Limit load analysis of a bolt with a chordal flaw

TWI Bulletin, May/June 1997

Annette Karstensen joined the Finite Element analysis team of TWI's Structural Integrity Department in 1996. Currently she is involved in research for the transfer of experimental data in fracture mechanics analysis.

Alan Smith works in the fracture assessment section of TWI's Structural Integrity Department. He is currently working on assessment of corroded structures.

Rob Phaal has worked on research and consulting projects relating to structural integrity assessment during his six years at TWI. His main focus has been on fracture mechanics, including development of software for performing fitness-for-purpose assessments.

For simplicity surface flaws are often assumed to be straight fronted ( ie chordal). However few limit load solutions for this type of configuration have been published. When ensuring an acceptably low risk level of fracture in bolts and round bars, the significance of plastic strain must be assessed. Annette Karstensen, Alan Smith and Rob Phaal report on finite element modelling carried out to investigate the significance of chordal flaws in round bars.

A common engineering approach to failure assessment ( Fig.1) is codified in documents such as BSI PD6493:1991, ^[1] which is based on a failure assessment diagram (FAD). When using a FAD, failure is determined both in terms of the likelihood of brittle fracture taking place and of plastic collapse of the surrounding ligament. When a structure is loaded to its collapse load, the deformation becomes unbounded as the load bearing capacity of the flawed component is reached. Plastic collapse load analysis determines the maximum load (limit load) sustainable by a structure made of a perfectly plastic material.

The applied stress to cause plastic collapse is known as the collapse stress. The plastic collapse load for a bolt or round bar with a chordal crack can be estimated from a simple loss of area calculation. However, such a calculation makes no allowance for secondary bending stresses induced in the bolt by the stress distribution associated with the change in position of the neutral axis.

This work assesses the significance of these secondary bending stresses in the collapse load of a bolt subjected to primary tension. It also assesses models for combined tension and bending. It is proposed that guidance on the use of the plastic collapse solution will be incorporated into the revised Annex J of PD6493. ^[2] (Re-designated Annex J in the revised document, which itself will be issued as a British Standard Guide with a new number).

Approach

Finite element analysis was used to determine the limit load of a bolt containing chordal cracks of various sizes, subjected to tension, combined tension and bending loading. The material was assumed to be elastic perfectly plastic. Plastic collapse occurs when the net section stress reaches the flow strength and plasticity spreads throughout the remaining ligament. Analysis was carried out using the finite element package ABAQUS V.5.5 (1995) on a SGI R4400 workstation running the IRIX 5.3 operating system. A quarter of the round bar was modelled, and symmetry conditions applied. Analysis was in 3-D, using 8 noded hybrid elements (ABAQUS type C3D8H). The length of the round bar was 60mm, while the radius (R) was l0mm, modelled, by 3200 elements with a total of 7886 degree of freedom. Figure 2 shows the FEA model, where the flow strength was assumed to be 500 MPa, and the crack depth (a) varied from a/2R = 0.1 to a/2R = 0.5. Two different boundary conditions were investigated - displacement controlled (restrained rotation) and load controlled (pin jointed).

Determination of collapse stress

The aim of this analysis was to determine if the assumption that the collapse load is proportional to the remaining cross-sectional area is conservative, for the full range of flaw depths ( ie the equations in the Tables are valid). 

For the case where the model was pin jointed, a uniform negative pressure was applied on the top boundary, which was free to rotate as the load increased and the crack opened. The rotation of the top boundary creates secondary bending on the ligament. When the bolt was loaded by controlled displacement the boundary was restrained from rotating, providing a uniaxial stress distribution in the uncracked bolt or bar.

The collapse stress is the applied stress which results in a limit load condition. This was determined from the analysis by increasing the load until the whole ligament had received yield, and the applied stress reached a plateau. Results of the analysis are shown in Table 1.

Table 1 Collapse stress for bolts/round bars containing chordal flaws, flow strength of 500MPa

A bar with a flaw to diameter ratio of a/2R = 0.38 was then analysed, using four different combinations of bending and tension load, for the pin jointed configuration. The collapse stresses in tension(P ^c _m) and in bending (P ^c _b) were obtained, where the index c indicates a combined case of tension and bending. For the first three combined bending and tension loading cases the ratio P ^c _m/P ^c _b was kept constant at the point of collapse. In the fourth case, pure tension was applied until approximately 80% of collapse, where the collapse criteria was calculated from the equation given in Fig.3. Bending was then applied until the plastic limit was obtained.

For the combined cases, collapse stresses are calculated by separating the applied stress into a tension component and a bending component. The tension component is then normalised by the predicted collapse stress for tension, assuming that it is proportional to the remaining area in the ligament.

The applied bending stress is normalised by the collapse stress. M _L is the moment to give collapse in bending, given by Miller. ^[3]

Results for the first three load cases can be seen in Table 2, for the fourth case in Table 3 and graphically in Fig 3.

Table 2 Collapse stress for combined tension and bending normalised by the collapse stress, (load cases 1-3)

Table 3 Collapse stress for combined bending and tension load case 4

Conclusion

Finite element analysis has shown that the limit load of a bolt or round bar with a chordal flaw can be predicted conservatively using a simple relationship between the area of the remaining ligament and the area of the uncracked bolt. This simple assumed area relationship is valid for the full range of crack depths analysed (0.1 ≤ a/2R ≤0.5) for cases where the bolt is constrained from rotation. However, when the loading on the bolt or bar is a pin joint the assumption is valid for 0.1 ≤ a/2R ≤ 0.4.

Analysis for combined tension and bending shows that the criterion provides a conservative estimate of plastic collapse, which can be applied across a broad range of industry sectors.

References