APPENDICES STRANDJACK WEDGES Frition oeffiients, miro slip and handling Report nr. DCT 2005-78 By: H.G.M.R. van Hoof Idnr : 501326 20 November, 2005 Researh strand jak wedges Appendies By H.G.M.R. van Hoof 1
Table of ontents TABLE OF CONTENTS...2 APPENDIX A ANCHORHEADS...3 APPENDIX B WEDGE PROPERTIES...5 APPENDIX C 18 MM DYFORM STRANDS...6 APPENDIX D OPERATION CYCLES STRANDJACK UNIT...7 D.1 JACK UP CYCLE (LIFTING THE LOAD)...7 D.2 JACK DOWN CYCLE (LOWERING THE LOAD)...10 APPENDIX E CALCULATIONS WEDGE MODEL...15 E.1 AREAS AND PROJECTED WEDGE AREAS A IN, A OUT, A INEFF EN A OUTEFF...16 E.2 HORIZONTAL WEDGE EQUILIBRIUM...17 E.3 VERTICAL WEDGE EQUILIBRIUM...18 E.4 WEDGE OPERATING RANGE...18 E.5 VERTICAL CABLE EQUILIBRIUM...19 APPENDIX F MALFUNCTION OF THE WEDGE...20 F.1 WEDGE OPERATION WITH : Μ KIN CONSTANT, INCREASING Μ KOUT...20 F.2 WEDGE OPERATION WITH : Μ KOUT CONSTANT, DECREASING Μ KIN...21 F.3 WEDGE OPERATION WITH: INCREASING Μ KOUT, DECREASING Μ KIN...21 APPENDIX G MICRO SLIP DURING JACK UP...23 Researh strand jak wedges Appendies By H.G.M.R. van Hoof 2
APPENDIX A Anhorheads An anhor head is drilled with a number of tapered holes suitable for aepting jak temporary wedges. An anhor head of the 830 SSL strand jak unit for example ontains 54 tapered holes (Figure 3). In Figure 1 a harateristi example of an anhor head is presented. The quantity of tapered holes of the anhor heads is saled to the required apaity. SECTION A-A The main speifiations of the anhor heads are presented in Table 1. Figure 1:Anhor head of 830 SSL strand jak unit Table 1: Speifiations Anhor heads. Soure: [ Mammoet] Anhor heads Speifiations SSL830 SSL550 SSL300 SSL100 Main dimensions(mm) : 520x190 450x190 354x100 200x150 Weight (kg) : 271 207 60 40 Material : 34CrNiMo6 34CrNiMo6 34CrNiMo6 34CrNiMo6 Wedges / holes (Qty) : 54 36 18 7 Surfae treatment : Nitroarburizing(duth: teniferen) Nitroarburizing(duth: teniferen) Nitroarburizing(duth: teniferen) Nitroarburizing(duth: teniferen) Layer thikness surfae treatment (mm) : 0.1-1.6 0.1-1.6 0.1-1.6 0.1-1.6 Solid lubriant : Molyote D321 R Molyote D321 R Molyote D321 R Molyote D321 R Optimal layer thikness lubriant (µm) : 15-25 15-25 15-25 15-25 Note that the surfae treatment of the anhor heads is nitro arburizing. Figure 2: 830 SSL anhor head with 54 wedges Figure 3: Tapered wedge seatings of an anhor head Researh strand jak wedges Appendies By H.G.M.R. van Hoof 3
In order to redue frition, the wedge seats of the anhor head are pre-treated with Molykote D321 R, a dry lubriant (see Figure 4). Figure 4: Molykote D321 R Pre-treated Anhor head Researh strand jak wedges Appendies By H.G.M.R. van Hoof 4
APPENDIX B WEDGE PROPERTIES The wedges used in the strand jak units ontain three wedge parts. The wedge is equipped with a frition or grip profile on the inside of eah part. The main dimensions of the wedge and wedge parts are presented in Figure 5 en Figure 6. DETAIL GRIP PROFILE WEDGE Figure 5: Assembled wedge and wedge part Figure 6:Detail grip profile The three wedge parts are assembled with an elasti rubber ring. This ring retains the three wedge segments/parts together during operation (Figure 7). Table 2: Speifiations TT 18/44 wedge (soures : Mammoet,KANIGEN ) Wedge Main speifiations TT 18/44 Wedge Figure 7: Assembled wedge with rubber ring Rubber ring Main dimensions(mm) : 25x 44x 18x80 Total weight (kg) : 0.450 Material : DIN 1.6523 (SAE 8620) Wedge parts (Qty) : 3 Weight wedge part (kg) : 0.150 Hardening (Carburizing) temperature ( C): 900-1000 Tempering temperature ( C): 190-200 Hardness of substrate after tempering (indiation*) (HV): 700-760 Surfae treatment : Eletroless Nikel plated (KANIGEN method) Proes temperature surfae treatment ( C) : 200-280 Hardness of surfae layer (indiation*) (HV): 500-550 Hardness substrate after surfae treatment (indiation*) (HV): 560-620 Layer thikness surfae treatment (µm) : 25-50 Solid lubriant : Molyote D321 R Optimal layer thikness lubriant (µm) : 5-20 Total ost (euro): 20 Manufaturer : TT Fijnmehania * Values for referene only, exat values have to be determined by material tests For main speifiations of the urrently used TT 18/44 wedge is referred to Table 2(In Table 2 several speifiations are noted with indiation, these values need to be reviewed by hardness tests). In order to redue frition, the wedges are pre-treated before operation with Molykote D321 R, a dry lubriant. Researh strand jak wedges Appendies By H.G.M.R. van Hoof 5
