V.AIJ 2005 UPT I

Verify the bending capacity and compression capacity of an I section beam.

Details

The member is a 5 m long simply-supported beam. The section used is an I300X150X8. The steel has a yield strength of 235 MPa, a modulus of elasticity of 205×(10)³ MPa, and a shear modulus of 79×(10)³ MPa.

The member is subject to the following loads:

a -10 kN concentrated force at mid-span in the global Y direction
a 3 kN concentrated force at mid-span in the global Z direction
a -40 kN axial force
a -0.2 kN·m concentrated moment at the left end

Validation

Section Properties

Dimensions:

Leg, D = 300 mm
Width, B = 150 mm
Thickness of web, t_w = 8 mm
Thickness of flange, t_f = 13 mm

Depth of web, d = D - 2t_f = 274 mm

Area, A_x = 6,092 mm²

Moment of inertia about major axis, $I_{z} = 94.08 {(10)}^{6} {mm}^{4}$

Moment of inertia about minor axis, $I_{y} = 7.324 {(10)}^{6} {mm}^{4}$

Major axis section modulus, $Z_{z} = 627.2 {(10)}^{3} {mm}^{3}$

Minor axis section modulus, $Z_{y} = 97.66 {(10)}^{3} {mm}^{3}$

Torsional constant, $J = 26.87 {(10)}^{3} {mm}^{4}$

Warping constant, $C_{W} = 150.6 {(10)}^{9} {mm}^{6}$

Radius of gyration about major axis, $r_{z} = \sqrt[]{\frac{I_{z}}{A}} = 124.3 mm$

Radius of gyration about minor axis, $r_{y} = \sqrt[]{\frac{I_{y}}{A}} = 34.67 mm$

Section Classification

For webs of beams in bending (Cl. 8.5):

\frac{d}{t_{w}} = \frac{274}{8} = 34.25 < 2.4 \sqrt[]{\frac{E}{F}} = 2.4 \sqrt[]{\frac{205,000}{235}} = 70.89

Depth to thickness ratio is "OK".

For flanges of beams in bending (Cl. 8.3):

\frac{B}{t_{f}} = \frac{150}{13} = 11.54 < 1.6 \sqrt[]{\frac{E}{F}} = 1.6 \sqrt[]{\frac{205,000}{235}} = 47.26

Depth to thickness ratio is "OK".

Allowable Compressive Stress

The slenderness ratio of compression members (Cl. 11.1):

λ_{y} = \frac{K_{y} \times L_{y}}{r_{y}} = \frac{1 \times 5,000}{34.67} = 144.2

Slenderness ratio dividing elastic and inelastic buckling (Cl no 5.5):

Λ = \sqrt{\frac{π^{2} \times E}{0.6 \times F}} = \sqrt{\frac{π^{2} \times 205,000}{0.6 \times 235}} = 119.8

ν= \frac{3}{2} + \frac{2}{3} × {(\frac{λ}{Ʌ})}^{2} = \frac{3}{2} + \frac{2}{3} × {(\frac{144.2}{119.8})}^{2} =2.466

λ > Λ, so the allowable compressive stress is given by Cl. 5.4:

f_{c} = \frac{0.277 \times F}{{(\frac{λ}{Ʌ})}^{2}} = \frac{0.277 \times 235}{{(\frac{144.2}{119.8})}^{2}} = 44.92 MPa

Allowable Flexural Stress

Distance between braced points of the compression flange, l_b = 5,000 mm.

The yield moment, $M_{y} = F \times Z_{z} = \frac{235 \times 627.2 {(10)}^{3}}{10^{6}} = 147.4 kN·m$

The modification factor for allowable flexural stress (Cl. 5.13):

C = 1

The elastic lateral-torsional buckling moment (Cl 5.12):

M_{e} = C \times \sqrt{\frac{π^{4} \times E \times I_{y} \times E \times I_{w}}{{l_{b}}^{4}} + \frac{π^{2} \times E \times I_{y} \times G \times J}{{l_{b}}^{2}}}

\frac{π^{4} \times E \times I_{y} \times E \times I_{w}}{{l_{b}}^{4}} = \frac{π^{4} \times 205,000 \times 7.324 \times 10^{6} \times 205,000 \times 150.6 \times 10^{9}}{{5,000}^{4}} = 7.224 \times 10^{15}

\frac{π^{2} \times E \times I_{y} \times G \times J}{{l_{b}}^{2}} = \frac{π^{2} \times 205,000 \times 7.324 \times 10^{6} \times 79,000 \times 26.87 \times 10^{3}}{{5,000}^{2}} = 1.258 \times 10^{15}

= 1.0 \times \sqrt{7.224 \times 10^{15} + 1.258 \times 10^{15}} = 140.6 kN·m

The maximum slenderness ratio to achieve the plastic strength (Cl. 5.12):

p λ_{b} = 0.3

The elastic slenderness limit:

e λ_{b} = \frac{1}{\sqrt{0.6}} = 1.291

Slenderness ratio of the flexural member,

λ_{b} = \sqrt{\frac{M_{y}}{M_{e}}} = 1.024

Here, pλ_b < λ_b ≤ eλ_b, so:

ν = \frac{3}{2} + \frac{2}{3} \times {(\frac{λ_{b}}{e λ_{b}})}^{2} = \frac{3}{2} + \frac{2}{3} \times {(\frac{1.024}{1.291})}^{2} = 1.919

f_{b} = \frac{(1 - 0.4 \times \frac{λ_{b} - p λ_{b}}{e λ_{b} - p λ_{b}}) \times F}{ν} = \frac{(1 - 0.4 \times \frac{1.024 - 0.3}{1.291 - 0.3}) \times 235}{1.919} = 86.68 MPa

