V. AIJ 2005 Check for MISES parameter 2

Determine the von Mises stress in a slender, cantilever beam subjected to a load at the end per the AIJ 2005 code.

Details

A cantilever beam of length 0.3 meters is subjected to a permanent joint load of 3 kN in the Y direction and 2 kN in the Z direction as well as a 0.008 kN·m torque applied at the end. Axial tension of 10 kN is also applied to the member. An H100x50x5 section is used from the Japanese steel tables.

Section properties

D = 100 mm, B = 50 mm, t_f = 7 mm, t_w = 5 mm
I_x = 15,000 mm⁴
A_x = 1185 mm², A_y = 500 mm², A_z = 467 mm²
Z_x = I_x/t_max = 15,000/7 = 2,143 mm³, Z_y = 5,920 mm³, Z_z = 37,400 mm³

The maximum of the left hand side of the von Mises stress equation apparently occurs at the fixed end of the beam. Section forces at the fixed end are as follows:

-10.0 kN (Tension)
0.6 kN·m (Bending-Y)
0.9 kN·m (Bending-Z)
-3.0 kN (Shear-Y)
-2.0 kN (Shear-Z)
-0.008 kN·m (Torsion

Material

FYLD = 300 MPa
E = 2.05E+05 MPa
G = E/2.6 MPa

Validation

From these section forces, σ_x and τ_xy at the section of the fixed end are calculated as follows:

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

= | \frac{- 10, 000}{1, 185} | + | \frac{600, 000}{5, 920} | + | \frac{- 900, 000}{37, 400} |

= 8.44 + 101.35 + 24.06 = 133.85 N/mm²

τ_{x y} = | \frac{M_{x}}{Z_{x}} | + \sqrt{{| \frac{F_{y}}{A_{y}} |}^{2} + {| \frac{F_{z}}{A_{z}} |}^{2}}

= | \frac{- 8, 000}{2, 143} | + \sqrt{{| \frac{- 3, 000}{500} |}^{2} + {| \frac{- 2, 000}{467} |}^{2}}

= 3.73 + \sqrt{6^{2} + {4.28}^{2}}

= 11.10 N/mm²

From σ_x and τ_xy, f_m is calculated:

f_{m} = \sqrt{σ_{x}^{2} + 3 τ_{x y}^{2}} = \sqrt{{(133.85)}^{2} + 3 {(11.10)}^{2}} = 135.22 N / {m m}^{2}

Since f_t = FYLD/1.5 = 300.0 MPa/15 = 200.0 N/mm² and k = 1 for permanent loading,

Ratio = 135.22/(200.0 · 1) = 0.676 < 1, So OK.

Results

Result Type	Reference	STAAD.Pro Result	Difference	Comments
von Mises Stress, f_m (N/mm²)	135.22	135.42	negligible

STAAD Input File

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ is typically installed with the program.

The following design parameters are used:

The standard AIJ calculation for the von Mises stresses is used per the MISES 1 parameter.
An unsupported length of 2 mm is specified using the UNL 0.002 parameter.
A steel yield strength of 300 GPa is specified using the FYLD 300000 parameter.

STAAD SPACE VERIFICATION FOR VON MISES STRESSES IN AIJ 2005
START JOB INFORMATION
ENGINEER DATE 18-AUG-10
END JOB INFORMATION
UNIT MMS KN
JOINT COORDINATES
1 0 0 0; 2 300 0 0;
MEMBER INCIDENCES
1 1 2;
UNIT METER KN
DEFINE MATERIAL START
ISOTROPIC STEEL
E 2.05e+08
POISSON 0.3
DENSITY 76.8195
ALPHA 1.2e-05
DAMP 0.03
END DEFINE MATERIAL
MEMBER PROPERTY JAPANESE
1 TABLE ST H100x50x5x7
UNIT MMS KN
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 FIXED
UNIT METER KN
LOAD 1 LC1
JOINT LOAD
2 FX 10 FY 3 FZ 2 MX 0.008
PERFORM ANALYSIS
LOAD LIST 1
PRINT MEMBER FORCES LIST ALL
PARAMETER 1
CODE JAPANESE 2005
UNL 0.002 ALL
MISES 1 ALL
TRACK 2 ALL
FYLD 300000 ALL
CHECK CODE ALL
FINISH

