V. CSA S16-19 - CLASS4 UPT Cantilever Beam with UDL

Determine the capacity of a cantilever beam using a user-provided section subject to a distributed load per the CSA S16-19 code.

Details

A 7.5 m long cantilever member is subjected to uniformly distributed load of 1 kN. The section is an I shape with 300 mm × 15 mm flanges and a 420 mm × 15 mm web. The compression flange is laterally unbraced along the length.

Material Properties

F_y = 650 MPa

Validation

Section Properties

D = 420 + 15 + 15 = 450 mm
A_g = 2×(300 × 15) + 420 × 15 = 15,300 mm²
I_x = $2 [\frac{300 {(15)}^{3}}{12} + 300 \times 15 {(\frac{420 + 15}{2})}^{2}] + \frac{15 {(420)}^{3}}{12} = 518.5 {(10)}^{6} {mm}^{4}$
I_y = $2 [\frac{15 {(300)}^{3}}{12}] + \frac{420 {(15)}^{3}}{12} = 67,620,000 {mm}^{4}$
S_x = $\frac{518.5 {(10)}^{6} 2}{450} = 2,305,000 {mm}^{3}$
S_y = $\frac{67,620,000 (2)}{300} = 450,800 {mm}^{3}$
J = $\frac{1}{3} [2 \times 300 {(15)}^{3} + 420 {(15)}^{3}] = 1,148,000 {mm}^{4}$
C_w = $\frac{{(420 + 15)}^{2} {(300)}^{3} 15}{24} = 3,193 {(10)}^{9} {mm}^{6}$

Section Classification

b_el = B / 2 = 150 mm

C_f = 0 (no compression force)

Flexure section classification per Table 2 of CSA S16-19:

Flange classification about major axis:
$\frac{b_{el}}{t_{f}} = \frac{150}{15} = 10 > \frac{200}{\sqrt{F_{y}}} = \frac{200}{\sqrt{650}} = 7.84$

Class 4
Flange classification about minor axis:
$\frac{b_{el}}{t_{f}} = \frac{150}{15} = 10 < \frac{340}{\sqrt{F_{y}}} = \frac{340}{\sqrt{650}} = 13.36$

Class 3
Web classification about major axis:
$\frac{h}{t_{w}} = \frac{420}{15} = 28 < \frac{1,100}{\sqrt{F_{y}}} (1 - 0.39 \frac{C_{f}}{ϕ C_{y}}) = 58.8$

Class 1
Web classification about minor axis:
$\frac{h}{t_{w}} = \frac{420}{15} = 28 < \frac{1,100}{\sqrt{F_{y}}} (1 - 0.39 \frac{C_{f}}{ϕ C_{y}}) = 58.8$

Class 1

Therefore, the section is Class 4 for flexure.

Bending Capacity

M_{fx} = \frac{1.0 {(7.5)}^{2}}{2} = 28.1 kN·m

For a Class 4, doubly symmetric section, the capacity is determined by Cl. 13.5.c. In this case, the compression flange is Class 4 but the web is Class 3, so Cl. 13.5.c.iii applies. Therefore, the effective section modulus, S_e, is used. The outstanding element length is limited to:

b_{el} = 200 t / \sqrt{F_{y}} = 200 \times 15 / \sqrt{650} = 118 mm

Therefore, b_eff = 2× b_el = 236 mm

I_{eff,x} = 2 [\frac{268 {(15)}^{3}}{12} + 268 \times 15 {(\frac{420 + 15}{2})}^{2}] + \frac{15 {(420)}^{3}}{12} = 427.7 {(10)}^{6} {mm}^{4}

S_{eff,x} = \frac{427.7 {(10)}^{6} 2}{450} = 1,901,000 {mm}^{3}

Major axis:

M_yx = S_eff,x×F_y = 1,901,000 × 650 (10)^-6 = 1,236 kN·m

M_rx = ϕM_yx = 0.9 × 1,236 = 1,112 kN·m

The ratio = 28.1 / 1,112 = 0.025

Minor axis:

M_yy = S_y×F_y = 450,800 × 650 (10)^-6 = 293.0 kN·m

M_ry = ϕM_yy = 0.9 × 293.0 = 263.7 kN·m

Check for lateral-torsional buckling per Cl. 13.6.h i).

