V. CSA S16-14 - Shear Capacity Combined Stresses

Reference

Kulak, G.L. and G.Y. Grondin. 2010. Limit States Design in Structural Steel. 9th Edition. Markham, ON:CISC. Example 8.6

Problem

The W250x73 column has the following concentrated loads

C_f = 900 kN

And a uniform moment of:

M_fx = 180 kN·m

The material used is G40.21 350W steel (F_y = 350 MPa) (FYLD). The beam is braced against lateral-torsional buckling only at the ends (LAT 0, default)

Calculations

Axial Capacity

Determine the buckling load of the column:

{(\frac{K L}{r})}_{x} = \frac{1.0 (3, 600)}{110} = 32.7

{(\frac{K L}{r})}_{y} = \frac{1.0 (3, 600)}{64.6} = 55.7

The largest slenderness ratio controls:

F_{e x} = \frac{π^{2} E}{{(\frac{K L}{r})}_{x}^{2}} = \frac{π^{2} (200, 000)}{{(55.7)}^{2}} = 635.6 kN

λ = \sqrt{\frac{F_{y}}{F_{e y}}} = \sqrt{\frac{350}{635.6}} = 0.742

C_{r} = ϕ A F_{y} {(1 + λ^{2 n})}^{- 1 / n} = 0.90 \times 9, 280 \times 350 {[1 + {0.742}^{(2 \times 1.34)}]}^{- 1 / 1.34} = 2, 216 \times {(10)}^{6} N = 2, 216 kN

Moment Modifier

Uniform moment:

ω₁ = 1.0

Calculate the elastic buckling load for the bending axis:

C_{e} = \frac{π^{2} \times 200, 000 \times 113 \times 10^{6}}{{(1.0 \times 3, 600)}^{2}} = 17, 211 kN

U_{1 x} = \frac{1.0}{1 - \frac{900}{17, 211}} = 1.06

Bending Capacity

ω₂ = 1.0

EI_yGJ = 200,000×38.8×76.92×575= 343.2×10²¹ MPa²·mm⁸

{(\frac{π E}{L})}^{2} I_{y} C_{w} = {(\frac{π \times 200, 000}{3, 600})}^{2} 38.8 \times 553 = 653.6 \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{1.0 \times π}{3,600} \sqrt{343.2 \times 10^{21} + 653.6 \times 10^{21}} = 871 \times 10^{6} N·mm = 871 kN·m

Plastic moment capacity:

M_p = Z_xF_y = 985×10³ × 350 = 345×10⁶ N·mm = 345 kN·m

M_u = 871 kN·m > 0.67×M_p = 0.67×345 = 231 kN·m

Therefore, the moment capacity is calculated as:

M_{r} = 1.15 ϕ M_{p} (1 - \frac{0.28 M_{p}}{M_{u}}) = 1.15 \times 0.9 \times 345 (1 - \frac{0.28 \times 345}{871}) = 317 kN·m

Check the capacity based on the strength cross-section:

M_rx = 0.9×985×10³ ×350 = 310 kN·m

This governs capacity.

Combined Stress Ratio

\frac{900}{2, 216} + \frac{0.85 \times 1.06 \times 180}{310} = 0.406 + 0.523 = 0.929 < 1.0

Comparison

Table 1. Verification Problem comparison
Criteria	Reference	STAAD.Pro	Difference
(KL/r)_x	33	32.624	negligible
(KL/r)_y	56	55.675	negligible
C_r (kN)	2,216	2,220	negligible
C_e (kN)	17,211	17,200	negligible
Stress Ratio	0.929	0.926	negligible

Note: Both the reference and hand calculations agree with the STAAD.Pro results that the section is adequate. The reference however neglects to account for section capacity checks, which control in this case.

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\Verification Models\09 Steel Design\Canada\CSA S16-14 - Shear Capacity Combined Stresses.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 17-Sep-13
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 0 3.6 0;
MEMBER INCIDENCES
1 1 2;
DEFINE MATERIAL START
ISOTROPIC STEEL
E 2.00e+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 TABLE ST W250X73
MEMBER PROPERTY CANADIAN
1 TABLE ST W250X73
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
*1 ENFORCED BUT MX MY MZ
1 PINNED
2 ENFORCED BUT FY MX MZ
*2 ENFORCED BUT FX FZ
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
JOINT LOAD
1 FY 900 MZ 180
2 FY -900 MZ -180
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PRINT SUPPORT REACTION ALL
PARAMETER 1
CODE CANADIAN
SSY 1 ALL
SSZ 1 ALL
FYLD 350000 ALL
FU 450000 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH

STAAD Output

                       STAAD.PRO CODE CHECKING - CSA S16-19 (v1.0)
                     ***********************************************
   ALL UNITS ARE IN KN   MET   UNLESS OTHERWISE Noted
|--------------------------------------------------------------------------|
|   MEMBER NO.:      1      SECTION: ST  W250X73          (CANADIAN SECTIONS)|
|  STATUS:        PASS    CRITICAL RATIO:  0.926         LOADCASE    1     |
|  LOCATION:      0.00    CRITICAL CONDITION:            Cl. 13.8.2        |
|--------------------------------------------------------------------------|
|  STRENGTH CHECK SUMMARY                                                  |
|   CRITICAL RATIO:  0.926(PASS)            LOAD CASE:     1               |
|   LOCATION      :   0.00                  CONDITION: Cl. 13.8.2          |
|--------------------------------------------------------------------------|
|  DESIGN FORCES              LC:       1      LOC:      0.00              |
|         FX:    900.00(C)    FY:     0.00      FZ:      0.00              |
|         MX:  0.000E+00      MY: 0.000E+00     MZ:  1.800E+02             |
|--------------------------------------------------------------------------|
|  SECTION PROPERTIES (LOC:   0.00)             UNIT:      CM              |
|   AXX:      92.800     AZZ:      72.136      AYY:         21.758         |
|   IXX:      57.500     IZZ:   11300.001      IYY:       3880.000         |
|   PZZ:     985.000     PYY:     463.000                                  |
|   SZZ:     893.281     SYY:     305.512      CW :     552900.312         |
|--------------------------------------------------------------------------|
|  MATERIAL PROPERTIES                          UNIT:      N MM            |
|   FYLD:     350.000     FU:   450.000      E: 199999.984  G:  76920.000  |
|--------------------------------------------------------------------------|
|  DESIGN PARAMETERS                                                       |
|   ACTUAL MEMBER LENGTH:     3.600        LZ:    3.600    LY:    3.600    |
|   KZ:  1.000        KY:  1.000      NSF:  1.000      SLF:  1.000         |
|--------------------------------------------------------------------------|
|  SECTION CLASS                                                           |
|   COMPRESSION    :            Class 1                                    |
|   FLEXURE           MAJOR               MINOR                            |
|   SECTION        : Class 2             Class 2                           |
|   FLANGE         : Class 2             Class 2                           |
|   WEB (CRITICAL) :            Class 1                                    |
|==========================================================================|
|  SLENDERNESS                                                             |
|   ACTUAL SLENDERNESS RATIO   :      55.675          LC   :        1      |
|   ALLOWABLE SLENDERNESS RATIO:     200.000          LOC  :      0.00     |
|--------------------------------------------------------------------------|
      STAAD SPACE                                              -- PAGE NO.    6
                       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   2.92E+03       0.000      Cl. 13.2          1     0.000    |
|  RUPTURE                                                                 |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00   3.13E+03       0.000      Cl. 13.2          1     0.000    |
|--------------------------------------------------------------------------|
|                       CHECKS FOR AXIAL COMPRESSION                       |
|--------------------------------------------------------------------------|
|  FLEX BUCKLING                                                           |
|            FORCE:    CAPACITY:   RATIO:    CRITERIA:      LC:    LOC:    |
|   MAJOR:   9.00E+02   2.71E+03     0.332  Cl. 13.3            1    0.000 |
|   MINOR:   9.00E+02   2.22E+03     0.406  Cl. 13.3            1    0.000 |
|  INTERMEDIATE RESULTS:                                                   |
|               Ag:(CM)     Fe:(N MM)       λ:        n:      KL/r:        |
|   MAJOR:      92.800        1854.621       0.434    1.34      32.624     |
|   MINOR:      92.800         636.808       0.741    1.34      55.675     |
|--------------------------------------------------------------------------|
|  FLEXURAL TORSIONAL BUCKLING                                             |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    9.00E+02   2.22E+03       0.406      Cl. 13.3          1     0.000    |
|  INTERMEDIATE RESULTS:                                                   |
