# V. CSA S16-01 - Cantilever with Biaxial Loading

A slender, cantilever beam subjected to a uniform load. Static analysis, 3D beam element.

## Reference

CISC Example 1, page 5-91, Limit State Design, CSA-S16.1-94

## Problem

A cantilever beam of length 4 meter is subjected to uniformly distributed load of 3 KN/Meter in both major and minor axis. Axial compression of 8 KN is also applied to the member. User defined steel section Sect_Class-4 from is assigned to the member.

Given

Design forces

• 8.0 KN (Compression)
• 6.0 KNm (Bending-Y)
• 6.0 KNm (Bending-Z)
• 6.0 KN (Shear-Y)
• 6.0 KN (Shear-Z)

Section Properties(Sect_Class-4):

• Area = 2766 mm2
• Depth of section, D = 150 mm
• Thickness of web Tw = 7 mm
• Width of flange Bf = 150 mm
• Thickness of flange Tf = 6 mm
• Moment of inertia about Z axis, Iz = 1086.96X104 mm4
• Moment of inertia about Y axis, Iy = 337.894X104 mm4
• Moment of inertia about X axis, Ix = 3.7378X104 mm4
• Warping constant, Cw = 1.752X1010 mm6

Member Length L = 2 m, Unbraced length = 100mm.

Material

• FYLD = 300 MPa
• E = 2.05E+05 MPa
• G = E/2.6 MPa

## Slenderness Ratio

Effective Length factor along Local Y-Axis = KY = 1

Effective Length factor along Local Z-Axis = KZ = 1

Slenderness ratio about Z axis, L/Rz = 31.9

Slenderness ratio about Y axis, L/Ry = 57.22

Maximum Slenderness Ratio, L/Rmax = 57.22

## Section Classification

Bf/Tf = 150*0.5/6 = 12.5 > 200/sqrt(Fy) = 11.54

Flange is Class 4.

d/Tw = (150-2.0*6)/7 = 19.714

(1100/sqrt(Fy))*(1-0.39*Cf/ϕ*Cy)=(1100/sqrt(300))*(1-0.39*8000/(0.9*2766*300)) = 63.24

Web is Class 1.

Overall section is Class 4 section.

## Check against axial compression (Clause 13.3.3)

Effective width, Beff = 200*Tf/sqrt(300) = 69.24

Effective area, Aeff = 69.24*6*4+(150-2*6)*7 = 2627.76 mm4.

Effective yield stress, FYLDeff =40000/( 0.5*Bf/Tf)4 =256 MPa.

As per Clause 13.3.3(a),

Elastic critical buckling, Fe = π4*E/ L_Rmax4 = 617.956 MPa.

Non-dimensional slenderness ratio, λ = sqrt(FYLD/Fe) =0.697

Axial compressive resistance, Cr = ϕ*Aeff*FYLD*(1+0.697^(2*1.34))^(-1/1.34) = 557886.104 N.

As per Clause 13.3.3(b),

Elastic critical buckling, Fe = π4*E/ L_Rmax4 = 617.956 MPa.

Effective non-dimensional slenderness ratio, λeff = sqrt(FYLDeff/Fe) = 0.644

Axial compressive resistance, Cr = ϕ*Area*FYLDeff*(1+0.644^(2*1.34))^(-1/1.34) = 521726.94 N.

Axial compressive resistance Min(557886.104, 521726.94) = 521726.94 N.

## Check against bending (Clause 13.5(c))

As the web of the section meets the requirement of Class 3 and flange exceeds Class 3 limit, flexural resistance should be calculated as per clause 13.5(c).iii.

Effective moment of inertia about Z axis,

Izeff =2*(2*69.24*63)/12 + 2*(2*69.24*6)*(150-6)*(150-6)/4 + (7*(150-2*6)3)/12 =10152591.12 mm4.

Effective section modulus about Z axis,

Szeff = 10152591.12*2/150 = 135367.88 mm3.

Effective moment of inertia about Y axis,

Iyeff =(2*6*(2*69.24)3)/12 +(0.5*(150-6)*73)/12 =2657648.856 mm4.

Effective section modulus about Y axis,

Syeff = 2657648.856/69.24 = 38383.144 mm3.

Major axis bending resistance if member is laterally supported,

Mrz1 = ϕ*Szeff*FYLD= 0.9*135367.88*300 =36549327.6 N-mm.

Minor axis bending resistance,

Mry = ϕ*Syeff*FYLD = 0.9*38383.144*300 = 10363448.88 N-mm.

If the member is laterally unsupported major axis bending resistance is determined by clause 13.6(b).

As the value of one of the end moments is 0.0, ω2 = 1.75.

Where, as per clause 13.6(a),

Mu = (1.75*3.14/2000)*sqrt(205000*337.894X104*78846.154*3.7378X104 + (3.14*205000/2000)4*337.894X104*1.752X10^10) =2.48X108

My = Sz*FYLD = (1086.96X104X2/150) *300 =43478400.

Since Mu > 0.65My,

Moment of resistance Mrz2 = 1.15*0.9*43478400*(1-0.28*43478400/2.48X108) =42791153.71 N-mm = 42.79 KN-m.

Mrz2 should not be more than Mrz1. Since, Mrz2 > Mrz1 in this example, Mrz2 = Mrz1.

