V.AIJ 2002 Check for MBG parameter

Verify the effect of the MBG parameter of the AIJ 2002 code on a doubly symmetric I shaped beam of span 6m, that is subjected to UDL of -20 KN/m on its span.

Details

Japanese H Shape Doubly Symmetric I Section H400X200X12X22 used in the following frame, E = 205,000 MPa, F_y = 235 MPa

where w = -20 kN/m and L = 4 m

Equation SSB-1.10 of JSME is used to evaluate Young's modulus, E, for equation 5.8 (YNG 1).

d = 400 mm, b_f = 200 mm, t_w = 12 mm, t_f = 22 mm

Verification

Maximum Moment, MZ1 = 43.714 kN·m

Minimum Moment, MZ2 = 43.714 kN·m

Span Moment = 46.286 kN·m

Unbraced length of compression flange, l_b = 6,000 mm

Note: Only the flanges will be considered (i.e., the web is ignored for calculating the section modulus). (MBG 1)

Area of compression flange = A_f = b_f × t_f = 4,400 mm²

C_{g} = \frac{b_{f} \times t_{f} \times (d - \frac{t_{f}}{2}) + b_{f} \times t_{f} \times \frac{t_{f}}{2}}{(b_{f} \times t_{f}) + (b_{f} \times t_{f})} = 200 m m

I_{z z} = 2 \times (b_{f} \times \frac{{t_{f}}^{3}}{12}) + 2 \times b_{f} \times t_{f} \times {(c_{g} - \frac{t_{f}}{2})}^{2} = 3.147 \times {(10)}^{8} {m m}^{4}

Elastic Section Modulus about Z axis on compression side Z_zc = I_zz/c_g = 1.573×10⁶ mm³

I_yc = t_f×b_f³/12 = 1.467×107 mm⁴

A_C = t_f×b_f = 4,400 mm²

r_i = √(I_yc/A_C) = 57.735 mm

Bending Moment Factor C = 1.75 – 1.05×(-MZ2/MZ1) + 0.3×(-MZ2/MZ1)2 = 3.198, but not more than 2.3

As Span moment > MZ1, C will be 1

Allowable slenderness ratio $Λ = \sqrt{\frac{π^{2} * E}{0.6 * F Y L D}} = 119.789$

f_{b 1} = 1.5 \times [1 - 0.4 \times \frac{{(\frac{l_{b}}{r_{i}})}^{2}}{C \times Λ^{2}}] \times \frac{f_{t}}{ϒ} = 109.501 N / {m m}^{2}

As per Eq. SSB-1.10 for YNG = 1:

f_{b 2} = 1.5 \times \frac{0.433 \times E}{\frac{l_{b} \times d}{A_{f}} \times ϒ} = 162.736 N / {m m}^{2}

f_bz1 = max(f_b1, f_b2) = 162.736 N/mm²

Allowable tensile bending stress, f_bzt = f_t = 156.67 N/mm²

Actual tensile bending stress about Z axis = 46.286/Z_ZC = 29.43 N/mm²

Utilization ratio = Actual bending stress/Allowable bending stress = 29.43/156.67 = 0.188

Allowable compressive bending stress, f_bzc = min(f_bz1, f_t) = 156.67 N/mm²

Actual compressive bending stress about Z axis = 46.286/Z_ZC = 29.43 N/mm²

Utilization ratio = Actual bending stress/Allowable bending stress = 29.43/156.67 = 0.188

Results

Table 1. Comparison of results
Result Type	Reference	STAAD.Pro	Difference
Actual bending tensile stress (N/mm²)	29.43	29.42	negligible
Allowable bending tensile stress (N/mm²)	156.67	156.67	none
Ratio	0.188	0.188	none
C	1	1	none
Actual bending compressive stress (N/mm²)	29.43	29.42	negligible
Allowable bending compressive stress (N/mm²)	156.67	156.67	none
Ratio	0.188	0.188	none

STAAD.Pro Input File

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\Japan\AIJ 2002 Check for MBG parameter.std is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 28-Apr-19
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 0 4 0; 3 6 4 0; 4 6 0 0;
MEMBER INCIDENCES
1 1 2; 2 2 3; 3 3 4;
START USER TABLE
TABLE 1
UNIT METER KN
ANGLE
ANGLE
0.2 0.2 0.018 0.0394179 0.0024 0.0024
END
DEFINE MATERIAL START
ISOTROPIC STEEL
E 2.05e+08
POISSON 0.3
DENSITY 76.8195
ALPHA 1.2e-05
DAMP 0.03
TYPE STEEL
STRENGTH RY 1.5 RT 1.2
END DEFINE MATERIAL
MEMBER PROPERTY JAPANESE
1 TO 3 TABLE ST H400X200X12X22
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 4 FIXED
LOAD 1 LOADTYPE Dead  TITLE LOAD CASE 1
MEMBER LOAD
2 UNI GY -20
PERFORM ANALYSIS
DEFINE ENVELOPE
1 ENVELOPE 1 TYPE STRENGTH
END DEFINE ENVELOPE
LOAD LIST ENV 1
PARAMETER 1
CODE JAPANESE 2002
TRACK 2 MEMB 2
MAIN 1 MEMB 2
MISES 1 MEMB 2
YNG 1 MEMB 2
MBG 1 MEMB 2
CB 1 MEMB 2
CHECK CODE MEMB 2
FINISH

