V.AIJ 2005 UPT Double Angle

Verify the bending capacity and compression capacity of a double angle section beam.

Details

The member is a 5 m long simply-supported beam. The section used is a 2x L100x100x13, with no spacing. 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 -5 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 = 100 mm
Width, B = 200 mm
Thickness, t = 13 mm

Outstanding leg length, d = D - t = 87 mm

Area, A_x = B×t + 2d×t = 4,862 mm²

Distance from heel to centroid, $c_{y} = \frac{\frac{1}{2} B t^{2} + 2 d t (t + \frac{d}{2})}{A} = \frac{\frac{1}{2} 200 {(13)}^{2} + 2 (87) (13) (13 + \frac{87}{2})}{4,862} = 29.76 mm$

Moment of inertia about major axis, $I_{z} = \frac{B t^{3}}{12} + B t (c_{y} - \frac{t}{2}) + 2 [\frac{t d^{3}}{12} + t d {(t + \frac{d}{2} - c_{y})}^{2}] = 4.487 {(10)}^{6} {mm}^{4}$

Moment of inertia about minor axis, $I_{y} = \frac{t B^{3}}{12} + \frac{2}{3} d t^{3} = 8.676 {(10)}^{6} {mm}^{4}$

Major axis section modulus, $Z_{z} = \frac{I_{z}}{D - c_{y}} = 63.81 {(10)}^{3} {mm}^{3}$

Minor axis section modulus, $Z_{y} = \frac{I_{y}}{D} = 86.76 {(10)}^{3} {mm}^{3}$

Torsional constant, $J = 273.9 {(10)}^{3} {mm}^{4}$

Warping constant, $C_{W} = 199.5 {(10)}^{9} {mm}^{6}$

Radius of gyration about major axis, $r_{z} = \sqrt[]{\frac{I_{z}}{A}} = 30.38 mm$

Radius of gyration about minor axis, $r_{y} = \sqrt[]{\frac{I_{y}}{A}} = 42.53 mm$

Section Classification

For webs of beams (Cl. 8.1):

\frac{d}{t_{w}} = 7.69 < 0.44 \sqrt[]{\frac{E}{F}} = 0.44 \sqrt[]{\frac{205,000}{235}} = 13.0

Depth to thickness ratio is "OK".

Allowable Compressive Stress

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

λ_{z} = \frac{K_{z} \times L_{z}}{r_{z}} = \frac{1 \times 5,000}{30.38} = 164.6

λ_{y} = \frac{K_{y} \times L_{y}}{r_{y}} = \frac{1 \times 5,000}{42.53} = 117.6

Therefore, λ = 164.6.

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{164.6}{119.8})}^{2} =2.758

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{164.6}{119.8})}^{2}} = 34.49 MPa

Allowable Flexural Stress

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

The yield moment, $M_{y} = F \times Z_{z} = \frac{235 \times 87.94 {(10)}^{3}}{10^{6}} = 20.67 kN·m$

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_{zz} \times E \times I_{w}}{{l_{b}}^{4}} + \frac{π^{2} \times E \times I_{zz} \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 4.847 \times 10^{6} \times 205,000 \times 199.5 \times 10^{6}}{{5,000}^{4}} = 6.333 \times 10^{12}

\frac{π^{2} \times E \times I_{y} \times G \times J}{{l_{b}}^{2}} = \frac{π^{2} \times 205,000 \times 4.847 \times 10^{6} \times 79,000 \times 273.9 \times 10^{3}}{{5,000}^{2}} = 8.488 \times 10^{15}

= 1.0 \times \sqrt{6.333 \times 10^{12} + 8.488 \times 10^{15}} = 88.59 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}}} = 0.483

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{0.483}{1.291})}^{2} = 1.593

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{0.483 - 0.3}{1.291 - 0.3}) \times 235}{1.593} = 136.6 MPa

