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V.AIJ 2005 Check for MBG parameter

Verify the effect of the MBG parameter of the AIJ 2005 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, Fy = 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, bf = 200 mm, tw = 12 mm, tf = 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, lb = 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 = Af = bf × tf = 4,400 mm2

C g = b f × t f × d - t f 2 + b f × t f × t f 2 b f × t f + ( b f × t f ) = 200   m m
I z z = 2 × b f × t f 3 12 + 2 × b f × t f × ( c g - t f 2 ) 2 = 3.147 × ( 10 ) 8   m m 4

Elastic Section Modulus about Z axis on compression side Zzc = Izz/cg = 1.573×106 mm3

Iyc = tf×bf3/12 = 1.467×107 mm4

AC = tf×bf = 4,400 mm2

ri = √(Iyc/AC) = 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. (Eq. 5.15)

Elastic critical slenderness ratio eλb = 1/√0.6 = 1.291 (Eq. 5.11)

Plastic slenderness ratio = pλb = 0.3 (Eq. 5.14)

My = fy × ZZ = 427.7 k·m

Me = 558.21 kN·m

(Eq. 5.16)

So, λb = √(My/Me) = 0.8753 (Eq. 5.10)

So, here pλb < λb < eλb

So, we calculate Allowable bending stress as per Eq. 5.8

ν = 3/2 + 2/3×(λb/eλb)2 = 1.8065

f b = 1 - 0.4 × λ b - p λ b e λ b - p λ b × f y ν = 99.88   N / m m 2

Allowable tensile bending stress, fbzt = ft = 156.667 N/mm2

Actual tensile bending stress about Z axis = 46.286/ZZC = 29.425 N/mm2

Utilization ratio = Actual bending stress/Allowable bending stress = 29.425/156.667 = 0.1878

Allowable compressive bending stress fbzc = min(fb, ft) = 99.88 N/mm2

Actual compressive bending stress about Z axis = 46.286/ZZC = 29.425 N/mm2

Utilization ratio = Actual bending stress/Allowable bending stress = 29.425/99.88 = 0.295

Results

Table 1. Comparison of results
Result Type Reference STAAD.Pro Difference Comments
Actual bending tensile stress (N/mm2) 29.425 29.42 Negligible  
Allowable bending tensile stress (N/mm2) 156.667 156.67 Negligible  
Ratio 0.1878 0.188 Negligible  
My (kN·m) 427.7 427.7 None  
Me (kN·m) 558.21 558.21 None  
pλb 0.3 0.3 None  
C 1 1 None  
Actual bending compressive stress (N/mm2) 29.425 29.42 Negligible  
Allowable bending compressive stress (N/mm2) 99.88 99.88 None  
Ratio 0.295 0.295 None  

STAAD.Pro Input File

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\AIJ\2005\AIJ 2005 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 2005
TRACK 2 MEMB 2
MAIN 1 MEMB 2
MISES 1 MEMB 2
MBG 1 MEMB 2
CB 1 MEMB 2
CHECK CODE MEMB 2
FINISH

STAAD.Pro Output

                        STAAD.Pro CODE CHECKING - (AIJ-2005)   V2.1
                        ***********************
ALL UNITS ARE IN - METE  KN   (U.N.O.)
|-----------------------------------------------------------------------------|
|   Member No:       2        Profile  : ST  H400X200X12X22(JAPANESE SECTIONS)  |
|  Ratio:    0.315  (PASS)  Reference: Eq.6.1            Loadcase:       1    |
|-----------------------------------------------------------------------------|
|  Location:    3.000  Criteria:     Stress      Load Case:     1  (Permanent)|
|  Px:(C)       15.749       Vy:         0.000          Vz:       0.000       |
|  Tx:           0.000       My:         0.000          Mz:     -46.286       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES AT DISTANCE: 3000.000 MM              (UNIT: MM)         |
|  Ax:     1.32200E+04       Az: 5.86652E+03            Ay: 4.80000E+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 :   205000.000          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    pλb:  0.000     |
|-----------------------------------------------------------------------------|
| Section Classification Results :    Width to Thickness Ratios               |
|-----------------------------------------------------------------------------|
|            Compression              |             Bending                   |
|-----------------------------------------------------------------------------|
|  LC#   |    Actual   |   Allowable  |   LC#   |    Actual   |   Allowable   |
|-----------------------------------------------------------------------------|
|      1 |    27.5000  |     47.2567  |      1  |    27.5000  |     70.8850   |
|-----------------------------------------------------------------------------|
| 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|   99.88| 0.295| Eq.5.8   | 
| 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 |   12.50|   90.45| 0.138| Eq.5.2   | 
| Comp+ Bend C | 3.000|   -    |     1| P |   -    |    -   | 0.315| 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 |   -    |    -   | 0.295| Eq.6.4   | 
| Von-Mises    | 0.000|   -    |     1| P |   36.17|  156.67| 0.231| Eq.5.24  | 
|                                                                             |
| INTERMEDIATE RESULTS (Bending)                                              |
|  Me: 5.5821E+02 KN-MET  My: 4.2770E+02 KN-MET   pλb:   0.300  C:  1.000     |
|                                                                             |
| INTERMEDIATE RESULTS (Von-Mises)                                            |
|  Sigmax:    28.972 N/mm2   Tou:    12.500 N/mm2  fm:    36.168 N/mm2        |
|-----------------------------------------------------------------------------|