# V. CSA S16-14 - 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 Lb = 6.0 m) (LT 6.0).

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

## Calculations

Assume a Class 2 section:

Mr = ϕMp = ϕZxFy

Thus,

Try a W460x82:

Zx = 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:

$b 2 t = 191 2 × 16.0 = 5.7 < 170 350 = 9.1$

Evaluate the slenderness effects of the beam web:

$h w = 460 − 2 ( 16.0 ) 9.9 = 43.2 ⁢ < 1 , 700 350 = 90.9$

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

Mr = 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
Section W460x82 * W460x82 none
Mr (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.

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\Canada\CSA S16-14 - 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:

$M 3 = w f L 2 8 − ( M 1 + M 2 ) 2 + ( M 1 − M 2 ) 2 2 w f L 2$
$256 = w f ( 11.0 ) 2 8 − ( 540 + 185 ) 2 + ( 540 − 185 ) 2 2 w f ( 11.0 ) 2$
Solving this yields wf = 40.033 kN/m

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 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
1 TABLE ST W530X82
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 FIXED BUT FX MY MZ
1 UNI GY -40.033
1 MZ 540
2 MZ -185
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PRINT SUPPORT REACTION ALL
PARAMETER 1
LAT 1 ALL
FYLD 350000 ALL
FU 450000 ALL
TRACK 2 ALL
CHECK CODE ALL
SELECT ALL
PERFORM ANALYSIS
FINISH


                       STAAD.PRO CODE CHECKING - S16-14 (v1.0)
*******************************************
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 : 0.00  CONDITION: Cl. 13.8
UNIT: KN   MET
STRENGTH CHECKS:
CRITICAL RATIO:  0.832(PASS)     LOAD CASE:     1  LOCATION : 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 (N MM):      350.000   FU (N MM):      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.:   0.000
ALLOWABLE SLENDERNESS RATIO:  300.000
SECTION CLASS:
COMPRESSION:     Class 4
FLEXURE      MAJOR          MINOR
SECTION:    Class 2        Class 2
FLANGE:     Class 2        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 - (          S16-14)   v1.0
********************************************
ALL UNITS ARE - KN   MET  (UNLESS OTHERWISE Noted)
COMPRESSION:    FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION:
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:
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 - (          S16-14)   v1.0
********************************************
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 2.25E+03 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
WARNING: The current units are in KN   METE. All the design results are in KN   MET  (unless otherwise noted).
STAAD SPACE                                              -- PAGE NO.    8
STAAD.PRO MEMBER SELECTION - S16-14 (v1.0)
**********************************************
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 : 0.00  CONDITION: Cl. 13.8
UNIT: KN   MET
STRENGTH CHECKS:
CRITICAL RATIO:  0.937(PASS)     LOAD CASE:     1  LOCATION : 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:     915745.312
SZZ:    1608.696 SYY:     194.764
IZZ:   37000.004 IYY:    1860.000
UNIT: NEW MM
MATERIAL PROPERTIES:      FYLD (N MM):      350.000   FU (N MM):      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.:   0.000
ALLOWABLE SLENDERNESS RATIO:  300.000
SECTION CLASS:
COMPRESSION:     Class 4
FLEXURE      MAJOR          MINOR
SECTION:    Class 1        Class 1
FLANGE:     Class 1        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 - (          S16-14)   v1.0
***********************************************
ALL UNITS ARE - KN   MET  (UNLESS OTHERWISE Noted)
COMPRESSION:    FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION:
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    1194.071      0.000    Cl. 13.3.5(a)      1        0.000
INTERMEDIATE:   Fe(N MM):       λ            n:
176.181    1.409        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:
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 - (          S16-14)   v1.0
***********************************************
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 2.26E+03 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 --------------------