Output

                        STAAD.Pro CODE CHECKING - (AIJ-2005)   V2.1
                        ***********************
ALL UNITS ARE IN - METE  KN   (U.N.O.)
|-----------------------------------------------------------------------------|
|   Member No:       1        Profile:ST  H100X50X5X7      (JAPANESE SECTIONS)  |
|  Ratio:    0.677  (PASS)  Reference: Eq.5.24           Loadcase:       1    |
|-----------------------------------------------------------------------------|
|  Location:    0.000  Criteria:     Stress      Load Case:     1  (Permanent)|
|  Px:(T)      -10.000       Vy:        -3.000          Vz:      -2.000       |
|  Tx:          -0.008       My:         0.600          Mz:      -0.900       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES AT DISTANCE:    0.000 MM              (UNIT: MM)         |
|  Ax:     1.18500E+03       Az: 4.66667E+02            Ay: 4.30000E+02       |
|  Iz:     1.87000E+06       Iy: 1.48000E+05             J: 1.53000E+04       |
|  Zz:     3.74000E+04       Zy: 5.92000E+03            Zx: 2.18571E+03       |
|  iZ:     3.97248E+01       iY: 1.11756E+01            Iw: 3.20013E+08       |
|-----------------------------------------------------------------------------|
| MATERIAL PROPERTIES   (Unit: N/mm2)                                         |
|  Fy:         300.000      E :   205000.000          G :     79000.000       |
|-----------------------------------------------------------------------------|
| DESIGN PROPERTIES                                                           |
|  Member Length:     0.300      Lz:     0.300 Ly:     0.300  UNL:      0.002 |
| DESIGN PARAMETERS                                                           |
|  Kz:   1.000     Ky:   1.000   NSF:  1.000   Cb:   0.000    pλb:  0.000     |
|-----------------------------------------------------------------------------|
| CRITICAL SLENDERNESS (Tension)                                              |
|  Actual :     26.844      Allowable :    400.000      Ratio :  0.067        |
|-----------------------------------------------------------------------------|
| Section Classification Results :    Width to Thickness Ratios               |
|-----------------------------------------------------------------------------|
|            Compression              |             Bending                   |
|-----------------------------------------------------------------------------|
|  LC#   |    Actual   |   Allowable  |   LC#   |    Actual   |   Allowable   |
|-----------------------------------------------------------------------------|
|   -    |      -      |       -      |      1  |    14.0000  |     62.7376   |
|-----------------------------------------------------------------------------|
| 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 |    8.44|  200.00| 0.042| Eq.5.1   | 
| Compression  |  -   |   -    |  -   | - |   -    |  189.47|  -   | Eq.5.3   | 
| Bending Z (T)| 0.000|   -0.90|     1| P |   24.06|  200.00| 0.120| Eq.5.1   | 
| Bending Z (C)| 0.000|   -0.90|     1| P |   24.06|  200.00| 0.120| Eq.5.7   | 
| Bending Y (T)| 0.000|    0.60|     1| P |  101.35|  200.00| 0.507| Eq.5.1   | 
| Bending Y (C)| 0.000|    0.60|     1| P |  101.35|  200.00| 0.507| Eq.5.1   | 
| Shear Z      | 0.000|   -2.00|     1| P |    4.29|  115.47| 0.037| Eq.5.2   | 
| Shear Y      | 0.000|   -3.00|     1| P |    6.98|  115.47| 0.060| Eq.5.2   | 
| Comp+ Bend C | 0.000|   -    |     1| P |   -    |    -   | 0.627| Eq.6.1   | 
| Comp+ Bend T | 0.000|   -    |     1| P |   -    |    -   | 0.627| Eq.6.2   | 
| Ten + Bend T | 0.000|   -    |     1| P |   -    |    -   | 0.669| Eq.6.3   | 
| Ten + Bend C | 0.000|   -    |     1| P |   -    |    -   | 0.585| Eq.6.4   | 
| Von-Mises    | 0.000|   -    |     1| P |  135.42|  200.00| 0.677| Eq.5.24  | 
|                                                                             |
| INTERMEDIATE RESULTS (Bending)                                              |
|  Me: 6.0918E+06 KN-MET  My: 1.1220E+01 KN-MET   pλb:   0.600  C:  1.750     |
|                                                                             |
| INTERMEDIATE RESULTS (Von-Mises)                                            |
|  Sigmax:   133.854 N/mm2   Tou:    11.848 N/mm2  fm:   135.418 N/mm2        |
|-----------------------------------------------------------------------------|