314.69%7D%7D\right ) 275 = 190.76 N/mm%5e2 $$ |
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Karakteristik basınç dayanımı
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| body | --uriencoded--$$ \normalsize P_n = F_%7Bcr%7D A_g = 190.76 \times 14910 \times 10%5e%7B-3%7D = 2844.2kN $$ |
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Karakteristik Eğilme Momenti Dayanımı
Yanal burulmalı burkulma sınır durumu
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| body | --uriencoded--$$ \normalsize L_p = 1.76 i_y \sqrt \frac %7BE%7D%7BF_y%7D = 3598mm \qquad L_r = 1.95i_%7Bts%7D \frac %7BE%7D%7B0.7F_y%7D \sqrt %7B\frac%7BJ_c%7D%7BW_%7Bex%7Dh_0%7D + \sqrt %7B \left ( \frac%7BJ_c%7D%7BW_%7Bex%7Dh_0%7D \right )%5e2 + 6.76 \left ( \frac%7B0.7 F_y%7D%7BE%7D \right ) %7D%7D = 15873mm \\ L_p < L_b = 6000 mm <L_r $$ |
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Moment dayanımı yönetmelik denklem 9.3 ile belirlenecektir.
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| body | --uriencoded--$$ \normalsize M_p = F_yW_%7Bpx%7D = 275 \times 1869 \times 10%5e%7B-3%7D = 514kNm \\ M_n = C_b \left [ M_p - (M_p - 0.7 F_yW_%7Bex%7D) \left ( \frac %7BL_b - L_p%7D%7BL_r - L_p%7D \right ) \right ] = 629 kNm \leq M_p = 514 kNm $$ |
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Karakteristik Eğilme Momenti Dayanımı
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| body | $$ \normalsize M_n = 514 kNm $$ |
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Karakteristik Eğilme Momenti Dayanımı:
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| body | $$ \normalsize M_n = 514 kNm $$ |
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Karakteristik Eksenel Basınç Dayanımı: | Mathinline |
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| body | $$ \normalsize P_n = 2844.2 kN $$ |
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GKT sonuçları| Mathinline |
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| body | --uriencoded--$$ \normalsize P_r = G + Q = 300 kN \qquad M_r = G + Q = 240 kNm \\ P_c = \frac %7BP_n%7D%7B\Omega%7D = \frac %7B2844.2%7D%7B1.67%7D = 1703.114 kN \\ M_%7Bc-major%7D = \frac %7BM_n%7D%7B\Omega%7D = \frac %7B514%7D%7B1.67%7D = 307.784 kN \\ \frac %7BP_r%7D%7BP_c%7D = \frac %7B300%7D%7B1703.1%7D = 0.17 <0.2 \\ \frac %7BP_r%7D%7B2P_c%7D + \left ( \frac %7BM_%7Brx%7D%7D%7BM_%7Bcx%7D%7D + \frac %7BM_%7Bry%7D%7D%7BM_%7Bcy%7D%7D \right ) = \frac %7B300%7D%7B2 \times 1703.1%7D + \left ( \frac %7B247%7D%7B307.784%7D + \frac %7B0%7D%7B143.289%7D \right ) = 0.866 $$ |
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YDKT sonuçları
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| body | --uriencoded--$$ \normalsize P_r = 1.2G + 1.6Q = 440 kN \qquad M_r = 1.2G + 1.6Q = 360 kNm \\ P_c = \phi P_n = 0.9 \times 2844.2 = 2559.78 kN \\ M_%7Bc-major%7D = \phi M_n = 0.9 \times 514 = 462.6 kNm \\ \frac %7BP_r%7D%7BP_c%7D = \frac %7B440%7D%7B2559.78%7D = 0.17 <0.2 \\ \frac %7BP_r%7D%7B2P_c%7D + \left ( \frac %7BM_%7Brx%7D%7D%7BM_%7Bcx%7D%7D + \frac %7BM_%7Bry%7D%7D%7BM_%7Bcy%7D%7D \right ) = \frac %7B440%7D%7B2 \times 2559.78%7D + \left ( \frac %7B360%7D%7B462.6%7D + \frac %7B0%7D%7B215.364%7D \right ) = 0.864 $$ |
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ideCAD Statik Sonuçları
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TS EN 1991-1-1
Etkileşim denklemi
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| body | --uriencoded--$$ \normalsize \frac %7BN_%7BEd%7D%7D%7B \chi_y N_%7BRk%7D/ \gamma_%7BM1%7D%7D + k_%7Byy%7D \frac %7BM_%7By,Ed%7D + \Delta M_%7By,Ed%7D%7D%7B \chi_%7BLT%7D M_%7By,Rk%7D/ \gamma_%7BM1%7D%7D + k_%7Byz%7D \frac %7BM_%7Bz,Ed%7D + \Delta M_%7Bz,Ed%7D%7D%7B M_%7Bz,Rk%7D/ \gamma_%7BM1%7D%7D \leq 1 \\ \frac %7BN_%7BEd%7D%7D%7B \chi_y N_%7BRk%7D/ \gamma_%7BM1%7D%7D + k_%7Bzy%7D \frac %7BM_%7By,Ed%7D + \Delta M_%7By,Ed%7D%7D%7B \chi_%7BLT%7D M_%7By,Rk%7D/ \gamma_%7BM1%7D%7D + k_%7Bzz%7D \frac %7BM_%7Bz,Ed%7D + \Delta M_%7Bz,Ed%7D%7D%7B M_%7Bz,Rk%7D/ \gamma_%7BM1%7D%7D \leq 1 $$ |
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Basınç dayanımının bulunması
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| body | --uriencoded--$$ \normalsize N_%7BRk%7D = N_%7Bc,rd%7D = Af_%7By%7D = 21798.8 \times 275 \times 10%5e%7B-3%7D = 4099.92kN $$ |
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Burkulma azaltma faktörü
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| body | --uriencoded--$$ \normalsize N_%7Bcr,y%7D = \frac %7B\pi%5e2 EI_y%7D%7B L_y%5e2%7D = \frac %7B\pi%5e2 (2 \times 10%5e8)25167,126 \times 10%5e%7B-8%7D%7D%7B 6%5e2%7D = 13799.421kN \qquad \overline%7B\lambda%7D_y = \sqrt \frac%7BAf_y%7D%7BN_%7Bcr,y%7D%7D = \sqrt \frac%7B14908.8 \times 275/1000%7D%7B13799.421%7D = 0.545 \\ \Phi_y = 0.5 \left [ 1+\alpha_z (\overline%7B\lambda%7D_y - 0.2)+ \overline%7B\lambda%7D_y%5e2 \right ] = 0.707 \qquad \chi_y = \frac %7B1%7D%7B \Phi +\sqrt %7B \Phi%5e2 - \overline%7B\lambda%7D_y%5e2 %7D%7D = 0.864 $$ |
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Etkileşim katsayıları
kyy = 1.042
kyz = 0.740
kzy = 0.977
kzz = 1.233
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Etkileşim Denklemi
| Mathinline |
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| body | --uriencoded--$$ \normalsize \frac %7BN_%7BEd%7D%7D%7B \chi_y N_%7BRk%7D/ \gamma_%7BM1%7D%7D + k_%7Byy%7D \frac %7BM_%7By,Ed%7D + \Delta M_%7By,Ed%7D%7D%7B \chi_%7BLT%7D M_%7By,Rk%7D/ \gamma_%7BM1%7D%7D + k_%7Byz%7D \frac %7BM_%7Bz,Ed%7D + \Delta M_%7Bz,Ed%7D%7D%7B M_%7Bz,Rk%7D/ \gamma_%7BM1%7D%7D \leq 1 \\ \frac %7B435%7D%7B 0.864 \times 4099.909/1%7D + 1.042 \frac %7B351%7D%7B0.866 \times 513.918 / 1%7D +0 = 0.930 \leq 1 $$ |
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ideCAD Statik Sonuçları
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