Graphical statics a forgotten tool for solving plane mechanical problems

Injury,Int. J. Care Injured31 (2000) S-B24-S-B36

Graphical statics a forgotten tool for solving plane mechanical problems F. Baumgart A0 Technical Commission,

Clavadelerstrasse,

CH-7270 Davos Platz

Summary’

1. Introduction

Graphical statics is an almost forgotten, intuitive drawing method for solving plane mechanical problems. It was already in use in the 19th century for biomechanical problems. It was still a standard method employed by civil engineers in the 1940s. Superceded by modern analytical methods, graphical statics disappeared almost completely. The method is restricted to plane static problems, but still remains a useful tool for visualizing, understanding and checking the actions of force groups occurring in modern biomechanical problems. After defining the basic mechanical terminology (body, motion, forces), the paper is written mainly as a teaching tool for immediate application. Many illustrative examples (sporting activities, functional forces in joints) help to clarify the difficult biomechanical content. For application, it must be assumed that the bodies investigated behave as rigid bodies under the action of the forces, but this does not prevent application of the method to deformable living bodies if specific static configurations of the bodies are considered. The application of the method requires a good anatomical knowledge and experience with the function of the musculoskeletal apparatus of living bodies. If reliable models are used, the method delivers quantitative results of sufficient accuracy. The paper may also help provide a better understanding of publications containing graphical solutions to bio-static problems.

Graphical statics was a standard method of solving plane mechanical problems used by civil engineers at least up until the 1940s. It was gradually replaced by alternative analytical methods, especially methods based on computer technology. Standard medical and biomechanical works frequently present graphical solutions or the graphical representation of static problems. There is still a need for researchers, developers, and readers involved in biomechanics to command an independent method for roughly checking the results of the complicated analytical evaluation of forces in or around bodies. Graphical statics has remained a useful tool for visualizing, understanding and checking the actions of force groups occurring in modem day biomechanical situations. However, its applications requires a full understanding of the basics of the method and the selection of appropriate models for the application. This paper aims to provide an understanding and a critical review of graphical solutions as presented in the literature and a tool to deal with the daily challenges of determining forces.

Keywords: rigid body, plane mechanical problems, functional biomechanics, internal forces, position plan, force plan Injury 2000, Vol. 31, Suppl. 2

1 Abstracts in German, French, Italian, Spanish, Japanese and Russian are printed at the end of this supplement.

2. Basic terminology To start with some basic definitions and terminology of mechanical science must be presented. There is no way of understanding graphical statics without a clear basic knowledge about the important mechanical terms. The following section gives the exact definitions and explanations for the terminology of mechanics used including biomechanics. 2.1 Mechanics Mechanics is the science of the motion of bodies under the influence of forces. Mechanics is a part of physics and so far a science based on experience. The definitions of the terms motion, body and force are given below.

0020- 1383/00/$ - see front matter Q 2000 Published by Elsevier Science Ltd. All rights reserved

Baumgart:

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Graphical statics

2.2 Biomechanics

2.6 Rigid body

Biomechanics is mechanics applied to biological bodies. This is true insofar as biomechanics and mechanics are based on the same laws which are independent of the material of the body Biomechanics is that part of mechanics dealing with bodies of a specific material, namely living substances.

Arigid body cannot be deformed by the action of a force. That means any distance between two arbitrary particles of the body is not changed by the action of forces. Therefore, the force may be moved freely along its direction line without changing any reaction of the rigid body on the force (Fig. 2). This rule is valid for a rigid body only. It is a necessary conditio sine qua non for the application of graphical statics.

Comment: As we will see later, for our purposes in this paper there is no difference between mechanical and biomechanical problems due to the fact that graphical statics works with the fiction of rigid bodies, that means it is independent of the material properties of the bodies. 2.3 Body

A body in mechanics is understood as a continuously connected assembly of material particles. The geometric condition of a body at a certain time can be described by its configuration, e.g. each particle of the body is located in a certain position in space. This position is uniquely determined by three coordinates in space. The particles are connected; two different particles can never be in the same position. Neighbours remain neighbours! (Fig. 1). 2.4 Motion Motion is the continuous transition of the configuration of a body to another configuration in space and in time. At each specified time the body is in a unique intermediate configuration. This phenomenon includes the special cases of rest and constant velocity of all particles of the body, which can also be investigated by the methods of statics. 2.5 Deformation If we compare only two specific configurations of the body, for instance before and after a loading process, we call it a deformation of the body.

2.7 Statics

Statics is a part of mechanics dealing with acceleration free motions (or with constant or zero velocity). It is one of the most important disciplines of civil engineering. All houses, bridges, dams, foundations, tunnels etc. are designed with the help of statics. Statics allow for the determination of the inner forces in structures caused by external loading and help to establish their safety factor. Currently, it is mostly understood as an analytical calculation method. 2.8 Graphical

statics

Graphical statics is a quantitative method to solve mechanical problems by graphical constructions. It is appropriate to visualize the interaction of forces on bodies and to “calculate” the forces by drawing on a sheet of paper. The basic subject of this method is the vector. 2.9 Vector In physical sciences the term vector is in use for the description of a directed physical or geometrical object which can be identified as having - a direction (can be symbolized by a straight line) - a sign of direction (can be symbolized by the tip of an arrow) - a value (can be symbolized by the length of an arrow) Therefore: A vector is symbolized by an arrow, which gives direction, sign and value’.

Fig. 1: Body, configuration and motion. The body “domino chip“ performs a motion. Its initial configuration C, is defined by all the coordinate triples x1, x2, x3 of all of its particles at a certain time. If we follow the body continuously in time through all of its configurations, represented by C1 to C,, from the yellow position to the red position, we talk about a motion. Be aware of the fact that time all the c, c_ \ different coloured parts represented here are certain positions of only one XI body!

T

Injury 2000, Vol. 31, Suppl. 2

Fig. 2: Deformation, rigid body. If we compare only two different configurations of a body, we

call this a deformation. The load group consists of two forces F acting on the upper and lower surface at the end of the cantilever “domino” beam. A rigid body (top) shows no deformation at all if we compare the drawn configuration with the unloaded initial configuration. The deformable body (bottom) shows an overall deformation. All particles undergo a displacement, except the particles in the rigidly fixed left surface. The particles on the right show the maximum values of deformation. In the case of a d very soft material, it can also be deh tected visually that the height h of the *domino” has changed under the two loads F.

S-B26 A typical vector is the velocity of a material particle (Fig. 3). In general a vector has three components, parallel to the three directions of three-dimensional space. In (conventional) graphical statics only the two inplane components of the vector are used. Three-dimensional problems are not appropriate for this method. Therefore, graphical statics is restricted to plane mechanical problems. 2.10 Addition

of vectors, the parallelogram

axiom.

Two vectors can be added geometrically as follows (Fig. 4): - Draw the first vector. - Draw the arrow of the second vector beginning at the tip of the first vector. Now connect the blunt end of the first vector to the tip of the arrow of the second vector forming a third arrow. The new vector is the geometrical sum of the two vectors. This rule is called the parallelogram axiom (see also Fig. 4). It is evident that this method can be used repeatedly in order to summarize an arbitrary number of vectors. The sums of the vectors appear then as the arrow which connects the beginning of the first vector to the tip of the last vector. It is called the resultant vector or the resultant. It is interesting that the resultant vector is independent of the sequence of the forces. The forces can be added arbitrarily. This method can be applied to all physical objects which have vector character. It goes without saying that only vectors of the same character (physical unit) can be summarized in such a way.

Fig. 3: Vector example: Velocity. The greyhound can serve as a model for a deformable body. The running dog demonstrates an arbitrary number of velocity vectors in its body, - The velocity vector v, shows more or less the average speed of its current centre of gravity. - The hind leg is in contact with the floor, showing necessarily zero speed vq of the particles in the contact area. - The front leg demonstrates that the speed vg of the paw is already going into relative motion backwards and downwards, that means the length of the vector is smaller than the average speed and is directed slightly downwards.

I

I

vector addition

parallelogram axiom

Fig. 4: Parallelogram axiom. The vector addition (left side) as described here is identical with the parallelogram axiom (right side). The black and the red vectors are added, the pink vector is the resultant vector.

2.11 Force 2.11.1 General description A force is a physical object able to act on bodies to accelerate and/or deform them. A force has the same physical character as the well known weight of a body (Fig. 5). Forces can be understood as vectors and can be summarized by the graphical method described.

1 The mathematical theory of vectors and tensors is very demanding. We will not go into it here. It explains the charac-

ter of a vector by certain properties of its triple of coordinates if the vector is transformed from one coordinate system to another. Tensors are physical or mathematical objects which have 3” coordinates (or components), n=1,2,3 etc.

The number n is the step of the tensor, n=l characterizes a vector. The stress state in a body is described by the stress tensor. It has 9 components in general and is a tensor of the second step, n=2. Due to its symmetry it only has 6 independent components.

G=49.05 N

I Fig. 5: Force. The man is carrying a mass of 5 kg in his hand. The weight of the mass transmits a force of 49.05 N to the hand of the man caused by the gravitational field of the earth. This result can be concluded either by comparison of the weight with a mass of known weight. The figure demonstrates also the application of the Eulerian cutting principle (see 2.11.4) making the internal contact force visible.

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Graphical statics

Fig, 6: Cutting principle of EULER. This mountain climber is hanging from the rock, sliding gently downhill using the rope technique. Cut a closed surface around him. The cut hits the ropes in two places. Put at the two cut surfaces the two red internal rope forces (in tension mode: Arrow tip points away from the cut surface). In the current centre of gravity of the man’s body, apply the weight force G (pink). If we know that the lower part of the rope is hanging free, we can conclude: The red force at the bottom is zero under normal circumstances. It is evident that the red force at the top of the cut must compensate for the weight in order to guarantee equilibrium. This force creates tension in the rope and has the value G if the climber moves down at a constant speed.

2.11.2 External forces act from outside a body. They are always “visible”. 2.11.3 Internal forces Internal forces are not “visible”. They can be transformed into “visible” objects if we “cut” a body into two pieces and separate the created parts now having two new “cut” surfaces. The inner forces appear now to be external forces acting on the two surfaces in opposite directions (Fig. 5). More about this aspect under 2.11.4. 2.11.4 EULERian cutting principle (Method of control surface) A simple procedure introduced by the Swiss physical and mathematical genius Leonhard Euler (1707-1783), [see Szab6] is a very important and useful method of thinking to make inner forces visible and accessible for physical description. It should be applied always at the beginning of an exact analysis of a mechanical problem. The application is very easy: If you “cut” a (drawn) mechanical system and move parts away from their original position, the cut

1 In hydrodynamics,

it is called a control surface.

Injury 2000, Vol. 31, Suppl. 2

must build a closed surface’ that means: In the case of a pure plane consideration, it must form a closed line. Practically, it has to be done like this: Remove all parts outside the closed surface and replace these parts by the former inner forces acting on these new cut surfaces. This allows us to make the inner forces visible. The internal forces are quasi changed to external forces, and these are now accessible to the investigator for an analysis. They appear as real physical facts now. It is always important to draw a closed cut around the part of the body under consideration. It is also important to check all “cut” surfaces to ensure that all inner forces (stresses) are included. Otherwise wrong results are predetermined. These inner forces also act on the removed parts of the body in the opposite direction. Therefore, if the removed parts are “puzzled together” with the rest of the body all inner forces eliminate themselves again and the body appears in its original configuration. The inner forces will disappear again. Comment: This simple principle will very often be underestimated. “Experienced” users do not draw closed cuts and work on incomplete open control surfaces neglecting or forgetting important inner forces. Wrong results are unavoidable. Therefore: A few more minutes of drawing and thinking time at the beginning can avoid an extensive, time-consuming search for the reasons for strange results afterwards. 2.11.5 Centre of gravity We know from experience that the weight of a body can be measured by comparing it with other known weights using a balance.

Direction of

Fig. 7: “Experimental” determination of the centre of gravity of a body. - Hang the body on an arbitrary fixed point (I’,) allowing free rotation. The vertical axis through the fixed point determines a line containing the centre of gravity (s). - Mark it appropriately on the body. Repeat this procedure a second time selecting a different fixation point (P2), not on the found axis. This procedure determines a second vertical axis containing the centre of gravity (s). - Mark the second axis appropriately. The intersection of these two axes defines the centre of gravity (s).

S-B28 If we cut a body into two arbitrary parts, we can detect that the sum of the weights of the two parts is identical with the earlier determined weight of the original body The weight is an additive property Despite the fact that the body weight is uniformly distributed over the whole body volume there is a specific point in the body where the total weight of a body can be assumed to be concentrated: The centre of gravity.1 This law allows us to replace a distributed weight of a body by a single force. Its weight force or, in brief, its weight.2 In the case of a rigid body, the assumption can serve as an exact model. The weight as a resultant gravitational force acts in a vertical direction in the centre of gravity. In the case of deformable bodies, it is a more or less good approximation depending on the stiffness of the body material. This centre can be determined by calculation or by measurement (Fig. 7). This point need not necessarily be identical with a real material point on the body A simple example for the validity of this statement is a hollow straight cylinder. It is evident that the centre of gravity must be positioned on the axis of the cylinder in the middle of its length. But the whole axis does not contain any material point of the cylinder! Therefore, the centre of gravity is not located in the material volume of this specific body.

2.11.6 Resulting

force Resulting force is the vector sum of a group of force vectors. It can be achieved by adding all force arrows graphically in arbitrary sequence always starting the next arrow at the tip of the last one. Finally, connecting the beginning of the first arrow to the tip of the last one will result in an arrow which represents the resultant force. The basis is given by the axiom of the addition of two vectors (see 2.10).

The moment is also a vector. It can be represented by an arrow which is perpendicular to the plane built by the pair of forces. Its length represents the value of the moment. The positive sign (tip of the arrow) is directed analogue to the forward motion of a screw with righthanded thread if the pair of forces were acting on it (Fig. 8). A moment acts on a body in such a way that the body would not translate but only start to rotate due to the fact that the pair of forces has no resultant force. Its only physical effect is..... the moment! 2.12 Translational

moment

If we translate a force perpendicular to its original direction (parallel displacement), we obviously change something regarding the effect of this force on its environment. If we would like to move a force F parallel to itself, say for a distance d , we can achieve this very easily by placing two opposite forces F (value and effect is zero!) at the distance d from the original force F. We see now that we have constructed a pair of forces F, having a moment F*d, and additionally a single force F at the distance d from the original position. Our conclusion: The original force can be replaced by the displaced force plus a correcting moment F+a which compensates the displacement of the force. This procedure is shown in Figure 10. We are able to perform this procedure in the reverse mode if we try to make a given moment M disappear. We translate the Force F perpendicularly exactly by the

M

‘t\

=FCY

2.11.7 Moment A moment can be represented by a pair of equal parallel forces F with opposite sign and a certain perpendicular distance a, having no resultant force. The value of a moment M is the product M = F a. (Fig. 8). This means that a pair of small forces with a large distance has the same value M as a pair of large forces having an adequate small distance.

1 In the case of a rigid body, it is a fixed point relative to the

body; in the case of a deformable body, it can change in position and time. 2 We will not enter into a detailed discussion of the differences between weight and mass. But: The mass of a body is a material property, the weight, caused by the action of the gravitational field on the mass, is a force.

Fig. 8: A pair of opposite forces F acting on a rigid body has no resultant force but a resultant moment of M = F a, where a is the perpendicular distance between the two forces. The moment can be symbolized by an arrow with a double tip which is perpendicular to the plane built by the two forces. The arrow is directed towards that side of the plane in which a corkscrew with a right-handed thread would move under the action of the two forces. The moment can be repesented by an infinite number of different force pairs because of the value of the force and the distance

Baumgart:

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Graphical statics

distance a=M/F to the right in such a way that the created translational moment and the original moment M compensate to zero (see also Figs 9 and 10). 2.13 Equilibrium If all the forces of a group are added and no resultant force remains, that is, if the tip of the last arrow exactly

meets the blunt end of the first arrow, the forces are in equilibrium. If these forces act at different points on the body, a resultant moment may remain which is able to rotate the body despite the fact that the vector sum of all forces disappears, We know already that we are able to compensate for this moment by an appropriate parallel translation of the resultant force. In order to determine the position of the resultant force, we have to draw a position plan. Practically, it is a picture of the force group which shows all forces in their correct position in the considered plane.

3 The methods of Graphical 3.1 Position

1 W=G . Fig. 9: This athletic woman has the weight G. The supporting opposite force G at the bottom is in line with the load G and compensates for the weight.

Statics

Plan

A Position Plan serves as the basic graphical drawing to determine the position of the resultant force. It needs a scale: 1 cm g y m. It shows the acting forces in their exact position and direction. The lengths (values) and the signs of the vectors are arbitrary (Fig. 11). However, it may be appropriate for reasons of better understanding to indicate also the right signs (tips) as far as they are given or expected by ex-

W direction

L

known

G

2

R

s

-- =200 N I

‘3 1 a-

G

Figure 10. Translation moment. Applying the EULERIan cutting principle, we can investigate the direct support forces acting on the hands of the sportswoman. We operate on a closed cut (blue line). We detect that obviously the vertical supporting force must compensate the weight G. But this support force G has a perpendicular lever arm a to the weight axis. It builds together with the weight a pair of forces resulting in a residual moment M = Ga . This will try to rotate the body clockwise. In order to reconstitute the original status of rest, we have to apply the supporting moment M = Ga (brown) counter-clockwise on the hands of the woman. If we move the brown supporting force G over the distance a towards the load line of G and place it now in this new position, we have to add a so-called translation moment M=Ga which acts clockwise in order to keep the equivalent action of the force regarding the new action point. Adding this to the brown moment (counter-clockwise) results in a zero moment. Then we have the same situation as is shown in Figure 9.

Injury 2000, Vol. 32, Suppl. 2

j

~ I

=50cm

direction unknown

Fig. 11: Position plan and force plan. Well knowing that this water skiing is a dynamic 3D problem, we assume that the man moves with constant velocity on a circular track. We work on a model using a vertical plane containing the centre of gravity of the man and the centre of the curve of motion. If we know the speed and the radius of his curve, we also know the red centrifugal force Z acting at the centre of gravity. The supporting yellow force W of the water acts on the ski, but it is unknown. The direction of the black holding force S of the driving cable is known. The graphical result shows that the black holding force S is small in comparison to the weight of the man. Due to its oblique position, the resultant pink force R is nearly parallel to the large water force (yellow). Therefore, the man has to transmit large forces through his legs, but only a small regulating force through his hand, provided he shifts his centre of gravity accordingly.

S-B30 3.2 Force Plan The second necessary tool is a Force Plan. It is the basic graphical picture for determination of the length and direction of the resultant force. It needs a scale: 1 cm ax N. In the force plan, all forces (arrows) are drawn in their original length and direction according to the defined scale factor. The force plan does not reflect the position (acting points) of the forces on the body. 3.3 “Funicular

Polygon”

The “Funicular-Polygon” is a plane graphical procedure for finding the position of the resultant force in order to make the resultant moment disappear. The resultant moment goes to zero if the “FunicularPolygon ” in the position plan also forms a closed loop. This guarantees that all acting forces intersect at one specific point. Then the body is in a state of complete equilibrium.

- Begin with the first arrow. Start the next force (arrow) at the tip of the first arrow. Add then the next force and so on and so on until the last force of the group is drawn. - Draw an arrow from the beginning of the first force to the end (tip) of the last force. This arrow is the resultant force. The example in Figure 12 shows an easy symmetrical case.l We will now discuss a more difficult case: Eccentric weightlifting! See Figure 13. The drawn assembly of connected arrows in Fig. 14 is the Force Plan. It needs a scale: lcm = . .. .. N to allow quantitative evaluation. The sequence of the forces is irrelevant. You may follow any arbitrary numbering and you will always get

I=, = IOOON

4 . Application

of the method,

weight excentric!

f2 = IOOON

examples

4.1 Parallel forces in a plane 4.1.1 Construction of the ,,Resultant Force” of a given known group of forces A group of forces in a plane can be summarized to a “resultant force” which is the vector sum of all forces (Fig. 12). In graphical statics that means: - Draw the forces as arrows with equivalent the values of the forces in a force plan.

Fl =lOOON

lengths to

Fig. 13: Eccentric weightlifting. Graphical statics allows for exact determination of the changes in the supporting forces A and B.

F2 =lOOON

Fig. 12: Example: Weightlifter. Loading forces. This weightlifter carries the two heavy weights of 1000N in a well centred, balanced way. His own weight of 800N must be added. It is very easy to conclude that the two supporting forces acting against his feet must have the value of 1400N each. The total resultant loading force (not drawn) obviously acts in the body axis and has the value of 2800N. 1 The world champion in weight-lifting might not be able to lift the weights used in this example. However, the round numbers are selected for simplification.

2 500

N

Fig. 14: Force plan. The two red forces F, and F, and the blue weight G are added as vectors. The pink arrow connects the beginning of the first force with the tip of the last force and represents the resultant force R regarding sign, direction and value (length), but not regarding position. Its length can now be determined by comparing the scale marker on the bottom right with the arrow of the resultant force.

Baumgart:

Graphical statics

S-B31

the same resultant force. This procedure is explained specifically in the text of the example in Figure 14. You have now found the resultant force as a vector arrow, but you still need to know where the starting point of the vector is located in the position plan of your force group. It is recommended to draw both plans on the same sheet of paper in order to facilitate the transference of parallel lines from one plan to the other one. The position plan also needs a scale, e.g. lcm = y m. In order to determine the position of the resultant force in the position plan, do the following (Figs 14 and 15): - select an arbitrary point P not too far away from the forces in the force plan, - draw all the lines from P to the corners of the force polygone. - Number the lines according to the numbers of the neighbouring forces: the line 1-2 lies between F, and F,. The line 1-R lies between Fl and the resultant force R. - Go to the position plan and start at F,, select an arbitrary point, and draw a parallel line to the line l-2 through this point. The line intersects F, at a point. Draw a parallel line to the line 2-3 through this point. It intersects a further Force F, at a third point etc.

Position plan I

I

F2=lOOON

n

A SC&: -

=^50cm

'

+

R

Fig. 15: Position plan. This drawing contains the forces in their right position and direction. The values are not necessarily to scale. Transfer the directions of the green help lines parallel from the force plan into this picture. - Begin with one force, say F,. - Draw the green lines l-2 and 1-R through one point of the red F, line. The line l-2 intersects the red F, line at a point. - Draw the green line 2-G through this point and elongate it until it intersects with the blue G axis. - Draw the green line G-R through this point and bring it to an intersection with the first green line 1-R. - This intersection point defines one point of the resultant force R. - Take the direction of the force R from the force plan and draw a parallel through this point This is the position of the resultant force. Remember: The green lines in the force plan which enclose a force, say 1-2,and 2-G for F2, must intersect in the position plan exactly on the axis of this force F,. This rule helps in final checking of the drawing!

Injury 2000, Vol. 31, Suppl. 2

Follow this method until you intersect the last force F, with the last line. In this case, the number of forces is n=2. Therefore, the last force was F,. - Draw the line 1-R through the start point and the line n-R through the last point. These two lines intersect at a point which is a point on the line of the resultant force. Draw a parallel to the resultant force in the force plan through this point and you have the location of the resultant force in the position plan. This drawing is called “Funicular Polygon”. The drawing in the force plan is called “force polygon” accordingly. 4.2 Stable support of a rigid body in a plane 4.2.1 Construction of unknown support forces A free rigid body will start to move if the resultant force or the resultant moment is not zero. Therefore, a stable support or connection to the environment is necessary. This aim can be achieved in the space by 6 different “legs” fixing the 6 degrees of freedom of a body to the environment (space). In a plane the number of such “legs” is reduced to 3 (degrees of freedom). This may be achieved by 3 forces or by two forces and a moment.1 Therefore, the second task of graphical statics is the determination of these unknown three supporting forces. If the resultant force is already determined, the task is as follows: The resultant force must be split into three different forces which do not intersect all in the same point.2 The method is easy: Intersect the resultant force direction with any of the three support forces, we call it now A. You get an intersection point Pl. Intersect the remaining two forces (B and C) and determine the intersection point P2. Draw the line Pl-P2. This is the so-called Culmann line H. Using the force plan, split the resultant R at the point Pl into the two components parallel to H and to A and move the component H from the point Pl to the point P2. Using the force plan again split the component H to the two other directions parallel to B and C. Now you have determined the three support forces A, B and C (see example: Fig. 16).

1 Be aware of the fact that graphical statics cannot solve the problem if a rigid body is supported by more than three nonparallel forces. 2 In such a case, the forces would not be able to prevent any rotation around this point.

S-B32

Eccentric weight position!!

_

Force plan B=lSIl N

lcm = ...N

Determination of the“.‘. position of the resultant force R i Determination of the ~twosupporting forces A and R

Fig. 16: Stable support of a “rigid” body in the plane. The three supporting forces A, B, C are unknown, the resultant external force G is given. Using the position plan, intersect the blue resultant force with the black force B of the right crutch (upper red circle). Intersect the red support force A and the left green crutch force C (lower red circle). Draw a (yellow) line through the two points. This is the socalled CULMANN straight line. Draw a parallel to the yellow line through one end point of the force G, and a parallel to the black force through the other end of G. The intersection of both lines determines the forces B and CULMANN (yellow). Draw a parallel to the red line and to the green line through the endpoints of the CULMANN force. The intersection point determines the two forces A and C. Assign the tips of the arrows of the forces A, B, and C in such a way that the arrows start at the tip of G and end at the begin of the force G in a natural sequence. Compare the lengths of the arrows with the ruler in the force plan and write down the numerical results. It has to be mentioned that this construction is only valid and applicable if we can confirm that the three forces A, B, C and the resultant force R act in the same plane. This is obviously only valid for a certain moment in time during the slow acceleration-free motion of the woman.

A= 1750 N

/

-

Fig. 18: Example: Position plan for the resolution of the resultant force into the support forces A and B. We have here the special case that only forces in a vertical direction are applied. Therefore, the feet of the weightlifter do not have to compensate for a horizontal force, the horizontal force can be set to zero. We only have to look at the two vertical forces A and B under the feet. - Take parallels from the two pink lines in the force plan, - transmit them to this drawing intersecting the pink resultant force vector in the same point. The two intersection points with the axes of A and B determine a new brown line. A parallel to this line must be transmitted back to the force plan running through the polar point P (see Figure 17). It intersects the resultant in the force plan dividing it into the forces A and B. The results show that there is a remarkable increase of supporting force A in the right leg, but still not yet an instability of stance. The force B has decreased accordingly.

4.3 Internal force in a rope Coming back to the example in Figure 6, but in a different phase. Figure 19 shows a mountain climber in a different situation from before: He is additionally loaded and the directions of the forces have changed. We apply the described method and apply the cutting principle. Position plan, force plan and results are shown and explained in Figure 20. Despite the fact that graphical statics is a very vivid method, checking the results is highly recommended. The force in the upper part of the rope shows 1640 N in tension. The weight of the climber plus the force transmitted from his partner results in 1600 N vertical load. From the slight angulation caused by the foot support an additional value can be expected. Therefore, the result seems to be reasonable.

2 500 N 4.4 Internal forces in the foot

Fig. 17: Example: Determination of support forces (reaction forces). Eccentric loading! In order to determine the supporting forces, which are the reactions to the given load configuration, we again use the force plan from Figure 14. We have to start with the position plan, see Figure 15.

The determination of internal or inner forces in a body can be handled by the same methods as we described above. We always have to apply the EULERIAN cutting principle first.

S-B33

Baumgart: Graphical statics

But the accuracy of our results depends very much on the choice of the mechanical model we use. Our imaginary cut through a body will hit several muscles, tendons or ligaments which may transmit forces or not. It needs a selection of such cut parts which do not transmit any force in the given motion phase. They can be neglected in the model and only the load transmitting parts considered. The anatomy of the foot is very complex, but some selected configurations of the loaded foot can be replaced by simple models. One case is the example of uphill walking as explained in Figure 21. Figure 22 shows the graphical solution of the problem according to the methods described. Fig. 19: Graphical determination

of the force in a rope.

First of all the resultant force of the two loads C and G were determined. - In the force plan, draw first the blue arrow for G, add the red arrow for C and get the pink resultant force R as the arrow running from the beginning of G to the tip of C. - Select an arbitrary pole P and draw the green lines to the relevant points in the force plan. - Transfer the green help lines parallel to the position plan (left) and find the intersection point between the two green dotted lines as a point of the resultant force R. Now the position of the resultant pink force R is known. Determine the intersection point between the yellow rope force (known direction) and the pink resultant force (little yellow circle), and draw the green connection line from the foot contact point (little red circle) to this intersection point. This is the direction of the foot contact force F. Transmit the yellow and the green lines parallel back to the force plan and let them run through the beginning and the end of the resultant force. The intersection of the green and the yellow line determines the lengths of the forces S and F. Draw the tips of the arrows (signs of forces S and F) in the force plan so that the arrows form a closed polygon track if followed in sequence.

4.5 Classical example of the hip forces in one leg stance loading 4.5.1 Description of the model One of the most shown examples in graphical statics is the determination of the hip joint force in single leg stance mode in gait analysis. Every time that it is shown, one can detect that the laws of graphical statics have been misunderstood or not understood at all. Therefore, the solution of this task should be repeated here. In the case of the single leg stance of a man, he has lifted one leg off the ground. That means the whole weight is transmitted through the other foot to the

Model:

Solid foot

Example: Heel off, walking uphill

Fig. 21: Example: Internal forces in the Achilles tendon and in

Fig. 20: The mountain climber is only supported by one foot touching the rock and is additionally loaded by an accidental load from one of his comrades hanging for the moment on the lower part of the rope. We look for the force which the upper part of the rope must transmit.

I+.4 y 2000, Vol. 31, Suppl. 2

the talo-tibia1 joint during uphill walking. The closed blue cut opens the approach to the internal joint force and to the tendon force in the Achilles tendon. The direction of the tendon force is given by the tendon axis. The direction of the joint force is arbitrary so far, but this force has to go through the axis of the joint. From the overall analysis of the whole body it is known that the ground reaction force R in the case of a single leg stance phase must be equilibrated by the weight G of the body. We assume slow, acceleration-free motion. The graphical solution is shown in the next figure. For those readers who like analytical methods: Obviously R*a =A*b is the condition for equilibrium of the moments of the two forces regarding the joint axis. It allows for determination of the Achilles tendon force A.

S-B34 ground. It is also clear that the centre of gravity of the body is positioned vertically exactly above the supporting foot. We use a closed cut through the hip joint, through the lateral abductor tendon, and then around the whole body. This whole part of the body has the weight of about 5/6 of the body weight G, assuming that one leg has l/6 of body weight (see Figure 23). The centre of gravity of the right leg can be located above the knee joint and slightly lateral to the centre of the joint. If we fix this partial weight force R at the right leg, we are able to determine the other weight force L of the left part by using the graphical method for decomposition of a force (G) into the two parallel forces L and R, as described above. We are interested in the hip joint force and in the force in the abductor tendon. If we now separate the two parts of the body, these two inner forces become visible and we are able to determine them using one of the two parts of the body

k

Independently from the part of the body we use for the graphical solution, the result must be the same. This is evident because the inner forces acting on the two parts have the same absolute value but opposite signs: They must compensate each other if we puzzle together the two parts of the whole body as a unit. 4.5.2 Classical model At first, we demonstrate the method on the classical figure of the left part (see Figure 24). The whole body weight G (pink, dotted) was already split into the two forces L and R by the little construction shown in the force plan. The method was as follows: Choose the arbitrary pole I’, Add the pink vector G and the known blue force R, Draw the yellow force vector. This is the resultant force L. Draw the three lines to the pole l? Transmit the two pink lines from the force plan into the position plan through an arbitrary point of the action line of the weight force G (and W).

position plan

force plan 1cm = )’ N

or

-=800N

Fig. 22: Graphical analysis of the forces in a rigid foot model. Using the example from the last figure, we start with the po-

sition plan. We know the direction of the forces G and A. This allows us to find the intersection point P of these two forces. The joint force F must go through the joint and through this intersection point in order to ensure that the resultant moment disappears. This procedure defines the black line which will be transmitted into the force plan, where we have already drawn the known ereen force R. Draw a p&allel to the red line through one end G and a parallel to the black line through the other end. Both lines intersect at a point S. Now we have determined the lengths of the forces A and F. In order to determine the sign of the forces, start with the sign of the green force G indicated by its arrow. Follow this arrow and assign consecutively an arrow to the black joint force F. Go ahead to the red line and assign in the same way an arrow to the red tendon force A. The tip of this last force in the force plan ends at the beginning of the first force G. This means: The sequence of the arrows in the force plan must form a closed loop! If this can be achieved, the force group G, F, T is in equilibrium. No resultant force remains.

-llW~Gl Fig. 23: Hip forces in “Single leg stance”. Body weight G (cyan arrow) acts exactly vertically above the supporting point. The supporting force W counteracts the body weight in the same line (black dotted) and provides equilibrium. The imaginary cut through the right hip joint divides the body into two parts: The right leg (weight l/6 G) and the rest of the body (5/6 G). The two single forces L and R act on two different vertical lines. These lines go through the two centres of gravity of the two parts of the body. The two forces together are equivalent to their resultant force: The body weight G. They can be used instead of it.

