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V.API Overlapping KJoint - Comp and Bend

Verify the design of an overlapping K joint with axial load, in-plane bending, and out-of-plane bending.

Details

Design of node 3 in the following model.

Chord member (2 & 3): D = 500 mm, T = 20 mm (Ac = 30,159 mm2, Zc = 3,480,380 mm3)

Brace member (5 & 7): d = 400 mm, t = 20 mm (Ab = 23,876 mm2, Zb = 2,160,790 mm3)

Table 1. Loads
Member Fx (kN) MIP (kN·m) MOP (kN·m)
Chord 49.99 (T) 17.58 4.96
Brace 72.51 (C) 0.727 6.254

Validation

Design Parameters

β = d/D = 0.8

So, 0.2 < β < 1.0 (OK)

γ = D/2T = 12.5

So, 10 < γ < 50 (OK)

θ = tan-1(10/5) = 63.4°

So, 30° < θ < 90° (OK)

Fy = 500 MPa (OK)

g = -76.2 mm

g/D = -76.2 / 500 = -0.1524

Determination of Qu

First, evaluate Qg. Since g/D < -0.50, use linear interpolation.

For g/D = -0.05: Qg = 0.13 + 0.65×ϕ×γ0.5 = 0.13 + 0.65×(1)×(12.5)0.5 = 2.428

Qu_ax = (16 + 1.2×γ)×β1.2×Qg = [16 + 1.2(12.5)](0.8)1.2(2.428) = 57.586

Qu_ipb = (5 + 0.7×γ)×β1.2 = [5 + 0.7(12.5)](0.8)1.2 = 10.52

Qu_opb = 2.5 + (4.5 + 0.2×γ)×β2.6 = 2.5 + [4.5 + 0.2(12.5)](0.8)2.6 = 6.419

Determination of Qf

From Table 4.3.2:

C1ax = 0.2, C2axi = 0.2, C3ax = 0.3

C1ipb = 0.2, C2ipb = 0, C3ipb = 0.4

C1opb = 0.2, C2opb = 0, C3opb = 0.4

Pyc = Ac×Fyc = 15,079.5 kN

Mpc = Zc×Fyc = 1,740.19 kN·m

Pc = 49.99 KN

Mc = √ (MIPC2+MOPC2) = 18.267 kN·m

A = ( F S × P c P y c ) 2 + ( F S × M c M p c ) 2 = ( 1.6 × 49.99 17 , 079.5 ) 2 + ( 1.6 × 18.267 1 , 740.19 ) 2 = 0.0176
Q f _ a x = 1 + C 1 a x ( F S × P c P y c ) − C 2 a x ( F S × M c M p c ) − C 3 a x A 2
Q f _ a x = 1 + 0.2 ( 1.6 × 49.99 15 , 079.5 ) − 0.2 ( 1.6 × 18.267 1 , 740.19 ) − 0.3 ( 0.0176 ) 2 = 0.9989
Q f_ipb = 1 + C1 ipb ( F S × P c P y c ) − C2 ipb ( F S × M c M p c ) − C3 ipb A 2
Q f_ipb = 1 + 0.2 ( 1.6 × 49.99 15 , 079.5 ) − 0 − 0.4 ( 0.0176 ) 2 = 1.001
Q f_opb = 1 + C1 opb ( F S × P c P y c ) − C2 opb ( F S × M c M p c ) − C3 opb A 2
Q f_ipb = 1 + 0.2 ( 1.6 × 49.99 15 , 079.5 ) − 0 − 0.4 ( 0.0176 ) 2 = 1.001

Axial Capacity

P a = Q u Q f F y c T 2 F S ⁢ sin ⁡ θ = 57.586 ( 0.9977 ) 500 ( 20 ) 2 1.6 sin ⁡ ( 63.4 ° ) 10 3 = 8 , 041.5  kN

Moment Capacity

In-plane bending:

M a = Q u Q f F y c T 2 d F S ⁢ sin ⁡ θ = 10.52 ( 1.001 ) 500 ( 20 ) 2 400 1.6 sin ⁡ ( 63.4 ° ) 10 6 = 588.85  kN⋅m

Out-of-plane bending:

M a = Q u Q f F y c T 2 d F S ⁢ sin ⁡ θ = 6.419 ( 1.001 ) 500 ( 20 ) 2 400 1.6 sin ⁡ ( 63.4 ° ) 10 6 = 359.31  kN⋅m

Brace Loads

As axial loads for through and overlapping brace are of different sign, resultant axial load will be that of through brace. (Page 65, API code).

