# Mechanics of Screw Joints Solved as Beams Placed in a Tangential Elastic Foundation
## 2 The Tangential (Tensile) Elastic Foundation  
In the case of a screw connection, it is assumed that the screw can be replaced by a beam and also that the screw is placed in the tangential elastic foundation over the length and circumference of the contact between the thread and the connected material. The screw (beam) is subjected to tension/pressure in this task.  
The proposed new model of elastic foundation assumes that the continuous tangent reaction in the foundation \(\mathrm{q}_{N}=\mathrm{q}_{N}(x,\,u,\,\dots)\) [Nm\({}^{-1}\)] is directly proportional to the axial displacement of the center line \(u=u(x)\) [m]. From equilibrium on the infinitesimal length section \(dx\) [m], the equation of equilibrium in the axial direction can be derived (see Figure 3).  
\[N+\mathrm{q}_{N}dx-(N+dN)=0\text{, i.e.,}\frac{dN}{dx} =\mathrm{q}_{N} \tag{1}\]  
Figure 2: Elastic foundation due to the loads (points 1 and 2 are moved to new positions 1\({}^{\bullet}\) and 2\({}^{\bullet}\)). (a) Bending according to [PERSON] (distributed reaction force in foundation \(q_{R}=kv\) in linear cases); (b) Tension/pressure—new tangential model according to [PERSON] (distributed reaction force \(q_{N}=k_{N}u\) in linear cases, where \(k_{N}\) [Pa] is elastic foundation stiffness, \(u\) [m] is axial displacement/elongation and \(F_{i}\) [N] is a tensile/compression force).  
Assuming small deformations in the screw (beam) and possible small/large deformations in the fasteners (e.g., thread tear, etc.), the relative deformation \(\epsilon\)[1] in tension/compression of the beam section is calculated as  
\[\epsilon= \frac{du}{dx}=\frac{N}{EA},\cdots,\text{ i.e.,}\cdots\frac{d^{2}u}{dx^{2}}= \frac{1}{EA}\frac{dN}{dx}, \tag{2}\]  
where \(A\) [m\({}^{2}\)] is the cross-section area of the beam (screw) and \(E\) [Pa] is the elastic modulus of the beam (screw). The relationship (2) is linear, however, the foundation and its deformation can generally be non-linear, which is a significant factor of the tangential elastic foundation.  
By substituting Equations (1) to (2), the relationship for the second order homogeneous differential equation is obtained  
\[\frac{d^{2}u}{dx^{2}}=\frac{\text{q}_{N}}{\text{EA}}, \tag{3}\]  
which is generally non-linear.  
If \(\text{q}_{N}=0\) (i.e., the beam is not placed in the tangential foundation), it is a common simple tension/pressure task.  
It is clear from the measurements that the dependence \(\text{F}=f(u_{\text{B}})\) and the reaction force of the tangential foundation \(\text{q}_{N}\) are non-linear. However, despite this fact, the differential Equations (1) to (3) can be solved in an analytical way, which is advantageous and fast, since the use of numerical methods, such as the finite element method, is no longer necessary. The solution procedure is based on the best possible and most practical approximation of the experiment and is as follows.  
It is assumed that \(\text{q}_{N}=f(u)\), or \(N=f(u)\), as indicated by the results of the experiment. Appropriate adjustment is introduced:  
\[\frac{dN}{dx}=\frac{du}{du}\frac{dN}{dx}=\frac{du}{dx}\frac{dN}{du}=\frac{N}{EA }\frac{dN}{du}, \tag{4}\]  
where Equation (2) is applied.  
Equation (1) can be adjusted to the following form:  
\[N\frac{dN}{du}= EA\text{ q}_{N}, \tag{5}\]  
which is a separable first order differential equation. This equation is generally non-linear but can be directly solved by integration, as will be shown below. Thus, the advantage is a reduction in the order of the differential equation (i.e., reduction in the degree of derivation), which is apparent from the comparison of Equations (5) and (3).  
Figure 3: Section of the beam (screw) resting on a tangential elastic foundation, where \(N\) [N] is a normal (axial, internal) force.