V.API Overlapping K Joint - Tens and Bend

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

References

Recommended Practice for Planning, Design and Constructing Fixed Offshore Platforms-Working Stress Design. American Petroleum Institute. 21st Edition (December 2000).

Details

Design of node 4 (between members 2 and 5) in the following model.

Chord member (2): D = 20 in, T = 1 in (A_c = 59.69 in², Z_c = 361.3 in³)

Brace member (5): d = 18 in, t = 1 in (A_b = 53.41², Z_b = 289.3³)

Brace member 10 overlaps with member 5.

Table 1. Loads
Member	Fx (kN)	MIP (kN·m)	MOP (kN·m)
Chord	499.21 (C)	268.18	0
Brace	322.34 (T)	101.72	0

Validation

Design Parameters

β = d/D = 0.9

So, 0.2 < β < 1.0 (OK)

γ = D/2T = 10

So, 10 < γ < 50 (OK)

θ = tan^-1(10/5) = 63.4°

So, 30° < θ < 90° (OK)

F_y = 50 ksi (OK)

g = -4 mm

g/D = -4 / 20 = -0.2

Determination of Q_u

First, evaluate Q_g:

For g/D < -0.05: Q_g = 0.13 + 0.65×ϕ×γ^0.5 = 0.13 + 0.65×(1)×(10)^0.5 = 2.185

Q_{u_ax} = (16 + 1.2×γ)×β^1.2×Q_g = [16 + 1.2(10)](0.9)^1.2(2.185) = 53.914

Q_{u_ipb} = (5 + 0.7×γ)×β^1.2 = [5 + 0.7(10)](0.9)^1.2 = 10.575

Determination of Q_f

From Table 4.3.2:

C1_ax = 0.2, C2_axi = 0.2, C3_ax = 0.3

C1_b = 0.2, C2_b = 0, C3_b = 0.4

P_yc = A_c×F_yc = 2,984.5 kips

M_pc = Z_c×F_yc = 18,067 in·k

P_c = -499.21 kips

M_c = √ (MIPC²+MOPC²) = 268.18 in·k

A = \sqrt{{(\frac{F S \times P_{c}}{P_{y c}})}^{2} + {(\frac{F S \times M_{c}}{M_{p c}})}^{2}} = \sqrt{{(\frac{2 \times 499.21}{2, 984.5})}^{2} + {(\frac{2 \times 268.18}{18, 067})}^{2}} = 0.3359

Q_{f_a x} = 1 + {C 1}_{a x} (\frac{F S \times P_{c}}{P_{y c}}) - {C 2}_{a x} (\frac{F S \times M_{c}}{M_{p c}}) - {C 3}_{a x} A^{2}

Q_{f_a x} = 1 + 0.2 (\frac{2 \times 499.21}{2, 984.5}) - 0.2 (\frac{2 \times 268.18}{18, 067}) - 0.3 {(0.3359)}^{2} = 0.893

Q_{f_b e n d} = 1 + {C 1}_{b} (\frac{F S \times P_{c}}{P_{y c}}) - {C 2}_{b} (\frac{F S \times M_{c}}{M_{p c}}) - {C 3}_{b} A^{2}

Q_{f_a x} = 1 + 0.2 (\frac{2 \times 499.21}{2, 984.5}) - 0 - 0.4 {(0.3359)}^{2} = 0.888

Axial Capacity

P_{a} = Q_{u} Q_{f} \frac{F_{y c} T^{2}}{F S \sin θ} = 53.914 (0.893) \frac{50 {(1)}^{2}}{2 \sin (63.4 °) 10^{3}} = 1, 346.1 kips

Moment Capacity

M_{a} = Q_{u} Q_{f} \frac{F_{y c} T^{2} d}{F S \sin θ} = 10.52 (1.001) \frac{50 {(1)}^{2} 18}{2 \sin (63.4 °) 10^{6}} = 4, 726 in⋅k

Out-of-plane bending is zero so moment capacity is not calculated.

Brace Loads

As axial loads for through and overlapping braces are of different signs, the resultant axial load will be the sum of both brace loads (reference p.65).

The gap distance, g = 4 in (segment QR).

Angle PQR = θ = 63.4°

The loaded segment length (green line PQ) = g/cos(θ) = 4/cos(63.4°) = 8.93 in

To determine the overlapping portion of the cross section of member 10 (i.e., the green portion of Section A-A), node that line segments OP and OB are equal to the radius of the pipe (use the nominal outer radius), r = 6 in.

cos(α) = (r - PQ)/r = (6 - 8.93) / 6 = -0.489

α = 119.3° = 2.082 rad

The arc length (green portion of Section A-A) = 2×α×r = 2(2.082)(6) = 25 in

The proportion of this arc to the total circumference = 25 / (2π×r) = 25 / (2π × 6) = 0.663

So the total axial load is the axial load of the through brace plus this ratio times the axial load of the overlapping brace (axial force values taken from STAAD.Pro post-processing workflow):

322.39 kip + 0.663 × 181.5 kip = 442.7 kip

According to p.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. Therefore, the total in-plane moment:

