# V.NZS3404 1997-Simply Supported Beam With Overhang

Verify the design strength of an I section per the NZS3404 1997 code.

## Reference

Kirke, Brian, and Iyad Hassan Al-Jamel. Steel Structures Design Manual to AS4100. 2004. p.124

## Details

From the reference:

The 250UC89.5 beam in Grade 300 steel shown below is continuous over the supports at B and D and is free at A and E. The beam section is restrained against lateral deflection at B and D, fully restrained against twist rotation at B and D, and is unrestrained at A and E. A downward concentrated load of 5P* acts at C and a downward concentrated load of P* acts at A and E. These loads act at the top flange, and are free to deflect laterally with the beam. Determine the maximum value of P*.

## Validation

Section Classification

Evaluate the slenderness effects of the flange:

$λ e f = B 2 t f f y 250 = 256 2(17.3) 280 250 = 7.83$

Section classification of the flange is compact.

Evaluate the slenderness effects of the web:

$λ e w = d t w f y 250 = 225.4 10.5 280 250 = 22.72$

Section classification of the web is compact

Effective Section Modulus, Zez = 1.23×106 mm3

The nominal section capacity in bending about Z axis, Msz = ϕfy×Zez

Msz = 280× 1.23 ×10-6 = 344.4 kN·m

ϕMsz = 0.9×344.4 = 309.96 kN·m

Effective Section Modulus, Zey = 575×103 mm3

The nominal section capacity in bending about Z axis, Msy = ϕfy×Zey

Msy = 280× 575×10-3 = 161 kN·m

ϕMsy = 0.9×161 = 144.9 kN·m

Member Bending Capacity of Cantilevers

End restraint arrangement = FU

A twist restraint factor, Kt (SKT) = 1.00

Minor axis rotation restraints = Both

Lateral rotation restraint factor, Kr (SKR) = 1.0

Load Height factor, Kl, (LHT) = 2 [Ref : Table 5.6.3(2)]

Effective length = 1×1×2×4,000 = 8,000 mm

The alpha value for a cantilever is taken from Table 5.6.2 in the code.

 $α m = 1.25$ [Ref. Table 5.6.2]

Reference buckling moment, Mo

$M o = π 2 E I y L e 2 G J + π 2 E I w L e 2$
$α s = 0.6 M s x M o a 2 + 3 - M s x M o a$
 $α s = 0.6 344.4 399 2 + 3 - 344.4 399 = 0.643$ [Ref : Clause 5.6.1.1 (c)]

Mbx = αmαsMsx ≤ Msx

 Mbz = 1.25 × 0.643 × 344.4 = 276.8 kN·m ≤ (Msz, Msy)Max. [Ref : Clause 5.6.1.1.1(a)]

ϕMbz = 0.9×276.8 = 249.1 kN·m

Member Bending Capacity of Main Span

End restraint arrangement = FF

A twist restraint factor, Kt (SKT) = 1.00

Minor axis rotation restraints = Both

Lateral rotation restraint factor, Kr (SKR) = 1.0

Load Height factor, Kl, (LHT) = 1.4 [Ref : Table 5.6.3(2)]

Effective length = 1×1×1.4×8,000 = 11,200 mm
$α m = 1.7 M m × ( M 1 * ) 2 + ( M 2 * ) 2 + ( M 3 * ) 2 ≤ 2.5$
$α m = 1.7 × 6P* ( P* ) 2 + ( 6P* ) 2 + ( P* ) 2 = 1.655$

Reference buckling moment, Mo

$M o = π 2 E I y L e 2 G J + π 2 E I w L e 2$
$α s = 0.6 M s x M o a 2 + 3 - M s x M o a$
 $α s = 0.6 344.4 271.9 2 + 3 - 344.4 271.9 = 0.528$ [Ref : Clause 5.6.1.1 (c)]

Mbx = αmαsMsx ≤ Msx

 Mbz = 1.655 × 0.528 × 344.4 = 300.95 kN·m ≤ (Msz, Msy)Max. [Ref : Clause 5.6.1.1.1(a)]

ϕMbz = 0.9×300.95 = 270.9 kN·m

The moments over the supports due to the cantilevers, M1 = M2 = 4×P*

The maximum moment in the middle span occurs at the 5P load:

$Mmax=(5P*)84-(M1+M2)=10P*-4P*=6P*$

So the maximum value of P* is derived from setting Mmax = ϕMbz and solving for P*:

P* = ϕMbz/6 = 270.9 / 6 = 45.1 kN

## Results

Table 1. Comparison of results
ϕMsz (kN·m) 309.96 309.96 none
ϕMsy (kN·m) 144.9 142.9313 1.4%
ϕMbz (kN·m) - Cantilever 249.1 249.4725 negligible
ϕMbz (kN·m) - Main span 270.9 269.5 negligible
P* (kN) 45.1† 44.9‡ 0.5%

†The reference gives a P* value of 44.33 kN (though to be noted this is also calculated for the AS 4100 code as opposed to the NZS 34043 1997 code).

‡For STAAD.Pro, the value of 44.9 kips gives a critical ratio of 1.00 for the member bending capacity.

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\New Zealand\NZS 3404 1997\NZS3404 1997-Simply Supported Beam With Overhang.std is typically installed with the program.

STAAD PLANE
*
*  INPUT FILE: NZS3404_Simply_Supported_Beam_with_Overhang.STD
*
* REFERENCE : Hand Calculation
*
*  OBJECTIVE : TO DETERMINE THE ADEQUACY OF UC SHAPE  PER
*              THE NZS3404-1997 CODE
*
START JOB INFORMATION
ENGINEER DATE 17-Feb-17
END JOB INFORMATION
INPUT WIDTH 79
*
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 4 0 0; 3 12 0 0; 4 16 0 0;
*
MEMBER INCIDENCES
1 1 2; 2 2 3; 3 3 4;
DEFINE PMEMBER
1 PMEMBER 1
2 PMEMBER 2
3 PMEMBER 3
*
*
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 FY 253200 FU 407800 RY 1.5 RT 1.2
END DEFINE MATERIAL
*
MEMBER PROPERTY AUSTRALIAN
1 TO 3 TABLE ST UC250X89.5
*
CONSTANTS
MATERIAL STEEL ALL
*
SUPPORTS
2 PINNED
3 FIXED BUT FX MY MZ
*
*SELFWEIGHT Y -1
2 CON GY -224.5
1 4 FY -44.9
PERFORM ANALYSIS
*
PARAMETER 1
CODE NZS3404 1997
TRACK 2 PMEMB 1 TO 3
SGR 2 PMEMB 1 TO 3
LHT 1 PMEMB 1 TO 3
PBRACE TOP 0 U 1 F PMEMB 1
PBRACE BOTTOM 0 U 1 F PMEMB 1
PBCRES ZZ 0 U 1 T PMEMB 1
PBCRES YY 0 U 1 T PMEMB 1
PBRACE TOP 0 F 1 U PMEMB 3
PBRACE BOTTOM 0 F 1 U PMEMB 3
PBCRES ZZ 0 T 1 U PMEMB 3
PBCRES YY 0 T 1 U PMEMB 3
CHECK CODE PMEMB 1 2 3
*
FINISH


