Copyrighted Materials Copyright @ 2009 DEStech Publications Retrieved from www.knovel.con CHAPTER 5 Filament Winding and Fiber Placement 1. FILAMENT WINDING 1.1.Introduction Filament winding is a process used to make composite structures such as pressure vessels, storage tanks or pipes. Composite pressure vessels offer light weight and high strength. Applications include oxygen tanks used in aircraft and by mountain climbers, compressed natural gas cylin- ders for vehicles, drive shafts for automobiles, and pipes for conducting corrosive liquids. Filament winding is a comparatively simple operation in which con- tinuous reinforcements in the form of rovings or monofilaments are wound over a rotating mandrel. Specially designed machines, traversing at speeds synchronized with the mandrel rotation, control the winding angles and the placement of the reinforcements. Structures may be plain cylinders or pipes or tubing, varying from a few centimeters to one or two meters in diameter. Spherical, conical, and geodesic shapes are within winding capability. End closures can be incorporated into the winding to produce pressure vessels and storage tanks. A schematic of a simple filament winding setup is shown in Figure 1.2(b)and repeated here as Figure 5.1. Figure 5.2 shows a photo of a fila- ment winding machine [repeat of Figure 1.2(c)]. The basic mechanism consists of pulling a roving (number of strands) of fibers from the creels. These are spread out using a bank of combs. The fibers then go through a bath of resin (for the case of wet winding). On exit from the bath of resin, the fibers are collimated into a band. The band 205
CHAPTER 5 1. FILAMENT WINDING 1.1. Introduction Filament winding is a process used to make composite structures such as pressure vessels, storage tanks or pipes. Composite pressure vessels offer light weight and high strength. Applications include oxygen tanks used in aircraft and by mountain climbers, compressed natural gas cylinders for vehicles, drive shafts for automobiles, and pipes for conducting corrosive liquids. Filament winding is a comparatively simple operation in which continuous reinforcements in the form of rovings or monofilaments are wound over a rotating mandrel. Specially designed machines, traversing at speeds synchronized with the mandrel rotation, control the winding angles and the placement of the reinforcements. Structures may be plain cylinders or pipes or tubing, varying from a few centimeters to one or two meters in diameter. Spherical, conical, and geodesic shapes are within winding capability. End closures can be incorporated into the winding to produce pressure vessels and storage tanks. A schematic of a simple filament winding setup is shown in Figure 1.2(b) and repeated here as Figure 5.1. Figure 5.2 shows a photo of a filament winding machine [repeat of Figure 1.2(c)]. The basic mechanism consists of pulling a roving (number of strands) of fibers from the creels. These are spread out using a bank of combs. The fibers then go through a bath of resin (for the case of wet winding). On exit from the bath of resin, the fibers are collimated into a band. The band 205
用 FIGURE 5.I Schematic of the filament winding process (courtesy of Wiley Interscience). FIGURE 5.2 The placement of fiber band on the mandrel.(www.gilgwang.com/english/ frp/grp01.html). 206
FIGURE 5.1 Schematic of the filament winding process (courtesy of Wiley Interscience). FIGURE 5.2 The placement of fiber band on the mandrel. (www.gilgwang.com/english/ frp/grp01.html). 206
Filament Winding 207 goes through a fiber feed and is then placed on the surface of a mandrel. The fiber feed traverses back and forth along the length of the mandrel. The mandrel is attached to a motor,which gives it rotational motion.The combined motion of the fiber feed and the rotation of the mandrel make the fiber bands spread over the surface of the mandrel.By covering the surface of the mandrel with many layers,one can build up the thickness of the part.The fiber orientation can be controlled by varying the speed of traverse of the fiber feed and the rotational speed of the mandrel.Fila- ment winding is usually used to make a composite structure in the form of surfaces of revolution,such as cylinders or spheres.The surfaces of these structures are usually convex due to the need to apply tension on the tows while these tows are placed on the surface of the mandrel.If the sur- face is concave,bridging of the fibers over the concave surface may oc- cur.As can be seen from these figures,the basic components of a filament winding system consist of a mandrel and devices to place the fiber tows on the surface of the mandrel to build up the thickness for the part. 1.2.The Winding Process The operation of filament winding is the reverse of the conventional machining process of milling on a lathe.In milling,one starts with a cy- lindrical surface and one removes the material from the surface one strip at a time.In filament winding,one deposits the material on the surface of the mandrel one strip at a time.The most basic form of filament winding is a two-degrees-of-freedom operation.This consists of the rotation of the mandrel and the linear movement of the feed along the axis of the