V. AS4100 1998 - Combined Section and Member In-Plane Capacity

Verify the bending capacity in major and minor axis bending, biaxial bending, and in-plane bending for a UC member subject an axial load per AS4100-1998.

Given

The member is 7 m long and part of a steel frame structure. The section is a 200UC52.2. The member is subject to an axial load of 143.9 kN.

Section Properties

Depth of section, D = 206 mm
Width of section, B = 204 mm
Thickness of web, t_w = 8 mm
Thickness of flange, t_f = 12.5 mm
Area, A_g = 6,660 mm²
Moment of inertia about major axis, I_zz = 52.8 (10)⁶ mm⁴
Moment of inertia about minor axis, I_yy = 17.7 (10)⁶ mm⁴
Elastic section modulus about major axis, Z_xx = 513 (10)³ mm³
Elastic section modulus about minor axis, Z_yy = 174 (10)³ mm³
Plastic section modulus about major axis, S_xx = 570 (10)³ mm³
Plastic section modulus about minor axis, S_yy = 264 (10)³ mm³
Radius of gyration, $r_{x} = \sqrt{\frac{I_{zz}}{A}} = 89.0 mm$
Radius of gyration, $r_{y} = \sqrt{\frac{I_{yy}}{A}} = 51.6 mm$

Material Properties

f_y = 300 MPa (flange); f_y = 320 MPa (web)

Validation

Combined Major Axis Section Moment Capacity

Nominal axial capacity: $ϕ N_{s} = 0.9 \times 6.66 \times 300 = 1,798 kN$

Nominal section capacity: $ϕ M_{sx} = 0.9 \times 570 \times 300 = 153.9 kN·m$

The section is compact, so the alternate method is applicable.

ϕ M_{rx} = ϕ 1.18 M_{sx} (1 - \frac{N}{ϕ N_{s}}) \leq ϕ M_{sx}

ϕ M_{rx} = 1.18 \times 153.9 (1 - \frac{143.9}{1,798}) = 167.1 kN·m > ϕ M_{sx}

ϕ M_{rx} = ϕ M_{sx} = 153.9 kN·m

Combined Minor Axis Section Moment Capacity

Nominal section capacity: $ϕ M_{sy} = ϕ S_{yy} F_{y} \leq ϕ 1.5 Z_{yy} F_{y}$

ϕ M_{sy} = 0.9 \times 264 \times 300 = 71.3 kN·m > 0.9 \times 1.5 \times 174 \times 300 = 70.5 kN·m

The section is compact, so the alternate method is applicable.

ϕ M_{ry} = ϕ 1.19 M_{sy} {(1 - \frac{N}{ϕ N_{s}})}^{2} \leq ϕ M_{sy}

ϕ M_{ry} = 1.19 \times 70.5 {(1 - \frac{143.9}{1,798})}^{2} = 71.0 kN·m > ϕ M_{sy}

ϕ M_{ry} = ϕ M_{sy} = 70.5 kN·m

Member In-plane Capacity Check

The section is compact, so the alternate method is applicable. The larger end moment = 111.1 kN·m and the smaller end moment = 108.1 kN·m.

β_{mx} = \frac{108.1}{111.1} = 0.973

ϕ M_{ix} = ϕ M_{s} {[1 - {(\frac{1 + β_{m}}{2})}^{3}] (1 - \frac{N}{ϕ N_{c}}) + 1.18 {(1 + \frac{1 + β_{m}}{2})}^{3} \sqrt{(1 - \frac{N}{ϕ N_{c}})}} \leq ϕ M_{rx}

ϕ M_{ix} = 153.9 {[1 - {(\frac{1 + 0.973}{2})}^{3}] (1 - \frac{143.9}{1,146}) + 1.18 {(1 + \frac{1 + 0.973}{2})}^{3} \sqrt{(1 - \frac{143.9}{1,146})}} = 168.4 kN·m > ϕ M_{rx}

ϕ M_{ix} = ϕ M_{rx} = 153.9 kN·m

Comparison

Table 1. Comparison of results
Result Type	Theory	STAAD.Pro	Difference
Combined major axis section moment capacity (kN·m)	153.9	153.9	none
Combined minor axis section moment capacity (kN·m)	153.9	153.9	none
Menber in-plane capacity check (kN·m)	153.9	153.9	none

