THREE DIMENSIONAL ANALYSIS OF BOLTED JOINT MEMBER STIFFNESS USING FRUSTA METHOD

Joint member stiffness has a great influence for both static and dynamic loading for a safe mechanical design of many types of bolted-jointed member connections. A three dimensional mathematical model for the jointed member system has been derived. This work provides a simple mathematical technique used to calculate different types a jointed member stiffness of a bolted connections. A finite element analysis is performed for the bolted joint and the model is used to predict stress field for a Grey Cast iron and aluminum alloy. Joint member stiffness is presented against member aspect ratio in dimensionless form. Results show good agreement as compared with those of used techniques of Montosh and Marschke


INTRODUCTION
In this work the bolted joint member stiffness has been studied analytically, several concepts have been used to determine the extra components of stiffness (these steps will be discussed in the next section). However, these steps was originally established according to the approach and concepts of conical geometry was used by Shigley (Frusta) model [1,2] . There is another approach called Roster's pressure-cone model for stiffness calculations with a variable cone angle. This method is quite complicated so we here choose a simpler method with a fixed predetermined cone angle α [3,4] .

Basic Concepts
The main concept of Frusta approach is that stress in the joints-member arrangements takes an approximate conical geometry shape (See Figure 1). This conical shape helps to derive the main governing equations based on circle and cone calculus . In general, stiffness is defined as = ; Where P is the compressive load raised from holding the joint to member. And δ is the deformation -deflection caused by this force. In this work the general form of stress distribution between the face and the head of the bolt is assumed to take conical form as shown in figure 1.

Derivation of the Governing Equations
Here We will be use the basic equations corresponding circle, cone and their combination which called frusta , here is main equations: The well-known general form of the circle equation is in two dimensions is: The cone equation These are the main two equations to be used to derive the main governing equations of bolt joint member stiffness. The fundamental equation used to determine the deformation δ is [1]: If we took a small deflection and a small displacement of a small element, equation [3] can be written as: It is to be noted that the circle and cone equations can be represented as : First we will find the joint stiffness in y-direction which is denoted as ky. For this purpose, we will use the circle equation (Eq. 5) Solving this equation with respect to x to obtain = − Taking the change of x with respect to y (i.e: we differentiate x with respect to y) , yields : We could write the differential form of deformation equation (Eq. 4). Substitute the value of obtained above to have the 1 st order differential formula of deformation equation to be = − (11) Note that, x here will be treated as a constant (horizontal distance along the width of the plates starting from centreline of the bolt). To find the value of the deformation δ, we should do single integral to solve the 1 st order differential equation above.
For this purpose, we should define the following conventions first:  The cone angle α: Whose value remains fixed from the definition of this approach.
( let α = 30°).  The plate thickness t: And it is used as the end limit the integral over z-axis , integral limits.  The bolt radius R: is the nominal value of the bolt radius . After we got the conventions cleared, we should solve Eq. 11 to find deformation, so we can relate it to stiffness by using Equation 3.
Here the variable of integration is y. And we can make an additional presence of y in the integration above by defining the term area [1] Using the basic concepts of calculus and integration , we can represent deformation equation as : Where K is constant and it was set to simplify algebraic operations, and = 0.334 +

+
Now we can find stiffness k from the general equation = as Now we find the stiffness in the direction of z (kz) : For this purpose, we will be using the cone equation (Eq. 6) We solve this equation with respect to x so we obtain = − (20) As we exactly did with stiffness on y, we do same thing here with z Let us recall the differential form of deformation equation (Eq. 4) and substitute dx as found in Eq. 23 in Eq 4 to find As we did earlier, we can solve Eq. 24 as a separable differential equation as Here, the variable of integration is z and it can be present in the integration by defining the term area as we last defined it We can see that after substituting the limits of integration, it is an improper integration where we can find some values after substituting t minus infinite ∞ so the answer will be zero.
Hence stiffness on this axis does not have a physical meaning. This is due to that this axis (z) is containing a region that is not included between the two plates.
So we can only find stiffness ky with x and z as two dimensional values to be substituted in Eq. 19. Now, we will use Eq. 19 to generate data curves to represent how stiffness varies with the position coordinates x, and z.
We will also show a simulation study trying to show how stress distributes in the members (plates) for some samples of materials.

RESULTS AND DISCUSSION
Now we will present a few resulted charts after applying Eq. 19 on some samples of geometries that are made of variety if materials. All the details will be shown separately for each sample. We will also include a simulation study.  We will study the stiffness here for this sample as single. Using Eq. 19, we generated the following data curves: editor@iaeme.com The shown graphs are for z = 0 cm, 1 cm, 2 cm, 3 cm, 4 cm, and 5 cm, respectively for Grey Cast Iron sample. For better visualization, we may put these data curves into a single chart as We can see the relation is linear and the difference between each value of z is the interval of applying the calculations. Now we will present a simulation study for the same sample but with applying a pre-load P of 70 N to see how stress distributes within the plate. See Figure 6. It can be noted that how stress distributes in the shape of a conic, which this is the basic assumption on which the approach of Frustum by which stiffness equations are derived. For a second case, Aluminium Alloy 1060-H12 plate, 10 mm hole diameter for bolt entrance at multiple fixed z coordinates is used in the presented analysis (see Figure 4). Now we will show how stress distribute within this sample plate for applying a pre-load P of 70 N. See Figure 8. As seen from figure 8, the same behavior we obtained from the previous sample is still the same but with larger values.

Case2: Two joined cast iron plates analysis
Now we may do the same simulation study but for two plates placed together and these plates will be given ductile and brittle one as follows:  Brittle-Brittle joint (Grey Cast Iron-Grey Cast Iron)  Ductile-Ductile joint (Aluminium Alloy 1060-H12-Aluminium Alloy 1060-H12)  Brittle-Ductile joint (Grey Cast Iron-Aluminium Alloy 1060-H12) Grey Cast Iron-Grey Cast Iron joint, 10 mm hole diameter for bolt entrance (see Figure 9). Each plate has the same dimensions as the one shown in Figure 4.

Case 3: Two joined Aluminium Alloy plates analysis
In this case, Aluminium Alloy 1060-H12-Aluminium Alloy 1060-H12 joint, 10 mm hole diameter for bolt entrance as shown in Figure 11 is used for analysis purpose. Each plate has the same dimensions as ones shown in Figure 4. After applying a pre-load of 70 N, the stress distribution within these plates is found as shown in Figure 12.

Case 4: cast iron and aluminium joined plates
In this case, a Grey Cast Iron and Aluminium Alloy 1060-H12 plates are joined by 10 mm hole diameter bolt as shown in see Figure 13. As the previous cases, the plates have the same dimensions as the one shown in Figure 4.  It should be noted that, these simulation studies were done using SolidWorks Simulation®. The bolt is made of Aluminium Alloy 1060 and its standard were:  For single plate, we used Formed hex screw, B18.2.3.2M -Formed hex screw, M10 x 1.5 x 20 --20WN  For two-plated joint, we used B18.2.3.2M -Formed hex screw, M10 x 1.5 x 30 --30WN

CONCLUSION
It was shown how stiffness varies with distance along the width of the plate linearly. And this behaviour confirms the validity of using Hook's law to relate stiffness to deformation.
Using simulation solutions, it is shown how stress distributed within the member plate which in turn confirmed the validity of the frustum approach to predict stiffness.
These results were obtained under the assumption that the members' material is the same (i.e.: homogeneous ) having same modulus of elasticity E .
Otherwise, these components can be applied for non-homogeneous members having number of n as several springs in series: