V. AS4100 1998 - Bending Capacity for Non-compact Section

Verify the bending capacity for a non-compact section per AS4100-1998.

Given

The member is 8 m long simply-supported beam. The section is a 900WB218. The member is subject to major axis bending due to uniform dead load of 4.17 kN/m and live load of 8 kN/m as well as concentrated dead load of 104 kN and live load of 140 kN at mid-span.

Section Properties

Overall depth of section, D = 910 mm
Width of section, B = 350 mm
Thickness of web, t_w = 12 mm
Thickness of flange, t_f = 25 mm
Area, A_g = 27,800 mm²
Moment of inertia about major axis, I_zz = 4,060 (10)⁶ mm⁴
Moment of inertia about minor axis, I_yy = 179 (10)⁶ mm⁴
Elastic section modulus about major axis, Z_xx = 8.92 (10)⁶ mm³
Elastic section modulus about minor axis, Z_yy = 1.02 (10)⁶ mm³
Plastic section modulus about major axis, S_xx = 9.96 (10)⁶ mm³
Plastic section modulus about minor axis, S_yy = 1.56 (10)⁶ mm³
Torsion constant, J = 4.02 (10)⁶ mm⁴
Warping moment, I_w = 35.05 (10)¹² mm⁴
Radius of gyration, $r_{x} = \sqrt{\frac{I_{zz}}{A}} = 382.2 mm$
Radius of gyration, $r_{y} = \sqrt{\frac{I_{yy}}{A}} = 80.2 mm$

Material Properties

E = 200,000 MPa
G = 80,000 MPa
f_y = 360 MPa (flange); f_y = 400 MPa (web)

Validation

Section Slenderness Check

\frac{b}{2} = \frac{(350 - 12)}{2} = 169 mm

Flange section slenderness parameter, $λ_{ef} = \frac{b}{t_{f}} \sqrt{\frac{f_{y}}{250}} = \frac{169}{25} \sqrt{\frac{360}{250}} = 8.112 > λ_{ep} = 8$ , therefore the flange is non-compact.

d = D - 2×t_f = 910 - 25×2 = 860 mm

Web section slenderness parameter, $λ_{ew} = \frac{d}{t_{w}} \sqrt{\frac{f_{y}}{250}} = \frac{860}{12} \sqrt{\frac{400}{250}} = 90.65 > λ_{ep} = 82$ , therefore the web is non-compact.

Section Bending Capacity Major Axis

The plastic section capacity, Z_cx, is the minimum of S_xx or 1.5×Z_xx = 1.5(8.92 × 10⁶) = 13.38 ×10⁶ mm³; so Z_cx = 9.96 ×10⁶ mm³.

From Table 5.2 of AS 4100, λ_ey = 115 and λ_ep = 82.

For sections where $λ_{ep} < λ < λ_{e y}$ , the effective section modulus is calculated as:

Z_{ez} = Z_{xx} + \frac{λ_{ey} - λ_{ew}}{λ_{ey} - λ_{ep}} (Z_{cx} - Z_{xx}) = [8.92 + \frac{115 - 90.65}{115 - 82} (9.96 - 8.92)] 10^{6} = 9.69 \times 10^{6} {mm}^{3}

M_{sx} = Z_{ez} \times f_{y} = 9.69 \times 360 = 3,488 kN·m

ϕ M_{sx} = 0.9 \times 3,488 = 3,139 kN·m

Section Bending Capacity Minor Axis

The plastic section capacity, Z_cy is the minimum of S_yy or 1.5×Z_yy = 1.5(1.02 × 10⁶) = 1.53 ×10⁶ mm³; so Z_cy = 1.53 ×10⁶ mm³.

