 # V.Deflection Check for a Steel Beam with Web Opening

This example illustrates the design procedure including a deflection check for a member with a web opening.

## Given

Design Method : ASD

Beam No : 5

Beam Section : W21X50

• d = 20.83 in
• tw = 0.38 in
• bf = 6.53 in
• tf = 0.535 in
• Z = 110 in3
• L = 30 ft
• E = 29,732 ksi

Opening Type: Rectangular

Number of openings : 1

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

• V u = 9.51 kip
• Mu = 344.36 kip-inch

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

Tee Properties :

• st = 5.415 in
• sb = 5.415 in
• Moment Area = $b f t f 2 2 + t w ( s t − t f ) ( s t t f 2 + t f ) = 6.4514 in 3$
• Area = bftf + tw(st - tf) = 5.34795 in2
• Cyt = Moment Area / Area = 1.20632 in.

• $I z z , t = I z z , b = b f t f 3 12 + b f t f ( C y − t f 2 ) 2 + t w ( s t − t f ) 3 12 + t w ( s t − t f ) ( c y − t f − s t − t f 2 ) 2 = 12.644 in 3$ (concentric opening)
• Izz = Izz,t + Izz,b = 25.2872 in3

## Calculations

$R t = I z z , b I x x = 0.5$
$R b = I z z , t I x x = 0.5$

Vt = FyRt = 4.755

Vb = FyRb = 4.755

$Δ 0 t = V t a 0 3 6 E I z z , t = 0.01686 in$
$Δ 0 b = V b a 0 3 6 E I z z , b = 0.01686 in$

Δ0 = Δ0b + Δ0b = 0.03373 in

L0 = distance from high moment end of opening to adjacent support $= L − s o p e n L − a 0 2 = 314 in$

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

L3 = L0 - L2 = 74 in

$Δ p 1 = Δ 0 L 2 L 0 = 0.02578 in$
$θ H = Δ 0 L 0 = 0.00011$
$θ T = V a 0 2 4 E I z z = 0.00126$
$Δ p 2 = 2 L 2 L 3 ( θ H + θ T ) L 0 + L 2 = 0.08799$

Δ = Δp1 + Δp2 = 0.11377 in (extra deflection due to opening)

displacement factor: Δfactor = 750

$Δ a l l o w = L Δ f a c t o r = 0.48 ⁢ in$

Ratio of Δactualallow = 0.66 (without web opening)

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

Total deflection considering the effect of web hole,

Δactual,wb= Δ + Δactual = 0.4306 in

$Ratio = Δ a c t u a l , w b Δ a l l o w = 0.897$

## Comparison

Table 1. Comparison of results
Result Type Hand Calculations STAAD.Pro Difference
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.

Δactual,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.

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
SELFWEIGHT X 1
SELFWEIGHT Y -1
4 5 FY -15
11 FY -35
8 TO 13 UNI Y -0.9
6 UNI GY -1.2
CALCULATE RAYLEIGH FREQUENCY
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
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.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  =   0.97 |
| 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 =   0.00 |
| 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  =   1.39 |
|                                                                          |
|                      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      |
|                                                                          |
|**************************************************************************|
|*                                                                        *|
|*                         DESIGN SUMMARY  (KIP-FEET)                     *|
|*                         --------------                                 *|
|*                                                                        *|
|         FX              MY             MZ           LOCATION             |
|        ======================================================            |
|          PASS        DEFLECTION       0.899              3               |
|         14.19 T          0.00         0.00            10.00              |
|*                                                                        *|
|**************************************************************************|
|                                                                          |
DEFLECTION OF STEEL BEAM WITH WEB OPENING                -- PAGE NO.    7
ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted)