V. AISC ASD - Deflection Check for a Steel Beam with Web Opening

Determine the deflection of a beam with a web opening.

Details

Design Method : ASD

Beam No : 5

Beam Section : W21X50

d = 20.83 in
t_w = 0.38 in
b_f = 6.53 in
t_f = 0.535 in
Z = 110 in³
L = 30 ft
E = 29,732 ksi

Opening Type: Rectangular

Number of openings : 1

Section location of Opening, s_open = 0.3
a₀ = 20 in
h₀ = 10 in
e = 0 in ( concentric opening )
F_y = 36 ksi

Loading at critical location:

V _u = 9.51 kip
M_u = 344.36 kip-inch

Location of maximum deflection from start of the member : L_critical = 120 in

Tee Properties :

s_t = 5.415 in
s_b = 5.415 in
Moment Area = $\frac{b_{f} t_{f}^{2}}{2} + t_{w} (s_{t} - t_{f}) (\frac{s_{t} t_{f}}{2} + t_{f}) = 6.4514 {in}^{3}$
Area = b_ft_f + t_w(s_t - t_f) = 5.34795 in²
C_yt = Moment Area / Area = 1.20632 in.
$I_{z z, t} = I_{z z, b} = \frac{b_{f} t_{f}^{3}}{12} + b_{f} t_{f} {(C_{y} - \frac{t_{f}}{2})}^{2} + \frac{t_{w} {(s_{t} - t_{f})}^{3}}{12} + t_{w} (s_{t} - t_{f}) {(c_{y} - t_{f} - \frac{s_{t} - t_{f}}{2})}^{2} = 12.644 {in}^{3}$ (concentric opening)
I_zz = I_zz,t + I_zz,b = 25.2872 in³

Validation

R_{t} = \frac{I_{z z, b}}{I_{x x}} = 0.5

R_{b} = \frac{I_{z z, t}}{I_{x x}} = 0.5

V_t = F_yR_t = 4.755

V_b = F_yR_b = 4.755

Δ_{0 t} = \frac{V_{t} a_{0}^{3}}{6 E I_{z z, t}} = 0.01686 in

Δ_{0 b} = \frac{V_{b} a_{0}^{3}}{6 E I_{z z, b}} = 0.01686 in

Δ₀ = Δ_0b + Δ_0b = 0.03373 in

L₀ = distance from high moment end of opening to adjacent support $= L - s_{o p e n} L - \frac{a_{0}}{2} = 314 in$

L₂ = distance from support to the point where deflection is maximum = L₀ - L_critical = 240 in

L₃ = L₀ - L₂ = 74 in

Δ_{p 1} = Δ_{0} \frac{L_{2}}{L_{0}} = 0.02578 in

θ_{H} = \frac{Δ_{0}}{L_{0}} = 0.00011

θ_{T} = \frac{V a_{0}^{2}}{4 E I_{z z}} = 0.00126

Δ_{p 2} = \frac{2 L_{2} L_{3} (θ_{H} + θ_{T})}{L_{0} + L_{2}} = 0.08799

Δ = Δ_p1 + Δ_p2 = 0.11377 in (extra deflection due to opening)

displacement factor: Δ_factor = 750

Δ_{a l l o w} = \frac{L}{Δ_{f a c t o r}} = 0.48 in

Ratio of Δ_actual/Δ_allow = 0.66 (without web opening)

Thus, Δ_actual= 0.3168 in (from analysis considering stiffness of unperforated member without web opening)

Total deflection considering the effect of web hole,

Δ_actual,wb= Δ + Δ_actual = 0.4306 in

Ratio = \frac{Δ_{a c t u a l, w b}}{Δ_{a l l o w}} = 0.897

Results

Table 1. Comparison of results
Result Type	Reference	STAAD.Pro	Difference	Comments
Interaction Ratio, R	0.897	0.899	< 1%
Total deflection, in	0.4306	0.4315*	< 1%

* STAAD.Pro does not directly report the deflection, but rather the ratio of deflection with length, dff, for comparison with a limiting displacement factor.

Δ_actual,wb = L / dff = 360 in / 834.32 = 0.4315 in

Note: The member length used for deflection calculations is defined using the start deflection joint number, DJ1, and end deflection joint number, DJ2. These are assigned to the member in the design code parameters. STAAD.Pro uses this to calculate the physical member length rather than the segment length of the analytical member for deflection checks.

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ is typically installed with the program.