For sections where $λ_{e p} < λ < λ_{e y}$ , the effective section modulus is calculated as:

Z_{ey} = Z_{yy} + \frac{λ_{ey} - λ_{ew}}{λ_{ey} - λ_{ep}} (Z_{cy} - Z_{yy}) = [1.02 + \frac{115 - 90.65}{115 - 82} (1.53 - 1.02)] 10^{6} = 1.53 \times 10^{6} {mm}^{3}

M_{sy} = Z_{ey} \times f_{y} = 1.53 \times 360 = 552.4 kN·m

ϕ M_{sy} = 0.9 \times 552.4 = 497.1 kN·m

Member Bending Capacity

Twist restraint factor,

k_{t} = 1 + (\frac{2 \times d}{l}) {(\frac{0.5 \times t_{f}}{t_{w}})}^{3} = 1 + (\frac{2 \times 860}{8000}) {(\frac{0.5 \times 25}{12})}^{3} = 1.24

From Table 5.6.3(2) of AS 4100, Load Height Factors, $k_{l} = 1.4$ .

From Table 5.6.3(3) of AS 4100, Lateral Rotation Restraint Factors, $k_{r} = 1.0$ .

Effective length of the member:

l_{e} = k_{t} k_{l} k_{r} l = 1.24 \times 1.4 \times 1.0 \times 8,000 = 13,900 mm

M_{o} = \sqrt{A (B + C)}

(Eqn 5.6.1.1(3) )

where

A	=	$\frac{π^{2} E I_{y}}{{l_{e}}^{2}} = \frac{π^{2} \times 200 \times 10^{3} 179 \times 10^{6}}{{(13,900)}^{2}} = 1.829 \times 10^{6} N$
B	=	$\frac{π^{2} E I_{w}}{{l_{e}}^{2}} = \frac{π^{2} \times 200 \times 10^{3} 35 \times 10^{12}}{{(13,900)}^{2}} = 357.6 \times 10^{9} {N·mm}^{2}$
C	=	$G \times J = 80 \times {(10)}^{3} 4.02 \times {(10)}^{6} = 321.6 \times {(10)}^{9} {N·mm}^{2}$

M_{o} = \sqrt{1.829 \times {(10)}^{6} [357.6 \times 10^{9} + 321.6 \times {(10)}^{9}]} = 1,116 \times {(10)}^{6} N·mm = 1,116 kN·m

The slenderness reduction factor is then:

ɑ_{s} = 0.6 [\sqrt{\frac{{M_{s}}^{2}}{{M_{o}}^{2}} + 3} - \frac{M_{s}}{M_{o}}] = 0.6 [\sqrt{\frac{{(3,488)}^{2}}{{(1,116)}^{2}} + 3} - \frac{3,488}{1,116}] = 0.268

(Eqn. 5.6.1.1(2) )

The moment shape factor is then:

M₁ = M₃ = 436.8 kN·m

M₂ = M_max = 805.6 kN·m

ɑ_{m} = \frac{1.7 M_{\max}}{\sqrt{{M_{2}}^{2} + {M_{3}}^{2} + {M_{4}}^{2}}} = \frac{1.7 \times 805.6}{\sqrt{{(436.8)}^{2} + {(805.6)}^{2} + {(436.8)}^{2}}} = 1.35

The member moment capacity:

ϕ M_{b} = ϕ ɑ_{m} ɑ_{s} M_{s} = 0.9 \times 1.35 \times 0.268 \times 3,488 = 1,135 kN·m

(Eqn 5.6.1.1(1) )

Comparison

Table 1. Comparison of results
Result Type	Theory	STAAD.Pro	Difference
Nominal section capacity major axis, M_sx (kN·m)	3,139	3,139	none
Nominal section capacity major axis, M_sy (kN·m)	497.1	495.7831	negligible
Member bending capacity, M_bx (kN·m)	1,135	1,134.4	negligible

STAAD Input

Tip: You can copy and paste this content directly into a .std file to run in STAAD.Pro.

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\Australia\AS4100 1998 - Bending Capacity for Non-Compact section.std is typically installed with the program.

