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, $J = \frac{2 B_{f} {T_{f}}^{3} + (D - T_{f}) {t_{w}}^{3}}{3} = \frac{2 \times 150 \times 6^{3} + (150 - 6) 7^{3}}{3} = 38.06 \times 10^{3} {mm}^{4}$

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

B_{f} / T_{f} = 150 \times 0.5 / 6 = 12.5 > \frac{200}{\sqrt{F_{y}}} = 11.54

Flange is Class 4.

d/T_w = (150 - 2.0×6)/7 = 19.7

\frac{1,100}{\sqrt{F_{y}}} (1 - 0.39 \frac{C_{f}}{ϕ C_{y}}) = \frac{1,100}{\sqrt{300}} (1 - 0.39 \frac{8,000}{0.9 \times 2,766 \times 300}) = 63.5

Web is Class 1.

Overall section is Class 4 section.

Check against axial compression (Clause 13.3.3)

Effective width, $B_{eff} = 200 \times \frac{T_{f}}{\sqrt{F_{y}}} = 69.24$

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

Effective yield stress, $F_{y,eff} = \frac{40,000}{{(0.5 \times B_{f} / T_{f})}^{4}} = 256 MPa$ .

As per Clause 13.3.3(a),

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

Non-dimensional slenderness ratio, $λ = \sqrt{F_{y} / F_{e}} = 0.697$

Axial compressive resistance, $C_{r} = ϕ A_{eff} \times F_{y} \times {[1 + {0.697}^{(2 \times 1.34)}]}^{(-1 / 1.34)} = 557.9 kN$ .

As per Clause 13.3.3(b),

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

Effective non-dimensional slenderness ratio, $λ = \sqrt{F_{y} / F_{e}} = 0.697$

Axial compressive resistance, $C_{r} = ϕ \times A \times F_{y,eff} \times {[1 + {0.644}^{(2 \times 1.34)}]}^{(-1 / 1.34)} = 521.8 kN$ .

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,

I_{z,eff} = 2 \times [\frac{2 \times 69.24 \times 6^{3}}{12} + 2 \times \frac{2 \times 69.24 \times 6 \times {(150 - 6)}^{2}}{4}] + \frac{7 \times {(150 - 2 \times 6)}^{3}}{12} = 10.15 \times {(10)}^{6} {mm}^{4}

Effective section modulus about Z axis,

S_{z,eff} = 10.15 \times {(10)}^{6} \times 2 / 150 = 135.4 \times {(10)}^{3} {mm}^{3}

Effective moment of inertia about Y axis,

I_{y,eff} = \frac{2 \times 6 \times {(2 \times 69.24)}^{3}}{12} + \frac{0.5 (150 - 6) 7^{3}}{12} = 2.658 \times {(10)}^{6} {mm}^{4}

Effective section modulus about Y axis,

S_{y,eff} = 2.658 \times {(10)}^{6} / 69.24 = 38.38 \times {(10)}^{3} {mm}^{3}

Major axis bending resistance if member is laterally supported,

M_{rz1} = ϕ S_{z,eff} F_{y} = 0.9 × 135.4 \times {(10)}^{3} \times 300 = 36.55 kN·m

Minor axis bending resistance,

M_{ry} = ϕ S_{y,eff} F_{y} = 0.9 × 38.38 \times {(10)}^{3} \times 300 = 10.36 kN·m

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

{(\frac{π E}{L})}^{2} I_{y} C_{w} = {(\frac{π \times 205,000}{2,000})}^{2} 3.38 \times 10^{6} \times 17.52 \times 10^{9} = 6.14 \times 10^{21} {MPa}^{2} \cdot {mm}^{8}

M_{u} = \frac{ω_{2} π}{L} \sqrt{E I_{y} G J + {(\frac{π E}{L})}^{2} I_{y} C_{w}}

= \frac{1.75 \times π}{2,000} \sqrt{23.87 \times 10^{21} + 6.14 \times 10^{21}} = 476.2 \times 10^{6} N·mm = 476.2 kN·m

M_{y} = S_{z} F_{y} = \frac{10.87 {(10)}^{6} \times 2}{150} 300 = 43.48 kN·m

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

M_{rz2} = 1.15 \times 0.9 \times 43.48 \times (1 - 0.28 \frac{43.48}{248}) = 42.79 kN·m

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 CONNECT Edition\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