STAAD.Pro Output

      STAAD SPACE                                              -- PAGE NO.    3
                        STAAD.Pro CODE CHECKING - (AIJ-2002)   V2.1
                        ***********************
ALL UNITS ARE IN - METE  KN   (U.N.O.)
|-----------------------------------------------------------------------------|
|   Member No:       2        Profile:ST  H400X200X12X22   (JAPANESE SECTIONS)  |
|  Ratio:    0.241  (PASS)  Reference: Eq.5.16           Loadcase:       1    |
|-----------------------------------------------------------------------------|
|  Location:    0.000  Criteria:     Stress      Load Case:     1  (Permanent)|
|  Px:(C)       15.749       Vy:        60.000          Vz:       0.000       |
|  Tx:           0.000       My:         0.000          Mz:      43.714       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES AT DISTANCE:    0.000 MM              (UNIT: MM)         |
|  Ax:     1.32200E+04       Az: 5.86667E+03            Ay: 4.27200E+03       |
|  Iz:     3.64000E+08       Iy: 2.94000E+07             J: 1.64000E+06       |
|  Zz:     1.82000E+06       Zy: 2.94000E+05            Zx: 7.45455E+04       |
|  iZ:     1.65934E+02       iY: 4.71583E+01            Iw: 1.05020E+12       |
|-----------------------------------------------------------------------------|
| MATERIAL PROPERTIES   (Unit: N/mm2)                                         |
|  Fy:         235.000      E :   204999.987          G :     79000.000       |
|-----------------------------------------------------------------------------|
| DESIGN PROPERTIES                                                           |
|  Member Length:     6.000      Lz:     6.000 Ly:     6.000  UNL:      6.000 |
| DESIGN PARAMETERS                                                           |
|  Kz:   1.000     Ky:   1.000   NSF:  1.000   Cb:   1.000                    |
|-----------------------------------------------------------------------------|
| Section Classification Results :    Width to Thickness Ratios               |
|-----------------------------------------------------------------------------|
|            Compression              |             Bending                   |
|-----------------------------------------------------------------------------|
|  LC#   |    Actual   |   Allowable  |   LC#   |    Actual   |   Allowable   |
|-----------------------------------------------------------------------------|
|      1 |    27.5000  |     47.9461  |      1  |    27.5000  |     71.7561   |
|-----------------------------------------------------------------------------|
| CHECKS                              |Env|     Stresses    |                 |
|              | Loc. | Demand |  L/C |Typ| Actual | Allow  |Ratio |   Ref.   |
|              |(MET) |(KN-MET)|      |   |(N/mm2) |(N/mm2) |      |          |
|--------------|------|--------|------|---|--------|--------|------|----------|
| Tension      |  -   |   -    |  -   | - |   -    |  156.67|  -   | Eq.5.1   | 
| Compression  | 0.000|   15.75|     1| P |    1.19|   57.70| 0.021| Eq.5.4   | 
| Bending Z (T)| 3.000|  -46.29|     1| P |   29.42|  156.67| 0.188| Eq.5.1   | 
| Bending Z (C)| 3.000|  -46.29|     1| P |   29.42|  156.67| 0.188| Eq.5.1   | 
| Bending Y (T)|  -   |   -    |  -   | - |   -    |  156.67|  -   | Eq.5.1   | 
| Bending Y (C)|  -   |   -    |  -   | - |   -    |  156.67|  -   | Eq.5.1   | 
| Shear Z      |  -   |   -    |  -   | - |   -    |   90.45|  -   | Eq.5.2   | 
| Shear Y      | 0.000|   60.00|     1| P |   14.04|   90.45| 0.155| Eq.5.2   | 
| Comp+ Bend C | 3.000|   -    |     1| P |   -    |    -   | 0.208| Eq.6.1   | 
| Comp+ Bend T | 3.000|   -    |     1| P |   -    |    -   | 0.180| Eq.6.2   | 
| Ten + Bend T | 3.000|   -    |     1| P |   -    |    -   | 0.188| Eq.6.3   | 
| Ten + Bend C | 3.000|   -    |     1| P |   -    |    -   | 0.188| Eq.6.4   | 
| Von-Mises    | 0.000|   -    |     1| P |   37.83|  156.67| 0.241| Eq.5.16  | 
|                                                                             |
| INTERMEDIATE RESULTS (Bending)                                              |
|  C:  1.000                                                                  |
|                                                                             |
| INTERMEDIATE RESULTS (Von-Mises)                                            |
|  Sigmax:    28.972 N/mm2   Tou:    14.045 N/mm2  fm:    37.831 N/mm2        |
|-----------------------------------------------------------------------------|