Theoretical modeling of nonlinear rotational stiffness for missile radial countersunk screw lap joints

doi:10.1007/s10483-026-3407-9

Abstract

Abstract:

Radial countersunk screw lap joints are widely employed to connect adjacent cabin sections in small- and medium-diameter missiles. However, the analysis and design of joint stiffness pose challenges because of geometric discontinuities, clearance, and friction nonlinearities. In this paper, a theoretical model for rapid and reliable prediction of nonlinear joint stiffness is developed. The joint is first discretized into multiple subjoints, each of which is defined to carry only a tensile or compressive load under external loading. The evolution of contact states is incorporated to simulate nonlinear tension and compression stiffness. By combining Bernoulli’s hypothesis with the ellipsoidal deformation theory, the joint rotational stiffness is derived. Finally, the effectiveness of the proposed stiffness prediction method is validated via experiments and detailed simulations. Furthermore, an orthogonal experimental design is used to analyze the importance of critical design parameters. The results indicate that the proposed theoretical model provides satisfactory accuracy. The specification and number of screws are the primary factors influencing the joint rotational stiffness, whereas the lap length and cabin thickness exert a secondary effect. This study presents an explicit theoretical mapping between the structural design parameters and joint nonlinear stiffness, thus facilitating improved design and optimization of jointed structures.

Key words: missile, radial countersunk screw lap joint, theoretical model, nonlinear stiffness, design optimization

2010 MSC Number:

O343.1

Shuo ZHANG, Ning GUO, Chao XU. Theoretical modeling of nonlinear rotational stiffness for missile radial countersunk screw lap joints. Applied Mathematics and Mechanics (English Edition), 2026, 47(7): 1625-1646.

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URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-026-3407-9

https://www.amm.shu.edu.cn/EN/Y2026/V47/I7/1625

Figures/Tables 30

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Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

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Fig. 12

Fig. 13

Fig. 14

Table 1

Structural design parameters of the specimens"

Assembly diameter D/mm	Lap length L₁, L₂/mm	Screw position $P h$ /mm	Cabin wall length H/mm	Cabin wall thickness $t sh$ /mm	Assembly clearance δ/mm
278	40, 30	15	68	4	0.1

Table 1

Fig. 15

Fig. 16

Table 2

Structural design parameters of the radial countersunk screw lap joints"

Group	Assembly diameter D/mm	Number of screws	Screw specification	Lap length L₁, L₂/mm	Shell length $L sh$ /mm	Thickness of the shell $t sh$ /mm
1	235	8	M6	30, 20	20	3
2	278	12	M6	40, 30	25	4
3	300	10	M6	30, 20	25	4
4	325	12	M5	40, 30	25	4
5	350	12	M6	40, 30	25	4

Table 2

Fig. 17

Table 3

Experimental results for nonlinear tension and compression stiffnesses"

Type	Load range/kN	Stage	Stiffness/ $(k N ⋅ m m − 1)$
Type	Load range/kN	Stage	Test 1	Test 2	Test 3	Average
Tension	0–1.5	1	54.29	55.81	55.58	55.23
Tension	4–6	3	15.23	15.17	14.92	15.11
Compression	$− 1.5 – 0$	1	57.16	56.32	55.38	56.29
Compression	$− 6 – − 4$	3	126.50	139.70	162.30	142.80

Table 3

Table 4

Comparison of the numerical simulation and theoretical calculation results (Stage 1)"

Group	Tension			Compression
Group	FEA/ $(k N ⋅ m m − 1)$	Theoretical result/ $(k N ⋅ m m − 1)$	Error/%	FEA/ $(k N ⋅ m m − 1)$	Theoretical result/ $(k N ⋅ m m − 1)$	Error/%
1	54.44	51.51	$− 5.38$	54.48	51.51	$− 5.45$
2	59.60	58.40	$− 2.01$	59.81	58.40	$− 2.36$
3	44.06	39.10	$− 11.26$	44.13	39.10	$− 11.40$
4	57.48	56.87	$− 1.06$	57.53	56.87	$− 1.15$
5	57.22	55.29	$− 3.37$	57.23	55.29	$− 3.39$

Table 4

Table 5

Comparison of the numerical simulation and theoretical calculation results (Stage 3)"

Group	Tension			Compression
Group	FEA/ $(k N ⋅ m m − 1)$	Theoretical result/ $(k N ⋅ m m − 1)$	Error/%	FEA/ $(k N ⋅ m m − 1)$	Theoretical result/ $(k N ⋅ m m − 1)$	Error/%
1	17.07	15.52	$− 9.08$	288.10	276.85	$− 3.90$
2	14.16	16.62	17.37	214.40	226.43	5.61
3	16.69	17.09	2.40	352.93	329.87	$− 6.53$
4	15.60	13.37	$− 14.29$	270.32	264.71	$− 2.08$
5	19.69	16.94	$− 13.97$	288.97	285.07	$− 1.35$

Table 5

Fig. 18

Fig. 19

Table 6

Comparison of the numerical simulation and theoretical calculation results"

Group	Stage 1			Stage 3
Group	FEA/ $(k N ⋅ m m − 1)$	Theoretical result/ $(k N ⋅ m m − 1)$	Error/%	FEA/ $(k N ⋅ m m − 1)$	Theoretical result/ $(k N ⋅ m m − 1)$	Error/%
1	3.27	3.02	$− 7.65$	3.72	3.43	$− 7.80$
2	8.01	7.57	$− 5.49$	9.46	10.68	12.90
3	7.31	6.93	$− 5.20$	8.64	7.18	$− 16.90$
4	10.24	8.37	$− 18.26$	12.49	10.80	$− 13.53$
5	11.88	12.34	3.87	14.39	16.96	17.86

Table 6

Table 7

Design of orthogonal experiments and theoretical results of the joint rotational stiffness"

Number	Factor A: SN	Factor B: SP	Factor C: LP L₂/mm	Factor D: CT/mm	Rotational stiffness/ $(M N ⋅ m ⋅ r a d − 1)$
Number	Factor A: SN	Factor B: SP	Factor C: LP L₂/mm	Factor D: CT/mm	Stage 1	Stage 3
1	A₁ (6)	B₁ (M4)	C₁ (20)	D₁ (3)	2.67	2.96
2	A₁	B₂ (M5)	C₂ (25)	D₂ (3.5)	3.35	4.59
3	A₁	B₃ (M6)	C₃ (30)	D₃ (4)	4.33	6.46
4	A₁	B₄ (M8)	C₄ (35)	D₄ (4.5)	6.16	9.44
5	A₂ (8)	B₁	C₂	D₃	1.99	3.11
6	A₂	B₂	C₁	D₄	2.72	3.86
7	A₂	B₃	C₄	D₁	5.61	8.89
8	A₂	B₄	C₃	D₂	6.22	9.17
9	A₃ (10)	B₁	C₃	D₄	1.84	3.58
10	A₃	B₂	C₄	D₃	4.49	8.34
11	A₃	B₃	C₁	D₂	6.65	7.26
12	A₃	B₄	C₂	D₁	5.73	8.36
13	A₄ (12)	B₁	C₄	D₂	4.03	7.12
14	A₄	B₂	C₃	D₁	6.44	9.37
15	A₄	B₃	C₂	D₄	6.57	9.08
16	A₄	B₄	C₁	D₃	9.16	10.05

Table 7

Table 8

Range analysis and optimal parameter combination for the joint rotational stiffness (Stage 1)"

Response and statistical metrics	Level	Factor A	Factor B	Factor C	Factor D
$K avg / (M N ⋅ m ⋅ r a d − 1)$	1	4.13	2.63	5.05	5.11
	2	4.14	4.25	4.41	5.06
	3	4.68	5.79	4.71	4.74
	4	6.30	6.57	5.07	4.32
Range/ $(M N ⋅ m ⋅ r a d − 1)$		2.17	3.94	0.66	0.79
Optimum		Level 4	Level 4	Level 4	Level 1

Table 8

Fig. 20

Table 9

Range analysis and optimal parameter combination for the joint rotational stiffness (Stage 3)"

Response and statistical metrics	Level	Factor A	Factor B	Factor C	Factor D
$K avg / (M N ⋅ m ⋅ r a d − 1)$	1	5.86	4.19	6.03	7.40
	2	6.26	6.54	6.29	7.04
	3	6.89	7.92	7.15	6.99
	4	8.91	9.26	8.45	6.49
Range/ $(M N ⋅ m ⋅ r a d − 1)$		3.04	5.06	2.42	0.90
Optimum		Level 4	Level 4	Level 4	Level 1

Table 9

Fig. 21

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