APPENDIX C 18 mm DYFORM strands The strand bundle ontains several Dyform strands with a maximum operation apaity of 167 kn/strand (speified by Mammoet). The quantity of strands depends on the used unit type.the Dyform strand ontains 7 twisted, high apaity, steel wires with a flattened side (see Figure 8). They are speially developed for heavy lifting and its length an be up to 1500 meters to meet the job requirements. DYFORM STRAND P P Cross setion P-P Flattened side(6x) Wire (7x) Figure 8: Dyform strand The main speifiations of the Dyform strand are presented in Table 3. Table 3: Main speifiations Dyform strand. Soure :[ Bridon wire Ltd ] Strand Main speifiations BS5896 (DYFORM) Nominal values Toleranes Nominal diameter (mm) : 18 +0.4-0.2 Mass (kg/m) : 1.75 +0.4% -2% Tensile strength (Rm) (N/mm 2 ) 1700 Surfae hardness of wires (HV) 430-480 Steel area (mm 2 ) 223 Breag load (Fm) (kn) 380 0.1% proof load (Fp 0.1) (kn) 323 Load at 1% elongation (Ft 1.0) (kn) 334 Wires (Qty) 7 Manufaturer : Carrington Wire Ltd / Bridon Wire Ltd Researh strand jak wedges Appendies By H.G.M.R. van Hoof 6
APPENDIX D Operation yles Strandjak unit In this appendix a review is presented on two main operation yles of the strand jak unit SSL to familiarize the reader with the general onepts of the Strand jak-unit SSL.: Jak up yle (lifting the load) Jak down yle (lowering the load) D.1 Jak up yle (lifting the load) In the next paragraph the Jak up yle of the hydrauli Strand jak units is desribed. This priniple onerns all types. START POSITION STEP 1 WEDGE UPPER ANCHOR HEAD RELEASE PIPE UPPER RELEASE CILINDER UPPER RELEASE PLATE WEDGE LOWER ANCHOR HEAD RELEASE PIPE LOWER RELEASE CILINDER LOWER RELEASE PLATE STROKE LOAD LOAD STROKE Figure 9: Start position and Step 1 of the Jak up yle The start position (see Figure 9): The 18 mm dyform ompat strands are installed through the unit and the load is onneted. Both upper and lower anhor heads are loked (red in Figure 9). Step 1 (see Figure 9):The piston of the jak extends and raises the upper anhor head inluding the loked strands and onneted load. During this movement the wedges in the lower anhor head are pulled up slightly by the movement of the strands; the lower anhor head is still losed. In ase of failure of the upper anhor head, the wedges in the lower anhor head seure the load. Researh strand jak wedges Appendies By H.G.M.R. van Hoof 7
STEP 2 STEP 3 STROKE -15 mm LOAD LOAD Figure 10: Step 2 and Step 3 of the Jak up yle Step 2 (see Figure 10): In top position the load is transferred from the upper anhor head to the lower anhor head by slightly retrating the piston, approx. 15 mm. During this load transfer both anhor heads remain loked. Step 3 (see Figure 10): After the transfer, the upper anhor head is opened hydraulially. The upper release ylinder lifts the release plate with the release tubes and shifts the wedges/grips up and of their seatings (Note the differene between left and right detail piture).the upper anhor head is now unloked (blue in Figure 10) allowing free passage of the strands. Researh strand jak wedges Appendies By H.G.M.R. van Hoof 8
STEP 4 STEP 5 -STROKE LOAD LOAD Figure 11: Step 4 and Step 5 of the Jak up yle Step 4 (see Figure 11): With the upper anhor head unloked, the piston of the jak retrats and returns to the starting position. The strands slide through the wedges in the upper anhor head. Step 5 (see Figure 11): At the end of step 4, the upper anhor head is loked again by the upper release ylinder retrating the release plate and release tubes. The jak up yle of step 1 to 5 is repeated till the required jak up distane is aomplished. Researh strand jak wedges Appendies By H.G.M.R. van Hoof 9
D.2 Jak down yle (lowering the load) In the next paragraph the Jak down yle of the hydrauli Strand jak units is desribed. This priniple onerns all types. START POSITION WEDGE UPPER ANCHOR HEAD RELEASE PIPE UPPER RELEASE CILINDER UPPER RELEASE PLATE STEP 1 WEDGE LOWER ANCHOR HEAD RELEASE PIPE LOWER RELEASE CILINDER LOWER RELEASE PLATE LOAD Figure 12: Start position and Step 1 of the Jak down yle The start position (see Figure 12): The 18 mm dyform ompat strands are installed through the unit and the load is onneted. Both upper and lower anhor heads are loked (red in Figure 12). Step 1 (see Figure 12): The upper anhor head is opened hydraulially. The upper release ylinder lifts the release plate with the release tubes and shifts the wedges/grips up and of their seatings (Note the differene between detail piture of the upper anhor head in Start position and Step1). The upper anhor head is now unloked (blue in Figure 12) allowing free passage of the strands. Researh strand jak wedges Appendies By H.G.M.R. van Hoof 10
STEP 2 STEP 3 LOAD LOAD Figure 13: Step 2 and Step 3 of the Jak down yle Step 2 (see Figure 13): With the upper anhor head unloked, the piston of the jak extends till approx. 15 mm before end of the ward stroke (ompare the position of the main hydrauli ylinder in step 1 and step 2). The strands slide through the wedges in the upper anhor head during the stroke. Step 3 (see Figure 13): In the position 15 mm before end of the ward stroke the upper anhor head is loked by the upper release ylinder retrating the release plate and release tubes (red in detail upper anhor head in see Figure 13 step 3). Researh strand jak wedges Appendies By H.G.M.R. van Hoof 11
STEP 4 STEP 5 LOAD LOAD Figure 14: Step 4 and Step 5 of the Jak down yle Step 4 (see Figure 14): The load is transferred from the lower anhor head to the upper anhor head by slightly extending the piston of the main ylinder further (approx. +15 mm) to the end of the stroke. During this load transfer both anhor heads remain loked (both red in Figure 14 step 4). During this movement the wedges in the lower anhor head are pulled up slightly by the movement of the strands; the lower anhor head is still losed. In ase of failure of the upper anhor head, the wedges in the lower anhor head seure the load. Step 5 (see Figure 14): After the transfer, the lower anhor head is opened hydraulially ( unloked ; blue in detail of lower anhor head step 5). The lower release ylinder lifts the release plate with the release tubes and shifts the wedges/grips up and of their seatings (Note the differene between left and right detail piture step 4 and step 5).The lower anhor head is now unloked (blue in Figure 14) allowing free passage of the strands. Researh strand jak wedges Appendies By H.G.M.R. van Hoof 12
STEP 6 Figure 15: Step 6 and Step 7 of the Jak down yle Step 6 (see Figure 15): The piston of the jak retrats and lowers the losed upper anhor head inluding the strands and load (movement is stroke) until approx. 15 mm before end of the inward stroke. The lower anhor head is still unloked (blue in Figure 15) during this movement and allows free passage of the strands. Step 7 (see Figure 15): In the position 15 mm before end of inward stroke, the lower anhor head is loked hydraulially by the lower release ylinder retrating the release plate and release tubes (red in detail lower anhor head in Figure 15 step 7). Researh strand jak wedges Appendies By H.G.M.R. van Hoof 13
Figure 16: Step 8 of the Jak down yle Step 8 (see Figure 16): After log the lower anhor head, the load is transferred from the upper anhor head to the lower anhor head by slightly retrating the piston of the main ylinder further to the end of the inward stroke, approx. -15 mm. During this load transfer both anhor heads remain loked. The jak down yle of step 1 to 8 is repeated until the required jak down distane is aomplished. Researh strand jak wedges Appendies By H.G.M.R. van Hoof 14
APPENDIX E Calulations wedge model In this appendix a mathematial model of the strand log system is presented. In the following onsiderations a single wedge part is examined and the spring fore of the prestress spring is negleted. STRAND WEDGE PART ANCHORHEAD Dstrand 2 x Ri k N CW CW S < L L w w F = L A able F L Figure 17 : Shemati presentation of mathematial wedge model µ k = Kineti oeffiient of frition between er wedge surfae and anhor head surfae µ = Kineti oeffiient of frition on inner wedge surfae and able surfae N = Normal fore on er wedge surfae N in = Normal fore on inner wedge surfae σ w = Compressive stress between wedge and able σ N = Compressive stress perpendiular wedge surfae and anhor head σ = Tensile stress in able as result of F L F L = Load fore α = Wedge angle τ = Frition shear stress between er wedge surfae and anhor head τ w = Frition shear stress between wedge and able S = Position of slip front (for detailed information is referred to Appendix G2.2) R i = Inner radius of wedge and therefore er radius of strand/able A able = Cross setion area of able From Figure 17, the following equations an be obtained: Researh strand jak wedges Appendies By H.G.M.R. van Hoof 15
τ = µ k σn (a) and w = µ σw τ (b) E.1 Areas and projeted wedge areas A in, A, A ineff en A eff The estimation of the theoretial er and inner surfae of a single wedge part (assuming ylindrial), A in and A, is formulated as : A in 2 = π R i S () and 3 A 2 = π R 3 S os ( α) (d) This aording to Figure 18. TOP VIEW WEDGE PART SIDE VIEW WEDGE PART Ri R S COS ( ) S Figure 18: Top view and side view wedge part in relation to A in and A The estimated projeted effetive er and inner surfae of a single wedge part, A ineff and A eff, an formulated as: S Aineff = Ri S 3 (e) and Aeff = R 3 (f) os( α) These equations an be derived from Figure 19. Researh strand jak wedges Appendies By H.G.M.R. van Hoof 16
TOP VIEW WEDGE PART Ri x 3 R Ri R x 3 R Ri Figure 19: Top view and side view wedge part in relation to A ineff and A eff The total normal fore N ating perpendiular on the effetive surfae of the wedge part beomes (see also Figure 17) : N S = σn Aeff = σn R 3 (g) os( α) The total normal fore N in ating perpendiular on the inner surfae of the wedge part beomes (see also Figure 17) : N in = σ A = σ R S 3 (h) w ineff w i E.2 Horizontal wedge equilibrium The total equilibrium of the horizontal fores on the wedgepart results in the following equation (aording to Figure 17): N os( α) = N τ sin( α) A (i) in eff After substitution of equations a, g, h and f in i S σn R 3 os( α) = σw R os( α) After solving and rearranging, we get σ R (1 µ tan( α)) = σ R or N k w i i S 3 µ σ k R σ N sin( α) R N (1 µ k tan( α)) Ri S os( α) (j) = σ w 3 Researh strand jak wedges Appendies By H.G.M.R. van Hoof 17
E.3 Vertial wedge equilibrium The total equilibrium of the vertial fores on the wedgepart results in the following equation (aording to Figure 17): A τ τ os( α) A σ sin( α) A 0 + (k) in w N = After substitution of equations b,, a, d in k, we get 2 π Ri S µ 3 σ N σ w 2 sin( α) π R 3 µ k σ S os N ( α) 2 os( α) π R 3 = 0 After solving and rearranging, this results in S os ( α) R µ σ = σ R ( µ + tan( α)) or i w N k R µ σw = σn ( µ k + tan( α)) (l) R i E.4 Wedge operating range Substituting equation j in l results in µ σ N R R After rearranging i (1 µ k tan( α)) = σ N R R i ( µ k + tan( α)) µ tan( α) µ k = and so 1+ µ tan( α) µ k + tan( α) µ = (m) 1 µ tan( α) k These two equations represent the borderline of the wedge operating range. Researh strand jak wedges Appendies By H.G.M.R. van Hoof 18
