V.AIJ 2005 UPT Channel

Verify the bending capacity and compression capacity of a channel section beam.

Details

The member is a 5 m long simply-supported beam. The section used is a C380X100X10.5. 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·m concentrated moment at the left end
a -3 kN·m concentrated moment at the right end

Validation

Section Properties

Dimensions:

Depth, D = 380 mm
Width, B = 100 mm
Thickness of web, t_w = 10.5 mm
Thickness of flange, t_f = 16 mm

Height of web, d = D - 2t_f = 348 mm

Area, A_x = 2B×t_f + d×t_w = 6,854 mm²

Distance from heel to centroid, $c_{z} = \frac{t_{f} B^{2} + \frac{1}{2} {t_{w}}^{2} d}{A} = \frac{16 {(100)}^{2} + \frac{1}{2} {(10.5)}^{2} (348)}{6,854} = 26.14 mm$

Moment of inertia about major axis, $I_{z} = \frac{d^{3} t_{w}}{12} + 2 [\frac{{t_{f}}^{3} B}{12} + t_{f} B {(\frac{t_{f}}{2} + \frac{d}{2})}^{2}] = 142.9 {(10)}^{6} {mm}^{4}$

Moment of inertia about minor axis, $I_{y} = \frac{{t_{w}}^{3} d}{12} + t_{w} d {(c_{z} - \frac{t_{w}}{2})}^{2} + 2 [\frac{t_{f} B^{3}}{12} + t_{f} B {(\frac{B}{2} - c_{z})}^{2}] = 6.117 {(10)}^{6} {mm}^{4}$

Major axis section modulus, $Z_{z} = \frac{I_{z}}{\frac{D}{2}} = 752.3 {(10)}^{3} {mm}^{3}$

Minor axis section modulus, $Z_{y} = \frac{I_{y}}{B - c_{z}} = 82.82 {(10)}^{3} {mm}^{3}$

Torsional constant, $J = \frac{1}{3} (d {t_{w}}^{3} + 2 B {t_{f}}^{3}) = 407.4 {(10)}^{3} {mm}^{4}$

Warping constant, $C_{W} = 144.3 {(10)}^{9} {mm}^{6}$

Radius of gyration about major axis, $r_{z} = \sqrt{\frac{I_{z}}{A}} = 144.4 mm$

Radius of gyration about minor axis, $r_{y} = \sqrt{\frac{I_{y}}{A}} = 29.87 mm$

Section Classification

For webs of beams (Cl. 8.5):

\frac{d}{t_{w}} = 33.14 < 2.4 \sqrt{\frac{E}{F}} = 2.4 \sqrt{\frac{205,000}{235}} = 70.89

Depth to thickness ratio is "OK".

Allowable Compressive Stress

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

λ = \frac{K_{y} \times L_{y}}{r_{y}} = \frac{1 \times 5,000}{29.87} = 167.4

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{167.4}{119.8})}^{2} =2.802

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{167.4}{119.8})}^{2}} = 33.35 MPa

Allowable Flexural Stress

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

The yield moment, $M_{y} = F \times Z_{z} = \frac{235 \times 752.3 {(10)}^{3}}{10^{6}} = 176.8 kN·m$

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

C = 1.75 - 1.05 \times (\frac{M_{2}}{M_{1}}) + 0.3 \times {(\frac{M_{2}}{M_{1}})}^{2} = 1.75 - 1.05 \times (\frac{3}{10}) + 0.3 \times {(\frac{3}{10})}^{2} = 1.462

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 6.117 \times 10^{6} \times 205,000 \times 144.3 \times 10^{9}}{{5,000}^{4}} = 5.791 \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 6.117 \times 10^{6} \times 79,000 \times 407.4 \times 10^{3}}{{5,000}^{2}} = 15.93 \times 10^{15}

= 1.462 \times \sqrt{5.791 \times 10^{15} + 15.93 \times 10^{15}} = 215.4 kN·m

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

p λ_{b} = 0.6 + 0.3 \times (\frac{- M_{2}}{M_{1}}) = 0.6 + 0.3 \times (\frac{- 3}{10}) = 0.51

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}}} = 0.906

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} = 1.828

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{0.906 - 0.3}{1.291 - 0.3}) \times 235}{1.828} = 102.5 MPa