mandrel.Two-axis filament winding machines can be used to wind pipes.Filament winding machines with more degrees of freedom exist. The availability of the additional degrees of freedom can be useful in winding at the end of the part,such as heads of pressure vessels,or the winding of shapes more complex than straight cylinders such as those with variation in cross section (i.e.cones)or spheres.For example,for the case of a four-axis winding machine,the basic movements are man- drel rotation and feed traverse.To these are added a cross-slide perpen- dicular to the mandrel axis and a fourth axis of motion,rotation of the feed eye.These latter permit more accurate fiber placement at the ends. Winding machines with more degrees of freedom up to the level of multi-degrees-of-freedom robots are available.To illustrate the concept of filament winding,only the simple operation of machines with two de- grees of freedom will be described in this chapter.Depending on the co- ordination between the rotational motion and the axial motion,different
goes through a fiber feed and is then placed on the surface of a mandrel. The fiber feed traverses back and forth along the length of the mandrel. The mandrel is attached to a motor, which gives it rotational motion. The combined motion of the fiber feed and the rotation of the mandrel make the fiber bands spread over the surface of the mandrel. By covering the surface of the mandrel with many layers, one can build up the thickness of the part. The fiber orientation can be controlled by varying the speed of traverse of the fiber feed and the rotational speed of the mandrel. Filament winding is usually used to make a composite structure in the form of surfaces of revolution, such as cylinders or spheres. The surfaces of these structures are usually convex due to the need to apply tension on the tows while these tows are placed on the surface of the mandrel. If the surface is concave, bridging of the fibers over the concave surface may occur. As can be seen from these figures, the basic components of a filament winding system consist of a mandrel and devices to place the fiber tows on the surface of the mandrel to build up the thickness for the part. 1.2. The Winding Process The operation of filament winding is the reverse of the conventional machining process of milling on a lathe. In milling, one starts with a cylindrical surface and one removes the material from the surface one strip at a time. In filament winding, one deposits the material on the surface of the mandrel one strip at a time. The most basic form of filament winding is a two-degrees-of-freedom operation. This consists of the rotation of the mandrel and the linear movement of the feed along the axis of the mandrel. Two-axis filament winding machines can be used to wind pipes. Filament winding machines with more degrees of freedom exist. The availability of the additional degrees of freedom can be useful in winding at the end of the part, such as heads of pressure vessels, or the winding of shapes more complex than straight cylinders such as those with variation in cross section (i.e. cones) or spheres. For example, for the case of a four-axis winding machine, the basic movements are mandrel rotation and feed traverse. To these are added a cross-slide perpendicular to the mandrel axis and a fourth axis of motion, rotation of the feed eye. These latter permit more accurate fiber placement at the ends. Winding machines with more degrees of freedom up to the level of multi-degrees-of-freedom robots are available. To illustrate the concept of filament winding, only the simple operation of machines with two degrees of freedom will be described in this chapter. Depending on the coordination between the rotational motion and the axial motion, different Filament Winding 207
208 FILAMENT WINDING AND FIBER PLACEMENT types of winding can be obtained.These are:polar,helical,circuit and pattern,layer,hoop,longitudinal,and combination. 1.2.1.Polar Winding This is also called planar winding.In this process,the mandrel remains stationary while a fiber feed arm rotates about the longitudinal axis,in- clined at the prescribed angle of the wind.The mandrel is indexed to ad- vance one fiber bandwidth for each rotation of the feed arm.This pattern is described as a single circuit polar wrap(Figure 5.3).The fiber bands lie adjacent to each other;a completed layer consists of two plies oriented at plus and minus the winding angle 1.2.2.Helical Winding In this process,the mandrel rotates continuously while the fiber feed carriage traverses back and forth.The carriage speed and mandrel rota- tion are regulated to generate the desired winding angle.The normal pat- tern is multi-circuit helical.After the first traverse,the fiber bands are not adjacent.Several circuits are required before the pattern repeats.A typi- cal 10-circuit pattern is shown in Figure 5.4. In the above configuration one needs to distinguish between the straight cylindrical part and the head (or dome).In the straight cylindri- cal part,the relation between the rotational displacement and axial dis- placement can be established.Refer to Figure 5.5.This figure shows the developed surface of the straight part of the cylinder.The dimension of the base is nD where D is the diameter of the mandrel.Let o be the wind- ing angle (angle between fiber path and the axis of the cylinder),b be the band width of the fiber band,and L be the axial distance traveled by the FIGURE 5.3 Planar winding