The following design parameters are used:

The steel grade of Fy = 350 MPa is given by SGR 8 (AS/NZS 3678 400).
The load height occuring at the top flange is specified by LHT 1.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 22-Mar-21
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 8.00002 0 0;
MEMBER INCIDENCES
1 1 2;
DEFINE PMEMBER
1 PMEMBER 1
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 AUSTRALIAN
1 TABLE ST WB900X218
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 FIXED BUT FX MY MZ
LOAD 1 LOADTYPE Dead  TITLE DEAD
MEMBER LOAD
1 UNI GY -4.17
1 CON GY -104
LOAD 2 LOADTYPE Dead  TITLE LIVE
MEMBER LOAD
1 UNI GY -8
1 CON GY -140
LOAD COMB 3 COMBINATION LOAD CASE 3
1 1.2 2 1.5 
PERFORM ANALYSIS
PARAMETER 1
CODE AUSTRALIAN
LHT 1 PMEMB 1
SGR 8 PMEMB 1
TRACK 2 PMEMB 1
PBRACE TOP 0 P 1 P PMEMB 1
PBRACE BOTTOM 0 P 1 P PMEMB 1
CHECK CODE PMEMB 1
FINISH

STAAD 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   WB900X218                (AUSTRALIAN SECTIONS)
   Governing Load Case:     3*
   Governing Criteria: AS-8.4.4.2  
   Governing Ratio:   0.710  (PASS)
   SECTION PROPERTIES
   =====================================================================================
    d:       909.9999 mm   bf:       350.0000 mm
   tf:        25.0000 mm   tw:        12.0000 mm
   Ag:     27800.0000 mm2   J:     4.0200E+06 mm4             Iw:    35.0493E+12 mm6
   Iz:     4.0600E+09 mm4  Sz:     9.9600E+06 mm3 (plastic)   Zz:     8.9231E+06 mm3 (elastic)
   rz:   382.1559E+00 mm
   Iy:   179.0000E+06 mm4  Sy:     1.5600E+06 mm3 (plastic)   Zy:     1.0229E+06 mm3 (elastic)
   ry:    80.2424E+00 mm
   MATERIAL PROPERTIES
   =====================================================================================
   Material Standard        :  AS 3678
   Nominal Grade            :  400
   Residual Stress Category :  HW (Heavily welded longitudinally)
   E (#)      :199947.000 MPa    [AS 4100 1.4]
   G          : 80000.000 MPa    [AS 4100 1.4]
   fy, flange :   360.000 MPa    [AS 4100 Table 2.1]
   fy, web    :   400.000 MPa    [AS 4100 Table 2.1]
   fu         :   480.000 MPa    [AS 4100 Table 2.1]
   SLENDERNESS
   =====================================================================================
   Actual slenderness   :  99.698
   Allowable slenderness: 400.000
      STAAD SPACE                                              -- PAGE NO.    4
   BENDING
   =====================================================================================
   Section Bending Capacity
   Critical Load Case:     3*          Critical Ratio:   0.257
   Critical Location:        4.000 m from Start.
   Mz* =  -805.6343E+00 KNm                      My* =     0.0000E+00 KNm
   Z-Axis Section Slenderness: Noncompact        Y-Axis Section Slenderness: Noncompact
   Zez =     9.6881E+06 mm3                      Zey =     1.5302E+06 mm3
   &#981;Msz =     3.1390E+03 KNm                     &#981;Msy =   495.7831E+00 KN[AS 4100 5.2.1]
   Member Bending Capacity
   Critical Load Case:     3*          Critical Ratio:   0.710
   Critical Location:        4.000 m from Start.
   Crtiical Segment/Sub-segment: 
   Location (Type):   0.00 m(P )-  8.00 m(P )
   Length:    8.00 m
   Mz* =  -805.6343E+00 KNm                      My* =     0.0000E+00 KNm
   kt   =      1.24                         [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   =     13.92 m                       [AS4100 5.6.3]
