V.CSA S16-09 - Select a Beam

Select a wide-flange shape for the given beam span.

Reference

Kulak, G.L. and G.Y. Grondin. 2010. Limit States Design in Structural Steel. 9th Edition. Markham, ON:CISC. Example 5.2

Problem

A beam must be selected to span 11.0 m. The end moment of the span are:

M₁ = -540 kN·m

M₂ = -185 kN·m

And the maximum moment in the span is:

M₃ = 256 kN·m

The beam is braced at 2.5 m from the center of each supporting column (thus L_b = 6.0 m) (LT 6.0).

The material used is G40.21 350W steel (F_y = 350 MPa) (FYLD)

Calculations

Assume a Class 2 section:

M_r = ϕM_p = ϕZ_xF_y

Thus,

Z_{x} \geq \frac{540 {(10)}^{6}}{0.9 \times 350} = 1, 714 \times {(10)}^{3} {mm}^{3}

Try a W460x82:

Z_x = 1,830×(10)³ mm³, tf = 16.0 mm, bf = 191.0 mm, tw = 9.9 mm, D = 460.0 mm

Evaluate the slenderness effects of the beam flanges:

\frac{b}{2 t} = \frac{191}{2 \times 16.0} = 5.7 < \frac{170}{\sqrt{350}} = 9.1

Evaluate the slenderness effects of the beam web:

\frac{h}{w} = \frac{460 - 2 (16.0)}{9.9} = 43.2 < \frac{1, 700}{\sqrt{350}} = 90.9

The assumptions are valid and this beam is sufficient to support the given moments.

M_r = 0.9×1,830×(10)³×350 [(10)^-6] = 576.5 kN·m

Ratio = 540 / 576.5 = 0.937

Comparison

Table 1. Comparison of results
Criteria	Reference	STAAD.Pro	Difference
Section	W460x82 *	W460x82	none
M_r (kN·m)	576.5 **	576	negligible
Critical Ratio	0.937 **	0.937	none

Notes:

*The reference also tries a W530x82, which is stiffer for the same weight. STAAD.Pro selects the shallower beam.
** The reference does not calculate the resisting moment capacity nor the critical ratio for either section size, but they are evaluated here for completeness.

STAAD.Pro Input File

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\Canada\S16 2009\CSA S16-09 - Select a Beam.STD is typically installed with the program.

In order to model this beam in STAAD.Pro, we need to determine the distributed load which would result in the same mid-span moment on a beam with the given end moments. From the AISC Streel Construction Manual, p 3-222:

M_{3} = \frac{w_{f} L^{2}}{8} - \frac{(M_{1} + M_{2})}{2} + \frac{{(M_{1} - M_{2})}^{2}}{2 w_{f} L^{2}}

256 = \frac{w_{f} {(11.0)}^{2}}{8} - \frac{(540 + 185)}{2} + \frac{{(540 - 185)}^{2}}{2 w_{f} {(11.0)}^{2}}

Solving this yields w_f = 40.033 kN/m

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 17-Mar-19
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 11 0 0;
MEMBER INCIDENCES
1 1 2;
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 TABLE ST W530X82
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 FIXED BUT FX MY MZ
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
1 UNI GY -40.033
JOINT LOAD
1 MZ 540
2 MZ -185
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PRINT SUPPORT REACTION ALL
PARAMETER 1
CODE CANADIAN 2009
LAT 1 ALL
FYLD 350000 ALL
FU 450000 ALL
TRACK 2 ALL
CHECK CODE ALL
SELECT ALL
PERFORM ANALYSIS
FINISH

STAAD.Pro Output

      STAAD SPACE                                              -- PAGE NO.    5
                       STAAD.PRO CODE CHECKING - S16-09 (v1.1)
                     *******************************************
 ALL UNITS ARE - KN   MET  (UNLESS OTHERWISE Noted)
  -------------------- START OF DESIGN OUTPUT OF MEMBER      1 --------------------
  MEMBER NO:     1    CRITICAL RATIO:  0.832(PASS)     LOAD:     1
  LOCATION (MET): 0.00  CONDITION: Cl. 13.8       
  SECTION: ST   W530X82                 (CANADIAN SECTIONS)
  UNIT: KN MET
  STRENGTH CHECKS:
  CRITICAL RATIO:  0.832(PASS)     LOAD CASE:     1  LOCATION (MET): 0.00  CONDITION: Cl. 13.8       
  DESIGN FORCES:   Fx:      0.00(T)    Fy:   252.45      Fz:      0.00
                   Mx:   0.00E+00      My:  0.00E+00     Mz:   5.40E+02
  UNIT: CM
  SECTION PROPERTIES:       AZZ:      55.594 AYY:      48.896 CW:    1340255.500
                            SZZ:    1806.818 SYY:     194.258
                            IZZ:   47700.008 IYY:    2030.000
  UNIT: NEW MM
  MATERIAL PROPERTIES:      FYLD:      350.000   FU:      450.000
  ACTUAL MEMBER LENGTH(MET):    11.000
  PARAMETERS:                  KZ:  1.000  KY:  1.000  NSF:  1.000  SLF:  1.000
  SLENDERNESS:   ACTUAL SLENDERNESS RATIO:     250.172  LOAD:     1   LOC.(MET):   0.000
                 ALLOWABLE SLENDERNESS RATIO:  300.000
  SECTION CLASS:
  COMPRESSION:     Class 4     
  FLEXURE:         Class 2     
  FLANGE:          Class 2     
  WEB:             Class 1     
  UNIT: KN MET
  TENSION:        FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  YIELDING:        0.000    3307.500      0.000    Cl. 13.2           1        0.000
  RUPTURE:         0.000    3543.750      0.000    Cl. 13.2           1        0.000
      STAAD SPACE                                              -- PAGE NO.    6
                       STAAD.PRO CODE CHECKING - (          S16-09)   v1.1
                       ********************************************
 ALL UNITS ARE - KN   MET  (UNLESS OTHERWISE Noted)
  COMPRESSION:    FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  MAJOR:           0.000    2253.274      0.000    Cl. 13.3.5(a)      1        0.000
  MINOR:           0.000     253.222      0.000    Cl. 13.3.5(a)      1        0.000
  INTERMEDIATE:   Ag:(CM)     KL/r:     Fe:(N MM)        λ          n:
  MAJOR:          89.688     51.609      759.622      0.679        1.340
  MINOR:          89.688    250.172       32.328      3.290        1.340
  FLEX TOR BUCK:  FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
                  0.000     854.388      0.000    Cl. 13.3.5(a)      1        0.000
  INTERMEDIATE:   Fe:(N MM) λ            n:
                 125.187    1.672        1.340
  SHEAR:          FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  MAJOR:         252.454    1042.826      0.242    Cl. 13.4.1.1       1        0.000
  MINOR:           0.000    1050.727      0.000    AISC G2-1          1        0.000
  INTERMEDIATE:   Aw:(CM)    Kv:          Ka:       Fcri:(N MM)  Fcre:(N MM)  Fs:(N MM)
  MAJOR:          48.896    5.349       0.048       237.748       345.654      231.000
  MINOR:          55.594
  UNIT: KN MET
  YIELDING:       FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  MAJOR:       -5.40E+02    6.49E+02      0.832    Cl. 13.5(a)        1        0.000
  MINOR:        0.00E+00    9.54E+01      0.000    Cl. 13.5(a)        1        0.000