Results

Table 1.
Result Type	Reference	STAAD.Pro	Difference	Comments
Allowable compressive stress (MPa)	33.35	33.34	negligible
Allowable flexural stress (MPa)	102.5	102.48	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 Channel.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
31 19 0 0; 32 24 0 0;
MEMBER INCIDENCES
16 31 32;
START USER TABLE
TABLE 1
UNIT METER KN
GENERAL
GE_H300X150X6.5X9
0.004678 0.3 0.0065 0.15 0.009 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
*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
WF_H300X100-200X7X9-12
0.005253 0.3 0.007 0.1 0.009 7.29942e-05 8.75797e-06 1.71399e-07 0.0021 -
0.0022 0.2 0.012
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
0.006854 0.38 0.0105 0.1 0.016 0.000142897 6.11657e-06 4.07351e-07 -
0.0261429 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
DA_L100X100X13LD
0.1 0.1 0.013 0 4.48744e-06 8.79409e-06 2.73893e-07 0.0294 0.00173333 -
0.00173333 1.94
TABLE 9
UNIT METER KN
TEE
TE_CT100X200X8X12
0.003177 0.1 0.2 0.012 0.008 1.84e-06 8.01e-06 1.3e-07 0.0173 0.000533333 -
0.0016
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
16 UPTABLE 4 CH_C380X100X10.5
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
31 PINNED
32 FIXED BUT FX MY MZ
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
JOINT LOAD
31 MZ 10
32 MZ -3
*148 FX 0.01 FY 0.01 MX 0.8
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:      16        Profile  : CH_C380X100X10(UPT)                    |
|  Ratio:    0.130  (PASS)  Reference: Eq.5.8            Loadcase:       1    |
|-----------------------------------------------------------------------------|
|  Location:    0.000  Criteria:     Stress      Load Case:     1  (Permanent)|
|  Px:(C)        0.000       Vy:         1.400          Vz:       0.000       |
|  Tx:           0.000       My:         0.000          Mz:      10.000       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES AT DISTANCE:    0.000 MM              (UNIT: MM)         |
|  Ax:     6.85400E+03       Az: 2.13333E+03            Ay: 3.99000E+03       |
|  Iz:     1.42897E+08       Iy: 6.11657E+06             J: 4.07351E+05       |
|  Zz:     7.52090E+05       Zy: 8.28163E+04            Zx: 2.54594E+04       |
|  iZ:     1.44391E+02       iY: 2.98732E+01            Iw: 1.44256E+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 (Tension)                                              |
|  Actual :    167.374      Allowable :    400.000      Ratio :  0.418        |
|-----------------------------------------------------------------------------|
| Section Classification Results :    Width to Thickness Ratios               |
|-----------------------------------------------------------------------------|
|            Compression              |             Bending                   |
|-----------------------------------------------------------------------------|
|  LC#   |    Actual   |   Allowable  |   LC#   |    Actual   |   Allowable   |
|-----------------------------------------------------------------------------|
|   -    |      -      |       -      |      1  |    33.1429  |     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  |  -   |   -    |  -   | - |   -    |   33.34|  -   | Eq.5.4   | 
| Bending Z (T)| 0.000|   10.00|     1| P |   13.30|  156.67| 0.085| Eq.5.1   | 
| Bending Z (C)| 0.000|   10.00|     1| P |   13.30|  102.48| 0.130| 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|    1.40|     1| P |    0.35|   90.45| 0.004| Eq.5.2   | 
| Comp+ Bend C | 0.000|   -    |     1| P |   -    |    -   | 0.130| Eq.6.1   | 
| Comp+ Bend T | 0.000|   -    |     1| P |   -    |    -   | 0.085| Eq.6.2   | 
| Ten + Bend T | 0.000|   -    |     1| P |   -    |    -   | 0.085| Eq.6.3   | 
| Ten + Bend C | 0.000|   -    |     1| P |   -    |    -   | 0.130| Eq.6.4   | 
| Von-Mises    | 0.000|   -    |     1| P |   13.31|  156.67| 0.085| Eq.5.24  | 
|                                                                             |
| INTERMEDIATE RESULTS (Bending)                                              |
|  Me: 2.1541E+02 KN-MET  My: 1.7674E+02 KN-MET   pλb:   0.510  C:  1.462     |
|                                                                             |
| INTERMEDIATE RESULTS (Von-Mises)                                            |
|  Sigmax:    13.296 N/mm2   Tou:     0.351 N/mm2  fm:    13.310 N/mm2        |
|-----------------------------------------------------------------------------|