E.5 Vertial able equilibrium When the equilibrium of the vertial fores on the able/strand in Figure 17 is onsidered, the following equation an be aquired: F L = 2 π R τ S (n) i w With F L able 2 i = σ A = σ π R (o) and w = µ σw τ (b) After substitution of o and b in equation n, the result is: 2 i σ π R = 2 π R µ σ S (p) After rearranging : i w S σ = 2 µ σw (q) R i Researh strand jak wedges Appendies By H.G.M.R. van Hoof 19
APPENDIX F Malfuntion of the wedge For the benefit of the analysis, three possible ases are desribed to larify the malfuntion of the wedge, aording to the mehanial wedge model: 1. Wedge operation with : µ Constant, inreasing µ k 2. Wedge operation with : µ k Constant, dereasing µ 3. Wedge operation with : Inreasing µ k, dereasing µ In the following paragraphs these load ases are desribed in detail. F.1 Wedge operation with : µ Constant, inreasing µ k When the wedge operation is started, a proper and normal operation of the wedge is assumed.this implies that the frition onditions are below the good/bad borderline in Figure 20. For example, fitive estimated start values of µ k and µ in operation are 0,2 and 0,5 (see start operation point A in Figure 20). 0.6 Frition oeffiients 0.4 Malfuntion of wedge (slipping of strand, situation 2) B ` µ k 0.2 A Wedge operating properly (situation 1) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.2 α =8.19 µ Figure 20:Wedge operation with inreasing µ k With inreased µ k, the operation point B is positioned in the red Malfuntion Area.The frition fore between wedge er surfae and tapered hole surfae is too high and onsequently gripping ation is less; the strand slips through the wedge. Researh strand jak wedges Appendies By H.G.M.R. van Hoof 20
F.2 Wedge operation with : µ k Constant, dereasing µ When the wedge operation is started, a proper and normal operation of the wedge is assumed. For example, fitive estimated start values of µ k and µ in operation are 0,2 and 0,5 (see start operation point A in Figure 21). 0.6 Frition oeffiients 0.4 Malfuntion of wedge (slipping of strand, situation 2) µ k 0.2 C A Wedge operating properly (situation 1) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.2 α =8.19 µ Figure 21: Wedge operation with dereasing µ When µ dereases during operation below the value 0,35 and the value µ k is onstant, the operation point of the wedge is moved from A to point C (see Figure 21 ). Operation point C is positioned in the red Malfuntion Area.The frition fore between wedge inner surfae and able surfae is too low, onsequently the gripping ation is less; the strand slips through the wedge. F.3 Wedge operation with: Inreasing µ k, dereasing µ In most pratial ases the values of the frition oeffiients will hange simultaneously, influened by environmental irumstanes (temperature, relative humidity et.). Researh strand jak wedges Appendies By H.G.M.R. van Hoof 21
A ombination of inreasing µ k and dereasing µ will lead to mal-funtion of the wedge onform Figure 22, the operation point of the wedge is moved from A to point D. Malfuntion of the wedge appears. 0.6 Frition oeffiients 0.4 Malfuntion of wedge (slipping of strand, situation 2) D 0.2 A µ k Wedge operating properly (situation 1) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.2 α =8.19 µ Figure 22: Wedge operation with dereasing µ and inreasing µ k Researh strand jak wedges Appendies By H.G.M.R. van Hoof 22
APPENDIX G Miro slip during jak up In every load yle (jak up and jak down) miro slip will our in the wedges of the upper and lower anhor head. This is a result of the differene in axial strain (as a result of the load fore F L ) between the able and wedge in onsequene of the differene in stiffness. The axial strain in the able is larger than in the wedge. Hene the strain in the loaded able an not be followed by the wedge what results in a relative displaement between the able er surfae and wedge inner surfae. Miro slip in ombination with a load fore results in wear of the inner frition surfae of the wedge and eventually in a redution of the inner frition oeffiient µ. Finally this will ause malfuntion of the wedge. When the piston of the jak extends, the losed upper anhor head is raised inluding the strands. The load fore F L and related strain in the strand inrease during this movement until the anhor head bears the total present load fore F L. The situation of the full loaded wedge in the anhor head is desribed shemati in Figure 23. F L = Load fore V = Spring fore (pre-stress) Figure 23: Loaded wedge during raising of upper anhor head Researh strand jak wedges Appendies By H.G.M.R. van Hoof 23
N in = Normal fore on inner wedge surfae µ = Kineti oeffiient of frition on inner wedge surfae Q = Length ar of wedge part α = Wedge angle σ w = Compressive stress between wedge and able σ = Tensile stress in able as result of F L ε = Strain in able S = Position of slip front L = Length of frition profile A able = Cross setion area of able E = Modulus elastiity of able τ w = Frition shear stress At the end of the trajetory of the inreasing load fore F L (the upper anhor head bears the total load fore F L ), the slip front has propelled to distane S. In the ross setion of the able on position S, the tensile stress σ = 0. Along the inner frition surfae of the wedge part, width Q and length L, exists a ompressive stress σ w between wedge and able as a result of the normal fore N in. The (pre-stress) spring fore V is negleted and the stiffness of the wedge is onsidered infinite in the following onsiderations. This ompressive stress σ w present on the 3 wedge parts is formulated as: σ w = N in 3 Q L The ompressive stress σ w is able to transmit an average frition shear stress τ f on the ontat area: τ = µ σ w w The load fore F L is ompensated by three inner frition surfaes, with an area of Q s and thus is derived for F L : F L = 3 Q s τ w = 3 Q s µ σ w 3 Q s µ N = 3 Q L in = s µ N L in A inreasing fore F L results in a inreasing value of s, as soon as s > L total slip and F L = F max = µ ours. N in If s > L : maroslip ours, the strand slips through the wedge If 0<F L <F max : miroslip ats along length s. For the tensile stress σ and strain ε in the able we an write σ = F A L able ε σ = E = E FL A able The length s an be alulated from the relation s = µ FL N in L (Eq. G.1) Researh strand jak wedges Appendies By H.G.M.R. van Hoof 24
The tensile stress aording to σ diminishes linear with values on positions x = 0 to x = s (see Figure 23 ) σ = 0 and (x 0) = σ (x = s) And analogially with the equations for the tensile stress σ 0 ε (x= 0) = = = 0 and E E A able ε (x= s) F = A L able A graph of the x- position against strain ε is provided in Figure 24. σ, equations for strain ε are σ FL = = E E A able Figure 24 Graph of position x, strain and tensile stress in able wedge With usage of Figure 24 the total able elongation dl in the wedge is derived. The elongation dl of the able in the wedge orresponds with area P 1 dl = P = ε (x= s) s 2 With σ L ε (x= s) = = and E E Aable F FL L s = µ N in After substitution, dl beomes Researh strand jak wedges Appendies By H.G.M.R. van Hoof 25
dl 1 2 = ε(x= s) s = 2 E F A 2 L able L µ N in Funtion ε (x ) an be desribed as (see Figure 24 ) F x L ε (x) = for 0 x s E Aable du ε dx du FL x = dx E A With the definition = (x) follows able To obtain the loal relative displaement funtion u(x) du FL x u(x) = [ ] dx = [ ] dx dx for 0 x s E A able This leads to the relative displaement funtion u(x) 2 FL x u(x) = + C for 0 x s 2 E A able With u (0) = 0 onsequently C = 0, relative displaement funtion u(x) beomes F u(x) = 2 E L x A 2 able for 0 x s (Eq. G.2) Researh strand jak wedges Appendies By H.G.M.R. van Hoof 26
Equation G-2 is presented graphially in Figure 25. Figure 25: Graph of relative displaement u(x) of the er able surfae against inner frition profile The magnitude of relative displaement u(x) (and thus wear) of the er able surfae against the frition profile depends mainly on the loation of the slip front S, assuming F L, A able, E onstant. Maximum slip is loated at the loaded let of the able at position S. Researh strand jak wedges Appendies By H.G.M.R. van Hoof 27