Baumgart:

Graphical statics

S-B35

- Intersect the left pink line with the blue force R. - Transmit the green line from the force plan into the position plan and draw it through this intersection point. - Intersect the green line with the right pink line in the position plan. This point is a point of the action line of the yellow resultant force L. 4.5.3 Alternative model Another possibility for the solution uses the closed cut around the right leg. We will show this just to demon-

Position plan

0

strate the flexibility of the method. The results are exactly the same (see Figure 25). The forces acting on this closed cut are the supporting force W, the weight force R of the right leg and the unknown inner forces A and F as described above. The addition of the two forces W and R in the force plan results in the equivalent force L = 5/6 G. Using the arbitrary pole P the green and pink lines in the force plan can be drawn. Transferring these lines to the position plan as described earlier, we are able to determine the action line of the resultant force L. It runs through the intersection point of the two pink lines (see the little triangle at the bottom of the position plan). The force L is the resultant force of the two forces R and W and will replace them completely. The determination of the inner forces A and F is described in the text of Figure 25. Obviously we can obtain the same results using one or other cut part of a body. It is just a question of “economy”: We are allowed to choose such a cut as will deliver

= ...N

Q

Position plan Icm = ...cm

Fig. 24: Classical example: Graphical determination of the hip joint force F using the upper part of the body. The resultant force acting on this closed cut is the yellow force L shown in the position plan. All other (dotted) forces are only used for to illustrate the description in the text. The hip joint force F (black) is known to go through the centre of the hip joint (red filled circle), but its direction is completely arbitrary so far. The abductor force A (red) is known regarding location and direction for anatomical reasons: The axis of the tendon defines it. The method is now as follows: In the position plan elongate the action line of the abductor force (red) and intersect it with the yellow action line of the resultant force L. The point Q is found (red circle). Draw the black line through this point and through the centre of the coxa. Transmit the red line and the black line into the force plan. Draw the red line through the beginning of the yellow force vector in the force plan. Draw the black line through the tip of the yellow force vector. The two lines intersect and determine the two force vectors F and A. (Comment: The last two steps can be done differently, choosing the other ends of the yellow vector. The figure will then appear below the force plan, and not above as drawn. The results are exactly the same.) The directions of the arrow tips must be chosen in such a way that the arrow tips followed in sequence around the closed figure always point in the same direction. In this way, we have now found that the hip joint force acts as a compression force against the acetabulum, whereas the abductor force F is a tensile force. This is in agreement with our expectations. It is recommended, based on experience, that results are always checked. Injury 2000, Vol. 31, Suppl. 2

IAj

Force plan

R=1/6

P

Fig. 25: Alternative solution using the right leg cut. The pink resultant force L is the only external force we have to consider, it replaces the forces W and R. The determination of the inner forces, the abductor force A and the joint force F, is the next task. Do the following: - In the position plan elongate the red abductor force line and intersect it with the pink action line of the resultant force L. It results in the point Q (red circle). - Draw the black line through this point and through the centre of the hip joint. This is the action line of the joint force F. - Transfer these two lines (red and black) into the force plan and draw them through the ends of the pink resultant force. The two lines intersect and build a triangle together with the pink force. The lengths of the two forces are now known. - Draw the arrow tips to the red and black force beginning with the given sign of direction by the pink force, followed by the red and finally by the black force in the same way. This builds a closed loop having no resultant force, this means: The three forces are in equilibrium. The equilibrium regarding the moments is fulfilled because the three forces intersect in the postion plan all at the same point Q. This means: There is no resultant moment and the body part will not start to rotate.

S-B36 the required results with a minimum of drawing and calculation effort. This minimizes at the same time the possibility of making a mistake.

5 Final remarks Originally this issue was presented in 1994 as an internal advanced educational topic in the A0 Research Institute, Davos, but it was never published. The aim of this paper is to remind ourselves of a very old but still valid method of determining unknown internal or external forces of bodies in a very plastic view. It is written partially as a teaching tool for immediate application. For the same reason, many illustrative examples should help to overcome the difficult biomechanical content. The paper should also help readers to better understand publications containing graphical solutions of bio-static problems. The accuracy of the results depends only on the accuracy of the models we use for graphical analysis. If we use wrong anatomical or functional models we cannot expect correct results. In any case the application of the method requires good anatomical knowledge and experience of the function of the musculo-skeletal apparatus of living bodies. The application of the method is restricted to plane static problems and to rigid bodies. This does not prevent application on deformable living bodies if we consider a specific static configuration of the body.

6 Acknowledgements The author thanks Dr. Jacques Cordey, Filisur, for the idea of bringing this issue to the attention of a larger public, Joy Buchanan, Hombrechtikon for discussions and editing, Dr. J&-g and Sabine Goldhahn, Davos, for discussions and critical comments, and Walter Seidl, Bochum for recommendations regarding the figures.

7 Relevant literature Szabo

S-B73

Abstracts

Graphische Statik - Ein vergessenes Werkzeug zur Liisung von mechanischen Problemen in einer Ebene

Die Belastung der Osteosyntheseplatte und die biologische Knochenreaktion bei Frakturbehandlung in vivo im Vergleich

F. Baumgart A0 Technische Kommission, CH-7270 Davos Platz

K. Stoffel, K. Klaue, SM. Perren A0 Research Institute, Clavadelerstr., Platz

Clavadelerstrasse,

CH-7270 Davos

Zusammenfassung

Zusammenfassung

Die graphische Statik ist eine fast vergessene, intuitive zeichnerische Methode zur Losung von mechanischen Problemen in einer Ebene. Sie wurde bereits im 19. Jahrhundert fur biomechanische Probleme verwendet. Noch in den 40er Jahren des 20. Jahrhunderts war sie eine von Bauingenieuren angewendete Standardmethode. Abgelost von modernen analytischen Methoden, verschwand die graphische Statik fast vollstandig. Die Methode ist auf ebene Statikprobleme beschrankt, aber sie ist immer noch ein niitzliches Werkzeug zur Veranschaulichung, zum Verstandnis und zur Kontrolle der Wirkung von Kraftegruppen, die bei modernen biomechanischen Problemen auftreten. Im Anschluss an die Definition der grundlegenden (K&per, mechanischen Terminologie Bewegung, Krafte) ist der Artikel hauptsachlich als Lehrmittel zur Anwendung gedacht. Zahlreiche unmittelbaren Anschauungsbeispiele (sportliche Aktivitaten, funktionelle Krafte in Gelenken) helfen bei der Erklarung des schwierigen biomechanischen Inhalts. Fur die Anwendung muss davon ausgegangen werden, dass die zu erforschenden K&per sich unter der Krafteeinwirkung wie starre Korper verhalten, was jedoch die Anwendung der Methode auf verformbare lebendige Kdrper nicht ausschliesst, wenn spezifische statische Konfigurationen der Korper beriicksichtigt werden. Die Anwendung der Methode erfordert gute anatomische Kenntnisse und Erfahrung mit der Funktionsweise der Skelettmuskulatur lebender K&-per. Wenn zuverlassige Modelle verwendet werden, liefert die Methode ausreichend exakte quantitative Resultate. Der Artikel kann ebenfalls zu einem besseren Verstandnis von Veroffentlichungen mit graphischen Losungen von bio-statischen Problemen beitragen.