                       STAAD.PRO CODE CHECKING - NZS-3404-1997 (v1.0)
**************************************************
AXIS NOTATION FOR ANY SECTION OTHER THAN ST ANGLE:-
---------     -------------     ---------------
X/x             Z/z          Longitudinal axis of section
Y/y             Y/y          Minor principal axis of section
Z/z             X/x          Major Principal axis of section
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 [NZS3404 Table 3.4]
3. (#) next to Young's modulus E indicates that its value is not 200000 MPa as per NZS3404 1.4.
DESIGN SUMMARY
--------------
Designation: ST   UC250X89.5               (AISC SECTIONS)
Governing Criteria: Cl.5.1
Governing Ratio:   0.720  (PASS)
Governing Location:   4.000 m from Start.
SECTION PROPERTIES
------------------
d:       260.0000 mm   bf:       256.0000 mm
tf:        17.3000 mm   tw:        10.5000 mm
Ag:     11400.0000 mm2   J:     1.0400E+06 mm4             Iw:   712.3514E+09 mm6
Iz:   143.0000E+06 mm4  Sz:     1.2300E+06 mm3 (plastic)   Zz:     1.1000E+06 mm3 (elastic)
rz:   111.9994E+00 mm
Iy:    48.4000E+06 mm4  Sy:   575.0001E+03 mm3 (plastic)   Zy:   378.1251E+03 mm3 (elastic)
ry:    65.1584E+00 mm
STAAD PLANE                                              -- PAGE NO.    4
*
MATERIAL PROPERTIES
-------------------
Material Standard        :  AS/NZS 3679.1
Residual Stress Category :  HR (Hot-rolled)
E (#)       : 204999.984 MPa         [NZS3404 1.4]
G           :  80000.000 MPa         [NZS3404 1.4]
fy, flange  :    280.000 MPa         [NZS3404 Table 2.1]
fy, web     :    320.000 MPa         [NZS3404 Table 2.1]
fu          :    440.000 MPa         [NZS3404 Table 2.1]
BENDING
-------
Critical Ratio     :   0.579
Critical Location  :   4.000 m from Start.
Mz* =   179.6000E+00 KNm
Section Slenderness: Compact
Zez =     1.2300E+06 mm3
ϕMsz =   309.9600E+00 KNm                [NZS3404 Cl.5.1    ]
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
My* =     0.0000E+00 KNm
Section Slenderness: Compact
Zey =   567.1876E+03 mm3
ϕMsy =   142.9313E+00 KNm                [NZS3404 Cl.5.1    ]
Member Bending Capacity
Critical Ratio     :   0.720
Critical Location  :   4.000 m from Start.
Crtiical Flange Segment:
Location (Type):   0.00 m(U )-  4.00 m(F )
Mz* =   179.6000E+00 KNm
kt   =      1.00                         [NZS3404 Table 5.6.3(1)]
kl   =      2.00                         [NZS3404 Table 5.6.3(2)]
kr   =      1.00                         [NZS3404 Table 5.6.3(3)]
le   =      8.00 m                       [NZS3404 5.6.3]
αm   =     1.250                         [NZS3404 5.6.1.1.1(b)(iii)]
Mo   =   399.9072E+00 KNm                [NZS3404 5.6.1.1.1(d)]
αsz  =     0.644                         [NZS3404 5.6.1.1.1(c)]
ϕMbz =   249.4725E+00 KNm (&lt;= ϕMsz)      [NZS3404 5.6.1.1.1(a)]
STAAD PLANE                                              -- PAGE NO.    5
*
SHEAR
-----
Section Shear Capacity (along Y-axis)
Critical Ratio     :   0.095
Critical Location  :   0.000 m from Start.
Vy*  =   -44.9000E+00 KN
ϕVvy =   471.7440E+00 KN                 [NZS3404 5.11.2]
Section Shear Capacity (along Z-axis)
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
Vz*  =     0.0000E+00 KN
ϕVvz =     1.3393E+03 KN                 [NZS3404 5.11.2]
STAAD PLANE                                              -- PAGE NO.    6
*
AXIAL
-----
Section Compression Capacity
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
N*    =     0.0000E+00 KN
Ae    =    11.4000E+03 mm2               [NZS3404 6.2.3 / 6.2.4]
kf    =     1.000                        [AS 4100 6.2.2]
An    =    11.4000E+03 mm2
ϕNs   =     2.8728E+03 KN                [NZS3404 6.2.1]
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
N*    =     0.0000E+00 KN
Unbraced Segment:
Location (Type):   0.00 m(U )-  4.00 m(T )
Lez   =      8.80 m
αb    =      0.00                        [NZS3404 Table 6.3.3(1)/6.3.3(2)]
λn,z  =    83.153                        [NZS3404 6.3.3]
λ,z   =    83.153                        [NZS3404 6.3.3]
ε,z   =     1.219                        [NZS3404 6.3.3]
αc,z  =     0.659                        [NZS3404 6.3.3]
ϕNcz  = 0.1892E+4 KN                     [NZS3404 6.3.3]
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
N*    =     0.0000E+00 KN
Unbraced Segment:
Location (Type):   0.00 m(U )-  4.00 m(T )
Ley   =      8.80 m
λn,y  =   142.929                        [NZS3404 6.3.3]
λ,y   =   142.929                        [NZS3404 6.3.3]
ε,y   =     0.782                        [NZS3404 6.3.3]
αc,y  =     0.318                        [NZS3404 6.3.3]
ϕNcy  = 0.9146E+3 KN                     [NZS3404 6.3.3]
Section Tension Capacity
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
N*    =     0.0000E+00 KN
kt    =      1.00                        [User defined]
An    =    11.4000E+03 mm2
ϕNt   =     2.8728E+03 KN                [NZS3404 7.2]
STAAD PLANE                                              -- PAGE NO.    7
*
COMBINED BENDING AND AXIAL
------------------------
Critical Ratio     :   0.579
Critical Location  :   4.000 m from Start.
ϕMrz  =   309.9600E+00 KNm               [NZS3404 8.3.2]
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
ϕMry  =   142.9313E+00 KNm               [NZS3404 8.3.3]
Section Combined Capacity (Biaxial)
Critical Ratio     :   0.466
Critical Location  :   4.000 m from Start.
γ     =     1.400                         [NZS3404 8.3.4]
Critical Ratio     :   0.579
Critical Location  :   4.000 m from Start.
ϕMiz  =   309.9600E+00 KNm               [NZS3404 8.4.2]
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
ϕMiy  =   142.9313E+00 KNm               [NZS3404 8.4.2]
Member Out-of-plane Capacity (Tension)
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
αbc   =      0.00
ϕNoz  =     0.0000E+00 KN                [NZS3404 8.4.4.1.2]
ϕMoz,t=     0.0000E+00 KNm               [NZS3404 8.4.4.1]
Member Out-of-plane Capacity (Compression)
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
ϕMoz,c=     0.0000E+00 KNm               [NZS3404 8.4.4.2]
Member Biaxial Capacity (Tension)
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
Member Biaxial Capacity (Compression)
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
********************************************************************************
AXIS NOTATION FOR ANY SECTION OTHER THAN ST ANGLE:-
---------     -------------     ---------------
X/x             Z/z          Longitudinal axis of section
Y/y             Y/y          Minor principal axis of section
Z/z             X/x          Major Principal axis of section
STAAD PLANE                                              -- PAGE NO.    8
*
MEMBER DESIGN OUTPUT FOR PMEMBER     2
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 [NZS3404 Table 3.4]
3. (#) next to Young's modulus E indicates that its value is not 200000 MPa as per NZS3404 1.4.
DESIGN SUMMARY
--------------
Designation: ST   UC250X89.5               (AISC SECTIONS)
Governing Criteria: Cl.5.1
Governing Ratio:   1.000  (PASS)
Governing Location:   4.000 m from Start.
SECTION PROPERTIES
------------------
d:       260.0000 mm   bf:       256.0000 mm
tf:        17.3000 mm   tw:        10.5000 mm
Ag:     11400.0000 mm2   J:     1.0400E+06 mm4             Iw:   712.3514E+09 mm6
Iz:   143.0000E+06 mm4  Sz:     1.2300E+06 mm3 (plastic)   Zz:     1.1000E+06 mm3 (elastic)
rz:   111.9994E+00 mm
Iy:    48.4000E+06 mm4  Sy:   575.0001E+03 mm3 (plastic)   Zy:   378.1251E+03 mm3 (elastic)
ry:    65.1584E+00 mm
MATERIAL PROPERTIES
-------------------
Material Standard        :  AS/NZS 3679.1
Residual Stress Category :  HR (Hot-rolled)
E (#)       : 204999.984 MPa         [NZS3404 1.4]
G           :  80000.000 MPa         [NZS3404 1.4]
fy, flange  :    280.000 MPa         [NZS3404 Table 2.1]
fy, web     :    320.000 MPa         [NZS3404 Table 2.1]
fu          :    440.000 MPa         [NZS3404 Table 2.1]
BENDING
-------
STAAD PLANE                                              -- PAGE NO.    9
*
Critical Ratio     :   0.869
Critical Location  :   4.000 m from Start.
Mz* =  -269.4000E+00 KNm
Section Slenderness: Compact
Zez =     1.2300E+06 mm3
ϕMsz =   309.9600E+00 KNm                [NZS3404 Cl.5.1    ]
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
My* =     0.0000E+00 KNm
Section Slenderness: Compact
Zey =   567.1876E+03 mm3