   αm   =     1.349                         [AS4100 5.6.1.1(a)(iii)]
   Mo   =     1.1120E+03 KNm                [AS4100 5.6.1.1(a)(iv)]
   αsz  =     0.268                         [AS4100 5.6.1.1(a)(iv)]
   ϕMbz =     1.1344E+03 KNm (&lt;= ϕMsz)      [AS4100 5.6.1.1(a)]
   SHEAR
   =====================================================================================
   Section Shear Capacity
   Critical Load Case:     3*          Critical Ratio:   0.111
   Critical Location:        0.667 m from Start.
   Vy*  =   224.0802E+00 KN
   ϕVvy  =     2.0266E+03 KN                [AS 4100 5.11.2]
   ϕVvmy =     2.0266E+03 KN                [AS 4100 5.12.3]
   Vz*  =     0.0000E+00 KN
   ϕVvz  =     3.4020E+03 KN                [AS 4100 5.11.2]
   ϕVvmz =     3.4020E+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    =    21.4645E+03 mm2               [AS 4100 6.2.3 / 6.2.4]
   kf    =     0.772                        [AS 4100 6.2.2]
   An    =    27.8000E+03 mm2
   ϕNs   =     6.9545E+03 KN                [AS 4100 6.2.1]
   Member Compression Capacity
   Lz    =      8.00 m
   Ly    =      8.00 m
   Lez   =      8.00 m
   Ley   =      8.00 m
   αb    =      0.50                        [AS 4100 Table 6.3.3(1)/6.3.3(2)]
   λn,z  =    22.073                        [AS 4100 6.3.3]
   αa,z  =     8.186                        [AS 4100 6.3.3]
   λ,z   =    26.166                        [AS 4100 6.3.3]
    h ,z   =     0.041                        [AS 4100 6.3.3]
    x ,z   =     6.660                        [AS 4100 6.3.3]
   αc,z  =     0.957                        [AS 4100 6.3.3]
   ϕNcz  = 0.6655E+4 KN                     [AS 4100 6.3.3]
   λn,y  =   105.125                        [AS 4100 6.3.3]
   αa,y  =    16.742                        [AS 4100 6.3.3]
   λ,y   =   113.496                        [AS 4100 6.3.3]
    h ,y   =     0.326                        [AS 4100 6.3.3]
    x ,y   =     0.917                        [AS 4100 6.3.3]
   αc,y  =     0.457                        [AS 4100 6.3.3]
   ϕNcy  = 0.3175E+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    =    27.8000E+03 mm2
   ϕNt   =     9.0072E+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:     3*          Critical Ratio:   0.257
   Critical Location:        4.000 m from Start.
   N* =     0.0000E+00 KN    Mz* =  -805.6343E+00 KNm      My* =     0.0000E+00 KNm
   ϕNs   =     6.9545E+03 KN                [AS 4100 8.3.1]
   ϕMsz  =     3.1390E+03 KNm
   ϕMsy  =   495.7831E+00 KNm
   ϕMrz  =     3.1390E+03 KNm               [AS 4100 8.3.2]
   ϕMry  =   495.7831E+00 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:     3*          Critical Ratio:   0.710
   Critical Location:        4.000 m from Start.
   N* =     0.0000E+00 KN    Mz* =  -805.6343E+00 KNm      My* =     0.0000E+00 KNm
   ϕMbz  =     1.1344E+03 KNm
   ϕNt   =     9.0072E+03 KN                [AS 4100 8.4.4.2]
   ϕMozt =     1.1344E+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:     3*          Critical Ratio:   0.619
   Critical Location:        4.000 m from Start.
   N* =     0.0000E+00 KN    Mz* =  -805.6343E+00 KNm      My* =     0.0000E+00 KNm
   ϕMtz  =     1.1344E+03 KNm               [AS 4100 8.4.5.2]
   ϕMry  =   495.7831E+00 KNm               [AS 4100 8.4.5.2]
      STAAD SPACE                                              -- PAGE NO.    7
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