In der Klinik ist man bestrebt, bei einer Plattenosteosynthese die Platte gegeniiber dem starksten Muskelzug anzubringen. Dadurch lasst sich eine optimal verteilte Kompression zwischen den frakturierten Enden erreichen (Zuggurtungsprinzip). Dies ist aber aus anatomisch-chirurgischen Grunden haufig nicht moglich. In einer ccin viva,) Studie von 8 Wochen wurden bei 16 Schafen nach einer standardisierten schragen Osteotomie der Tibia vier verschiedene Operationsmodelle (dem Knochen angepasste und uberbogene Platte mit oder ohne Verwendung einer interfragmentaren Zugschraube) unter Ausschluss des Zuggurtungsprinzips uberpriift: es wurde die Spannung auf der Plattenoberflache mittels Dehnungsmessstreifen bei unterschiedlithen Ganggeschwindigkeiten auf einer Rollgehbahn registriert. Diese Messungen wurden wahrend der ganzen Versuchszeit durchgefiihrt. Zur Stabilitatsbeurteilung wurden regelmassig Riintgenaufnahmen angefertigt und nebst polychromen Sequenzmarkierungen zur Untersuchung des Heilungsmodus wurden mikroradiographische Untersuchungen durchgefiihrt. Moglithe Beziehungen/Wechselwirkungen zwischen Plattenspannung und Knochenheilung wurden untersucht. Die Implantatbelastung unter Biegespannung liess sich in der Kombination Platteniiberbiegung zusammen mit einer Zugschraube am Starksten reduzieren. Die Verwendung einer Zugschraube reduziert sowohl bei der uberbogenen als such oberflachenkonform angepassten Platte die Oberflachenspannung auf der Platte. Unter Ausschluss des Zuggurtungsprinzipes ist die alleinige Verwendung einer planen Platte am Starksten auf Biegung und Torsion belastet. Eine direkte Knochenheilung konnte nur in der Gruppe mit einer oberflachenkonform angepassten Platte mit einer Zugschraube nachgewiesen werden. Die iiberbiegung gab in Kombination mit einer Zugschraube eine relativ instabile Fixation. Ein residueller Spalt knapp unter der Platte lasst eine c
Schhisselworter: starre KGrper, mechanische Probleme in einer Ebene, funktionelle Biomechanik, innere Krafte, Positionsplan, Kraftplan

Inju y 2000, Vol. 31, Suppl. 2

Abstracts

La statique graphique: un outil intkessant dClaissC pour rCsoudre des problemes de mkcanique bidimensionnelle

S-B77

mais

La charge fonctionnelle des plaques de fixation in vivo et sa corrklation avec la consolidation osseuse

F. Baumgart Commission technique AO, Clavadelerstrasse, CH-7270 Davos Platz

K. Stoffel, K. Klaue, S.M. Perren Institut de recherche AO, Clavadelerstrasse, CH-7270 Davos Platz

R&urn6

Resume

La statique graphique, une methode intuitive largement oubliee, permet la resolution de problemes de mecanique bidimensionnelle. Utilisee d&s le XIXe siecle en biomecanique, elle est toujours employee en genie civil dans les an&es 40 comme methode standard. Mais, avec le developpement de methodes analytiques modernes, la statique graphique est tombee dans l’oubli. Bien qu’applicable qu’aux problemes statiques bidimensionnelles, elle reste un outil interessant pour mieux visualiser, comprendre et valider les actions de groupes de forces rencontres dans les problemes modernes de biomecanique. Apres avoir defini les termes mecaniques appropries (corps, mouvement, force), cet article se presente principalement comme un outil de formation permettant une application immediate. Plusieurs exemples (activites sportives, forces fonctionnelles articulaires) illustrent sa capacite a rendre plus lisible des problemes de biomecanique complexe. Lorsqu’on applique cette methode, on accepte l’hypothese que les corps etudies et soumis a une force agissent en tant que corps rigides. Ceci n’est pas en contradiction avec son application aux corps vivants deformables si l’on etudie leurs configurations statiques specifiques. La methode demande une bonne connaissance de l’anatomie fonctionnelle de l’appareil locomoteur des corps vivants. En se servant de modeles fiables, elle permet l’acquisition de don&es quantitatives d’une precision satisfaisante. Le but de cet article est de favoriser une meilleure comprehension des publications presentant des solutions graphiques de problemes de biomecanique statique.

En pratique clinique, on s’efforce d’appliquer la plaque de fixation sur le c&e oppose a la force musculaire la plus forte. Ceci permet une distribution optimale de la compression au niveau des extremites des fragments (principe de la fixation par bande de tension). Cependant, pour des raisons anatomiques ou chirurgicales, cette position est parfois impossible. Dans une etude in vivo d’une duke de 8 semaines, nous avons effect& une osteotomie oblique standardisee sur les tibias de 16 moutons afin d’etudier 4 modeles differents de fixation par bande de tension (une plaque contournee ajustee ou une plaque trop pliee, avec ou sans vis de compression interfragmentaire). La force appliquee a la surface de la plaque, enregistree par des jauges appropriees, etait mesuree pour differentes allures de marche sur un tapis rouiant. Des mesures rep&es effect&es tout au long de la periode d’etude et des contrbles radiographiques reguliers etaient faits afin d’evaluer la stabilite de la fixation. Des microradiographies sequentielles a marquage polychrome ont permis de suivre l’evolution de la guerison. Nous avons cherche des rapports et/au des interactions existantes entre la plaque de fixation et le processus de consolidation osseuse. La plus petite charge de flexion subie par la plaque etait observee pour la fixation associant une plaque trop pliee et une vis de compression. Avec l/insertion d’une vis de compression, on observait une reduction de la tension appliquee a la surface de la plaque, qu’elle soit contournee ou trop pliee. Les forces de flexion et de torsion etaient plus importantes pour les plaques droites utilisees seules sans application du principe de bande de tension. Une consolidation osseuse directe n’etait observee que dans le groupe traite par plaque contour&e et vis de compression. Un pliage excessif de la plaque associe a une vis de compression ne permettait qu’une fixation instable relative. Une beance residuelle sous la plaque etait associee a une cccompression dynamique,) car, lors de la mise en charge, la vis glissait vers l’osteotomie entrainant une resorption osseuse sous la plaque et des signes de decollement au niveau des vis. Dans les modeles oti une plaque contournee ou une plaque trop pliee etait montee sans vis de compression, l’histologie montrait des signes de deplacement secondaire des fragments, une formation evidente de cals, une resorption osseuse au niveau des extremites des fragments et

Mots cl&.: corps rigide, problemes de mecanique bidimensionnelle, biomecanique fonctionnelle, forces internes, position bidimensionnelle, force bidimensionnelle

Injury 2000, Vol. 31, Suppl. 2

Abstracts

S-B81

Statica grafica - Uno strumento dimenticato risolvere i problemi di meccanica piana F. Baumgart A0 Technical Commission, CH-7270 Davos Platz

per

Clavadelerstr.,

Carico funzionale delle placche nella fissazione in vivo e sua correlazione nella guarigione dell’osso K. Stoffel, K. Klaue, S.M. Perren A0 Research Institute, Clavadelerstr., CH-7270 Davos Platz

