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

1 – stringer

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 6 m in a normal 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:

3.1 Beam_Example_3.1.spr;
report – 3.1 Beam_Example_3.1.doc — отчет.

Initial data:

а = 1,125 m	Spacing of stringers
q_н = (0,77 + 20) kN/m² ×1,125 m = 23,37 kN/m	Total characteristic load;
q₁ = 1,05×0,77 kN/m² ×1,125 m = 0,91 kN/m	Design permanent load;
q₂ = 1,2×20 kN/m² ×1,125 м = 27 kN/m	Design temporary load;
R_y = 23 kN/cm2,	Steel grade C235;
l = 6 m	Beam span;
[ f ] = 1/250×6,0 m = 24 mm	Limit deflection;
γ_c = 1	Service factor;
W_x = 596,364 cm³	Selected I-beam No.33 GOST 8239-89;
I_x = 9840 cm⁴, S_x= 339 cm³, t_w = 7 mm.

SCAD Results. STEEL Postprocessor:

[Element No. 1] Forces

N
Max. 0 T
Snap 0 m

Max. 0 Т
Snap 0 m

M_y
Max. 0 T*m
Snap 0 m

Max. 12,8 T*m
Snap 3 m

M_z
Max. 0 T*m
Snap 0 m

Max. 0 T*m
Snap 0 m

M_k
Max. 0 T*m
Snap 0 m

Max. 0 T*m
Snap 0 m

Q_z
Max. 8,54 Т
Snap 0 m

Max. -8,54 T
Snap 6 m

Q_y
Max. 0 Т
Snap 0 m

Max. 0 T
Snap 0 m

Length of the bar 6 m
Length of the flexible part 6 m
Loading L1 - "1"

[Element No. 1] Deflections

Max. -23,23 mm
Snap 3 m

Length of the bar 6 m
Length of the flexible part 6 m
Loading L1 - "1"

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

Steel: C235

Member length 6 m
Limit slenderness for members in compression: 180
Limit slenderness for members in tension: 300
Service factor 1
Importance factor 1
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 33

Results	Check	Utilization factor
Sec.5.12	Strength under action of bending moment M_y	0,92
Sec.5.12,5.18	Strength under action of lateral force Q_z	0,31
Sec.5.24,5.25	Strength under combined action of longitudinal force and bending moments, no plasticity	0,92
Sec.5.15	Stability of in-plane bending	0,92
Sec.6.15,6.16	Limit slenderness in XoY plane	0,72
Sec.6.15,6.16	Limit slenderness in XoZ plane	0,15

Utilization factor 0,92 - Strength under action of bending moment M_y

Manual calculation:

1. Design bending moment and shear force:

\[M_{\max } =\frac{q_{\Sigma } l^{2}}{8}=\frac{\left( {0.91+27} \right)\cdot 6.0^{2}}{8}=125.593 \quad kNm; \] \[ Q_{\max } =\frac{q_{\Sigma } l}{2}=\frac{\left( {0.91+27} \right)\cdot 6.0}{2}=83,73\quad kN. \];

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

\[ W=\frac{M_{\max } }{R_{y} }=\frac{125.593\cdot 100}{23}=546.057 \quad cm^{3}. \]

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

\[ f_{\max } =\frac{5}{384}\cdot \frac{q_{н} l^{4}}{EI_{x} }=\frac{5}{384}\cdot \frac{23,37\cdot 6^{4}}{2,06\cdot 10^{5}\cdot 10^{3}\cdot 9840\cdot 10^{-8}}=19,46 \quad mm. \]

4. Check of the maximum shear stresses:

\[ \tau_{\max } =\frac{Q_{\max } S_{x} }{I_{x} t_{w} }=\frac{83,73\cdot 339}{9840\cdot 0,7}=4,12577 \quad kN/cm^{2} \quad \quad < R_{s} \gamma_{c} =0,58\cdot 23=13,34 \quad kN/cm^{2}. \]

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	4,126/13,34 = 0,309	546,06/596,36 = 0,916	–	19,46/24 = 0,81
SCAD	0,309	0,916	0,916	23,23/1,1945/24 = 0,81
Deviation, %	0,0	0,0	0,0	0,0

Comments:

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.