Joint Stiffness

Introduction

To determine the performance of a bolted joint it is necessary to calculate the joint stiffness. That is the deflection of the joint under a bolt loading condition When geometry of the bolted joint is an annulus with an OD less than 2,5 x the bolt diameter the joint stiffness can be conveniently calculated using k = EA/l . When the joint includes a non-metallic gasket or material with a low modulus of elasticity then this item may have such a low stiffness value that the stiffness of the metal parts of the connection will have very low impact on the overall stiffness.

It is very difficult to obtain theoretical values of joint stiffness because a number of factors affect the result including.

The effect of the angle of the screw thread
Screw hole size
Surface finish of contacting faces
Flatness of contacting faces
Use of Washer
Selected size of contacting surface (d and D)
Condition of surfaces between clamped plates

Any of these factors can have a significant effect on the resulting joint stiffness

The notes below identify methods which approximate the loading regime which has been found to occur by various studies using different methods.

Overall stiffness evaluation for multi-component joints

A bolted joint can include a number of separate parts. The relationship between the individual stiffness values of the parts and the total stiffness of the clamped joint is as shown below..

The sketch below shows a bolted surface. It has been shown using using ultrasonics and FEA that the pressure in a bolted surface is greatest under the bolt head and reduces as the distance from the bolt interface increases. A method of be approximating the force distribution is based on use of the cone geometry as shown. Various cone half-apex angles are used but for systems using a washer-face annulus and with hardened steel, cast iron or aluminium an angle of 25^o to 35^o considered reasonable. These notes relate to half apex angle of 30^o

Referring to the contraction of a small element dx of cone subject to a load P as shown.

The area is determined as follows

Substituting this into the equation for contraction of an element..

To obtain the total deflection the equation above is integrated between 0 and t.

Using integration manipulation techniques this results in

The spring rate of the truncated cone is therefore ..

The bolt hole diameter is d, the diameter of the applied pressure e.g.the washer diameter, is and the plate thickness is t.

For a half apex angle of 30^o which provides a reasonable value for many engineering metals, the equation simplifies to..

Important Note: Widely different stiffness values result from different studies. It has also been proved that different loading conditions and surface conditions also affect the resulting stiffness.

The variation in half apex angle α can be notice from the different studies completed. Differences result from various reason including D/d ratios differences and surface finish difference

Originator	Technique	Surface Type	α
Link 3 below	Ultrasonics	Ground	41^o
Link 3 below	Ultrasonics	Turned /Ground	68^o
Shigley and Mischke	Analysis	Any	30^o
Ito	Ultrasonics	Ground	70^o
Gould and Mikic	Analysis	Smooth	38^o

Table A

Showing stiffness values for steel plate E = 207 kN/mm² and a half apex angle α of 30 ^o calculated using equation X above..

d= bolt dia and D = 1,5 d = approximately as would result from a typical bolt or cap screw .

-		Plate Thickness (mm)
		6	12	20	40	50	60	80	100
d	D	k = Stiffness (MN/mm ) based on...E = 207 kN/mm² and α = 30^o
6	9	2.757	2.102	1.83	1.619	1.576	1.547	1.511	1.489
8	12	4.245	3.098	2.622	2.254	2.178	2.127	2.063	2.024
10	15	6.009	4.236	3.503	2.934	2.817	2.739	2.639	2.579
12	18	8.047	5.514	4.47	3.66	3.493	3.381	3.239	3.153
16	24	12.945	8.49	6.661	5.245	4.953	4.757	4.508	4.356
20	30	18.932	12.017	9.191	7.006	6.556	6.253	5.868	5.634
24	36	26.008	16.095	12.055	8.94	8.3	7.867	7.32	6.986
30	45	38.658	23.236	16.974	12.163	11.175	10.509	9.664	9.149
36	54	53.75	31.605	22.638	15.768	14.36	13.41	12.207	11.474

Table B

Showing stiffness values for steel plate E = 207 kN/mm² and a half apex angle α of 60 ^o calculated using Equation X above. This table is provided to identify the variation is stiffness using two different, but still relevant, values of α

d= bolt dia and D = 1,5 d = approx as would result from a typical bolt or cap screw .

-		Plate Thickness (mm)
		6	12	20	40	50	60	80	100
d	D	k = Stiffness (MN/mm ) based on...E = 207 kN/mm² and α = 60^o
6	9	5.627	4.929	4.642	4.423	4.378	4.349	4.312	4.289
8	12	8.108	6.886	6.382	5.995	5.917	5.864	5.798	5.759
10	15	10.88	8.996	8.216	7.616	7.494	7.412	7.31	7.248
12	18	13.939	11.255	10.142	9.284	9.11	8.993	8.845	8.757
16	24	20.909	16.216	14.27	12.764	12.456	12.25	11.99	11.833
20	30	29	21.76	18.758	16.431	15.955	15.635	15.231	14.988
24	36	38.203	27.878	23.602	20.284	19.604	19.146	18.569	18.219
30	45	54.079	38.121	31.526	26.407	25.355	24.647	23.753	23.211
36	54	72.426	49.629	40.231	32.938	31.437	30.426	29.148	28.373

Bolted joint comprising two equal plates

The force distribution on a typical bolted joint comprising two plates is shown below:

It is necessary to obtain a value for the stiffness for each part of the joint. In the case above for each of the two plates.

If the plates are of the same material and similar thickness then the overall stiffness of the joint is..

FEA work has been completed (Wileman ,Choudury & Green)for similar 2 part bolted joint with typical washers (D_w = 1,5d) and joint members of the same material and an approximate curve fit curve for the resulting relationship has been derived..

Values for A and B are provided in the following table...

	Poisson Ratio	Youngs Modulus	A	B
Material		GPa
Steel	0,291	207	0,78715	0,62873
Aluminium	0,334	71	0,79670	0,63816
Copper	0,326	119	0,79568	0,63553
Gray Cast Iron	0,211	100	0,77871	0,61616
General Case			0,78952	0,62914

Example comparing stiffness value using two methods.

Consider two plates 20mm thick clamped with a 10mm bolt using the table with a half apex angle α of 30 ^o

k for each plate from Table A above = 3 503 kN/mm .... 1/ k _t = 1/k + 1/k .
Therefore k _t = k/2 = 1 7513 kN/mm.

Using the formula above with A = 0,78715,B = 0,62873,d = 10, l=40 and E = 207000
k _t = E.d.A exp (Bd/l) = 1 906 kN/mm

Sites Providing Information On Bolt Loading

Engineers edge bolt preload calculator ..A very easy to use calculator for bolted flanges -very useful and convenient
DAN_notes Loads in an elastic bolted assembly..Very detailed notes on bolt loading-similar to notes above.
ExcelCalcs.com calculation Detailed Bolted Joint Calculations.xls..An exellent exel based worksheet showing all equations used