ϕMsy =   142.9313E+00 KNm                [NZS3404 Cl.5.1    ]
Member Bending Capacity
Critical Ratio     :   1.000
Critical Location  :   4.000 m from Start.
Crtiical Flange Segment:
Location (Type):   0.00 m(F )-  8.00 m(F )
Mz* =  -269.4000E+00 KNm
kt   =      1.00                         [NZS3404 Table 5.6.3(1)]
kl   =      1.40                         [NZS3404 Table 5.6.3(2)]
kr   =      1.00                         [NZS3404 Table 5.6.3(3)]
le   =     11.20 m                       [NZS3404 5.6.3]
αm   =     1.655                         [NZS3404 5.6.1.1.1(b)(iii)]
Mo   =   270.1554E+00 KNm                [NZS3404 5.6.1.1.1(d)]
αsz  =     0.525                         [NZS3404 5.6.1.1.1(c)]
ϕMbz =   269.5068E+00 KNm (&lt;= ϕMsz)      [NZS3404 5.6.1.1.1(a)]
SHEAR
-----
Section Shear Capacity (along Y-axis)
Critical Ratio     :   0.294
Critical Location  :   4.000 m from Start.
Vy*  =   112.2500E+00 KN
ϕVvy =   381.8150E+00 KN                 [NZS3404 5.11.2]
Section Shear Capacity (along Z-axis)
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
Vz*  =     0.0000E+00 KN
ϕVvz =     1.3393E+03 KN                 [NZS3404 5.11.2]
STAAD PLANE                                              -- PAGE NO.   10
*
AXIAL
-----
Section Compression Capacity
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
N*    =     0.0000E+00 KN
Ae    =    11.4000E+03 mm2               [NZS3404 6.2.3 / 6.2.4]
kf    =     1.000                        [AS 4100 6.2.2]
An    =    11.4000E+03 mm2
ϕNs   =     2.8728E+03 KN                [NZS3404 6.2.1]
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
N*    =     0.0000E+00 KN
Unbraced Segment:
Location (Type):   0.00 m(U )-  8.00 m(U )
Lez   =      8.00 m
αb    =      0.00                        [NZS3404 Table 6.3.3(1)/6.3.3(2)]
λn,z  =    75.593                        [NZS3404 6.3.3]
λ,z   =    75.593                        [NZS3404 6.3.3]
ε,z   =     1.352                        [NZS3404 6.3.3]
αc,z  =     0.711                        [NZS3404 6.3.3]
ϕNcz  = 0.2043E+4 KN                     [NZS3404 6.3.3]
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
N*    =     0.0000E+00 KN
Unbraced Segment:
Location (Type):   0.00 m(U )-  8.00 m(U )
Ley   =      8.00 m
λn,y  =   129.936                        [NZS3404 6.3.3]
λ,y   =   129.936                        [NZS3404 6.3.3]
ε,y   =     0.831                        [NZS3404 6.3.3]
αc,y  =     0.372                        [NZS3404 6.3.3]
ϕNcy  = 0.1068E+4 KN                     [NZS3404 6.3.3]
Section Tension Capacity
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
N*    =     0.0000E+00 KN
kt    =      1.00                        [User defined]
An    =    11.4000E+03 mm2
ϕNt   =     2.8728E+03 KN                [NZS3404 7.2]
STAAD PLANE                                              -- PAGE NO.   11
*
COMBINED BENDING AND AXIAL
------------------------
Critical Ratio     :   0.869
Critical Location  :   4.000 m from Start.
ϕMrz  =   309.9600E+00 KNm               [NZS3404 8.3.2]
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
ϕMry  =   142.9313E+00 KNm               [NZS3404 8.3.3]
Section Combined Capacity (Biaxial)
Critical Ratio     :   0.822
Critical Location  :   4.000 m from Start.
γ     =     1.400                         [NZS3404 8.3.4]
Critical Ratio     :   0.869
Critical Location  :   4.000 m from Start.
ϕMiz  =   309.9600E+00 KNm               [NZS3404 8.4.2]
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
ϕMiy  =   142.9313E+00 KNm               [NZS3404 8.4.2]
Member Out-of-plane Capacity (Tension)
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
αbc   =      0.00
ϕNoz  =     0.0000E+00 KN                [NZS3404 8.4.4.1.2]
ϕMoz,t=     0.0000E+00 KNm               [NZS3404 8.4.4.1]
Member Out-of-plane Capacity (Compression)
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
ϕMoz,c=     0.0000E+00 KNm               [NZS3404 8.4.4.2]
Member Biaxial Capacity (Tension)
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
Member Biaxial Capacity (Compression)
Critical Ratio     :   0.000
Critical Location  :   0.000 m from Start.
********************************************************************************