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. A user-defined steel section Sect_Class-4 is assigned to the member.

Given

Design forces

8.0 kN (Compression)
6.0 kN·m (Bending-Y)
6.0 kN·m (Bending-Z)
6.0 kN (Shear-Y)
6.0 kN (Shear-Z)

Section Properties(Sect_Class-4):

Area = 2,766 mm²
Depth of section, D = 150 mm
Thickness of web T_w = 7 mm
Width of flange B_f = 150 mm
Thickness of flange T_f = 6 mm
Moment of inertia about Z axis, I_z = 10.87×10⁶ mm⁴
Moment of inertia about Y axis, I_y = 3.38×10⁶ mm⁴
Moment of inertia about X axis, I_x = 37.38×10³ mm⁴
Warping constant, C_w = 17.52×10⁹ mm⁶
Torsion constant,

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

Material

Fy (FYLD) = 300 MPa
E = 205,000 MPa
G = 78,850 MPa

Calculations

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

Flange is Class 4.

d/T_w = (150 - 2.0×6)/7 = 19.7

Web is Class 1.

Overall section is Class 4 section.

Check against axial compression (Clause 13.3.3)

Effective width,

Effective area, A_eff = 69.24 × 6 × 4 + (150 - 2 × 6) × 7 = 2,628 mm⁴ .

Effective yield stress, .

As per Clause 13.3.3(a),

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

Non-dimensional slenderness ratio,

Axial compressive resistance, .

As per Clause 13.3.3(b),

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

Effective non-dimensional slenderness ratio,

Axial compressive resistance, .

Axial compressive resistance = min(557.9, 521.8) = 521.8 kN.

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,

Effective section modulus about Z axis,

Effective moment of inertia about Y axis,

Effective section modulus about Y axis,

Major axis bending resistance if member is laterally supported,

Minor axis bending resistance,

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),

EI_yGJ = 205,000×38.8(10)⁶×78,850×38.06(10)³= 23.87×10²¹ MPa²·mm⁸

Since M_u > 0.65×M_y = 28.26, Moment of resistance

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

M_rz2 = 36.55 kN-m

Comparison

Table 1. Comparison of results
Criteria	Hand Calculation	STAAD.Pro Result	Comments
Axial compressive resistance (kN)	521.7	521.9	negligible
Major axis bending resistance (kN-m)	36.55	36.57	negligible
Minor axis bending resistance (kN-m)	10.36	10.38	negligible

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro 2024\Samples \Verification Models\09 Steel Design\Canada\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
LOAD 1 LC1
MEMBER LOAD
1 UNI GY -3
1 UNI GZ -3
JOINT LOAD
2 FX -8
PERFORM ANALYSIS
LOAD LIST 1
PRINT MEMBER FORCES ALL
PARAMETER 1
CODE CANADIAN 2001
CB 0 ALL
TRACK 2 ALL
FYLD 300000 ALL
CHECK CODE ALL
FINISH

STAAD Output

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-13.8.3C        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