On the structural stability of timber members to Eurocode

Figures & data

Figure 1. A column subjected to axial force, N_Ed and transversal force H_Ed.

Table 1. Material properties of glulam timber with strength class GL32c.

Property	fc,0,k	fmk	E0.05	G0.05	kmod	E0, mean
Value	24.5 MPa	32 MPa	11.2 GPa	540 MPa	0.8	13.7 GPa

Figure 2. A continuous beam on three supports. The dashed line is the assumed buckling curve of the beam.

Figure 3. Discrete bracing of the beam (Fig. 2) at the upper level of the beam (compressive edge) with lateral supports.

Figure 4. A simply supported beam. The dashed line is the assumed buckling curve of the beam.

Figure 5. Discrete bracing of the beam (Fig. 4) at the upper level of the beam (compressive edge) with lateral supports.

Table 2. The effect of beam dimensions on the beam stability, Fig. 2, with different interaction formulae.

Beam dimensions (mm)	Flexural buckling around y-y Eq. (1)(1) $$\frac{{\sigma }_{c,0,d}}{k_{c,y}{f}_{c,0,d}}+\frac{{\sigma }_{m,y,d}}{f_{m,,d}}+k_m\frac{{\sigma }_{m,z,d}}{f_{m,z,d}}\le 1$$(1)	Flexural buckling around y-y Eq. (17)(17) $$\frac{{\sigma }_{c,0,d}}{f_{c,0,d}}+\frac{{\sigma }_{m,y,d}}{f_{m,y,d}}\left(\frac{{\nu }_y}{{\nu }_y-1}\right)+k_m\frac{{\sigma }_{m,z,d}}{f_{m,z,d}}\left(\frac{{\nu }_y}{{\nu }_y-1}\right)\le 1\mathrm{ }$$(17)	Flexural buckling around y-y Eq. (22)(22) $$\frac{{\sigma }_{c,0,d}}{f_{c,0,d}}+\frac{{\sigma }_{m,y,d,\mathit{II}}}{f_{m,y,d}}+k_m\frac{{\sigma }_{m,z,\mathrm{d},\mathit{II}}}{f_{m,z,d}}\le 1$$(22)	Slender-ness ratio λrel,y Eq. (3)(3) $${\lambda }_{\mathrm{rel},y}=\frac{{\lambda }_y}{\pi }\sqrt{\frac{f_{c,0,k}}{E_{0,05}}}$$(3)	Flexural buckling around z-z Eq. (2)(2) $$\frac{{\sigma }_{c,0,d}}{k_{c,z}{f}_{c,0,d}}+k_m\frac{{\sigma }_{m,y,d}}{f_{m,y,d}}+\frac{{\sigma }_{m,z,d}}{f_{m,z,d}}\le 1$$(2)	Flexural buckling around z-z Eq. (18)(18) $$\frac{{\sigma }_{c,0,d}}{f_{c,0,d}}+k_m\frac{{\sigma }_{m,y,d}}{f_{m,y,d}}\left(\frac{{\nu }_z}{{\nu }_z-1}\right)+\frac{{\sigma }_{m,z,d}}{f_{m,z,d}}\left(\frac{{\nu }_z}{{\nu }_z-1}\right)\le 1$$(18)	Flexural buckling around z-z Eq. (23)(23) $$\frac{{\sigma }_{c,0,d}}{f_{c,0,d}}+k_m\frac{{\sigma }_{m,y,d,\mathit{II}}}{f_{m,y,d}}+\frac{{\sigma }_{m,z,d,\mathit{II}}}{f_{m,z,d}}\le 1$$(23)	Slender-ness ratio λrel,z Eq. (4)(4) $${\lambda }_{\mathrm{rel},z}=\frac{{\lambda }_z}{\pi }\sqrt{\frac{f_{c,0,k}}{E_{0,05}}}$$(4)
140 × 315	1.083	1.15	1.189	0.696	0.870	0.884	0.963	0.368
140 × 450	0.625	0.633	0.681	0.487	0.526	0.529	0.568	0.368
66 × 540	1.014	1.036	1.103	0.406	0.924	0.956	0.933	0.781
56 × 630	0.959	0.975	1.039	0.348	0.945	0.957	0.890	0.921
115 × 405	0.891	0.917	0.975	0.541	0.745	0.756	0.806	0.448
165 × 315	0.919	0.962	1.006	0.696	0.736	0.744	0.815	0.313
66 × 450	1.325	1.384	1.452	0.487	1.179	1.272	1.210	0.781
165 × 270	1.185	1.292	1.298	0.812	0.922	0.935	1.037	0.313
165 × 495	0.461	0.463	0.500	0.443	0.390	0.392	0.421	0.313
140 × 720	0.318	0.320	0.343	0.304	0.281	0.281	0.297	0.368
115 × 360	1.068	1.120	1.172	0.609	0.879	0.897	0.959	0.448
90 × 720	0.495	0.499	0.534	0.304	0.447	0.445	0.462	0.573

Table 3. The effect of beam dimensions on the beam stability, Fig. 4, with different interaction formulae.