|   Ag:(CM)    92.800   Fe:(N MM)      636.808  λ:     0.741  n: 1.34      |
|--------------------------------------------------------------------------|
|                             CHECKS FOR SHEAR                             |
|--------------------------------------------------------------------------|
|  SHEAR ALONG Y AXIS                                                      |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00   4.52E+02       0.000    Cl. 13.4.1.1        1     0.000    |
|  INTERMEDIATE RESULTS:                                                   |
|   Aw:(CM)      Kv:       Ka:     Fcri:(N MM)  Fcre:(N MM)  Fs:(N MM)     |
|     21.758    5.360     0.070       480.943      1414.474    231.000     |
|--------------------------------------------------------------------------|
|  SHEAR ALONG Z AXIS                                                      |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00   1.36E+03       0.000    AISC G2-1           1     0.000    |
|  INTERMEDIATE RESULTS:                                                   |
|   Aw:(CM)                                                                |
|     72.136                                                               |
|--------------------------------------------------------------------------|
      STAAD SPACE                                              -- PAGE NO.    7
                       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:       |
|    -1.80E+02  3.103E+02      0.580    Cl. 13.5(a)         1     0.000    |
|  INTERMEDIATE RESULTS: Mp:  3.45E+02   Se(CM):  0.00E+00    My:  0.00E+00|
|--------------------------------------------------------------------------|
|  YIELDING ALONG MINOR AXIS                                               |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00  1.458E+02      0.000    Cl. 13.5(a)         1     0.000     |
|  INTERMEDIATE RESULTS: Mp:  1.62E+02   Se(CM):  0.00E+00    My:  0.00E+00|
|--------------------------------------------------------------------------|
|  LATERAL TORSIONAL BUCKLING (MAJOR AXIS)                                 |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|   -1.80E+02  3.103E+02      0.580   Cl. 13.6(a)(i)       1     0.000     |
|  INTERMEDIATE RESULTS:                                                   |
|   Iyc(CM):    w 2:    w 3:    b x(CM):  Mu:       rt:   Myr:     Lu:       Lyr:  |
|  0.00E+00   1.00  0.00     0.00  8.712E+02  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   : 0.493  Cl. 13.9.2      1   0.00   0.00E+00  2.92E+03  |
|   MEMBER STRENGTH: 0.580  Cl. 13.9.3      1   0.00                       |
|                        Mfz:         Mfy:          Mrz:         Mry:      |
|   C/S STRENGTH   :   -1.80E+02      0.00E+00     3.10E+02      1.46E+02  |
|--------------------------------------------------------------------------|
|  FLEXURE AND AXIAL COMPRESSION                                           |
|                    RATIO:   CRITERIA:   LC:   LOC:     Cf:        Cr:    |
|   C/S STRENGTH   : 0.828  Cl. 13.8.2      1   0.000 9.00E+02  2.92E+03   |
|   MEMBER STRENGTH: 0.852  Cl. 13.8.2      1   0.000 9.00E+02  2.71E+03   |
|   LTB STRENGTH   : 0.926  Cl. 13.8.2      1   0.000 9.00E+02  2.22E+03   |
|                        Mfz:      Mfy:      Mrz:      Mry:    w 1z:    w 1y:    |
|   C/S STRENGTH   : -1.80E+02  0.00E+00  3.10E+02  1.46E+02  1.00   0.60  |
|   MEMBER STRENGTH: -1.80E+02  0.00E+00  3.10E+02  1.46E+02  0.60   1.00  |
|   LTB STRENGTH   : -1.80E+02  0.00E+00  3.10E+02  1.46E+02  1.00   0.60  |
|                        b :         Cez:     Cey:         U1z:     U1y:     |
|   C/S STRENGTH   :    0.60     1.72E+04  5.91E+03     1.06     1.00      |
|   MEMBER STRENGTH:    0.85     1.72E+04  5.91E+03     1.06     0.71      |
|   LTB STRENGTH   :    0.85     1.72E+04  5.91E+03     1.06     0.71      |
|--------------------------------------------------------------------------|
|  SHEAR CAPACITY MODIFICATION  ( SHEAR AND FLEX )                         |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|     0.00E+00  4.523E+02      0.000    Cl. 14.6            1     0.000    |
|--------------------------------------------------------------------------|
|  BIAXIAL FLEXURE                                                         |
|   RATIO:  CRITERIA:  LC:   LOC:     Mfz:      Mrz:       Mfy:      Mry:  |
|   0.580  Cl. 13.8     1    0.00  -1.80E+02  3.10E+02  0.00E+00  1.46E+02 |
|==========================================================================|
      STAAD SPACE                                              -- PAGE NO.    8