In our post Tensile stresses in preloaded bolts, we showed how a preload on high-strength bolts is shared between the bolt itself and the connecting elements. The stiffness of these two parts is involved in this distribution, being higher the higher the stiffness.

Although the calculation of the bolt stiffness is not complicated because it can be simulated as that of a cylinder of length equal to the thickness of the elements it joins, the stiffness of these connecting elements is not easy, requiring the use of finite element simulations or models extracted from these simulations that approximate this stiffness.

This constant corresponds to a cylindrical element of resistant diameter ( d ), with an effective fastening length L , given by the thickness of connecting elements, under a tensile stress ( \sigma ) produced by a force ( F ) and made of a material having a modulus of elasticity ( E ).

F = \sigma A

\sigma = E  \epsilon = E  \Large \frac{\delta_b}{L}

A = \Large \frac{\pi d^2}{4}

F = E  \Large \frac {\delta_b}{L}\normalsize A = K_b  \delta_b

K_b = \Large \frac{E  A}{L}\normalsize = \Large \frac{E \pi d^2}{4L}

Where:

• F : tensile force
• \sigma : tensile stress
• A : bolt section
• E : module of elasticity
• \epsilon : unit strain
• \delta_b : bolt strain
• L : bolt fastening length
• d : bolt resistant diameter
• K_b: bolt stiffness constant

The set of parts to be joined can be considered as springs of different stiffness arranged in series, therefore the total stiffness constant ( K_c ) can be determined as:

\Large \frac {1}{K_c} \normalsize = \Large \frac {1}{K_1} \normalsize + \Large \frac {1}{K_2} \normalsize +...+ \Large \frac {1}{K_i} \normalsize +...+\Large \frac {1}{K_n} \normalsize

When one of the connecting elements has a stiffness ( K_J ) significantly lower than the others, as in the case of a seal between two metallic elements, the stiffness of the joint will depend mainly on this one, ignoring the others:

\Large \frac{1}{K_c}\normalsize \approx \Large \frac{1}{K_J}

These constants are obtained by experimentation, since the compression of these elements extends progressively from the bolt head to the nut base, so the compression section is not uniform.

Because of this, simple models, that have equivalent elastic stiffness close to those observed experimentally, are used.

Based on ultrasonic techniques for detecting deformation profiles in bolted joints, he proposes the use of the elastic stiffness of a truncated hollow cone of angle \alpha .

When no values of such angle \alpha are available, a value of 26.5651º is usually taken, to obtain a tangent of 0.5.

It is usually used in joints with intermediate sealing rings, placing the bolts at a distance where the cones intersect in the plane of the seal to ensure the compression along the entire length of the joint.

The cross-section ( A_M ) along the thickness of the connecting parts is:

A_M(x) = \Large \frac{\pi}{4}\normalsize \left[ d(x)^2-d_0^2 \right]

Where:

d(x) = \begin{cases} 2x\tan\alpha+d_w & \forall x \in [0, \frac{L}{2}]  \\ -2x\tan\alpha+d_w+2L\tan\alpha & \forall x \in [\frac{L}{2}, L] \end{cases}

Being:

• d_0 : bolt hole diameter
• d_w : distance between bolt head faces

The stiffness ( K_i ) for an element of thickness between distances l_{i-1} and l_i in the interval [0, \frac{L}{2}] , will be:

dK_i = \Large \frac{E A_M(x)}{dx}

\Large \frac{1}{dK_i}\normalsize = \Large \frac{dx}{E A_M(x)}

\Large \frac{1}{K_i}\normalsize = \large \int_{l_{i-1}}^{l_i} \Large \frac{dx}{E \frac{\pi}{4} [d(x)^2-d_0^2]}

\Large \frac{1}{K_i}\normalsize =\Large \frac{4}{E\pi} \large \int_{l_{i-1}}^{l_i} \Large \frac{dx}{(2x\tan\alpha+d_w)^2-d_0^2}

\Large \frac{1}{K_i}\normalsize =\Large \frac{4}{E\pi} \large \int_{l_{i-1}}^{l_i} \Large \frac{dx}{4x^2\tan^2\alpha+4xd_w\tan\alpha-d_0^2}

\Large \frac{1}{K_i}\normalsize =\Large \frac{4}{E\pi} \large \int_{l_{i-1}}^{l_i} \Large \frac{dx}{Ax^2+Bx+C}

Where:

A = 4\tan^2\alpha

B = 4d_w\tan\alpha

C = d_w^2-d_0^2

D = \sqrt {B^2-4AC}

And knowing that, for A \neq 0 and D > 0 :

\large \int \Large \frac{dx}{Ax^2+Bx+C} \normalsize = \Large \frac{1}{D}\normalsize \ln \Large \left[ \frac{2Ax+B-D}{2Ax+B+D}\right] \normalsize + Cte

We have that:

K_i = \Large \frac{E\pi D}{4 \ln \Large \left[ \frac{(2 A l_i + B-D)(2 A l_{i-1}+B+D)}{(2Al_i+B+D)(2Al_{i-1}+B-D)} \right]}

By analogy, the same expression is obtained for the stiffness constant of a joint element in the interval [\frac{L}{2}, L] , where:

A = 4\tan^2\alpha

B = 4(d_w + 2L\tan\alpha)\tan\alpha

C = (d_w + 2L\tan\alpha)^2-d_0^2

D = \sqrt {B^2-4AC}