Results

Table 1.
Result Type	Reference	STAAD.Pro	Difference	Comments
Allowable compressive stress (MPa)	34.49	34.48	negligible
Allowable flexural stress (MPa)	136.6	136.62	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 Double Angle.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
29 12 0 8; 30 17 0 8;
MEMBER INCIDENCES
15 29 30;
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
TABLE 4
UNIT METER KN
CHANNEL
CH_C380X100X10.5
0.006939 0.38 0.0105 0.1 0.016 0.000145 5.35e-06 3.99e-07 0.0241 0.00399 -
0.00213333
TABLE 5
UNIT METER KN
ANGLE
AG_L175X175X15
0.175 0.175 0.015 0.0342 0.00175 0.00175
TABLE 6
UNIT METER KN
TUBE
TB_TUB1501506
0.003363 0.15 0.15 0.006 1.15e-05 1.15e-05 1.79e-05 0.0018 0.0018
TB_RHS200X100X9
0.004867 0.2 0.1 0.009 2.35e-05 7.82e-06 1.93e-05 0.0036 0.0018
TABLE 7
UNIT METER KN
PIPE
PP_PIP165.2X7.1
0.1652 0.151 0.001763 0.001763
TABLE 8
UNIT METER KN
DOUBLE ANGLE
L100X100X13_LD
0.1 0.1 0.013 0 4.48744e-06 8.79409e-06 2.73893e-07 0.029762 0.00173333 -
0.00173333 0
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
15 UPTABLE 8 L100X100X13_LD
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
29 PINNED
30 FIXED BUT FX MY MZ
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
15 CON GY -5
15 CON GZ 3
JOINT LOAD
30 FX -40
29 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:      15        Profile  : L100X100X13_LD(UPT)                    |
|  Ratio:    0.923  (PASS)  Reference: Eq.6.3            Loadcase:       1    |
|-----------------------------------------------------------------------------|
|  Location:    2.500  Criteria:     Stress      Load Case:     1  (Permanent)|
|  Px:(C)       40.000       Vy:         2.500          Vz:      -1.500       |
|  Tx:           0.200       My:        -3.750          Mz:      -6.250       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES AT DISTANCE: 2500.000 MM              (UNIT: MM)         |
|  Ax:     4.86200E+03       Az: 1.73333E+03            Ay: 1.73333E+03       |
|  Iz:     4.48744E+06       Iy: 8.79409E+06             J: 2.73893E+05       |
|  Zz:     6.12519E+04       Zy: 8.79409E+04            Zx: 2.10687E+04       |
|  iZ:     3.03803E+01       iY: 4.25293E+01            Iw: 1.99536E+08       |
|-----------------------------------------------------------------------------|
| 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 :    164.580      Allowable :    200.000      Ratio :  0.823        |
|-----------------------------------------------------------------------------|
| Section Classification Results :    Width to Thickness Ratios               |
|-----------------------------------------------------------------------------|
|            Compression              |             Bending                   |
|-----------------------------------------------------------------------------|
|  LC#   |    Actual   |   Allowable  |   LC#   |    Actual   |   Allowable   |
|-----------------------------------------------------------------------------|
|      1 |     7.6923  |     12.9956  |      1  |     7.6923  |     12.9956   |
|-----------------------------------------------------------------------------|
| 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 |    8.23|   34.48| 0.239| Eq.5.4   | 
| Bending Z (T)| 2.500|   -6.25|     1| P |  102.04|  156.67| 0.651| Eq.5.1   | 
| Bending Z (C)| 2.500|   -6.25|     1| P |   37.24|  156.67| 0.238| Eq.5.1   | 
| Bending Y (T)| 2.500|   -3.75|     1| P |   42.64|  156.67| 0.272| Eq.5.1   | 
| Bending Y (C)| 2.500|   -3.75|     1| P |   42.64|  136.62| 0.312| Eq.5.8   | 
| Shear Z      | 0.000|   -1.50|     1| P |    0.87|   90.45| 0.010| Eq.5.2   | 
| Shear Y      | 0.000|    2.50|     1| P |    1.44|   90.45| 0.016| Eq.5.2   | 
| Comp+ Bend C | 2.500|   -    |     1| P |   -    |    -   | 0.788| Eq.6.1   | 
| Comp+ Bend T | 2.500|   -    |     1| P |   -    |    -   | 0.871| Eq.6.2   | 
| Ten + Bend T | 2.500|   -    |     1| P |   -    |    -   | 0.923| Eq.6.3   | 
| Ten + Bend C | 2.500|   -    |     1| P |   -    |    -   | 0.550| Eq.6.4   | 
| Von-Mises    | 2.500|   -    |     1| P |  137.82|  156.67| 0.880| Eq.5.24  | 
|                                                                             |
| INTERMEDIATE RESULTS (Bending)                                              |
|  Me: 8.8679E+01 KN-MET  My: 2.0666E+01 KN-MET   pλb:   0.300  C:  1.000     |
|                                                                             |
| INTERMEDIATE RESULTS (Von-Mises)                                            |
|  Sigmax:   136.453 N/mm2   Tou:    11.175 N/mm2  fm:   137.819 N/mm2        |
|-----------------------------------------------------------------------------|