Mrz2 = 36549327.6 N-mm = 36.549 KN-m

## Comparison

Table 1. Comparison of results
Axial compressive resistance (kN) 521.73 521.9 none
Major axis bending resistance (kN-m) 36.549 36.57 none
Minor axis bending resistance (kN-m) 10.363 10.38 none

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\Canada\S16 2001\CSA S16-01 - Cantilever with Biaxial Loading.STD is typically installed with the program.

``````STAAD SPACE VERIFICATION CISC 1994 HANDBOOK EXAMPLE PAGE 5-91
START JOB INFORMATION
ENGINEER DATE 16-FEB-10
END JOB INFORMATION
* CISC EXAMPLE 1 PAGE 5-91, LIMIT STATES DESIGN, CSA-S16.1-94
* SIMPLY SUPPORTED BEAM WITH UNIFORM LOAD
* LIVE LOAD DEFLECTION OF L/300
UNIT MMS KN
JOINT COORDINATES
1 0 0 0; 2 2000 0 0;
MEMBER INCIDENCES
1 1 2;
START USER TABLE
TABLE 1
UNIT METER NEWTON
WIDE FLANGE
SECT_CLASS-4
0.002766 0.15 0.007 0.15 0.006 1.08696e-05 3.37894e-06 3.7378e-08 -
0.00105 0.0018
END
UNIT METER KN
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 2.05e+08
POISSON 0.3
ISOTROPIC STEEL
E 2.05e+08
POISSON 0.3
DENSITY 76.8195
ALPHA 1.2e-05
DAMP 0.03
END DEFINE MATERIAL
MEMBER PROPERTY AMERICAN
1 UPTABLE 1 SECT_CLASS-4
UNIT MMS KN
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 FIXED
UNIT METER KN
1 UNI GY -3
1 UNI GZ -3
2 FX -8
PERFORM ANALYSIS
PRINT MEMBER FORCES ALL
PARAMETER 1
CB 0 ALL
TRACK 2 ALL
FYLD 300000 ALL
CHECK CODE ALL
FINISH
``````

The design output from STAAD.Pro:

```                       STAAD.PRO CODE CHECKING - (CAN/CSA-S16-01  ) V2.1
********************************************
ALL UNITS ARE - KNS  MET  (UNLESS OTHERWISE Noted)
MEMBER     TABLE       RESULT/   CRITICAL COND/     RATIO/     LOADING/
FX            MY             MZ       LOCATION
=======================================================================
1 ST   SECT_CLASS-4             (UPT)
PASS     CSA-13.8.3B        0.760         1
8.00 C         -6.00           6.00        0.00
VERIFICATION CISC 1994 HANDBOOK EXAMPLE PAGE 5-91        -- PAGE NO.    5
ALL UNITS ARE - KNS  MET  (UNLESS OTHERWISE Noted)
MEMBER     TABLE       RESULT/   CRITICAL COND/     RATIO/     LOADING/
FX            MY             MZ       LOCATION
=======================================================================
MEMBER PROPERTIES (UNIT = CM)
-----------------------------
CROSS SECTION AREA =  2.77E+01   MEMBER LENGTH =  2.00E+02
IZ =  1.09E+03   SZ =  1.45E+02   PZ =  1.63E+02
IY =  3.38E+02   SY =  4.51E+01   PY =  6.92E+01
IX =  3.74E+00   CW =  1.75E+04
EFFECTIVE MEMBER PROPERTIES FOR CLASS-4 SECTION(UNIT = CM)
----------------------------------------------------------
EFFECTIVE CROSS SECTION AREA =  2.63E+01
EFFECTIVE IZ =  1.02E+03   EFFECTIVE SZ =  1.35E+02
EFFECTIVE IY =  2.66E+02   EFFECTIVE SY =  3.85E+01
EFFECTIVE YIELD STRESS = 256.0  MPA
COMPRESSIVE CAPACITIES FOR CLASS 4 SECTION(UNIT = MPA)
------------------------------------------------------
BASED ON EFFECTIVE AREA
CR1 =  7.098E+02   CR2 =  5.582E+02   CRZ =  6.705E+02
CTORFLX =  5.582E+02
BASED ON EFFECTIVE YIELD STRENGTH
CR1 =  6.373E+02   CR2 =  5.219E+02   CRZ =  6.084E+02
CTORFLX =  5.219E+02
MATERIAL PROPERTIES (UNIT = MPA)
--------------------------------
FYLD = 300.0   FU = 345.0   E =  2.05E+05   G =  7.88E+04
SECTION CAPACITIES (UNIT - KN,M)
---------------------------------
CR1 =  6.373E+02   CR2 =  5.219E+02   SECTION CLASS 4
CRZ =  6.084E+02   CTORFLX =  5.219E+02
TENSILE CAPACITY     =  7.300E+02   COMPRESSIVE CAPACITY =  5.219E+02
FACTORED MOMENT RESISTANCE : MRY =  1.038E+01   MRZ =  3.657E+01
MU =  2.486E+02
FACTORED SHEAR RESISTANCE  : VRY =  1.871E+02   VRZ =  3.208E+02
MISCELLANEOUS INFORMATION
--------------------------
NET SECTION FACTOR FOR TENSION =  1.000
KL/RY =   57.222   KL/RZ =   31.904   ALLOWABLE KL/R =  200.000
UNSUPPORTED LENGTH OF THE COMPRESSION FLANGE (M) =  2.000
OMEGA-1 (Y-AXIS) = 1.00   OMEGA-1 (Z-AXIS) = 1.00   OMEGA-2 = 1.75
SHEAR FORCE (KNS) : Y AXIS =  6.000E+00   Z AXIS =  6.000E+00
SLENDERNESS RATIO OF WEB (H/W) =  1.97E+01
```