Results

Table 1.
Result Type	Reference	STAAD.Pro	Difference	Comments
Allowable compressive stress (MPa)	44.92	44.92	none
Allowable flexural stress (MPa)	86.68	86.74	negligible

STAAD.Pro Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\Japan\AIJ 2005 UPT I.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 22-Oct-18
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
3 1 0 2; 4 6 0 2;
MEMBER INCIDENCES
2 3 4;
START USER TABLE
TABLE 1
UNIT METER KN
GENERAL
GE_H300X150X6.5X9
0.004678 0.3 0 0.15 0 7.21e-05 5.08e-06 9.95e-08 0.000480667 6.77333e-05 -
0.00195 0.0018 0 0 1.07545e-07 0.256
TABLE 2
UNIT METER KN
WIDE FLANGE
WF_H300X150X6.5X9
0.004678 0.3 0.0065 0.15 0.009 7.21e-05 5.08e-06 9.95e-08 0.00195 0.0018
WF2_H300X150X6.5X9
0.004678 0.3 0.0065 0.15 0.009 7.21e-05 5.08e-06 9.95e-08 0.00195 -
0.0018 0.15 0.009
TABLE 3
UNIT METER KN
ISECTION
IS_I300X150X8
0.3 0.008 0.3 0.15 0.013 0.15 0.013 0.0024 0.0026 2.69e-07
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
2 UPTABLE 3 IS_I300X150X8
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
3 PINNED
4 FIXED BUT FX MY MZ
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
2 CON GY -10
2 CON GZ 3
JOINT LOAD
4 FX -40
3 MX 0.2
PERFORM ANALYSIS
PARAMETER 1
CODE JAPANESE 2005
TRACK 2 ALL
CHECK CODE ALL
PRINT MEMBER PROPERTIES ALL
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  : IS_I300X150X8 (UPT)                    |
|  Ratio:    0.621  (PASS)  Reference: Eq.6.1            Loadcase:       1    |
|-----------------------------------------------------------------------------|
|  Location:    2.500  Criteria:     Stress      Load Case:     1  (Permanent)|
|  Px:(C)       40.000       Vy:         5.000          Vz:      -1.500       |
|  Tx:           0.200       My:        -3.750          Mz:     -12.500       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES AT DISTANCE: 2500.000 MM              (UNIT: MM)         |
|  Ax:     6.09200E+03       Az: 2.60000E+03            Ay: 2.40000E+03       |
|  Iz:     9.40786E+07       Iy: 7.32419E+06             J: 2.69000E+05       |
|  Zz:     6.27191E+05       Zy: 9.76559E+04            Zx: 2.06923E+04       |
|  iZ:     1.24270E+02       iY: 3.46737E+01            Iw: 1.50822E+11       |
|-----------------------------------------------------------------------------|
| MATERIAL PROPERTIES   (Unit: N/mm2)                                         |
|  Fy:         235.000      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    pλb:  0.000     |
|-----------------------------------------------------------------------------|
| CRITICAL SLENDERNESS (Compression)                                          |
|  Actual :    144.202      Allowable :    200.000      Ratio :  0.721        |
|-----------------------------------------------------------------------------|
| Section Classification Results :    Width to Thickness Ratios               |
|-----------------------------------------------------------------------------|
|            Compression              |             Bending                   |
|-----------------------------------------------------------------------------|
|  LC#   |    Actual   |   Allowable  |   LC#   |    Actual   |   Allowable   |
|-----------------------------------------------------------------------------|
|      1 |    34.2500  |     47.2567  |      1  |    34.2500  |     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|   40.00|     1| P |    6.57|   44.92| 0.146| Eq.5.4   | 
| Bending Z (T)| 2.500|  -12.50|     1| P |   19.93|  156.67| 0.127| Eq.5.1   | 
| Bending Z (C)| 2.500|  -12.50|     1| P |   19.93|   86.74| 0.230| Eq.5.8   | 
| Bending Y (T)| 2.500|   -3.75|     1| P |   38.40|  156.67| 0.245| Eq.5.1   | 
| Bending Y (C)| 2.500|   -3.75|     1| P |   38.40|  156.67| 0.245| Eq.5.1   | 
| Shear Z      | 0.000|   -1.50|     1| P |    0.58|   90.45| 0.006| Eq.5.2   | 
| Shear Y      | 0.000|    5.00|     1| P |    2.08|   90.45| 0.023| Eq.5.2   | 
| Comp+ Bend C | 2.500|   -    |     1| P |   -    |    -   | 0.621| Eq.6.1   | 
| Comp+ Bend T | 2.500|   -    |     1| P |   -    |    -   | 0.330| Eq.6.2   | 
| Ten + Bend T | 2.500|   -    |     1| P |   -    |    -   | 0.372| Eq.6.3   | 
| Ten + Bend C | 2.500|   -    |     1| P |   -    |    -   | 0.475| Eq.6.4   | 
| Von-Mises    | 2.500|   -    |     1| P |   68.05|  156.67| 0.434| Eq.5.24  | 
|                                                                             |
| INTERMEDIATE RESULTS (Bending)                                              |
|  Me: 1.4083E+02 KN-MET  My: 1.4739E+02 KN-MET   pλb:   0.300  C:  1.000     |
|                                                                             |
| INTERMEDIATE RESULTS (Von-Mises)                                            |
|  Sigmax:    64.896 N/mm2   Tou:    11.827 N/mm2  fm:    68.053 N/mm2        |
|-----------------------------------------------------------------------------|