  INTERMEDIATE:   Mp:        Se:         My:
  MAJOR:        7.21E+02    0.00E+00    0.00E+00
  MINOR:        1.06E+02    0.00E+00    0.00E+00
  UNIT: KN MET
  INTERACTION:                RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  FLEXURE AND AXIAL TENSION:
  C/S STRENGTH:                0.832    Cl. 13.9.1         1       0.000
  FLEXURE AND AXIAL COMPRESSION:
  C/S STRENGTH:                0.707    Cl. 13.8.2         1       0.000
  MEMBER STRENGTH:             0.707    Cl. 13.8.2         1       0.000
  FLEX AND SHEAR:              0.715    Cl. 14.6(a)        1       0.000
  BIAXIAL FLEX:                0.832    Cl. 13.8           1       0.000
  INTERMEDIATE:
  FLEXURE AND AXIAL TENSION:
  C/S STRENGTH:   Tf:   0.00E+00  Tr:   3.31E+03  Mfz:  -5.40E+02  Mrz:   6.49E+02
      STAAD SPACE                                              -- PAGE NO.    7
                       STAAD.PRO CODE CHECKING - (          S16-09)   v1.1
                       ********************************************
 ALL UNITS ARE - KN   MET  (UNLESS OTHERWISE Noted)
  FLEXURE AND AXIAL COMPRESSION:
             Cf:       Mfz:       Mfy:     Cr:     Mrz:     Mry:     b :    Cez:     Cey:    w 1z:  w 1y: U1z: U1y:
  C/S STR: 0.00E+00 -5.40E+02  0.00E+00 2.83E+03 6.49E+02 9.54E+01 0.85 7.98E+03 3.39E+02 0.74 0.60 1.00 1.00
  MEM STR: 0.00E+00 -5.40E+02  0.00E+00 2.53E+02 6.49E+02 9.54E+01 0.85 7.98E+03 3.39E+02 0.60 0.74 1.00 1.00
  FLEX SHEAR:  Mfz:  -5.40E+02  Mrz:   6.49E+02  Vfy:   2.52E+02  Vry:   1.04E+03
  BIAX FLEX:   Mfz:  -5.40E+02  Mrz:   6.49E+02  Mfy:   0.00E+00  Mry:   9.54E+01
  -------------------- END OF DESIGN OUTPUT OF MEMBER      1 --------------------
   ***NOTE:OMEGA1 AND OMEGA2 ARE CALCULATED USING C/S MOMENT OF ANALYTICAL MEMBERS
    45. SELECT ALL
 STEEL DESIGN    
      STAAD SPACE                                              -- PAGE NO.    8
                       STAAD.PRO MEMBER SELECTION - S16-09 (v1.1)
                     **********************************************
 ALL UNITS ARE - KN   MET  (UNLESS OTHERWISE Noted)
  -------------------- START OF DESIGN OUTPUT OF MEMBER      1 --------------------
  MEMBER NO:     1    CRITICAL RATIO:  0.937(PASS)     LOAD:     1
  LOCATION (MET): 0.00  CONDITION: Cl. 13.8       
  SECTION: ST   W460x82                 (CANADIAN SECTIONS)
  UNIT: KN MET
  STRENGTH CHECKS:
  CRITICAL RATIO:  0.937(PASS)     LOAD CASE:     1  LOCATION (MET): 0.00  CONDITION: Cl. 13.8       
  DESIGN FORCES:   Fx:      0.00(T)    Fy:   252.45      Fz:      0.00
                   Mx:   0.00E+00      My:  0.00E+00     Mz:   5.40E+02
  UNIT: CM
  SECTION PROPERTIES:       AZZ:      61.120 AYY:      43.956 CW:    1340255.500
                            SZZ:    1608.696 SYY:     194.764
                            IZZ:   37000.004 IYY:    1860.000
  UNIT: NEW MM
  MATERIAL PROPERTIES:      FYLD:      350.000   FU:      450.000
  ACTUAL MEMBER LENGTH(MET):    11.000
  PARAMETERS:                  KZ:  1.000  KY:  1.000  NSF:  1.000  SLF:  1.000
  SLENDERNESS:   ACTUAL SLENDERNESS RATIO:     260.108  LOAD:     1   LOC.(MET):   0.000
                 ALLOWABLE SLENDERNESS RATIO:  300.000
  SECTION CLASS:
  COMPRESSION:     Class 4     
  FLEXURE:         Class 1     
  FLANGE:          Class 1     
  WEB:             Class 1     
  UNIT: KN MET
  TENSION:        FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  YIELDING:        0.000    3276.000      0.000    Cl. 13.2           1        0.000
  RUPTURE:         0.000    3510.000      0.000    Cl. 13.2           1        0.000
      STAAD SPACE                                              -- PAGE NO.    9
                     STAAD.PRO MEMBER SELECTION - (          S16-09)   v1.1
                     ***********************************************
 ALL UNITS ARE - KN   MET  (UNLESS OTHERWISE Noted)
  COMPRESSION:    FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  MAJOR:           0.000    2261.248      0.000    Cl. 13.3.5(a)      1        0.000
  MINOR:           0.000     253.374      0.000    Cl. 13.3.5(a)      1        0.000
  INTERMEDIATE:   Ag:(CM)     KL/r:     Fe:(N MM)        λ          n:
  MAJOR:          96.728     58.319      594.890      0.767        1.340
  MINOR:          96.728    260.107       29.905      3.421        1.340
  FLEX TOR BUCK:  FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
                  0.000    1279.603      0.000    Cl. 13.3.5(a)      1        0.000
  INTERMEDIATE:   Fe:(N MM) λ            n:
                 194.448    1.342        1.340
  SHEAR:          FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  MAJOR:         252.454     946.776      0.267    Cl. 13.4.1.1       1        0.000
  MINOR:           0.000    1155.168      0.000    AISC G2-1          1        0.000
  INTERMEDIATE:   Aw:(CM)    Kv:          Ka:       Fcri:(N MM)  Fcre:(N MM)  Fs:(N MM)
  MAJOR:          43.956    5.347       0.042       290.187       514.950      231.000
  MINOR:          61.120
  UNIT: KN MET
  YIELDING:       FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  MAJOR:       -5.40E+02    5.76E+02      0.937    Cl. 13.5(a)        1        0.000
  MINOR:        0.00E+00    9.54E+01      0.000    Cl. 13.5(a)        1        0.000
  INTERMEDIATE:   Mp:        Se:         My:
  MAJOR:        6.41E+02    0.00E+00    0.00E+00
  MINOR:        1.06E+02    0.00E+00    0.00E+00
  UNIT: KN MET
  INTERACTION:                RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  FLEXURE AND AXIAL TENSION:
  C/S STRENGTH:                0.937    Cl. 13.9.1         1       0.000
  FLEXURE AND AXIAL COMPRESSION:
  C/S STRENGTH:                0.796    Cl. 13.8.2         1       0.000
  MEMBER STRENGTH:             0.796    Cl. 13.8.2         1       0.000
  FLEX AND SHEAR:              0.802    Cl. 14.6(a)        1       0.000
  BIAXIAL FLEX:                0.937    Cl. 13.8           1       0.000
  INTERMEDIATE:
  FLEXURE AND AXIAL TENSION:
  C/S STRENGTH:   Tf:   0.00E+00  Tr:   3.28E+03  Mfz:  -5.40E+02  Mrz:   5.76E+02
      STAAD SPACE                                              -- PAGE NO.   10
                     STAAD.PRO MEMBER SELECTION - (          S16-09)   v1.1
                     ***********************************************
 ALL UNITS ARE - KN   MET  (UNLESS OTHERWISE Noted)
  FLEXURE AND AXIAL COMPRESSION:
             Cf:       Mfz:       Mfy:     Cr:     Mrz:     Mry:     b :    Cez:     Cey:    w 1z:  w 1y: U1z: U1y:
  C/S STR: 0.00E+00 -5.40E+02  0.00E+00 3.05E+03 5.76E+02 9.54E+01 0.85 6.19E+03 3.11E+02 0.74 0.60 1.00 1.00
  MEM STR: 0.00E+00 -5.40E+02  0.00E+00 2.53E+02 5.76E+02 9.54E+01 0.85 6.19E+03 3.11E+02 0.60 0.74 1.00 1.00
  FLEX SHEAR:  Mfz:  -5.40E+02  Mrz:   5.76E+02  Vfy:   2.52E+02  Vry:   9.47E+02
  BIAX FLEX:   Mfz:  -5.40E+02  Mrz:   5.76E+02  Mfy:   0.00E+00  Mry:   9.54E+01
  -------------------- END OF DESIGN OUTPUT OF MEMBER      1 --------------------