STAAD PLANE Deflection of Steel Beam with Web Opening
START JOB INFORMATION
ENGINEER DATE 18-May-05
END JOB INFORMATION
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 30 0 0; 3 0 20 0; 4 10 20 0; 5 20 20 0; 6 30 20 0; 7 0 35 0;
8 30 35 0; 9 7.5 35 0; 10 22.5 35 0; 11 15 35 0; 12 5 38 0; 13 25 38 0;
14 10 41 0; 15 20 41 0; 16 15 44 0;
MEMBER INCIDENCES
1 1 3; 2 3 7; 3 2 6; 4 6 8; 5 3 4; 6 4 5; 7 5 6; 8 7 12; 9 12 14;
10 14 16; 11 15 16; 12 13 15; 13 8 13; 14 9 12; 15 9 14; 16 11 14;
17 11 15; 18 10 15; 19 10 13; 20 7 9; 21 9 11; 22 10 11; 23 8 10;
MEMBER PROPERTY AMERICAN
1 3 4 TABLE ST W14X90
2 TABLE ST W10X49
5 6 7 TABLE ST W21X50
8 TO 13 TABLE ST W18X35
14 TO 23 TABLE ST L40404
MEMBER TRUSS
14 TO 23
MEMBER RELEASE
5 START MZ
UNIT INCHES KIP
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 29000
POISSON 0.3
DENSITY 0.000283
ISOTROPIC STEEL
E 29732.7
POISSON 0.3
DENSITY 0.000283
ALPHA 1.2e-005
DAMP 0.03
END DEFINE MATERIAL
CONSTANTS
BETA 90 MEMB 3 4
MATERIAL MATERIAL1 MEMB 1 TO 4 6 TO 23
MATERIAL STEEL MEMB 5
UNIT FEET KIP
SUPPORTS
1 FIXED
2 PINNED
PRINT MEMBER INFORMATION LIST 1 5 14
PRINT MEMBER PROPERTIES LIST 1 2 5 8 14
LOAD 1 DEAD AND LIVE LOAD
SELFWEIGHT X 1
SELFWEIGHT Y -1
JOINT LOAD
4 5 FY -15
11 FY -35
MEMBER LOAD
8 TO 13 UNI Y -0.9
6 UNI GY -1.2
CALCULATE RAYLEIGH FREQUENCY
LOAD 2 WIND FROM LEFT
MEMBER LOAD
1 2 UNI GX 0.6
8 TO 10 UNI Y -1
* 1/3 RD INCREASE IS ACCOMPLISHED BY 75% LOAD
LOAD COMB 3 75 PERCENT DL LL WL
1 0.75 2 0.75
PERFORM ANALYSIS
LOAD LIST 3
UNIT INCHES KIP
PARAMETER
CODE AISC
TRACK 2 MEMB 5
DFF 750 
DJ1 3 MEMB 5
DJ2 6 MEMB 5
*WEB OPENINGS
*********************
RHOLE 0.3 MEMB 5
RDIM 20.0 10.0 MEMB 5
*********************
CHECK CODE MEMB 5 
FINISH

STAAD Output

                       STAAD.PRO CODE CHECKING - (AISC 9TH EDITION)   v1.0
                       ********************************************
  |--------------------------------------------------------------------------|
  |                                                    Y        PROPERTIES   |
  |*************                                       |        IN INCH UNIT |
  |            *  |=============================|   ===|===     ------------ |
  |MEMBER   5  *  |  AISC SECTIONS              |      |        AX =  14.70  |
  |            *  | ST  W21X50                  |      |   --Z  AY =   6.94  |
  |DESIGN CODE *  |                             |      |        AZ =   4.69  |
  | AISC-1989  *  ===============================   ===|===     SY =   7.63  |
  |            *                                                SZ =  94.62  |
  |            *  |&lt;---LENGTH (FT)=   10.00 --->|               RY =   1.30  |
  |*************                                                RZ =   8.18  |
  |                                                                          |
  |                 94.4 (KIP-FEET)                                          |
  |PARAMETER        |                                       L3  STRESSES     |
  |IN KIP  INCH     |                               L3  L3      IN KIP  INCH |
  |---------------  +                           L3              -------------|
  | KL/R-Y=  92.20  |                       L3                  FA  =  21.60 |
  | KL/R-Z=  14.67  +                   L3                      fa  =   0.94 |
  | UNL   = 120.00  |               L3                          FCZ =  19.35 |
  | CB    =   1.00  +                                           FTZ =  21.60 |
  | CMY   =   0.85  |       L3                                  FCY =  27.00 |
  | CMZ   =   0.85  +   L3                                      FTY =  27.00 |
  | FYLD  =  36.00  |L0         L0                              fbz =  11.97 |
  | NSF   =   1.00  +---+---+---+---+---+---+---+---+---+---|   fby =   0.00 |
  | DFF   = 750.00  -5.2                                        Fey =  18.01 |
  | dff=    834.32               ABSOLUTE MZ ENVELOPE           Fez = 711.71 |
  | (KL/R)max =  92.20              (WITH LOAD NO.)             FV  =  14.40 |
  |                                                             fv  =   1.33 |
  |                                                                          |
  |                      MAX FORCE/ MOMENT SUMMARY  (KIP-FEET)               |
  |                      -------------------------                           |
  |                                                                          |
  |                AXIAL      SHEAR-Y    SHEAR-Z    MOMENT-Y    MOMENT-Z     |
  |                                                                          |
  |      VALUE     -14.2         9.6        0.0         0.0        94.4      |
  |   LOCATION       0.0         0.0        0.0         0.0        10.0      |
  |    LOADING         3           3          0           0           3      |
  |                                                                          |
  |**************************************************************************|
  |*                                                                        *|
  |*                         DESIGN SUMMARY  (KIP-FEET)                     *|
  |*                         --------------                                 *|
  |*                                                                        *|
  |*       RESULT/     CRITICAL COND/     RATIO/         LOADING/           *|
  |         FX              MY             MZ           LOCATION             |
  |        ======================================================            |
  |          PASS        DEFLECTION       0.899              3               |
  |         13.81 T          0.00       -94.36            10.00              |
  |*                                                                        *|
  |                                                                          |
  |--------------------------------------------------------------------------|
      DEFLECTION OF STEEL BEAM WITH WEB OPENING                -- PAGE NO.    7
   OUTPUT FOR WEB OPENING
   ----------------------
   SECTION    LOAD        MZ/           FY/                IR
                      MOM. CAP. (Mm)  SHR. CAP. (Vm) ((MZ/Mm)^3+(FY/Vm)^3)^0.33
   =======================================================================
    0.300       3         344.36          9.51
                         3618.00         38.44             0.25
   ======================= END WEB OPENING OUTPUT ========================
    79. FINISH
             *********** END OF THE STAAD.Pro RUN ***********         
               **** DATE= OCT 27,2022   TIME= 15:35:54 ****
      DEFLECTION OF STEEL BEAM WITH WEB OPENING                -- PAGE NO.    8
         ************************************************************