Analysis of a Top Truss Chord from Unequal Angles

Objective: Check the mode for calculating truss members

Task: Check the top truss chord section from two unequal angles L160x100x9. The truss panel length is 2,58 m. The top truss chord is restrained out of the plane through the panel.

Source: Steel Structures: Student Handbook / [Kudishin U.I., Belenya E.I., Ignatieva V.S and others] - 13-th ed. rev. - M.: Publishing Center "Academy", 2011. p. 280.

Compliance with the codes: SNiP II-23-81*, SP 16.13330, DBN B.2.6-163:2010.

Initial data file:

7.1.sav;
report — Kristall-7.1.doc

Initial data:

N = 535 kN	Design compressive force;
R_y = 24 kN/cm²	Steel grade C245;
γ_c = 0,95	Service factor;
g = 12 mm	Thickness of the gusset plate;
l_y = 2,58 m, l_z = 5,16 m	Effective lengths of the bar;
i_y = 2,851 cm, А = 45,74 cm²	Geometric properties of
i_z = 7,745 cm	the top chord section from two angles 160х100х9.

KRISTALL parameters:

Steel: C245
Group of structures according to the table 50* of SNiP II-23-81* 3

Importance factor 1
Type of element – Truss member
Panel length 2.58 m

Length between out-of-plane restraints – 5.16 m

Section:

Profile: Unequal angle GOST 8510-86* L160x100x9

Manual calculation (SNiP II-23-81*):

1. Strength check

\[ \frac{N}{A}=\frac{535}{45,74}=11,69655 \quad kN/cm^{2} \quad \quad < R_{y} \gamma_{c}=24\cdot 0,95=22,8 \quad kN/cm^{2}. \]

2. Slenderness of the truss member:

\[ {\lambda}_{y} =\frac{l_{ef,y} }{i_{y} }=\frac{2,58\cdot 100}{2,851}=90,49456; \] \[ {\lambda}_{z} =\frac{l_{ef,z} }{i_{z} }=\frac{5,16\cdot 100}{7,745}=66,6236. \]

3. Conditional slenderness of the truss member:

\[ \bar{{\lambda }}_{y} =\frac{l_{ef,y} }{i_{y} }\sqrt {\frac{R_{y} }{E}} =\frac{2,58\cdot 100}{2,851}\sqrt {\frac{240}{2,06\cdot 10^{5}}} =3,0888; \] \[ \bar{{\lambda }}_{z} =\frac{l_{ef,z} }{i_{z} }\sqrt {\frac{R_{y} }{E}} =\frac{5,16\cdot 100}{7,745}\sqrt {\frac{240}{2,06\cdot 10^{5}}} =2,274. \]

4. Buckling coefficients:

\[ \begin{array}{l} \varphi_{y} =1,47-13,0\frac{R_{y} }{E}-\left( {0,371-27,3\frac{R_{y} }{E}} \right)\bar{{\lambda }}_{y} +\left( {0,0275-5,53\frac{R_{y} }{E}} \right)\bar{{\lambda }}_{y}^{2} = \\ =1,47-\frac{13,0\cdot 240}{2,06\cdot 10^{5}}-\left( {0,371-\frac{27,3\cdot 240}{2,06\cdot 10^{5}}} \right)\cdot 3,0888+\left( {0,0275-\frac{5,53\cdot 240}{2,06\cdot 10^{5}}} \right)\cdot 3,0888^{2}=0,60805 \\ \end{array} \] \[ \varphi_{z} =1-\left( {0,073-5,53\frac{R_{y} }{E}} \right)\bar{{\lambda }}_{z} \sqrt {\bar{{\lambda }}_{z} } =1-\left( {0,073-\frac{5,53\cdot 240}{2,06\cdot 10^{5}}} \right)\cdot 2,274\sqrt {2,274} =0,77176. \]

5. Strength of the truss member from the condition of providing the general stability under axial compression:

\[ N_{b,y} =\varphi_{y} AR_{y} \gamma_{c} =0,60805\cdot 45,74\cdot 24\cdot 0,95=634,118 \quad kN; \] \[ N_{b,z} =\varphi_{z} AR_{y} \gamma_{c} =0,77176\cdot 45,74\cdot 24\cdot 0,95=804,847 \quad kN. \]

6. Limit slenderness of the truss member:

\[ \left[ \lambda \right]_{y} =180-60\alpha_{y} =180-60\cdot \frac{N}{\varphi _{y} AR_{y} \gamma_{c} }=180-60\cdot \frac{535}{634,118}=129,3785; \] \[ \left[ \lambda \right]_{z} =180-60\alpha_{z} =180-60\cdot \frac{N}{\varphi _{z} AR_{y} \gamma_{c} }=180-60\cdot \frac{535}{804,847}=140,1166. \]

Comparison of solutions:

Factor	Source	Manual calculation	KRISTALL
Strength of member	535/45,8/22,8=0,512	11,6966/22,8 = 0,513	0,513
Stability of member in the truss plane	21,4/22,8=0,938	535/634,118 = 0,844	0,844
Stability of member out of the truss plane	not defined	535/804,847 = 0,665	0,665
Slenderness of member	not defined	90,4946/129,3785 = 0,7 66,6236/140,1166 = 0,4755	0,7

Comments:

1. In the source the buckling coefficient for the conditional slenderness of the bar of 3.09 was mistakenly taken as 0.546 instead of 0.6081, which caused the differences in the results of the stability analysis.

2. When checking the slenderness of the truss member the value of the factor was taken as the larger one calculated for the slenderness of the element in two principal planes of inertia of the section.