## 2 Forming Limits of the Tube-Nosing Process  
Tube nosing involves pressing the end of a tube into a spherical or conical die to create a rounded nose. Figure 1 shows an example of tube nosing using a conical die. A tube with specific dimensions is pressed against the die under axial force, causing local bending and contraction near the die inlet. The advancing punch draws the tube inward over the die profile until it eventually inverts, forming a deformed nose. However, if the forming operation is inadequate, axisymmetric or non-axisymmetric buckling may occur, as shown in Figure 2. These defects are unacceptable and depend on tube dimensions, material properties, die geometry, and friction conditions. Failure modes in nosing processes can be grouped into three categories: column buckling, axial buckling, and circumferential buckling of the nosed part. Euler buckling is a phenomenon that occurs when a tube is nosed and is long and relatively thick. If the tube material has high strength, it can resist the flow through the die cavity, leading to buckling. This usually happens in the initial stages of deformation when the axial compressive stress on the tube wall exceeds the strength of the tube material. Euler buckling happens when the tube collapses due to instability caused by compressive loading. The critical buckling load, \(F_{cr}\), at which the elastic instability of the tube occurs, is defined as the Euler load, and it is given by the following expression:  
\[F_{cr}=cEI\left(\frac{\pi}{L}\right)^{2} \tag{1}\]  
where \(c\) is the end condition factor and depends on the tube supporting system, \(E\) is the elastic modulus of tube material, \(I\) is the area moment of inertia, and \(L\) is the tube length. Euler buckling formulae Equation (1) gives reliable predictions of the global buckling load if the end-conditions are known. The corresponding critical buckling stress \(\sigma_{cr1}\) is calculated as follows:  
\[\sigma_{cr1}=\frac{F_{cr}}{A}=cE\left(\frac{\pi r_{g}}{L}\right)^{2} \tag{2}\]  
where, \(A\) is the cross-sectional area of tube and \(r_{g}\) is the radius of gyration. For a tubular material:  
\[r_{g}^{2}=0.5r^{2},\ I=\pi r^{3}t,\ A=2\pi rt\]  
where, \(r\) is the initial mean radius of tube and \(t\) is the initial tube wall thickness. For the tubular material under buckling conditions, Euler critical buckling stress and Euler critical load can be expressed as follows:  
\[\sigma_{cr1}=0.5 cE\left(\frac{\pi r}{L}\right)^{2} \tag{3}\]  
\[F_{cr}=ctEI\left(\frac{\pi r}{L}\right)^{3} \tag{4}\]  
Figure 1: Diagram of the tube-nosing process. Figure 2: Common failure modes that limit the tube-nosing process; (**a**) column or “Euler buckling”, (**b**) axial buckling, (**c**) circumferential buckling. In the plastic range, tangent modulus (\(E_{t}\)) of a material is substituted instead of elastic modulus (\(E\)) in Equations (3) and (4). The tangent modulus is a function of the material true stress-true strain relation. Using Hollomon strain hardening expression, the equivalent stress can be expressed as:  
\[\overline{\sigma}=K(\overline{\epsilon})^{n} \tag{5}\]  
where \(K\) is the material strength coefficient and \(n\) the strain hardening exponent. The slope of the true stress-true strain curve at the critical buckling stress can be obtained as follows:  
\[E_{t}=\frac{d\overline{\sigma}}{d\overline{\epsilon}}=nK(\overline{\epsilon}) ^{n-1} \tag{6}\]  
The tube to be nosed must be taken as short as possible to avoid Euler buckling. If altering the design length of the required part is not possible, an inside solid mandrel and/or external supporting sleeve (tube holder) may be used to avoid buckling. For large shell nosing, local annealing prior to the nosing process for the tube part to be nosed may be adopted. Axial buckling: Under the applied compressive load the tubes may buckle along the axial direction in two ways. This mode of failure normally takes place during the deforming stage when excessive compressive loading is induced through the tube wall.