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 - Combined Section and Member In-Plane Capacity.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 16-Apr-21
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 0 7 0; 3 12 7 0; 4 12 0 0; 5 0 12 0; 6 12 12 0;
MEMBER INCIDENCES
1 1 2; 2 2 3; 3 3 4; 4 2 5; 5 3 6; 6 5 6;
DEFINE PMEMBER
1 PMEMBER 4
4 PMEMBER 5
6 PMEMBER 6
2 PMEMBER 7
5 PMEMBER 8
3 PMEMBER 9
DEFINE MATERIAL START
ISOTROPIC STEEL
E 2.05e+08
POISSON 0.3
DENSITY 76.8195
ALPHA 1.2e-05
DAMP 0.03
G 7.88462e+07
TYPE STEEL
STRENGTH RY 1.5 RT 1.2
END DEFINE MATERIAL
MEMBER PROPERTY AUSTRALIAN
4 5 TABLE ST UC150X30.0
1 3 TABLE ST UC200X52.2
6 TABLE ST UB250X25.7
2 TABLE ST UB310X40.4
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 4 FIXED
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
6 CON GY -40
2 CON GY -80
JOINT LOAD
5 6 FY -20
2 3 FY -50
5 FX 10
2 FX 30
PERFORM ANALYSIS
PARAMETER 1
CODE AUSTRALIAN
TRACK 2 PMEMB 9
CHECK CODE PMEMB 9
FINISH