M_{u} = ω_{2} C_{H} C_{B} \frac{\sqrt{E I_{y} G J}}{L_{c}}

where

X	=	$\frac{π}{L_{c}} \sqrt{\frac{E C_{w}}{G J}} = \frac{π}{7,500} \sqrt{\frac{205,000 \times 3,193 {(10)}^{9}}{76,920 \times 1,148,000}} = 1.142$ 0.5 ≤ X ≤ 2.5 for a distributed load
ω₂	=	5.38 + 8.71 X = 5.38 + 8.71 (1.142) = 15.77
C_H	=	0.49 - 0.27 X + 0.06 X² = 0.49 - 0.27 (1.142) + 0.06 (1.142)² = 0.26
C_B	=	1.48 + 0.16 X = .48 + 0.16 (1.142) = 1.66

M_{u} = 15.77 (0.26) (1.66) \frac{\sqrt{205,000 × 67,620,000 × 76,920 \times 1,148,000}}{7,500 × {(10)}^{6}} = 1,004 kN·m

0.67×M_y = 0.67 (1,236) = 827.8 kN·m < M_u

M_r = 1.15ϕM_y[1 - 0.28 M_y / M_u] = 1.15 × 0.9 × 1,236 [ 1 - 0.28×(1,236 / 1,004) ] = 838.3 kN·m

The ratio = 28.1 / 838.3 = 0.034

Results

Result Type	Reference	STAAD.Pro	Difference
M_yx (kN·m)	1,236	1,230	negligible
M_rx (kN·m)	1,112	1,110	negligible
Ratio of M_fx/M_yx	0.025	0.025	none
M_yy (kN·m)	293.0	290.0	negligible
M_ry (kN·m)	263.7	261.1	negligible
M_u (kN·m)	1,004	1,006	negligible
M_r (kN·m)	838.3	837.9	negligible
ω₂	15.77	15.77	none
Critical ratio	0.034	0.034	none

STAAD.Pro Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\Canada\CSA S16-19 - CLASS4 UPT Cantilever Beam with UDL.STD is typically installed with the program.

The following design parameters are used:

The cantilever is braced along the tension flange only specified using CAN 2.
The load acts on the tension side of the bending member, thus LHT 1.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 3-Dec-21
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 7.5 0 0;
MEMBER INCIDENCES
1 1 2;
START USER TABLE
TABLE 1
UNIT METER KN
WIDE FLANGE
UPTI
0.0153 0.45 0.015 0.3 0.015 0.000518535 6.76181e-05 1.1475e-06 0.00675 -
0.006 0.3 0.015
END
DEFINE MATERIAL START
ISOTROPIC STEEL
E 2.05e+08
POISSON 0.3
DENSITY 76.92
ALPHA 1.2e-05
DAMP 0.03
TYPE STEEL
STRENGTH FY 350000 FU 450000 RY 1.5 RT 1.2
G 7.692e+07
END DEFINE MATERIAL
MEMBER PROPERTY CANADIAN
1 UPTABLE 1 UPTI
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 FIXED
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
1 UNI GY -1
PERFORM ANALYSIS
PARAMETER 1
CODE CANADIAN
CAN 2 ALL
LHT 1 ALL
FYLD 650000 ALL
FU 450000 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH

STAAD.Pro Output

                       STAAD.PRO CODE CHECKING - CSA S16-19 (v1.0)
                     ***********************************************
   ALL UNITS ARE IN KN   MET   UNLESS OTHERWISE Noted
|--------------------------------------------------------------------------|
|   MEMBER NO.:      1      SECTION: ST  UPTI             (UPT)              |
|  STATUS:        PASS    CRITICAL RATIO:  0.034         LOADCASE    1     |
|  LOCATION:      0.00    CRITICAL CONDITION:            Cl. 13.8.4        |
|--------------------------------------------------------------------------|
|  STRENGTH CHECK SUMMARY                                                  |
|   CRITICAL RATIO:  0.034(PASS)            LOAD CASE:     1               |
|   LOCATION      :   0.00                  CONDITION: Cl. 13.8.4          |
|--------------------------------------------------------------------------|
|  DESIGN FORCES              LC:       1      LOC:      0.00              |
|         FX:      0.00(T)    FY:     7.50      FZ:      0.00              |
|         MX:  0.000E+00      MY: 0.000E+00     MZ:  2.813E+01             |
|--------------------------------------------------------------------------|
|  SECTION PROPERTIES (LOC:   0.00)             UNIT:      CM              |
|   AXX:     153.000     AZZ:      90.000      AYY:         63.000         |
|   IXX:     114.750     IZZ:   51853.500      IYY:       6761.811         |
|   PZZ:    2619.000     PYY:     698.625                                  |
|   SZZ:    2304.600     SYY:     450.787      CW :    3198759.000         |
|--------------------------------------------------------------------------|
|  MATERIAL PROPERTIES                          UNIT:      N MM            |
|   FYLD:     650.000     FU:   450.000      E: 204999.984  G:  76920.000  |
|--------------------------------------------------------------------------|
|  DESIGN PARAMETERS                                                       |
|   ACTUAL MEMBER LENGTH:     7.500        LZ:    7.500    LY:    7.500    |
|   KZ:  1.000        KY:  1.000      NSF:  1.000      SLF:  1.000         |
|--------------------------------------------------------------------------|
|  SECTION CLASS                                                           |
|   COMPRESSION    :            Class 4                                    |
|   FLEXURE           MAJOR               MINOR                            |
|   SECTION        : Class 4             Class 3                           |
|   FLANGE         : Class 4             Class 3                           |
|   WEB (CRITICAL) :            Class 1                                    |
|==========================================================================|
|  SLENDERNESS                                                             |
|   ACTUAL SLENDERNESS RATIO   :     112.817          LC   :        1      |
|   ALLOWABLE SLENDERNESS RATIO:     300.000          LOC  :      0.00     |
|--------------------------------------------------------------------------|
      STAAD SPACE                                              -- PAGE NO.    4
                       STAAD.PRO CODE CHECKING - CSA S16-19 (v1.0)
                     ***********************************************
   ALL UNITS ARE IN KN   MET   UNLESS OTHERWISE Noted
   Member :     1       Contd.
|--------------------------------------------------------------------------|
|                         CHECKS FOR AXIAL TENSION                         |
|--------------------------------------------------------------------------|
|  YIELDING                                                                |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00   8.95E+03       0.000      Cl. 13.2          1     0.000    |
|  RUPTURE                                                                 |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00   5.16E+03       0.000      Cl. 13.2          1     0.000    |
|--------------------------------------------------------------------------|
|                       CHECKS FOR AXIAL COMPRESSION                       |
|--------------------------------------------------------------------------|
|  FLEX BUCKLING                                                           |
|            FORCE:    CAPACITY:   RATIO:    CRITERIA:      LC:    LOC:    |
|   MAJOR:   0.00E+00   5.81E+03     0.000  Cl. 13.3.4(a)       1    0.000 |
|   MINOR:   0.00E+00   1.67E+03     0.000  Cl. 13.3.4(a)       1    0.000 |
|  INTERMEDIATE RESULTS:                                                   |
|               Ag:(CM)     Fe:(N MM)       λ:        n:      KL/r:        |
|   MAJOR:     129.731        1219.039       0.730    1.34      40.740     |
|   MINOR:     129.731         158.965       2.022    1.34     112.817     |
|--------------------------------------------------------------------------|
|  FLEXURAL TORSIONAL BUCKLING                                             |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00   1.67E+03       0.000      Cl. 13.3.4        1     0.000    |
|  INTERMEDIATE RESULTS:                                                   |
|   Ag:(CM)   129.731   Fe:(N MM)      158.965  λ:     2.022  n: 1.34      |
|--------------------------------------------------------------------------|
