V. AS4100 1998 - Welded section subject to bending

Verify the bending capacity of a welded girder per the AS 4100 1998 code.

Details

The member is 21 m in length, with pinned supports at both ends. The section has top & bottom flanges of 450 mm × 60 mm and a web of 1,390 mm × 32 mm. The plates are all grade 280 steel. The member is subject to major axis bending. The section is partially restrained at the supports and at the third points along the span. The point loads are applied at the top flange.

Material Properties

E = 199,947 MPa
G = 80,000 MPa
F_y = 280 MPa
F_u = 440 MPa

Validation

Section Properties

Depth of section, D = 1,390 + 2 × 60 = 1,510 mm
Cross-sectional area, A_g = 1,390 × 32 + 2 × 450 × 60 = 98,480 mm²
Moment of inertia about major axis:
$I_{zz} = 2 [\frac{b_{f} {t_{f}}^{3}}{12} + b_{f} t_{f} {(\frac{d_{w} + t_{f}}{2})}^{2}] + \frac{t_{w} {d_{w}}^{3}}{12} = 2 [\frac{450 {(60)}^{3}}{12} + 450 \times 60 {(\frac{1,390 + 60}{2})}^{2}] + \frac{32 {(1,390)}^{3}}{12} = 35.6 {(10)}^{9} {mm}^{4}$
Moment of inertia about minor axis:
$I_{yy} = \frac{d_{w} {t_{w}}^{3}}{12} + 2 \frac{t_{f} {b_{f}}^{3}}{12} = \frac{1,390 \times {(32)}^{3}}{12} + 2 \frac{60 {(450)}^{3}}{12} = 915 {(10)}^{6} {mm}^{4}$
Elastic section modulus, $Z_{xx} = \frac{I_{xx}}{\frac{D}{2}} = \frac{35.6 {(10)}^{9}}{1,510 / 2} = 47.1 {(10)}^{6} {mm}^{3}$
Plastic section modulus,
$S_{xx} = 2 [\frac{{d_{w}}^{2} t_{w}}{8} + b_{f} t_{f} (\frac{d_{w} - t_{f}}{2})] = 2 [\frac{{(1,390)}^{2} 32}{8} + 450 \times 60 (\frac{1,390 - 60}{2})] = 54.6 {(10)}^{6} {mm}^{3}$

Warping constant:

I_{w} = \frac{I_{y} {(d_{w} + t_{f})}^{2}}{4} = \frac{915 {(10)}^{6} {(1,390 + 60)}^{2}}{4} = 481 {(10)}^{12} {mm}^{6}

(AS 4100-1998, Section H4)

Torsional constant:

J = \frac{2 b_{f} {t_{f}}^{3} + d_{w} {t_{w}}^{3}}{3} = \frac{2 (450) {(60)}^{3} + 1,390 {(32)}^{3}}{3} = 80.0 {(10)}^{6} {mm}^{4}

(AS 4100-1998, Section H4)

Section Slenderness Ratio

Flange section slenderness parameter:

\frac{b}{2} = \frac{b_{f} - t_{w}}{2} = \frac{450 - 32}{2} = 209 mm

λ_{ef} = \frac{b}{t_{f}} \sqrt{\frac{f_{y}}{250}} = \frac{209}{60} \sqrt{\frac{280}{250}} = 3.69 < λ_{ep} = 8

Hence, the flange is compact.

Web section slenderness parameter:

λ_{ew} = \frac{d_{w}}{t_{w}} \sqrt{\frac{f_{y}}{250}} = \frac{1,390}{32} \sqrt{\frac{280}{250}} = 46.0 < λ_{ep} = 82

Hence, the web is compact.

Section Bending Capacity

Z_cx is the lesser of S_xx and 1.5×Z_xx = 1.5(47.1×10⁶) = 70.7(10⁶).

M_{sx} = Z_{cx} f_{y} = 54.6 {(10)}^{6} 280 {(10)}^{-6} = 15,290 kN·m

ϕ M_{sx} = 0.9 \times 15,290 = 13,760 kN·m

Member Bending Capacity

The length between braced points, l = 7,000 mm

k_{t} = 1 + (\frac{2 d}{l}) {(\frac{0.5 t_{f}}{t_{w}})}^{3} = 1 + (\frac{2 \times 1,390}{7,000}) {(\frac{0.5 \times 60}{32})}^{3} = 1.33

Top flange load height position: k_l = 1.4. No lateral rotational restraint: k_r = 1.0.

The effective length, l_e = k_tk_lk_rl = 1.33 × 1.4 ×1.0 ×7,000 = 13,000 mm.

The reference buckling moment:

M_{o} = \sqrt{\frac{π^{2} E I_{y}}{{l_{e}}^{2}} [G J + \frac{π^{2} E I_{w}}{{l_{e}}^{2}}]} = \frac{3,800}{79.2} \sqrt{\frac{1 \times 250}{250}}

= \sqrt{\frac{π^{2} \times 199,947 \times 915 {(10)}^{6}}{{(13,000)}^{2}} [80,000 \times 80 {(10)}^{6} + \frac{π^{2} \times 199,947 \times 481 {(10)}^{12}}{{(13,000)}^{2}}]} {(10)}^{-6} = 11,330 kN·m

The moment modification factor:

ɑ_{m} = \frac{1.7 M_{m}}{\sqrt{{(M_{2})}^{2} + {(M_{3})}^{2} + {(M_{4})}^{2}}}

where

M_M	=	the maximum design moment in the segment, = M₃ = 6,142.5 kN·m (from STAAD.Pro analysis)
M₂, = M₄	=	the design moment at the quarter points in the segment, = 5,862.5 kN·m (from STAAD.Pro analysis)

ɑ_{m} = \frac{1.7 \times 6,142.5}{\sqrt{2 \times {(5,862.5)}^{2} + {(6,142.5)}^{2}}} = 1.012

The slenderness reduction factor per AS 4100 Eq. 5.6.1.1(2):

ɑ_{s} = 0.6 [\sqrt{\frac{{M_{s}}^{2}}{{M_{o}}^{2}} + 3} - \frac{M_{s}}{M_{o}}] = 0.6 [\sqrt{\frac{{(15,290)}^{2}}{{(11,330)}^{2}} + 3} - \frac{15,290}{11,330}] = 0.508

ϕ M_{b} = ϕ ɑ_{m} ɑ_{s} M_{s} = 0.9 \times 1.012 \times 0.508 \times 15,290 = 7,074 kN·m

Results

Result Type	Reference	STAAD.Pro	Difference	Comments
Nominal section capacity major axis, M_sx(kN·m)	13,760	13,760.9	negligible
Member bending capacity, M_bx (kN·m)	7,074	7,072	negligible

STAAD.Pro Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\Australia\AS4100 1998 - Welded section subject to bending.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 21-Jul-21
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 7.00001 0 0; 3 14 0 0; 4 21 0 0;
MEMBER INCIDENCES
1 1 2; 2 2 3; 3 3 4;
DEFINE PMEMBER
1 TO 3 PMEMBER 1
START USER TABLE
TABLE 2
UNIT METER KN
WIDE FLANGE
1510X450X60X32
0.09848 1.51 0.032 0.45 0.06 0.0355616 0.000915046 7.99825e-05 0.04832 -
0.036 0.45 0.06
END
DEFINE MATERIAL START
ISOTROPIC STEEL
E 1.99947e+08
POISSON 0.3
DENSITY 76.8191
ALPHA 6.5e-06
DAMP 0.03
G 7.7221e+07
TYPE STEEL
STRENGTH RY 1.5 RT 1.2
END DEFINE MATERIAL
MEMBER PROPERTY
1 TO 3 UPTABLE 2 1510X450X60X32
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
4 FIXED BUT FX MZ
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
1 TO 3 UNI GY -40
JOINT LOAD
2 3 FY -250
MEMBER LOAD
1 TO 3 CON GY -250
PERFORM ANALYSIS
PARAMETER 1
CODE AUSTRALIAN
TRACK 2 PMEMB 1
LHT 1 PMEMB 1
IST 4 PMEMB 1
PBRACE TOP 0 P 0.333 P 0.666 P 1 P PMEMB 1
PBRACE BOTTOM 0 P 0.333 P 0.666 P 1 P PMEMB 1
CHECK CODE PMEMB ALL
FINISH

STAAD.Pro Output

                    STAAD.Pro CODE CHECKING - (  AS4100-1998   ) V2.3           
                    ****************************************************
   MEMBER DESIGN OUTPUT FOR PMEMBER     1
   DESIGN Notes
   ------------
   1. (*) next to a Load Case number signifies that a P-Delta analysis has not been performed for
      that particular Load Case; i.e. analysis does not include second-order effects.
   2. ϕ = 0.9 for all the calculations [AS4100 Table 3.4]
   3. (#) next to Young's modulus E indicates that its value is not 200000 MPa as per AS4100 1.4.
   DESIGN SUMMARY
   =====================================================================================
   Designation: ST   1510X450X60X32           (UPT)
   Governing Load Case:     1*
   Governing Criteria: AS-8.4.4.2  
   Governing Ratio:   0.869  (PASS)
   SECTION PROPERTIES
   =====================================================================================
    d:      1510.0000 mm   bf:       450.0000 mm
   tf:        60.0000 mm   tw:        32.0000 mm
   Ag:     98480.0000 mm2   J:    79.9825E+06 mm4             Iw:   480.9710E+12 mm6
   Iz:    35.5616E+09 mm4  Sz:    54.6068E+06 mm3 (plastic)   Zz:    47.1015E+06 mm3 (elastic)
   rz:   600.9199E+00 mm
   Iy:   915.0460E+06 mm4  Sy:     6.4308E+06 mm3 (plastic)   Zy:     4.0669E+06 mm3 (elastic)
   ry:    96.3934E+00 mm
   MATERIAL PROPERTIES
   =====================================================================================
   Material Standard        :  AS/NZS 3679.1
   Nominal Grade            :  300
   Residual Stress Category :  LW (Lightly welded longitudinally)
   E (#)      :199947.000 MPa    [AS 4100 1.4]
   G          : 80000.000 MPa    [AS 4100 1.4]
   fy, flange :   280.000 MPa    [AS 4100 Table 2.1]
   fy, web    :   280.000 MPa    [AS 4100 Table 2.1]
   fu         :   440.000 MPa    [AS 4100 Table 2.1]
   SLENDERNESS
   =====================================================================================
   Actual slenderness   : 217.857
   Allowable slenderness: 400.000
      STAAD SPACE                                              -- PAGE NO.    4
   BENDING
   =====================================================================================
   Section Bending Capacity
   Critical Load Case:     1*          Critical Ratio:   0.446
   Critical Location:       10.500 m from Start.
   Mz* =    -6.1425E+03 KNm                      My* =     0.0000E+00 KNm
   Z-Axis Section Slenderness: Compact           Y-Axis Section Slenderness: Compact
   Zez =    54.6068E+06 mm3                      Zey =     6.1003E+06 mm3
   ϕMsz =    13.7609E+03 KNm                     ϕMsy =     1.5373E+03 KN[AS 4100 5.2.1]
   Member Bending Capacity
   Critical Load Case:     1*          Critical Ratio:   0.869
   Critical Location:       10.500 m from Start.
   Crtiical Segment/Sub-segment: 
   Location (Type):   6.99 m(P )- 13.99 m(P )
   Length:    6.99 m
   Mz* =    -6.1425E+03 KNm                      My* =     0.0000E+00 KNm
   kt   =      1.33                         [AS4100 Table 5.6.3(1)]
   kl   =      1.40                         [AS4100 Table 5.6.3(2)]
   kr   =      1.00                         [AS4100 Table 5.6.3(3)]
   le   =     13.00 m                       [AS4100 5.6.3]
   αm   =     1.012                         [AS4100 5.6.1.1(a)(iii)]
   Mo   =    11.3341E+03 KNm                [AS4100 5.6.1.1(a)(iv)]
   αsz  =     0.508                         [AS4100 5.6.1.1(a)(iv)]
   ϕMbz =     7.0722E+03 KNm (&lt;= ϕMsz)      [AS4100 5.6.1.1(a)]
   SHEAR
   =====================================================================================
   Section Shear Capacity
   Critical Load Case:     1*          Critical Ratio:   0.152
   Critical Location:        0.583 m from Start.
   Vy*  =     1.0217E+03 KN
   ϕVvy  =     6.7254E+03 KN                [AS 4100 5.11.2]
   ϕVvmy =     6.7254E+03 KN                [AS 4100 5.12.3]
   Vz*  =     0.0000E+00 KN
   ϕVvz  =     8.1648E+03 KN                [AS 4100 5.11.2]
   ϕVvmz =     8.1648E+03 KN                [AS 4100 5.12.3]
      STAAD SPACE                                              -- PAGE NO.    5
   AXIAL
   =====================================================================================
   Section Compression Capacity
   Critical Load Case:     1*          Critical Ratio:   0.000
   Critical Location:        0.000 m from Start.
   N*    =     0.0000E+00 KN
   Ae    =    92.7036E+03 mm2               [AS 4100 6.2.3 / 6.2.4]
   kf    =     0.941                        [AS 4100 6.2.2]
   An    =    98.4800E+03 mm2
   ϕNs   =    23.3613E+03 KN                [AS 4100 6.2.1]
   Member Compression Capacity
   Lz    =     21.00 m
   Ly    =     21.00 m
   Lez   =     21.00 m
   Ley   =     21.00 m
   αb    =      1.00                        [AS 4100 Table 6.3.3(1)/6.3.3(2)]
   λn,z  =    35.883                        [AS 4100 6.3.3]
   αa,z  =    16.856                        [AS 4100 6.3.3]
   λ,z   =    52.739                        [AS 4100 6.3.3]
    h ,z   =     0.128                        [AS 4100 6.3.3]
    x ,z   =     2.142                        [AS 4100 6.3.3]
   αc,z  =     0.847                        [AS 4100 6.3.3]
   ϕNcz  = 0.1979E+5 KN                     [AS 4100 6.3.3]
   λn,y  =   223.694                        [AS 4100 6.3.3]
   αa,y  =     9.070                        [AS 4100 6.3.3]
   λ,y   =   232.764                        [AS 4100 6.3.3]
    h ,y   =     0.715                        [AS 4100 6.3.3]
    x ,y   =     0.628                        [AS 4100 6.3.3]
   αc,y  =     0.133                        [AS 4100 6.3.3]
   ϕNcy  = 0.3109E+4 KN                     [AS 4100 6.3.3]
   ϕNc   = N/A                              [AS 4100 6.3.3 / AS 4600 3.4.1(b)]
   Section Tension Capacity
   Critical Load Case:     1*          Critical Ratio:   0.000
   Critical Location:        0.000 m from Start.
   N*    =     0.0000E+00 KN
   kt    =      1.00                        [User defined]
   An    =    98.4800E+03 mm2
   ϕNt   =    24.8170E+03 KN                [AS 4100 7.2]
      STAAD SPACE                                              -- PAGE NO.    6
   COMBINED BENDING AND AXIAL
   =====================================================================================
   Section Combined Capacity
   Critical Condition: Cl 8.3.2
   Critical Load Case:     1*          Critical Ratio:   0.446
   Critical Location:       10.500 m from Start.
   N* =     0.0000E+00 KN    Mz* =    -6.1425E+03 KNm      My* =     0.0000E+00 KNm
   ϕNs   =    23.3613E+03 KN                [AS 4100 8.3.1]
   ϕMsz  =    13.7609E+03 KNm
   ϕMsy  =     1.5373E+03 KNm
   ϕMrz  =    13.7609E+03 KNm               [AS 4100 8.3.2]
   ϕMry  =     1.5373E+03 KNm               [AS 4100 8.3.3]
   Member Combined Capacity - In-plane
   Critical Load Case: N/A             Critical Ratio: N/A
   Critical Location: N/A
   Member Combined Capacity - Out-of-plane(compression)
   Critical Load Case: N/A             Critical Ratio: N/A
   Critical Location: N/A
   Member Combined Capacity - Out-of-plane(tension)
   Critical Load Case:     1*          Critical Ratio:   0.869
   Critical Location:       10.500 m from Start.
   N* =     0.0000E+00 KN    Mz* =    -6.1425E+03 KNm      My* =     0.0000E+00 KNm
   ϕMbz  =     7.0722E+03 KNm
   ϕNt   =    24.8170E+03 KN                [AS 4100 8.4.4.2]
   ϕMozt =     7.0722E+03 KNm               [AS 4100 8.4.4.2]
   Member Combined Capacity - Biaxial(compression)
   Critical Load Case: N/A             Critical Ratio: N/A
   Critical Location: N/A
   Member Combined Capacity - Biaxial(tension)
   Critical Load Case:     1*          Critical Ratio:   0.821
   Critical Location:       10.500 m from Start.
   N* =     0.0000E+00 KN    Mz* =    -6.1425E+03 KNm      My* =     0.0000E+00 KNm
   ϕMtz  =     7.0722E+03 KNm               [AS 4100 8.4.5.2]
   ϕMry  =     1.5373E+03 KNm               [AS 4100 8.4.5.2]
      STAAD SPACE                                              -- PAGE NO.    7
   ********************************************************************************
   ************** END OF TABULATED RESULT OF DESIGN **************