CALCS  PROC075 Bolted Joint Prising 
CI  Critical Interface 
CZ  Compression Zone (thru flanges IWO bolt) 
EQV  Equivalent To 
ETB  Engineer's Theory of Bending 
FDP  Fully Developed Prising 
LP  Loading Phase 
PCF  Full Contact Prising 
PCP  Partial Contact Prising 
PFA  Active Flange Prising 
PFL  Limit Flange Prising 
AB ABm  CSA bolt nominal, minor 
AF  CSA flange 
ACI  CSA at CI 
BO  bolt orientation 1 HEAD or 1 NUT on flange end 
DB DBm  dia bolt nominal, minor 
DBH  dia bolt head A/F 
DFh  dia flange hole 
DW  dia 'washer' 
dB dBe dBp  bolt extension, under external force PE, under pretension PBp 
dBerr  bolt displacement error check 
dFB  flange displacement at bolt position B 
E  Young's Modulus 
fbB ftB  bolt bending stress, tension 
fbF  flange bending stress 
fdCI  direct stress at CI for PCF (compression & bending) 
IB IBm  2MA bolt, minor 
IF  flange 2MA 
KB  bolt axial stiffness 
KCZ  CZ axial stiffness 1 frustum, 2 cyl, 3 hole 
KBp  factor bolt pretension force 
Ke  =e/r eccentricity factor PCP 
Kp  =PB/PBp>1 FDP 
LB LBm  length bolt (under tension), proportion screw (minor dia) 
LFr  length of frustum (truncated cone) 0 origin, 1 start, 2 end 
MB  bolt bending moment 
MF  moment at flange contact point FDP 180620 
PB PBp  bolt force, =KBp*FyB*AB bolt pretension force 
PE PEi  external prising force, interval 
PCPa  angle 
PCZ  force thru CZ 
pitch  of bolt thread 
R  layer radius (dimensional limit): R1 at CI to Rn for head or nut 
RCI  compression radius at CI 
RB RBm  bolt nom. radius DB/2, minor 
RCI  radius at CI 
RFH  flange Hole radius 
Rk  Radius of kern across CI 
rfB rsB  bolt head rotation under flexural & shear loading 181120 
TBH  thickness bolt head (or nut) 
TF  thickness flange 
TW  'washer' thickness, between flange and bolt head (may be multiple layers) 
WF  width flange 
XCZ  position of PCZ 
XE  position of external force 
XF  position of flange edge 

Figure 1 shows the characteristic arrangement of the bolted joint considered in the analysis. This shows two flanges, clamped together by a bolt and pretensioned by controlled tightening of the nut. The external force is applied to the joint (as shown acting on the upper flange and towards the left of the figure), in a direction parallel to the bolt axis.
Figure2 shows the basic displacement response of a common form of bolted joint to prising action. The flange in way of the external force is prised upwards, being resisted by bending and shear action of the flange. The part of the flange in way of the bolt is held down by bolt tension, causing a downwards force to act between the head and upper surface of the flange. The bolt head is rotated with the upper surface of the flange. First contact across the joint, between the lower surface of the upper flange and the upper surface of the lower flange, is shown to the right of the bolt axis. Under less significant loading, contact is across an area in way of the bolt. This prising force causes some combination of bolt tension, flexure of the flanges together with compression acting through the thicknesses of the connected flanges. This compression is caused by equal & opposite forces acting between each of the bolt head and the contacting flange outer surface. The nut and second flange are loaded similarly. The contact plane between the two flanges is identified as the Critical Interface. The external reactive force (not shown in Figure1, applied to the lower flange of the joint) is a vector equal and opposite to the external force applied (to the upper flange) and acts along the same axis. The lower flange is chosen to be the thicker of the two and consequently having higher flexural stiffness. The lower flange thickness then adds to bolt length and then bolt tensile loading flexibility. Solutions are provided for lower levels of loading using formulae for contact between two bodies. Under higher levels of applied loading, the solution is equivalent to the 'beam model' solution of ref.[4]. As the external force increases, distinct load paths (each different to the others) develop through the joint. Each path transforms from the preceding one at a particular phase transition force. Transformation occurs when a limit (relating to loadgeometry relations) is reached and the next load path takes over. There are four Loading Phases and three phase transition forces. The analyses of all phases are based on linearelastic principles and failure modes are reported for: loss of full contact; loss of partial contact; flange length limit; bolt tension yielding; flange flexural yielding; bolt tearthrough flange (block shear); flange bending SCF iwo bolt hole. 
Figure3.1 Phase1 PCF Full Contact Prisingoccurs at low levels of external loading when bolt pretension is sufficient to maintain full contact across the critical interface. 

Figure3.2 Phase2 PCP Partial Contact Prisingtransforms from Phase1 when bolt pretension is insufficient to maintain full contact across the critical interface and separation occurs at the external force side of the joint. 

Figure3.3 Phase3 PFA Active Flange Prisingtransforms from Phase2 when the contact zone across the Critical Interface is insufficient (approaching zero area) and fully developed prising ensues with contact occuring outside of the Critical Interface. 

Figure3.4 Phase4 PFL Limit Flange Prisingtransforms from Phase3 when the contact point has moved to the edge of the flange. This limits any further progression of the contact point position as loading is increased further. 
In the analysis of this kind of bolted joint, the compressive zone through the thicknesses of the two connected flanges is normally developed from frustum (that is truncated cone) and cylindrical shapes. The conical inclination from the axis is taken to be 30deg as [3]. Note that other authors have chosen different forms of the throughthickness external boundary, however, 30deg provides a reasonable lower bound volume. Two full frustum shapes (one emanating from the bolt head & the other from the nut surface contacting with respective flanges) develop and meet at the same maximum diameter at half bolt length. Where the edges of flanges (or diameters of washers) limit full development of the frustums, the maximum radius is taken to be from the bolt axis to the limiting edge. The resulting frustums are then less than half the bolt length and a cylindrical shape (with the same limited diameter) connects the two. This is shown in Figure4, where the width of flange (not length in this instance) is the limiting dimension and this could result from the pitch between bolts connecting flanges.
The Critical Interface (between the two flanges) is not necessarily at the bolt half length position. The outer radius of the CI may then be less than the common maximum of the frustums. Engineer's Theory of Bending is considered under axial force & moment loading at the CI, caused by some particular external force & bolt pretension. Full Contact is defined at the CI and requires the unloaded annular shape to be maintained under the action of the external force. The radius of a kern is obtained, according to short prism compressive analysis [2]. This defines a limiting radius for the effective position of a compressive force acting across the CI. Providing the external loading causes an effective position of compressive force across the CI to be less than the kern radius, then the requirement for PCF has been satisfied. 
The analysis for Partial Contact Prising further develops the principles as for PCF. PCP occurs when an effective position of compressive force across the CI is greater than the kern radius and less than the CI outer radius. The size of sector removed from the annulus and compressive stress at the CI are taken from ref.[2]. The elevation view of Figure5 shows the analytic model while the plan view shows a segment across which contact is lost during separation at the Critical Interface. Stiffness of the compressive zone through the thicknesses of the flanges is calculated using formulae from ref.[3] for the frustums. The stiffness of the compressive zone is taken as if there is full contact across the CI annulus. As the contact area diminishes, the final PCP loading is obtained in the analysis before the zero or negative area discontinuity is encountered. 

Prising by the external force elevates bolt tension in excess of the specified pretension. Contact between the two flanges is removed to a position outside outer the radius of the Critical Interface. Flexural displacement of the flange is then the dominant response of the bolted joint, as shown in Figure6.
Additional loading is induced in the shank of the bolt, with maximun bending stress immediately under and due to the head (or nut) conforming with flange rotational displacement at the same position. Bolt tension stiffness is a necessary constituent of the analysis, described below. It is noted that inclusion of bolt bending stiffness would reduce bolt tension. However, is not included because the reduction would normally not be significant. The plan view shows the flange boundaries and the force PCZ at a position greater than the Critical Interface outer radius and less than the flange edge. The width of flange WF may be to the edges (providing it is narrow relative to the thickness) or an effective width based on the pitch between bolts. The analytic model for the 'FDP  Flange Active Length' loading phase is shown in the elevation of Figure6. The flange is represented by the member PBF with 2MA IF . The external force PE acts on and causes bending and shear in the flange. The reaction force PB acts in the bolt having axial stiffness KB. The force PCZ and moment MF result from the flange having builtin support at F. This is the point of contact between the two flanges. The object of the analysis is to obtain a solution that reduces MF to zero. The moment MF is then an analytic device used to obtain the required zero inclination of the flange at the contact point. The builtin support also fully constrains vertical displacement at F. An equation is formulated for moment equalibrium at F, expressed in terms of flange flexural & bolt tension stiffnesses and loading by forces PE & PB. The formulation ensures compatibilty of displacements. The solution to this nonlinear equation MF=f(PE,PB,KB,IF,XE,XCZ) is obtained in terms of the length XCZ that gives MF=0. Figure7 shows an idealised impression of flange shear deformation (flexural displacement is omitted from the view for clarity). In this, the rectangular shape of the flange section is deformed into two parallelogram shapes. Contact between the flanges is affected only in way of the external force and there is consequenty no movement of the contact point established by the flexural analysis. Notice the bolt angle caused by shear deformation of the flange under the bolt head. The LH & RH parts of the head lower surface conform with the rotational displacements of the comtacting parts of the flange upper surface. This causes an effective rotation of the bolt head, half that of the flange shear deformation angle. Bolt shank bending stress, immediately under the head, is estimated from this rotation and included in the results. 
The analytic model for the loading condition with contact point (between flanges and) at the upper flange edge is shown in Figure8. This is a derivation of the Phase 3 model. The differences are that the length XF is known and mo monent MF or rotational displacement need be considered. Consistent with this, a simple support supplants the builtin support (as Figure6) at position F. Inspection of Figure6 informs the flange is a beam PBF under loading PE, with point supports at B & F. This is a statically determinate loading condition and the solution obtained using basic beam theory.
Analytic steps for Loading Phase 4 at some elevated level of external loading PE are as follows: i) the analysis for LP3 is executed; ii) the result is compared against the flange limit iii) if XCZ>XF then Loading Phase 4 is identified as valid iv) if XCZ>XF then XCZ is reduced to XF v) if XCZ>XF then bolt force is obtained by taking moments about position F. 