V.API X Joint

Verify the design of a simple X joint with axial load and in-plane bending.

Details

Design of node 3 between members 2 and 6 in the following model.

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

Brace member (6): d = 475 mm, t = 15 mm (A_b = 21,677 mm², Z_b = 2,416,700 mm³)

Table 1. Loads
Member	Fx (kN)	MIP (kN·m)	MOP (kN·m)
Chord	49.99 (T)	17.34	4.97
Brace	66.082 (C)	0.921	6.255

Validation

Design Parameters

β = d/D = 0.95

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)

F_y = 500 MPa (OK)

Determination of Q_u

Q_β = 0.3/[β(1 - 0.833×β)] = 1.513

Q_{u_ax} = [2.8 + (12 + 0.1×γ)×β]×Q_β = [2.8 + (12 + 0.1×12.5)×0.95](1.513) = 23.289

Q_{u_ipb} = (5 + 0.7×γ)×β^1.2 = [5 + 0.7(12.5)](0.95)^1.2 = 12.929

Q_{u_opb} = 2.5 + (4.5 + 0.2×γ)×β^2.6 = 2.5 + [4.5 + 0.2(12.5)](0.95)^2.6 = 8.626

Determination of Q_f

From Table 4.3.2:

C1_ax = 0.2, C2_axi = 0, C3_ax = 0.35

For C1, C2, and C3, interpolate between values for β ≤ 0.9 and β = 1.0.

C1_b = 0.2, C2_b = 0, C3_b = 0.4

P_yc = A_c×F_yc = 15,079.5 kN

M_pc = Z_c×F_yc = 1,740.19 kN·m

P_c = 49.99 kN

M_c = √ (MIPC²+MOPC²) = 18.04 kN·m

A = \sqrt{{(\frac{F S \times P_{c}}{P_{y c}})}^{2} + {(\frac{F S \times M_{c}}{M_{p c}})}^{2}} = \sqrt{{(\frac{1.6 \times 49.99}{17, 079.5})}^{2} + {(\frac{1.6 \times 18.04}{1, 740.19})}^{2}} = 0.0174

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{1.6 \times 49.99}{15, 079.5}) - 0 - 0.35 {(0.0174)}^{2} = 1.0

Q_{f_ipb} = 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_ipb} = 1 + 0.2 (\frac{1.6 \times 49.99}{15, 079.5}) - 0 - 0.4 {(0.0174)}^{2} = 1.001

Q_{f_opb} = 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_opb} = 1 + 0.2 (\frac{1.6 \times 49.99}{15, 079.5}) - 0 - 0.4 {(0.0174)}^{2} = 1.001

Axial Capacity

P_{a} = Q_{u} Q_{f} \frac{F_{y c} T^{2}}{F S \sin θ} = 23.289 (1.0) \frac{500 {(20)}^{2}}{1.6 \sin (63.4 °) 10^{3}} = 3, 255.7 kN

Moment Capacity

In-plane bending:

M_{a} = Q_{u} Q_{f} \frac{F_{y c} T^{2} d}{F S \sin θ} = 12.929 (1.001) \frac{500 {(20)}^{2} 475}{1.6 \sin (63.4 °) 10^{6}} = 858.53 kN⋅m

Out-of-plane bending:

M_{a} = Q_{u} Q_{f} \frac{F_{y c} T^{2} d}{F S \sin θ} = 8.626 (1.001) \frac{500 {(20)}^{2} 475}{1.6 \sin (63.4 °) 10^{6}} = 572.8 kN⋅m

Interaction Ratio

P/P_a + (M/M_a)_ipb²+ (M/M_a)_opb = (66.082 / 3,255.7) + (0.921 / 858.53)² + (6.255 / 572.8) = 0.031

Results

Table 2. Comparison of results
Result Type	Reference	STAAD.Pro	Difference
Q_{u_ax}	23.829	23.921	negligible
Q_{u_ipb}	12.929	12.934	negligible
Q_{u_opb}	8.626	8.627	negligible
Q_{f_ax}	1.0	1.0	none
Q_{f_ipb}	1.001	1.001	none
Q_{f_opb}	1.001	1.001	none
Axial capacity (kN)	3,255.7	3,250,61	negligible
In-plane Bending Capacity (kN·m)	858.53	858.33	negligible
Out-of-plane Bending Capacity (kN·m)	572.8	572.55	negligible
Interaction ratio	0.031	0.031	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 XJoint.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 TABLE ST W6X12
*5 6 7 TABLE ST HSSP16X0.625
*2 3 TABLE ST HSSP20x0.5
2 3 TABLE ST PIPE OD 0.5 ID 0.46
5 TO 7 TABLE ST PIPE OD 0.475 ID 0.445
*MEMBER PROPERTY EUROPEAN
*2 3 TABLE ST 610x20CHS
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 =  475.01  t = 15.01                                      
  Angle (THETA) =  63.4 deg  GAP   =    0.00   Fyc = 499.9                      
  BETA          =  0.95      GAMMA =   12.51   TAU =  0.75                      
      STAAD SPACE                                              -- PAGE NO.    7
JOINT CLASS :           X                                                       
  FACTORS :                                                                     
  --------------------------------------------------------------------          
  Joint   Load   Strength     Chord Load    C1        C2        C3              
  Class   Cond   factor (Qu)  factor (Qf)                                       
  --------------------------------------------------------------------          
    X     AX      23.291        1.000      -0.000    0.000    0.350             
    X     IPB     12.934        1.001       0.200    0.000    0.400             
    X     OPB      8.627        1.001       0.200    0.000    0.400             
----------------------------------------------------------------------          
CAPACITY CHECKS        BRACE LOAD    LC   CAPACITY   RATIO   STATUS             
   (Cl. 4.3)             (KN, m )         (KN, m )                              
----------------------------------------------------------------------          
AXIAL        :            -66.08      1     3250.61    0.020   PASS             
IP BENDING   :              0.92      1      858.33    0.001   PASS             
OP BENDING   :              6.26      1      572.55    0.011   PASS             
INTERACTION  :                        1                0.031   PASS             
----------------------------------------------------------------------          
CRITICAL     :                        1                0.031   PASS             
----------------------------------------------------------------------