Axial load = 72.511 kN (C)

According to Page 65 of the API code, the resultant in-plane and out-plane bending moments will be algebraic summation of moments of the through brace and overlapping brace.

In-plane moment = 0.727 + (-0.89) = -0.163 kN·m

Out-of-plane moment = 6.254 + 6.254 = 12.508 kN·m

Interaction Ratio

P/Pa + (M/Ma)ip2+ (M/Ma)op = (72.511 / 8,014.5) + (0.163 / 588.85)2 + (12.508 / 359.31) = 0.044

Results

Table 2. Comparison of results
Result Type Reference STAAD.Pro Difference Comments
Qu_ax 57.586 57.617 negligible
Qu_ipb 10.52 10.523 negligible
Qu_opb 6.419 6.419 none  
Qf_ax 0.9989 0.999 negligible
Qf_ipb 1.001 1.001 none
Qf_opb 1.001 1.001 none  
Axial capacity (kN) 8,041.5 8,029.9 negligible
In-plane Bending Capacity (kN·m) 588.86 588.1 negligible
Out-of-plane Bending Capacity (kN·m) 359.31 358.74 negligible  
Interaction ratio 0.044 0.044 none  

STAAD.Pro Input File

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\US\API\API Overlapping KJoint - Comp and Bend.std is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 12-Mar-19
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 0 10 0; 3 5 10 0; 4 10 10 0; 5 10 0 0; 6 5 0 0;
MEMBER INCIDENCES
1 1 2; 2 2 3; 3 3 4; 4 4 5; 5 3 5; 6 3 1; 7 3 6;
DEFINE MATERIAL START
ISOTROPIC STEEL
E 2.05e+008
POISSON 0.3
DENSITY 76.8195
ALPHA 1.2e-005
DAMP 0.03
TYPE STEEL
STRENGTH FY 253200 FU 407800 RY 1.5 RT 1.2
END DEFINE MATERIAL
MEMBER PROPERTY AMERICAN
1 4 7 TABLE ST W6X12
5 6 TABLE ST PIPE OD 0.4 ID 0.36
2 3 TABLE ST PIPE OD 0.5 ID 0.46
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 5 FIXED
6 PINNED
LOAD 1 LOADTYPE Dead  TITLE LOAD CASE 1
JOINT LOAD
2 FX -50
MEMBER LOAD
2 3 UNI GY -5
2 3 CMOM Y 5
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PARAMETER 1
CODE API
*FSJ 2.6 ALL
FYLD 500000 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH

STAAD.Pro Output

Note: The generated .PUN file is required to generated this output for K or X joints. Otherwise, the program assumes all joints are Y joints. Refer to D1.J.8.2 Joint File Format for details. 
======================================================================          
NODE NO :      3   CHORD NO:     2   BRACE NO:     6                            
======================================================================          
DESIGN DATA : (Units    : N , mm  )                                             
  Chord Memb :      D =  500.00  T = 19.99                                      
  Brace Memb :      d =  400.00  t = 19.99                                      
  Angle (THETA) =  63.4 deg  GAP   =  -76.20   Fyc = 499.9                      
  BETA          =  0.80      GAMMA =   12.51   TAU =  1.00                      
      STAAD SPACE                                              -- PAGE NO.    7
      ***WARNING - JNT      3: OVERLAP                                          
      OVERLAP PROVIDED =  76.20 IS LESS THAN THE NOMINAL VALUE OF 0.25*BETA*D.  
JOINT CLASS :          KO                                                       
  Overlap Brace :       5   Angle =   63.43 deg.   Diam. =  400.00              
  FACTORS :                                                                     
  --------------------------------------------------------------------          
  Joint   Load   Strength     Chord Load    C1        C2        C3              
  Class   Cond   factor (Qu)  factor (Qf)                                       
  --------------------------------------------------------------------          
    K     AX      57.617        0.999       0.200    0.200    0.300             
    K     IPB     10.523        1.001       0.200    0.000    0.400             
    K     OPB      6.419        1.001       0.200    0.000    0.400             
----------------------------------------------------------------------          
CAPACITY CHECKS        BRACE LOAD    LC   CAPACITY   RATIO   STATUS             
   (Cl. 4.3)             (KN, m )         (KN, m )                              
----------------------------------------------------------------------          
AXIAL        :            -72.51      1     8029.90    0.009   PASS             
IP BENDING   :             -0.16      1      588.09    0.000   PASS             
OP BENDING   :             12.51      1      358.74    0.035   PASS             
INTERACTION  :                        1                0.044   PASS             
----------------------------------------------------------------------          
CRITICAL     :                        1                0.044   PASS             
----------------------------------------------------------------------
Parent topic: V. API
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