V.AIJ 2005 UPT Tee

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

Details

The member is a 2 m long simply-supported beam. The section used is a CT100x200x8x12. 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 -2 kN·m concentrated moment at the left end
a 2 kN·m concentrated moment at the right end

Validation

Section Properties

Dimensions:

Leg, D = 100 mm
Width, B = 200 mm
Thickness of web, t_w = 8 mm
Thickness of flange, t_f = 12 mm

Depth of web, d = D - t_f = 88 mm

Area, A_x = 3,104 mm²

Moment of inertia about major axis, $I_{z} = 1.844 {(10)}^{6} {mm}^{4}$

Moment of inertia about minor axis, $I_{y} = 8.004 {(10)}^{6} {mm}^{4}$

Major axis section modulus, $Z_{z} = 22.31 {(10)}^{3} {mm}^{3}$

Minor axis section modulus, $Z_{y} = 80.04 {(10)}^{3} {mm}^{3}$

Torsional constant, $J = 130 {(10)}^{3} {mm}^{4}$

Warping constant, $C_{W} = 107.8 {(10)}^{9} {mm}^{6}$

Radius of gyration about major axis, $r_{z} = \sqrt[]{\frac{I_{z}}{A}} = 24.07 mm$

Radius of gyration about minor axis, $r_{y} = \sqrt[]{\frac{I_{y}}{A}} = 50.21 mm$

Section Classification

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

\frac{d}{t_{w}} = \frac{88}{8} = 11 < 0.53 \sqrt[]{\frac{E}{F}} = 0.53 \sqrt[]{\frac{205,000}{235}} = 15.65

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 2,000}{24.07} = 83.11

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{83.11}{119.8})}^{2} =1.821

λ < Λ, so the allowable compressive stress is given by Cl. 5.3:

f_{c} = \frac{[1 - 0.4 {(\frac{λ}{Ʌ})}^{2}] \times F}{ν} = \frac{[1 - 0.4 {(\frac{83.11}{119.8})}^{2}] \times 235}{1.821} = 104.2 MPa

Allowable Flexural Stress

Distance between braced points of the compression flange, l_b = 2,000 mm.

The yield moment, $M_{y} = F \times Z_{z} = \frac{235 \times 80.04 {(10)}^{3}}{10^{6}} = 18.8 kN·m$

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 1.844 \times 10^{6} \times 205,000 \times 107.8 \times 10^{9}}{{2,000}^{4}} = 50.86 \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 1.844 \times 10^{6} \times 79,000 \times 13 \times 10^{3}}{{2,000}^{2}} = 0.958 \times 10^{15}

= 1.0 \times \sqrt{50.86 \times 10^{15} + 0.958 \times 10^{15}} = 98.03 kN·m

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

p λ_{b} = 0.6

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}}} = 0.43

Here, λ_b < pλ_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{0.43}{1.291})}^{2} = 1.573

f_{b} = \frac{F}{ν} = \frac{235}{1.573} = 149.4 MPa

Results

Table 1.
Result Type	Reference	STAAD.Pro	Difference	Comments
Allowable compressive stress (MPa)	104.2	104.24	negligible
Allowable flexural stress (MPa)	149.4	149.04	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 Tee.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
127 26 0 20; 128 28 0 20;
MEMBER INCIDENCES
64 127 128;
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
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
64 UPTABLE 9 TE_CT100X200X8X12
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
127 PINNED
128 FIXED BUT FX MY MZ
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
64 CON GY -10
JOINT LOAD
127 MZ 2
128 MZ -2
PERFORM ANALYSIS
UNIT MMS KN
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 - MMS   KN   (U.N.O.)
|-----------------------------------------------------------------------------|
|   Member No:      64        Profile  : TE_CT100X200X8(UPT)                    |
|  Ratio:    0.867  (PASS)  Reference: Eq.5.24           Loadcase:       1    |
|-----------------------------------------------------------------------------|
|  Location:  999.999  Criteria:     Stress      Load Case:     1  (Permanent)|
|  Px:(C)        0.000       Vy:         5.000          Vz:       0.000       |
|  Tx:       0.000E+00       My:     0.000E+00          Mz:  -3.000E+03       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES AT DISTANCE:  999.999 MM              (UNIT: MM)         |
|  Ax:     3.17700E+03       Az: 1.60000E+03            Ay: 5.33333E+02       |
|  Iz:     1.84000E+06       Iy: 8.01000E+06             J: 1.30000E+05       |
|  Zz:     2.22491E+04       Zy: 8.01000E+04            Zx: 1.08333E+04       |
|  iZ:     2.40658E+01       iY: 5.02120E+01            Iw: 1.07813E+08       |
|-----------------------------------------------------------------------------|
| MATERIAL PROPERTIES   (Unit: N/mm2)                                         |
|  Fy:         235.000      E :   205000.000          G :     79000.000       |
|-----------------------------------------------------------------------------|
| DESIGN PROPERTIES                                                           |
|  Member Length:  1999.999      Lz:  1999.999 Ly:  1999.999  UNL:   1999.999 |
| DESIGN PARAMETERS                                                           |
|  Kz:   1.000     Ky:   1.000   NSF:  1.000   Cb:   0.000    pλb:  0.000     |
|-----------------------------------------------------------------------------|
| CRITICAL SLENDERNESS (Tension)                                              |
|  Actual :     83.105      Allowable :    400.000      Ratio :  0.208        |
|-----------------------------------------------------------------------------|
| Section Classification Results :    Width to Thickness Ratios               |
|-----------------------------------------------------------------------------|
|            Compression              |             Bending                   |
|-----------------------------------------------------------------------------|
|  LC#   |    Actual   |   Allowable  |   LC#   |    Actual   |   Allowable   |
|-----------------------------------------------------------------------------|
|   -    |      -      |       -      |      1  |    11.0000  |     15.6538   |
|-----------------------------------------------------------------------------|
| 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  |  -   |   -    |  -   | - |   -    |  104.21|  -   | Eq.5.3   | 
| Bending Z (T)| 1.000|   -3.00|     1| P |  134.84|  156.67| 0.861| Eq.5.1   | 
| Bending Z (C)| 0.000|    2.00|     1| P |   89.89|  156.67| 0.574| Eq.5.1   | 
| Bending Y (T)|  -   |   -    |  -   | - |   -    |  156.67|  -   | Eq.5.1   | 
| Bending Y (C)|  -   |   -    |  -   | - |   -    |  149.04|  -   | Eq.5.7   | 
| Shear Z      |  -   |   -    |  -   | - |   -    |   90.45|  -   | Eq.5.2   | 
| Shear Y      | 0.000|    5.00|     1| P |    9.38|   90.45| 0.104| Eq.5.2   | 
| Comp+ Bend C | 0.000|   -    |     1| P |   -    |    -   | 0.574| Eq.6.1   | 
| Comp+ Bend T | 1.000|   -    |     1| P |   -    |    -   | 0.861| Eq.6.2   | 
| Ten + Bend T | 1.000|   -    |     1| P |   -    |    -   | 0.861| Eq.6.3   | 
| Ten + Bend C | 0.000|   -    |     1| P |   -    |    -   | 0.574| Eq.6.4   | 
| Von-Mises    | 1.000|   -    |     1| P |  135.81|  156.67| 0.867| Eq.5.24  | 
|                                                                             |
| INTERMEDIATE RESULTS (Bending)                                              |
|  Me: 9.8026E+01 KN-MET  My: 1.8824E+01 KN-MET   pλb:   0.600  C:  1.000     |
|                                                                             |
| INTERMEDIATE RESULTS (Von-Mises)                                            |
|  Sigmax:   134.837 N/mm2   Tou:     9.375 N/mm2  fm:   135.811 N/mm2        |
|-----------------------------------------------------------------------------|