 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 Tw = 7 mm
• Width of flange Bf = 150 mm
• Thickness of flange Tf = 6 mm
• Moment of inertia about Z axis, Iz = 10.87×106 mm4
• Moment of inertia about Y axis, Iy = 3.38×106 mm4
• Moment of inertia about X axis, Ix = 37.38×103 mm4
• Warping constant, Cw = 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

## 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

$B f / T f = 150 × 0.5 / 6 = 12.5 > 200 F y = 11.54$

Flange is Class 4.

d/Tw = (150 - 2.0×6)/7 = 19.7

$1,100 F y ( 1 - 0.39 C f ϕ C y ) = 1,100 300 ( 1 - 0.39 8,000 0.9 × 2,766 × 300 ) = 63.5$

Web is Class 1.

Overall section is Class 4 section.

## Check against axial compression (Clause 13.3.3)

Effective width, $B eff = 200 × T f F y = 69.24$

Effective area, Aeff = 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, $λ = F y / F e = 0.697$

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, $λ = F y / F e = 0.697$

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),

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

$M u = ω 2 π L E I y G J + ( π E L ) 2 I y C w$

Since Mu > 0.65×My = 28.26, Moment of resistance

$M rz2 = 1.15 × 0.9 × 43.48 × ( 1 - 0.28 43.48 248 ) = 42.79 kN·m$

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

Mrz2 = 36.55 kN-m

## Comparison

Table 1. Comparison of results
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 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


                       STAAD.PRO CODE CHECKING - (CAN/CSA-S16-01  ) V2.1
********************************************
ALL UNITS ARE - KNS  MET  (UNLESS OTHERWISE Noted)
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)
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