STAAD Output

                    STAAD.Pro CODE CHECKING - (  AS4100-1998   ) V2.3           
                    ****************************************************
   MEMBER DESIGN OUTPUT FOR PMEMBER     9
   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   UC200X52.2               (AUSTRALIAN SECTIONS)
   Governing Load Case:     1*
   Governing Criteria: SLENDERNESS 
   Governing Ratio:   0.754  (PASS)
   SECTION PROPERTIES
   =====================================================================================
    d:       206.0000 mm   bf:       204.0000 mm
   tf:        12.5000 mm   tw:         8.0000 mm
   Ag:      6660.0000 mm2   J:   325.0000E+03 mm4             Iw:   165.6820E+09 mm6
   Iz:    52.8000E+06 mm4  Sz:   570.0000E+03 mm3 (plastic)   Zz:   512.6214E+03 mm3 (elastic)
   rz:    89.0389E+00 mm
   Iy:    17.7000E+06 mm4  Sy:   264.0000E+03 mm3 (plastic)   Zy:   173.5294E+03 mm3 (elastic)
   ry:    51.5525E+00 mm
   MATERIAL PROPERTIES
   =====================================================================================
   Material Standard        :  AS/NZS 3679.1
   Nominal Grade            :  300
   Residual Stress Category :  HR (Hot-rolled)
   E (#)      :204999.984 MPa    [AS 4100 1.4]
   G          : 80000.000 MPa    [AS 4100 1.4]
   fy, flange :   300.000 MPa    [AS 4100 Table 2.1]
   fy, web    :   320.000 MPa    [AS 4100 Table 2.1]
   fu         :   440.000 MPa    [AS 4100 Table 2.1]
   SLENDERNESS
   =====================================================================================
   Actual slenderness   : 135.784
   Allowable slenderness: 180.000
      STAAD SPACE                                              -- PAGE NO.    4
   BENDING
   =====================================================================================
   Section Bending Capacity
   Critical Load Case:     1*          Critical Ratio:   0.722
   Critical Location:        0.000 m from Start.
   Mz* =   111.0612E+00 KNm                      My* =     0.0000E+00 KNm
   Z-Axis Section Slenderness: Compact           Y-Axis Section Slenderness: Compact
   Zez =   570.0000E+03 mm3                      Zey =   260.2941E+03 mm3
   ϕMsz =   153.9000E+00 KNm                     ϕMsy =    70.2794E+00 KN[AS 4100 5.2.1]
   Member Bending Capacity
   Critical Load Case:     1*          Critical Ratio:   0.722
   Critical Location:        0.000 m from Start.
   Crtiical Segment/Sub-segment: 
   Location (Type):   0.00 m(F )-  7.00 m(F )
   Length:    7.00 m
   Mz* =   111.0612E+00 KNm                      My* =     0.0000E+00 KNm
   kt   =      1.00                         [AS4100 Table 5.6.3(1)]
   kl   =      1.00                         [AS4100 Table 5.6.3(2)]
   kr   =      1.00                         [AS4100 Table 5.6.3(3)]
   le   =      7.00 m                       [AS4100 5.6.3]
   αm   =     2.435                         [AS4100 5.6.1.1(a)(iii)]
   Mo   =   154.9263E+00 KNm                [AS4100 5.6.1.1(a)(iv)]
   αsz  =     0.570                         [AS4100 5.6.1.1(a)(iv)]
   ϕMbz =   153.9000E+00 KNm (&lt;= ϕMsz)      [AS4100 5.6.1.1(a)]
   SHEAR
   =====================================================================================
   Section Shear Capacity
   Critical Load Case:     1*          Critical Ratio:   0.110
   Critical Location:        0.000 m from Start.
   Vy*  =    31.3075E+00 KN
   ϕVvy  =   284.7744E+00 KN                [AS 4100 5.11.2]
   ϕVvmy =   284.7744E+00 KN                [AS 4100 5.12.3]
   Vz*  =     0.0000E+00 KN
   ϕVvz  =   826.2000E+00 KN                [AS 4100 5.11.2]
   ϕVvmz =   826.2000E+00 KN                [AS 4100 5.12.3]
      STAAD SPACE                                              -- PAGE NO.    5
   AXIAL
   =====================================================================================
   Section Compression Capacity
   Critical Load Case:     1*          Critical Ratio:   0.269
   Critical Location:        0.000 m from Start.
   N*    =   143.8336E+00 KN
   Ae    =     6.6600E+03 mm2               [AS 4100 6.2.3 / 6.2.4]
   kf    =     1.000                        [AS 4100 6.2.2]
   An    =     6.6600E+03 mm2
   ϕNs   =     1.7982E+03 KN                [AS 4100 6.2.1]
   Member Compression Capacity
   Lz    =      7.00 m
   Ly    =      7.00 m
   Lez   =      7.00 m
   Ley   =      7.00 m
   αb    =      0.00                        [AS 4100 Table 6.3.3(1)/6.3.3(2)]
   λn,z  =    86.121                        [AS 4100 6.3.3]
   αa,z  =    18.714                        [AS 4100 6.3.3]
   λ,z   =    86.121                        [AS 4100 6.3.3]
    h ,z   =     0.237                        [AS 4100 6.3.3]
    x ,z   =     1.175                        [AS 4100 6.3.3]
   αc,z  =     0.637                        [AS 4100 6.3.3]
   ϕNcz  = 0.1146E+4 KN                     [AS 4100 6.3.3]
   λn,y  =   148.744                        [AS 4100 6.3.3]
   αa,y  =    12.969                        [AS 4100 6.3.3]
   λ,y   =   148.744                        [AS 4100 6.3.3]
    h ,y   =     0.441                        [AS 4100 6.3.3]
    x ,y   =     0.764                        [AS 4100 6.3.3]
   αc,y  =     0.298                        [AS 4100 6.3.3]
   ϕNcy  = 0.5353E+3 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    =     6.6600E+03 mm2
   ϕNt   =     1.7982E+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:   0.722
   Critical Location:        0.000 m from Start.
   N* =   143.8336E+00 KN    Mz* =   111.0612E+00 KNm      My* =     0.0000E+00 KNm
   ϕNs   =     1.7982E+03 KN                [AS 4100 8.3.1]
   ϕMsz  =   153.9000E+00 KNm
   ϕMsy  =    70.2794E+00 KNm
   ϕMrz  =   153.9000E+00 KNm               [AS 4100 8.3.2]
   ϕMry  =    70.2794E+00 KNm               [AS 4100 8.3.3]
   Member Combined Capacity - In-plane
   Critical Load Case:     1*          Critical Ratio:   0.722
   Critical Location:        0.000 m from Start.
   N* =   143.8336E+00 KN    Mz* =   111.0612E+00 KNm      My* =     0.0000E+00 KNm
   ϕNcz  =     1.1463E+03 KN                [AS 4100 8.4.2.2]
   ϕMiz  =   153.9000E+00 KNm               [AS 4100 8.4.2.2]
   ϕNcy  =   535.3019E+00 KN                [AS 4100 8.4.2.2]
   ϕMiy  =    70.2794E+00 KNm               [AS 4100 8.4.2.2]
   Member Combined Capacity - Out-of-plane(compression)
   Critical Load Case:     1*          Critical Ratio:   0.722
   Critical Location:        0.000 m from Start.
   N* =   143.8336E+00 KN    Mz* =   111.0612E+00 KNm      My* =     0.0000E+00 KNm
   ϕMbz  =   153.9000E+00 KNm
   ϕNcy  =   535.3019E+00 KN
   ϕMozc =   153.9000E+00 KNm               [AS 4100 8.4.4.1]
   Member Combined Capacity - Out-of-plane(tension)
   Critical Load Case: N/A             Critical Ratio: N/A
   Critical Location: N/A
   Member Combined Capacity - Biaxial(compression)
   Critical Load Case:     1*          Critical Ratio:   0.633
   Critical Location:        0.000 m from Start.
   N* =   143.8336E+00 KN    Mz* =   111.0612E+00 KNm      My* =     0.0000E+00 KNm
   ϕMcz  =   153.9000E+00 KNm               [AS 4100 8.4.5.1]
   ϕMiy  =    70.2794E+00 KNm               [AS 4100 8.4.5.1]
   Member Combined Capacity - Biaxial(tension)
   Critical Load Case: N/A             Critical Ratio: N/A
   Critical Location: N/A
      STAAD SPACE                                              -- PAGE NO.    7
   ********************************************************************************