V.AIJ 2002 Check for MISES parameter

Verify the von Mises stress calculations per the AIJ 2002 code using a cantilever beam.

Details

Japanese Angle section L250X250X35, E = 205,000 MPa, F_y = 200 MPa

The member is a 5 m cantilever with the following applied loads at the end:

Fx = 10 kN
Vz = 5 kN
Vy = 5 kN
T = 5 kN·m

Verification

The maximum of the left-hand side of the von Mises stress equation apparently occurs at the fixed support of the beam. Section forces at the fixed end are calculated as below.

FX = -10 KN
FY = -5 KN
FZ = -5 KN
MX = -5 KN-m
MY = 25 KN-m
MZ = -25 KN-m

The standard AIJ calculation is used (MISES 1). From these section forces, σ_x and τ_xy at the section of the fixed end are calculated:

σ_{x} = |\frac{F_{x}}{A_{x}}| + |\frac{M_{y}}{Z_{y}}| + |\frac{M_{z}}{Z_{z}}|

= |\frac{- 10,000}{16,260}| + |\frac{25,000,000}{838,660}| + |\frac{- 25,000,000}{355,900}|

=0.615 + 29.809 + 70.244 = 100.668 N/mm²

τ_{x y} = |\frac{M x}{Z x}| + \sqrt{{|\frac{F y}{A y}|}^{2} + {|\frac{F z}{A z}|}^{2}}

= |\frac{5,000,000}{189,700}| + \sqrt{{|\frac{5,000}{5,833}|}^{2} + {|\frac{5,000}{5,833}|}^{2}}

= 27.57 N/mm²

From σ_x and τ_xy , f_m is calculated:

f_{m} = \sqrt{σ_{x}^{2} + 3 τ_{x y}^{2}} = 111.42 N / {m m}^{2}

For permanent loading, k = 1 and f_t = f_y/1.5 = 133.33 N/mm²

Therefore, the ratio = f_m/k×f_t = 111.419/(1 × 133.33) = 0.836

Results

Table 1. Comparison of results
Result Type	Reference	STAAD.Pro	Difference
f_m (N/mm²)	111.42	111.42	none
f_t (N/mm²)	133.33	133.33	none
σ_x (N/mm²)	100.668	100.669	negligible
τ_xy (N/mm²)	27.569	27.57	negligible
Ratio	0.836	0.836	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 MISES parameter.std is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 27-Apr-15
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 5 0 0;
MEMBER INCIDENCES
1 1 2;
DEFINE MATERIAL START
ISOTROPIC STEEL
E 2.05e+008
POISSON 0.3
DENSITY 76.8195
ALPHA 1.2e-005
DAMP 0.03
END DEFINE MATERIAL
MEMBER PROPERTY JAPANESE
1 TABLE ST L250X250X35
*1 TABLE ST H100X50X5
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 FIXED
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
JOINT LOAD
2 FX 10 FY 5 FZ 5 MX 5
PERFORM ANALYSIS PRINT STATICS LOAD
PRINT MEMBER PROPERTIES ALL
PRINT ANALYSIS RESULTS
PARAMETER 1
CODE JAPANESE 2002
*TMP 1 ALL
*KZ 1.5 ALL
FYLD 200000 ALL
MISES 1 ALL
*MBG -1 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH

STAAD.Pro Output

                        STAAD.Pro CODE CHECKING - (AIJ-2002)   V2.1
                        ***********************
ALL UNITS ARE IN - METE  KN   (U.N.O.)
|-----------------------------------------------------------------------------|
|   Member No:       1        Profile:ST  L250X250X35      (JAPANESE SECTIONS)  |
|  Ratio:    0.836  (PASS)  Reference: Eq.5.16           Loadcase:       1    |
|-----------------------------------------------------------------------------|
|  Location:    0.000  Criteria:     Stress      Load Case:     1  (Permanent)|
|  Px:(T)      -10.000       Vy:        -5.000          Vz:      -5.000       |
|  Tx:          -5.000       My:        25.000          Mz:     -25.000       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES AT DISTANCE:    0.000 MM              (UNIT: MM)         |
|  Ax:     1.62600E+04       Az: 5.83333E+03            Ay: 5.83333E+03       |
|  Iz:     3.79328E+07       Iy: 1.48256E+08             J: 6.63950E+06       |
|  Zz:     3.55901E+05       Zy: 8.38661E+05            Zx: 1.89700E+05       |
|  iZ:     4.83000E+01       iY: 9.54873E+01            Iw: 2.99365E+10       |
|-----------------------------------------------------------------------------|
| MATERIAL PROPERTIES   (Unit: N/mm2)                                         |
|  Fy:         200.000      E :   205000.000          G :     79000.000       |
|-----------------------------------------------------------------------------|
| DESIGN PROPERTIES                                                           |
|  Member Length:     5.000      Lz:     5.000 Ly:     5.000  UNL:      5.000 |
| DESIGN PARAMETERS                                                           |
|  Kz:   1.000     Ky:   1.000   NSF:  1.000   Cb:   0.000                    |
|-----------------------------------------------------------------------------|
| CRITICAL SLENDERNESS (Tension)                                              |
|  Actual :    103.520      Allowable :    400.000      Ratio :  0.259        |
|-----------------------------------------------------------------------------|
| Section Classification Results :    Width to Thickness Ratios               |
|-----------------------------------------------------------------------------|
|            Compression              |             Bending                   |
|-----------------------------------------------------------------------------|
|  LC#   |    Actual   |   Allowable  |   LC#   |    Actual   |   Allowable   |
|-----------------------------------------------------------------------------|
|   -    |      -      |       -      |      1  |     7.1429  |     14.1421   |
|-----------------------------------------------------------------------------|
| CHECKS                              |Env|     Stresses    |                 |
|              | Loc. | Demand |  L/C |Typ| Actual | Allow  |Ratio |   Ref.   |
|              |(MET) |(KN-MET)|      |   |(N/mm2) |(N/mm2) |      |          |
|--------------|------|--------|------|---|--------|--------|------|----------|
| Tension      | 0.000|  -10.00|     1| P |    0.62|  133.33| 0.005| Eq.5.1   | 
| Compression  |  -   |   -    |  -   | - |   -    |   77.53|  -   | Eq.5.3   | 
| Bending Z (T)| 0.000|  -25.00|     1| P |   70.24|  133.33| 0.527| Eq.5.1   | 
| Bending Z (C)| 0.000|  -25.00|     1| P |   70.24|  133.33| 0.527| Eq.5.1   | 
| Bending Y (T)| 0.000|   25.00|     1| P |   29.81|  133.33| 0.224| Eq.5.1   | 
| Bending Y (C)| 0.000|   25.00|     1| P |   29.81|  133.33| 0.224| Eq.5.1   | 
| Shear Z      | 0.000|   -5.00|     1| P |    0.86|   76.98| 0.011| Eq.5.2   | 
| Shear Y      | 0.000|   -5.00|     1| P |    0.86|   76.98| 0.011| Eq.5.2   | 
| Comp+ Bend C | 0.000|   -    |     1| P |   -    |    -   | 0.750| Eq.6.1   | 
| Comp+ Bend T | 0.000|   -    |     1| P |   -    |    -   | 0.750| Eq.6.2   | 
| Ten + Bend T | 0.000|   -    |     1| P |   -    |    -   | 0.755| Eq.6.3   | 
| Ten + Bend C | 0.000|   -    |     1| P |   -    |    -   | 0.746| Eq.6.4   | 
| Von-Mises    | 0.000|   -    |     1| P |  111.42|  133.33| 0.836| Eq.5.16  | 
|                                                                             |
| INTERMEDIATE RESULTS (Bending)                                              |
|  C:  0.000                                                                  |
|                                                                             |
| INTERMEDIATE RESULTS (Von-Mises)                                            |
|  Sigmax:   100.669 N/mm2   Tou:    27.570 N/mm2  fm:   111.420 N/mm2        |
|-----------------------------------------------------------------------------|