Strength and Stiffness Analysis of Stringers for a Complex Stub Girder System

1 – stringer
2 – secondary beam

Objective: Check the mode for the beam analysis in the “Steel” postprocessor of SCAD.

Task: Select a rolled I-beam for the stringers with a span of 4,5 m in a complex stub girder system. The top chord of the stringers is continuously restrained by the floor plate.

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. 183.

Compliance with the codes: SNiP II-23-81*, SP 16.13330.2011, DBN B.2.6-163:2010.

Initial data file:

3.2 Beam_Example_3.2.spr;
report – 3.2 Beam_Example_3.2.doc

Initial data:

а = 1,0 m	Spacing of stringers;
q_н = (0,77 + 20) kN/m² ×1 = 20,77 kN/m	Total characteristic load;
q₁ = 1,05×0,77 kN/m² ×1 m = 0,8085 kN/m	Design permanent load;;
q₂ = 1,2×20 kN/m² ×1 = 24 kN/m	Design temporary load;
R_y = 23 kN/cm²,	Steel grade C235;
l = 4,5 м	Beam span;
[ f ] = 1/250×4,5 m = 18 mm	Limit deflection;
γ_c = 1	Service factor;
W_x = 288,33 cm³	Selected I-beam No.24 GOST 8239-89.
I_x = 3460 cm⁴.

SCAD Results. STEEL Postprocessor:

[Element No. 1] Forces

Max. 0 T
Snap 0 m

Max. 0 Т
Snap 0 m

M_y

Max. 0 T*m
Snap 0 m

Max. 6,4 T*m
Snap 2,25 m

M_z

Max. 0 T*m
Snap 0 m

M_k

Max. 0 T*m
Snap 0 m

Q_z

Max. 5,69 T
Snap 0 m

Max. -5,69 T
Snap 4,5 m

Q_y

Max. 0 T
Snap 0 m

Length of the bar 4,5 m
Length of the flexible part 4,5 m
Loading L1 - "1"

[Element No. 1] Deflections

Max. -18,58 mm
Snap 2,25 m

Length of the bar 4,5 m
Length of the flexible part 4,5 m
Loading L1 - "1"

Analysis complies with SNiP II-23-81*
Structural member beam

Steel: C235

Importance factor 1
Service factor 1
Limit slenderness for members in compression: 180
Limit slenderness for members in tension: 300
Member length 4,5 m
Effective length factor XoZ -- 1
Effective length factor XoY -- 1
Length between out-of-plane restraints 0,01 m

Section

Profile: I-beam with sloped inner flange surfaces GOST 8239-89 24

Results	Check	Utilization factor
Sec.5.12	Strength under action of bending moment M_y	0,95
Sec.5.12,5.18	Strength under action of lateral force Q_z	0,35
Sec.5.24,5.25	Strength under combined action of longitudinal force and bending moments, no plasticity	0,95
Sec.5.15	Stability of in-plane bending	0,95
Sec.6.15,6.16	Limit slenderness in XoY plane	0,63
Sec.6.15,6.16	Limit slenderness in XoZ plane	0,15

Utilization factor 0,95 - Strength under action of bending moment M_y

Manual calculation:

1. Design bending moment acting in the beam span:

\[ M_{\max } =\frac{q_{\Sigma } l^{2}}{8}=\frac{\left( {0,8085+24} \right)\cdot 4,5^{2}}{8}=62,7965 \quad kNm. \]

2. Necessary beam section modulus assuming that the deformations of steel are elastic:

\[ W=\frac{M_{\max } }{R_{y} }=\frac{62,7965\cdot 100}{23}=273,028 \quad cm^{3}. \]

3. Maximum deflection occurring in the middle of the beam span:

\[ f_{\max } =\frac{5}{384}\cdot \frac{q_{н} l^{4}}{EI_{x} }=\frac{5}{384}\cdot \frac{20,77\cdot 4,5^{4}}{2,06\cdot 10^{5}\cdot 10^{3}\cdot 3460\cdot 10^{-8}}=15,56 \quad mm. \]

Comparison of solutions:

Factor	Strength under action of lateral force	Strength under action of bending moment	Stability of in-plane bending under moment	Maximum deflection
Manual calculation	not defined	273,028/288,33 = 0,947	not defined	15,56/18 = 0,864
SCAD	0,352	0,947	0,947	18,58/1,1944/18 = 0,864
Deviation, %	0,0	0,0	0,0	0,0

Comments:

The check of tangential stresses was not performed in the manual calculation due to the absence of weakenings and a relatively large thickness of the beam webs.
The check of the general stability of the beam was not performed in the manual calculation, because the compressed beam chord is restrained against lateral displacements out of the bending plane by a welded floor plate.
The check of the beam strength taking into account the development of the limited plastic deformations was not performed, because according to the codes this calculation is possible only when the beam web has stiffeners. In the initial data of the example the stringer was specified without any intermediate stiffeners.