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V.CSA S16-09 - Beam Bending

Determine the uniformly factored load that the member can resist.

Reference

Kulak, G.L. and G.Y. Grondin. 2010. Limit States Design in Structural Steel. 9th Edition. Markham, ON:CISC. Example 5.1

Problem

A W310x52 beam spans 7.3 m. Both ends of the beam are supported by columns connected standard web angle connections.

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

Calculations

The nominal yield strength is used for the actual yield strength for a Group 2 section, per Table 3 and Table 4 of Appendix A of the reference.

Section Classification

Evaluate the slenderness effects of the beam flanges:

b 2 t = 167 2 × 13.2 = 6.3 < 170 350 = 9.1

Evaluate the slenderness effects of the beam web:

h w = 318 2 ( 13.2 ) 7.6 = 38.4 < 1 , 700 350 = 90.9

Both the beam flanges and web are less than the local buckling limit, so the capacity of the column is evaluated using Eq. 5.7.

Bending Capacity

Factored moment resistance is:

M r = ϕ Z x F y = 0.90 × 841 × ( 10 ) 3 × 350 = 265 × ( 10 ) 3  N⋅mm = 265  kN⋅m

Equate this to the bending moment due to a uniformly distributed load and solve for the load:

M r w f L 2 8
w f = 8 M r L 2 = 8 × 265 ( 7.3 ) 2 = 39.8  kN/m

Comparison

Table 1. Comparison of results
Criteria Reference STAAD.Pro Difference
Mr (kN·m) 265 265 none
wf (kN/m) 39.8 39.76* negligible
Note: (*) STAAD.Pro does calculate the allowable uniform load on the beam. Instead, this is done by applying a uniform load incrementally until the critical ratio is ≈ 1.0. The, this uniform load is divided by the resulting critical ratio to normalize the distributed load capacity of the beam. Thus, 39.8/1.001 = 39.67 kN/m

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 - Beam Bending.STD is typically installed with the program.

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 7.3 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 W310X52
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 -39.8
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
FINISH

STAAD.Pro Output

                       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:  1.001(FAIL)     LOAD:     1
  LOCATION (MET): 3.65  CONDITION: Cl. 13.8      
  SECTION: ST   W310X52                 (CANADIAN SECTIONS)
  UNIT: KN MET
  STRENGTH CHECKS:
  CRITICAL RATIO:  1.001(FAIL)     LOAD CASE:     1  LOCATION (MET): 3.65  CONDITION: Cl. 13.8      
  DESIGN FORCES:   Fx:      0.00(T)    Fy:     0.00      Fz:      0.00
                   Mx:   0.00E+00      My:  0.00E+00     Mz:  -2.65E+02
  UNIT: CM
  SECTION PROPERTIES:       AZZ:      44.088 AYY:      23.165 CW:     237980.875
                            SZZ:     748.428 SYY:     123.353
                            IZZ:   11900.002 IYY:    1030.000
  UNIT: NEW MM
  MATERIAL PROPERTIES:      FYLD:      350.000   FU:      450.000
  ACTUAL MEMBER LENGTH(MET):     7.300
  PARAMETERS:                  KZ:  1.000  KY:  1.000  NSF:  1.000  SLF:  1.000
  SLENDERNESS:   ACTUAL SLENDERNESS RATIO:     185.766  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    2101.050      0.000    Cl. 13.2           1        0.000
  RUPTURE:         0.000    2251.125      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    1587.513      0.000    Cl. 13.3.5(a)      1        0.000
  MINOR:           0.000     322.454      0.000    Cl. 13.3.5(a)      1        0.000
  INTERMEDIATE:   Ag:(CM)     KL/r:     Fe:(N MM)        λ          n:
  MAJOR:          65.224     54.653      677.375      0.719        1.340
  MINOR:          65.224    185.766       58.630      2.443        1.340
  FLEX TOR BUCK:  FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
                  0.000    1023.539      0.000    Cl. 13.3.5(a)      1        0.000
  INTERMEDIATE:   Fe:(N MM) λ            n:
                 253.108    1.176        1.340
  SHEAR:          FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  MAJOR:        -145.270     502.453      0.289    Cl. 13.4.1.1       1        7.300
  MINOR:           0.000     833.263      0.000    AISC G2-1          1        0.000
  INTERMEDIATE:   Aw:(CM)    Kv:          Ka:       Fcri:(N MM)  Fcre:(N MM)  Fs:(N MM)
  MAJOR:          23.165    5.348       0.044       326.992       653.857      231.000
  MINOR:          44.088
  UNIT: KN MET
  YIELDING:       FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  MAJOR:        2.65E+02    2.65E+02      1.001    Cl. 13.5(a)        1        3.650
  MINOR:        0.00E+00    5.95E+01      0.000    Cl. 13.5(a)        1        0.000
  INTERMEDIATE:   Mp:        Se:         My:
  MAJOR:        2.94E+02    0.00E+00    0.00E+00
  MINOR:        6.62E+01    0.00E+00    0.00E+00
  UNIT: KN MET
  INTERACTION:                RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  FLEXURE AND AXIAL TENSION:
  C/S STRENGTH:                1.001    Cl. 13.9.1         1       3.650
  FLEXURE AND AXIAL COMPRESSION:
  C/S STRENGTH:                0.851    Cl. 13.8.2         1       3.650
  MEMBER STRENGTH:             0.851    Cl. 13.8.2         1       0.000
  FLEX AND SHEAR:              0.729    Cl. 14.6(a)        1       3.042
  BIAXIAL FLEX:                1.001    Cl. 13.8           1       3.650
  INTERMEDIATE:
  FLEXURE AND AXIAL TENSION:
  C/S STRENGTH:   Tf:   0.00E+00  Tr:   2.10E+03  Mfz:   2.65E+02  Mrz:   2.65E+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  2.65E+02  0.00E+00 2.05E+03 2.65E+02 5.95E+01 0.85 4.52E+03 3.91E+02 0.60 0.60 1.00 1.00
  MEM STR: 0.00E+00  2.65E+02  0.00E+00 3.22E+02 2.65E+02 5.95E+01 0.85 4.52E+03 3.91E+02 0.60 0.60 1.00 1.00
  FLEX SHEAR:  Mfz:   2.58E+02  Mrz:   2.65E+02  Vfy:   2.42E+01  Vry:   5.02E+02
  BIAX FLEX:   Mfz:   2.65E+02  Mrz:   2.65E+02  Mfy:   0.00E+00  Mry:   5.95E+01
  -------------------- END OF DESIGN OUTPUT OF MEMBER      1 --------------------
   ***NOTE:OMEGA1 AND OMEGA2 ARE CALCULATED USING C/S MOMENT OF ANALYTICAL MEMBERS