Beam dimensions (mm)	Flexural buckling around y-y Eq. (1)(1) $$\frac{{\sigma }_{c,0,d}}{k_{c,y}{f}_{c,0,d}}+\frac{{\sigma }_{m,y,d}}{f_{m,,d}}+k_m\frac{{\sigma }_{m,z,d}}{f_{m,z,d}}\le 1$$(1)	Flexural buckling around y-y Eq. (17)(17) $$\frac{{\sigma }_{c,0,d}}{f_{c,0,d}}+\frac{{\sigma }_{m,y,d}}{f_{m,y,d}}\left(\frac{{\nu }_y}{{\nu }_y-1}\right)+k_m\frac{{\sigma }_{m,z,d}}{f_{m,z,d}}\left(\frac{{\nu }_y}{{\nu }_y-1}\right)\le 1\mathrm{ }$$(17)	Flexural buckling around y-y Eq. (22)(22) $$\frac{{\sigma }_{c,0,d}}{f_{c,0,d}}+\frac{{\sigma }_{m,y,d,\mathit{II}}}{f_{m,y,d}}+k_m\frac{{\sigma }_{m,z,\mathrm{d},\mathit{II}}}{f_{m,z,d}}\le 1$$(22)	Slender-ness ratio λrel,y Eq. (3)(3) $${\lambda }_{\mathrm{rel},y}=\frac{{\lambda }_y}{\pi }\sqrt{\frac{f_{c,0,k}}{E_{0,05}}}$$(3)	Flexural buckling around z-z Eq. (2)(2) $$\frac{{\sigma }_{c,0,d}}{k_{c,z}{f}_{c,0,d}}+k_m\frac{{\sigma }_{m,y,d}}{f_{m,y,d}}+\frac{{\sigma }_{m,z,d}}{f_{m,z,d}}\le 1$$(2)	Flexural buckling around z-z Eq. (18)(18) $$\frac{{\sigma }_{c,0,d}}{f_{c,0,d}}+k_m\frac{{\sigma }_{m,y,d}}{f_{m,y,d}}\left(\frac{{\nu }_z}{{\nu }_z-1}\right)+\frac{{\sigma }_{m,z,d}}{f_{m,z,d}}\left(\frac{{\nu }_z}{{\nu }_z-1}\right)\le 1$$(18)	Flexural buckling around z-z Eq. (23)(23) $$\frac{{\sigma }_{c,0,d}}{f_{c,0,d}}+k_m\frac{{\sigma }_{m,y,d,\mathit{II}}}{f_{m,y,d}}+\frac{{\sigma }_{m,z,d,\mathit{II}}}{f_{m,z,d}}\le 1$$(23)	Slender-ness ratio λrel,z Eq. (4)(4) $${\lambda }_{\mathrm{rel},z}=\frac{{\lambda }_z}{\pi }\sqrt{\frac{f_{c,0,k}}{E_{0,05}}}$$(4)
140 × 315	1.107	1.197	1.125	0.819	0.870	0.884	0.918	0.368
140 × 450	0.630	0.639	0.650	0.573	0.526	0.529	0.546	0.368
66 × 540	1.020	1.049	1.057	0.478	0.924	0.956	0.901	0.781
56 × 630	0.963	0.982	0.998	0.409	0.945	0.957	0.862	0.921
115 × 405	0.900	0.935	0.928	0.637	0.745	0.756	0.773	0.448
165 × 315	0.939	0.993	0.952	0.819	0.736	0.744	0.777	0.313
66 × 450	1.335	1.417	1.384	0.573	1.179	1.272	1.162	0.781
165 × 270	1.237	1.375	1.223	0.955	0.922	0.935	0.985	0.313
165 × 495	0.463	0.466	0.479	0.521	0.390	0.392	0.405	0.313
140 × 720	0.320	0.320	0.331	0.358	0.281	0.281	0.289	0.368
115 × 360	1.083	1.153	1.112	0.716	0.879	0.897	0.917	0.448
90 × 720	0.497	0.500	0.515	0.358	0.447	0.445	0.449	0.573