|                             CHECKS FOR SHEAR                             |
|--------------------------------------------------------------------------|
|  SHEAR ALONG Y AXIS                                                      |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    7.50E+00   2.43E+03       0.003    Cl. 13.4.1.1        1     0.000    |
|  INTERMEDIATE RESULTS:                                                   |
|   Aw:(CM)      Kv:       Ka:     Fcri:(N MM)  Fcre:(N MM)  Fs:(N MM)     |
|     63.000    5.353     0.056       610.909      1228.900    429.000     |
|--------------------------------------------------------------------------|
|  SHEAR ALONG Z AXIS                                                      |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00   3.16E+03       0.000    AISC G2-1           1     0.000    |
|  INTERMEDIATE RESULTS:                                                   |
|   Aw:(CM)                                                                |
|     90.000                                                               |
|--------------------------------------------------------------------------|
      STAAD SPACE                                              -- PAGE NO.    5
                       STAAD.PRO CODE CHECKING - CSA S16-19 (v1.0)
                     ***********************************************
   ALL UNITS ARE IN KN   MET   UNLESS OTHERWISE Noted
   Member :     1       Contd.
|--------------------------------------------------------------------------|
|                            CHECKS FOR BENDING                            |
|--------------------------------------------------------------------------|
|  YIELDING ALONG MAJOR AXIS                                               |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    -2.81E+01  1.110E+03      0.025    Cl. 13.5(c)         1     0.000    |
|  INTERMEDIATE RESULTS: Mp:  0.00E+00   Se(CM):  1.90E+03    My:  1.23E+03|
|--------------------------------------------------------------------------|
|  YIELDING ALONG MINOR AXIS                                               |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00  2.611E+02      0.000    Cl. 13.5(c)         1     0.000     |
|  INTERMEDIATE RESULTS: Mp:  0.00E+00   Se(CM):  4.46E+02    My:  2.90E+02|
|--------------------------------------------------------------------------|
|  LATERAL TORSIONAL BUCKLING (MAJOR AXIS)                                 |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|   -2.81E+01  8.379E+02      0.034   Cl. 13.6(b)(ii)      1     0.000     |
|  INTERMEDIATE RESULTS:                                                   |
|   Iyc(CM):    w 2:    w 3:    b x(CM):  Mu:       rt:   Myr:     Lu:       Lyr:  |
|  0.00E+00  15.77  0.00     0.00  1.006E+03  0.00  0.00E+00  0.00   0.00  |
|--------------------------------------------------------------------------|
|                          CHECKS FOR INTERACTION                          |
|--------------------------------------------------------------------------|
|  FLEXURE AND AXIAL TENSION                                               |
|                    RATIO:   CRITERIA:   LC:   LOC:     Tf:        Tr:    |
|   C/S STRENGTH   : 0.025  Cl. 13.9.1      1   0.00   0.00E+00  5.16E+03  |
|   MEMBER STRENGTH: 0.034  Cl. 13.9.3      1   0.00                       |
|                        Mfz:         Mfy:          Mrz:         Mry:      |
|   C/S STRENGTH   :   -2.81E+01      0.00E+00     1.11E+03      2.61E+02  |
|--------------------------------------------------------------------------|
|  FLEXURE AND AXIAL COMPRESSION                                           |
|                    RATIO:   CRITERIA:   LC:   LOC:     Cf:        Cr:    |
|   C/S STRENGTH   : 0.025  Cl. 13.8.4      1   0.000 0.00E+00  7.59E+03   |
|   MEMBER STRENGTH: 0.025  Cl. 13.8.4      1   0.000 0.00E+00  1.67E+03   |
|   LTB STRENGTH   : 0.034  Cl. 13.8.4      1   0.000 0.00E+00  1.67E+03   |
|                        Mfz:      Mfy:      Mrz:      Mry:    w 1z:    w 1y:    |
|   C/S STRENGTH   : -2.81E+01  0.00E+00  1.11E+03  2.61E+02  0.60   0.60  |
|   MEMBER STRENGTH: -2.81E+01  0.00E+00  1.11E+03  2.61E+02  0.60   0.60  |
|   LTB STRENGTH   : -2.81E+01  0.00E+00  8.38E+02  2.61E+02  0.60   0.60  |
|                        b :         Cez:     Cey:         U1z:     U1y:     |
|   C/S STRENGTH   :   N/A     1.87E+04  2.43E+03     1.00     1.00        |
|   MEMBER STRENGTH:   N/A     1.87E+04  2.43E+03     1.00     1.00        |
|   LTB STRENGTH   :   N/A     1.87E+04  2.43E+03     1.00     1.00        |
|--------------------------------------------------------------------------|
|  SHEAR CAPACITY MODIFICATION  ( SHEAR AND FLEX )                         |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|     7.50E+00  2.432E+03      0.003    Cl. 14.6            1     0.000    |
|--------------------------------------------------------------------------|
|  BIAXIAL FLEXURE                                                         |
|   RATIO:  CRITERIA:  LC:   LOC:     Mfz:      Mrz:       Mfy:      Mry:  |
|   0.034  Cl. 13.8     1    0.00  -2.81E+01  8.38E+02  0.00E+00  2.61E+02 |
|==========================================================================|
      STAAD SPACE                                              -- PAGE NO.    6