types of winding can be obtained. These are: polar, helical, circuit and pattern, layer, hoop, longitudinal, and combination. 1.2.1. Polar Winding This is also called planar winding. In this process, the mandrel remains stationary while a fiber feed arm rotates about the longitudinal axis, inclined at the prescribed angle of the wind. The mandrel is indexed to advance one fiber bandwidth for each rotation of the feed arm. This pattern is described as a single circuit polar wrap (Figure 5.3) .The fiber bands lie adjacent to each other; a completed layer consists of two plies oriented at plus and minus the winding angle. 1.2.2. Helical Winding In this process, the mandrel rotates continuously while the fiber feed carriage traverses back and forth. The carriage speed and mandrel rotation are regulated to generate the desired winding angle. The normal pattern is multi-circuit helical. After the first traverse, the fiber bands are not adjacent. Several circuits are required before the pattern repeats. A typical 10-circuit pattern is shown in Figure 5.4. In the above configuration one needs to distinguish between the straight cylindrical part and the head (or dome). In the straight cylindrical part, the relation between the rotational displacement and axial displacement can be established. Refer to Figure 5.5. This figure shows the developed surface of the straight part of the cylinder. The dimension of the base is πD where D is the diameter of the mandrel. Let α be the winding angle (angle between fiber path and the axis of the cylinder), b be the band width of the fiber band, and L be the axial distance traveled by the 208 FILAMENT WINDING AND FIBER PLACEMENT FIGURE 5.3 Planar winding
DOME 11 DOME 12 10 AXIS POLAR PORRTA POLAR PORRT B 1510139 FIGURE 5.4 An example of a helical winding pattern. TTD FIGURE 5.5 Developed envelope with fiber path. 209
209 FIGURE 5.4 An example of a helical winding pattern. FIGURE 5.5 Developed envelope with fiber path
210 FILAMENT WINDING AND FIBER PLACEMENT fiber feed corresponding to one rotational revolution.The relation be- tween the rotational distance and axial distance can be written as: 元D L=- (5.1) tano If h represents the length of the straight part of the cylinder to be built, the number of revolutions required for the fiber feed to travel this dis- tance is given as: h htano. n三 (5.2) LπD Equation(5.2)gives the number of revolutions.This can be a whole number or a decimal number.One needs to convert this into the number of degrees(by multiplying n by 360)in order to determine the number of degrees of revolution. Since filament winding is a continuous process,the fiber feed has to re- verse its motion to go back to the other end.Also it is essential that ten- sion be maintained in the fibers to ensure good properties of the final product.One also needs to identify the location of the fiber feed(point A in Figure 5.6)and the point of separation between the fiber band and the surface of the mandrel (point B). When the point B reaches the end of the straight part of the cylinder, this point will move over the surface of the head of the component to be built (i.e.a vessel).The fiber feed (point A)starts to go into reverse.It takes some time before point B touches the end of the straight part of the cylinder again(point B).The number of degrees of rotation of the man- Traverse B Mandrel FIGURE 5.6 Relative position of the fiber feed (point A)and point of separation (point B)between fiber band and mandrel surface
fiber feed corresponding to one rotational revolution. The relation between the rotational distance and axial distance can be written as: L D = π tanα (5.1) If h represents the length of the straight part of the cylinder to be built, the number of revolutions required for the fiber feed to travel this distance is given as: n h L h D = = tanα π (5.2) Equation (5.2) gives the number of revolutions. This can be a whole number or a decimal number. One needs to convert this into the number of degrees (by multiplying n by 360) in order to determine the number of degrees of revolution. Since filament winding is a continuous process, the fiber feed has to reverse its motion to go back to the other end. Also it is essential that tension be maintained in the fibers to ensure good properties of the final product. One also needs to identify the location of the fiber feed (point A in Figure 5.6) and the point of separation between the fiber band and the surface of the mandrel (point B). When the point B reaches the end of the straight part of the cylinder, this point will move over the surface of the head of the component to be built (i.e. a vessel). The fiber feed (point A) starts to go into reverse. It takes some time before point B touches the end of the straight part of the cylinder again (point B′). The number of degrees of rotation of the man- 210 FILAMENT WINDING AND FIBER PLACEMENT FIGURE 5.6 Relative position of the fiber feed (point A) and point of separation (point B) between fiber band and mandrel surface
Filament Winding 211 angular Br advance B1 FIGURE 5.7 Relative angular position of the point of separation at the end and at the be- ginning of a circuit. drel during this time is termed the dwell angle.A dwell angle exists at both ends of the cylinder. 1.2.3.Circuit and Pattern When the point B has gone one complete cycle and returns to the same axial position along the length of the cylinder and goes in the same direc- tion,a circuit has been completed.Due to the complexity of the motion, there is no guarantee that after one circuit,the point B at the end of one circuit will have the same angular position as its position at the beginning of the circuit(B )Figure 5.7 illustrates this point. It takes a number of circuits before the point B can return to its position at the beginning.When this happens,one has a pattern.This can be illus- trated in the following example. Example 5.1 It is desirable to wind a 30 cm diameter by 100 cm long cylinder at a 30 wind angle. The fiber band width is assumed to be 0.6 cm and the dwell angle is 180.Determine the number of circuits required to make a pattern. Solution First,define the reference circle as the circle of the cross-sectional area of the cylinder at one end,say,the left end.Assume that winding starts from a point B on that circle. In one circuit,the feed moves twice the length of the mandrel.This means forward once and backward once along the length of the mandrel.When a pattern is complete, a set of circuits has been made and the fiber path returns to the initial position
drel during this time is termed the dwell angle. A dwell angle exists at both ends of the cylinder. 1.2.3. Circuit and Pattern When the point B has gone one complete cycle and returns to the same axial position along the length of the cylinder and goes in the same direction, a circuit has been completed. Due to the complexity of the motion, there is no guarantee that after one circuit, the point B1′ at the end of one circuit will have the same angular position as its position at the beginning of the circuit (B1). Figure 5.7 illustrates this point. It takes a number of circuits before the point B can return to its position at the beginning. When this happens, one has a pattern. This can be illustrated in the following example. Filament Winding 211 FIGURE 5.7 Relative angular position of the point of separation at the end and at the beginning of a circuit. Example 5.1 It is desirable to wind a 30 cm diameter by 100 cm long cylinder at a 30° wind angle. The fiber band width is assumed to be 0.6 cm and the dwell angle is 180°. Determine the number of circuits required to make a pattern. Solution First, define the reference circle as the circle of the cross-sectional area of the cylinder at one end, say, the left end. Assume that winding starts from a point B on that circle. In one circuit, the feed moves twice the length of the mandrel. This means forward once and backward once along the length of the mandrel. When a pattern is complete, a set of circuits has been made and the fiber path returns to the initial position
212 FILAMENT WINDING AND FIBER PLACEMENT Equation(5.2)gives the number of revolutions required for the fiber feed to move a distance h which is equal to the length of the cylinder.For a circuit,two cylinder lengths need to be traveled.The corresponding number of revolutions will therefore be: 2n=2htand=(2)(100 cm)tan 30=1.23 (a) πD π(30cm) The corresponding number of degrees is: (1.23)(360)=441° (b) In addition to the number of degrees in Equation (b),one has to add two times the dwell angle in order to obtain the total number of degrees required to make a circuit. This gives: 0=441+2(180)=801° (c) If one subtracts the above number by a whole multiple of 360,one would obtain the angular advance of the starting point(new point B)as compared to starting point B on the reference circle.This angular advance is 801-(360)(2)=81.This is shown in Figure 5.7. In order to make a pattern,one needs to have a multiple of the advance angles such that this multiple will be equal to a multiple of 360.This can be expressed as: (m)(81)=(n)(360) (d) where m and n are integers and should be as small as possible. Equation (d)shows that m and n can be quite large before the equation is satisfied. This may not be practical.In order to reduce the numbers m and n,one needs to adjust the operation to make the advance angle a good whole number.One good whole num- ber close to 81 is 90.This can be done by adjusting the dwell angle to be 180+9/2= 184.5° (This can be done by adjusting the machine setting.)If this is done,Equation (d)be- comes: (m)(90)=(n)(360)or m=4 (e) One can select m=4 and n=1.What this means is that it takes 4 times the advance an- gle (or 4 circuits)to make a pattern. Note:In the pattern calculated above,the fiber band will go back exactly to the same position on the reference circle as at the beginning of the winding process.This may not be desirable because if one continues this process,the fiber will follow the same path as before and one may not be able to cover the whole surface of the mandrel.It is desirable to advance the position B one bandwidth distance along the circumfer- ential direction after one pattern.This distance in angular value can be calculated to be (note that the circumferential coverage of a bandwidth b is b/cos a):
212 FILAMENT WINDING AND FIBER PLACEMENT Equation (5.2) gives the number of revolutions required for the fiber feed to move a distance h which is equal to the length of the cylinder. For a circuit, two cylinder lengths need to be traveled. The corresponding number of revolutions will therefore be: 2 2 2 100 30 30 n 1 23 h D == = tan ( )( tan ( . α π π cm) cm) (a) The corresponding number of degrees is: (1.23)(360) = 441° (b) In addition to the number of degrees in Equation (b), one has to add two times the dwell angle in order to obtain the total number of degrees required to make a circuit. This gives: θ = 441 + 2(180) = 801° (c) If one subtracts the above number by a whole multiple of 360°, one would obtain the angular advance of the starting point (new point B1′ ) as compared to starting point B1 on the reference circle. This angular advance is 801 − (360)(2) = 81°. This is shown in Figure 5.7. In order to make a pattern, one needs to have a multiple of the advance angles such that this multiple will be equal to a multiple of 360°. This can be expressed as: (m)(81) = (n)(360) (d) where m and n are integers and should be as small as possible. Equation (d) shows that m and n can be quite large before the equation is satisfied. This may not be practical. In order to reduce the numbers m and n, one needs to adjust the operation to make the advance angle a good whole number. One good whole number close to 81 is 90. This can be done by adjusting the dwell angle to be 180 + 9/2 = 184.5°. (This can be done by adjusting the machine setting.) If this is done, Equation (d) becomes: ( )( ) ( )( ) m n m n 90 360 = = or 4 (e) One can select m = 4 and n = 1. What this means is that it takes 4 times the advance angle (or 4 circuits) to make a pattern. Note: In the pattern calculated above, the fiber band will go back exactly to the same position on the reference circle as at the beginning of the winding process. This may not be desirable because if one continues this process, the fiber will follow the same path as before and one may not be able to cover the whole surface of the mandrel. It is desirable to advance the position B1′ one bandwidth distance along the circumferential direction after one pattern. This distance in angular value can be calculated to be (note that the circumferential coverage of a bandwidth b is b/cos α):
Filament Winding 213 △0= b (360)= 0.6 -(360)=2.65 (⑤ πDcosa π(30)cos(30 This advanced angular value is accumulated over 4 circuits.The value for each circuit is 2.65/4 =0.66.This angle is then divided by two dwell angles.The dwell angle is then adjusted to be:184.5+0.66/2 184.8. 1.2.4.Layer A pattern may consist of fiber intersections (fiber crossovers-see Figure 5.4)at certain sections.Crossovers may occur at more than one section,depending on the wind angle.A layer is defined as a set of pat- terns that completely cover the surface of the mandrel with fibers. From Figure 5.5,it can be seen that the relation between the circumfer- ential coverage S and the bandwidth b can be written as: b S=- (5.3) cosO In order to make a layer,the whole circumferential distance nD has to be covered.The number of circuits per layer C can be calculated as: πDπDCOs C= (5.4) S b Example 5.2 Continue with Example 5.1 and determine the number of circuits required to make a layer. Solution Fora=30°，one has(from Equation5.4): c=m30)cos30=136 0.60 There are 136 circuits to make up a layer.Recall from Example 5.1 that it takes 4 cir- cuits to make a pattern.The number of patterns per layer is then 136/4=34. 1.2.5.Hoop Winding Hoop or circumferential layers are wound close to 90.The feed ad- vances one bandwidth per revolution.The layer is considered a single
1.2.4. Layer A pattern may consist of fiber intersections (fiber crossovers—see Figure 5.4) at certain sections. Crossovers may occur at more than one section, depending on the wind angle. A layer is defined as a set of patterns that completely cover the surface of the mandrel with fibers. From Figure 5.5, it can be seen that the relation between the circumferential coverage S and the bandwidth b can be written as: S b = cosα (5.3) In order to make a layer, the whole circumferential distance πD has to be covered. The number of circuits per layer C can be calculated as: C D S D b = = ππ α cos (5.4) 1.2.5. Hoop Winding Hoop or circumferential layers are wound close to 90°. The feed advances one bandwidth per revolution. The layer is considered a single Filament Winding 213 ∆θ πα π == = b Dcos ( ) . ( )cos( ) 360 (). 0 6 30 30 360 2 65 (f) This advanced angular value is accumulated over 4 circuits. The value for each circuit is 2.65/4 = 0.66°. This angle is then divided by two dwell angles. The dwell angle is then adjusted to be: 184.5 + 0.66/2 = 184.8°. Example 5.2 Continue with Example 5.1 and determine the number of circuits required to make a layer. Solution For α = 30°, one has (from Equation 5.4): C = = π( )cos . 30 30 0 60 136 There are 136 circuits to make up a layer. Recall from Example 5.1 that it takes 4 circuits to make a pattern. The number of patterns per layer is then 136/4 = 34.
214 FILAMENT WINDING AND FIBER PLACEMENT ply.Hoop layers may also be applied as doublers or localized stiffeners at strategic points along the cylinder. 1.2.6.Longitudinal Winding Longitudinal winding applies to low angle wrap which is either planar or helical.For closed pressure vessels,the minimum angle is determined by the polar openings at each end. 1.2.7.Combination Winding Longitudinals are reinforced with hoop layers.The customary practice with pressure vessels is to place the bulk of the hoop wraps in the outer layer.A balance of hoop and longitudinal reinforcement can also be achieved by winding at two or more helical angles. 1.2.8.Wet/dry Winding In addition to the classification of different winding patterns,one also distinguishes the type of winding depending on whether fibers are wet- ted with liquid resin in-situ or prepregs are used.These are referred to as wet winding or dry winding.In wet winding,the resin is applied during the winding stage (Figure 5.1).The alternate dry winding method uti- lizes the pre-impregnated B-staged rovings.Wet winding tends to be messy due to the possible dripping of the wet resin.Dry winding is cleaner but the raw materials (prepregs)are more expensive than the tows. 1.3.End Closures End closures for pressure vessels are either mechanically fastened to the cylindrical portion or are integrally wound.If end closures are fas- tened to the cylindrical portion,both the end closures and cylindrical portion need to have flanges.Integrally wound end closures can provide better pressure containment than mechanically fastened heads.The fiber path yields a balance of meridional and circumferential forces and is con- sistent with winding conditions so that no slippage occurs.The head con- tours and related polar bosses are critical in vessel design. One common contour follows the geodesic isotensoid.This contour is normally adapted to helical winding.The fiber path is taken as tangential to the polar boss(Figure 5.8).The geodesic path is the path of shortest distance between two points on a curved surface.The reason to select the
ply. Hoop layers may also be applied as doublers or localized stiffeners at strategic points along the cylinder. 1.2.6. Longitudinal Winding Longitudinal winding applies to low angle wrap which is either planar or helical. For closed pressure vessels, the minimum angle is determined by the polar openings at each end. 1.2.7. Combination Winding Longitudinals are reinforced with hoop layers. The customary practice with pressure vessels is to place the bulk of the hoop wraps in the outer layer. A balance of hoop and longitudinal reinforcement can also be achieved by winding at two or more helical angles. 1.2.8. Wet/dry Winding In addition to the classification of different winding patterns, one also distinguishes the type of winding depending on whether fibers are wetted with liquid resin in-situ or prepregs are used. These are referred to as wet winding or dry winding. In wet winding, the resin is applied during the winding stage (Figure 5.1). The alternate dry winding method utilizes the pre-impregnated B-staged rovings. Wet winding tends to be messy due to the possible dripping of the wet resin. Dry winding is cleaner but the raw materials (prepregs) are more expensive than the tows. 1.3. End Closures End closures for pressure vessels are either mechanically fastened to the cylindrical portion or are integrally wound. If end closures are fastened to the cylindrical portion, both the end closures and cylindrical portion need to have flanges. Integrally wound end closures can provide better pressure containment than mechanically fastened heads. The fiber path yields a balance of meridional and circumferential forces and is consistent with winding conditions so that no slippage occurs. The head contours and related polar bosses are critical in vessel design. One common contour follows the geodesic isotensoid. This contour is normally adapted to helical winding. The fiber path is taken as tangential to the polar boss (Figure 5.8). The geodesic path is the path of shortest distance between two points on a curved surface. The reason to select the 214 FILAMENT WINDING AND FIBER PLACEMENT