101.72 in·k + 12.61 in·k = 114.33 in·k

Interaction Ratio

P/P_a + (M/M_a)² = (442.7 / 1,346.1) + (114.33 / 4,726)² = 0.329

Results

Table 2. Comparison of results
Result Type	Reference	STAAD.Pro	Difference
Q_{u_ax}	53.91	53.926	negligible
Q_{u_ipb}	10.575	10.575	none
Q_{f_ax}	0.893	0.893	none
Q_{f_ipb}	0.888	0.888	none
Axial capacity (kip)	1,346.1	1,346.18	negligible
In-plane Bending Capacity (in·k)	4,726	4,724.22	negligible
Brace Load, Axial (kip)	442.7	442.72	negligible
Brace Load, In-plane Bending (in·k)	114.33	114.33	none
Interaction ratio	0.329	0.329	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 - Tens 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 INCHES KIP
JOINT COORDINATES
1 0 0 0; 2 0 787.402 0; 3 787.402 787.402 0; 4 393.701 787.402 0;
5 787.402 0 0; 6 393.701 0 0; 7 708.661 0 0; 8 551.181 0 0; 9 78.7402 0 0;
10 236.22 0 0;
MEMBER INCIDENCES
1 1 2; 2 2 4; 3 4 3; 4 5 3; 5 1 4; 6 4 5; 7 4 6; 8 4 7; 9 4 8; 10 9 4; 11 10 4;
DEFINE MATERIAL START
ISOTROPIC STEEL
E 29732.7
POISSON 0.3
DENSITY 0.000283
ALPHA 1.2e-005
DAMP 0.03
TYPE STEEL
STRENGTH FY 36.7236 FU 59.1464 RY 1.5 RT 1.2
END DEFINE MATERIAL
MEMBER PROPERTY AMERICAN
1 TO 4 7 TABLE ST PIPE OD 20 ID 18
5 6 8 9 11 TABLE ST PIPE OD 18 ID 16
10 TABLE ST PIPE OD 12 ID 10
CONSTANTS
MATERIAL STEEL ALL
*START USER TABLE
*TABLE 1
*UNIT INCHES KIP
*PRISMATIC
*UPRISMATIC-PIPE(1)
*6.09218 280.735 280.735 561.471 3.04623 3.04623 20 20
*TABLE 2
*UNIT INCHES KIP
*PRISMATIC
*UPRISMATIC-PIPE(2)
*5.48931 205.369 205.369 410.738 2.74481 2.74481 18 18
*END
*DEFINE MATERIAL START
*ISOTROPIC ASTM_STEEL
*E 29000
*POISSON 0.3
*DENSITY 8.80613e-006
*ALPHA 2.16e-005
*DAMP 0.03
*G 11153.8
*END DEFINE MATERIAL
*MEMBER PROPERTY AMERICAN
*1 TO 4 7 UPTABLE 1 UPRISMATIC-PIPE(1)
*5 6 8 TO 11 UPTABLE 2 UPRISMATIC-PIPE(2)
*CONSTANTS
*MATERIAL ASTM_STEEL ALL
SUPPORTS
1 5 FIXED
6 TO 10 PINNED
DEFINE REFERENCE LOADS
LOAD R1 LOADTYPE Dead  TITLE LOAD GROUP 1
JOINT LOAD
2 FX 500
END DEFINE REFERENCE LOADS
LOAD 1 LOADTYPE Dead  TITLE LOAD CASE 1
REFERENCE LOAD
R1 1.0 
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PARAMETER 1
CODE API
FYLD 50 ALL
FSJ 2 MEMB 2 3 5 6 8 10
TRACK 2 MEMB 2 3 5 6 8 10
CHECK CODE MEMB 2 3 5 6 8 10
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.

          STAAD.Pro - API JOINT CHECKS TO 21st edition.
          ---------------------------------------------------
                     (Includes Supplements 1-3)
          ---------------------------------------------------
======================================================================          
NODE NO :      4   CHORD NO:     2   BRACE NO:     5                            
======================================================================          
DESIGN DATA : (Units    : kip,inch)                                             
  Chord Memb :      D =   20.00  T =  1.00                                      
  Brace Memb :      d =   18.00  t =  1.00                                      
  Angle (THETA) =  63.4 deg  GAP   =   -4.00   Fyc =  50.0                      
  BETA          =  0.90      GAMMA =   10.00   TAU =  1.00                      
      ***WARNING - JNT      4: OVERLAP                                          
      OVERLAP PROVIDED =   4.00 IS LESS THAN THE NOMINAL VALUE OF 0.25*BETA*D.  
JOINT CLASS :          KO                                                       
  Overlap Brace :      10   Angle =   68.20 deg.   Diam. =   12.00              
  FACTORS :                                                                     
  --------------------------------------------------------------------          
  Joint   Load   Strength     Chord Load    C1        C2        C3              
  Class   Cond   factor (Qu)  factor (Qf)                                       
  --------------------------------------------------------------------          
    K     AX      53.926        0.893       0.200    0.200    0.300             
    K     IPB     10.575        0.888       0.200    0.000    0.400             
----------------------------------------------------------------------          
CAPACITY CHECKS        BRACE LOAD    LC   CAPACITY   RATIO   STATUS             
   (Cl. 4.3)             (kip,in)         (kip,in)                              
----------------------------------------------------------------------          
AXIAL        :            442.72      1     1346.18    0.329   PASS             
IP BENDING   :            114.33      1     4724.22    0.024   PASS             
INTERACTION  :                        1                0.329   PASS             
----------------------------------------------------------------------          
CRITICAL     :                        1                0.329   PASS             
----------------------------------------------------------------------