Riassunto La statica grafica e un metodo intuitivo di disegno quasi dimenticato, per risolvere i problemi di meccanica piana. Era gia usato nel XIX secolo per i problemi biomeccanici. Negli anni ‘40 era ancora un metodo standard utilizzato dagli ingegneri civili. Superata dai moderni metodi analitici, la statica grafica P quasi completamente scomparsa. I1 metodo I? applicabile solo ai problemi di statica piana, ma rimane tuttora uno strumento utile per visualizzare, capire e controllare le azioni dei gruppi di forze the si verificano nei moderni problemi biomeccanici. Dopo aver definito la terminologia meccanica di base (corpo, movimento, forze), il lavoro P inteso soprattutto come strumento di insegnamento per un’applicazione immediata. Molti esempi illustrativi (attivita sportive, forze funzionali nelle articolazioni) aiutano a chiarire i difficili concetti biomeccanici. Per l’applicazione, bisogna tenera presente the i corpi oggetto dello studio si comportano come corpi rigidi sotto l’azione delle forze, ma the quest0 non impedisce l’applicazione de1 metodo ai corpi viventi deformabili, se si considerano delle specifiche configurazioni statithe dei corpi. L’applicazione de1 metodo richiede delle buone conoscenze anatomiche e dimestichezza con le funzioni dell’apparato muscolo-scheletrico dei corpi viventi. Se si usano dei modelli affidabili, il metodo offre dei risultati quantitativi sufficientemente accurati. Questo lavoro pub anche aiutare a fornire una migliore comprensione delle pubblicazioni contenenti soluzioni grafiche a problemi bio-statici. Parole chiave: corpo rigido, problemi di meccanica piana, biomeccanica funzionale, forze interne, piano della posizione, piano di forza.

Ir+lry 2000, Vol. 31, Sllppl. 2

Riassunto Nella pratica clinica si cerca di applicare le placche per l’osteosintesi dalla parte opposta rispetto al muscolo the esercita la trazione pih forte. Cib permette di ottenere una distribuzione ottimale della compressione fra le estremith dei frammenti (principio delle placche in compressione). Spesso, tuttavia, quest0 P impossibile per ragioni anatomiche o chirurgiche. In uno studio ccin viva,, durato 8 settimane, si P eseguita un’osteotomia obliqua standardizzata sulla tibia di 16 pecore con quattro diversi modelli di placche in funzione di tirante (una placca modellata e una placca iperflessa con 0 senza vite in compressione interframmentaria). La trazione sulla superficie della placca e stata registrata per mezzo di estensimetri per differenti velocita di cammino su un cilindro orizzontale. Queste misurazioni son0 state eseguite per tutto l’esperimento. A intervalli regolari sono state eseguite delle radiografie per controllare la stabilita ed inoltre il process0 di guarigione P stato seguito mediante etichettature policrome sequenziali e microradiografie. Sono stati studiati i possibili rapporti e/o le interazioni fra la trazione sulla placca e la guarigione ossea. 11 carico dell’impianto sotto deformazione da flessione P stato molto ridotto nell’associazione della placca in iperflessione con una vite in compressione interframmentaria. L’introduzione di una vite in compressione interframmentaria riduce la deformazione superficiale della placca, sia the questa sia modellata o iperflessa. Le forze di flessione e di torsione sono maggiori se si usa una placca retta da sola, senza applicare il principio de1 tirante. La guarigione diretta dell’osso P stata osservata solo nel gruppo con placca modellata e vite in compressione interframmentaria. L’iperflessione della placca combinata con una vite in compressione interframmentaria ha prodotto solo una fissazione relativamente instabile. Una fessura residua immediatamente sotto la placca permette la wompressione dinamica)>, poiche le viti scivolano verso l’osteotomia quando caricate, provocando il riassorbimento osseo sotto la placca e dei segni di allentamento della vite. I modelli con le placche modellate o iperflesse, ma senza vite in compressione interframmentaria, all’esame istologico si sono rivelati molto instabili con segni di spostamento secondario dei frammenti, un’evidente formazione di callo, riassorbimento all’estremita dei frammenti e sotto la placca, rimodellamento Haversiano ritardato e diminuito, e punti di corrosione sia della testa delle viti, sia

Abstracts

La estitica de gdfica: una herramienta olvidada para resolver problemas mecPnicos en el plano F. Baumgart Comision Tecnica de AO, Clavadelerstrasse, CH-7270 Davos Platz

S-B85

Carga funcional de las placas en la fijaci6n in vivo de fracturas y su correlaci6n en la consolidaci6n 6sea K. Stoffel, K. Klaue, S.M. Perren Instituto de Investigation de AO, Clavadelerstr., 7270 Davos Platz

CH-

Resumen La estdtica grafica es un metodo de dibujo intuitivo, casi olvidado, para resolver problemas mecanicos en el plano. Ya se utilizaba en el siglo XIX para 10s problemas biomecanicos. En 10s anos cuarenta era todavia el metodo estandar utilizado por 10s ingenieros civiles. Sustituida por 10s metodos analiticos modernos, la estatica grafica desaparecio casi por complete. El metodo ha quedado limitado a problemas estaticos en el plano pero sigue siendo una herramienta util a la hora de visualizar, entender y verificar las acciones de grupos de fuerzas que se dan en 10s problemas biomecanicos modernos. Tras definir la terminologia mecanica basica (cuerpo, movimiento, fuerzas), el trabajo se ha escrito principalmente a modo de herramienta de ensenanza para aplication inmediata. La inclusion de muchos ejemplos ilustrativos (actividades deportivas, fuerzas funcionales en las articulaciones) ayudan a aclarar el dificil contenido biomecanico. Para su aplicacion, es necesario asumir que 10s cuerpos investigados se comportan coma cuerpos rigidos bajo la action de las fuerzas, pero esto no implica que no pueda aplicarse el metodo a cuerpos vivos deformables siempre que se consideren configuraciones estaticas especificas de estos cuerpos. La aplicacion de1 metodo requiere poseer un buen conocimiento anatomico y experiencia en el funcionamiento de1 aparato musculoesqueletico de 10s cuerpos vivos. Si se utilizan modelos fiables, el metodo proporciona unos resultados cuantitativos de una precision suficiente. Este trabajo puede tambien ayudar a la comprension de publicaciones que contienen soluciones graficas a problemas bioestaticos. Palabras clave: cuerpo rigido, problemas mecanicos en el plano, biomecanica funcional, fuerzas internas, plano de posiciones, plano de fuerzas

kjury

2000, Vol. 31, Suppl. 2

Resumen En la practica clinica se intenta aplicar una placa de fijacion en el lado opuesto al de mayor fuerza de traccidn muscular. Con ello se consigue una distribution optima de la compresion entre 10s extremos de1 fragmento (principio de la colocacion de placas con banda de tension). No obstante, con mucha frecuencia esto no es posible debido a razones anatomicas o quirurgicas. En un estudio ‘in vivo’ de 8 semanas de duration, se realizaron osteotomias oblicuas estandarizadas en tibias de 16 ovejas y se evaluaron cuatro modelos diferentes de aplicacion de placas de tension (placas perfiladas y placas dobladas en exceso, con o sin tornillo de traction interfragmentario). Se registrb la tension en la superficie de la placa mediante un medidor de deformaciones, para distintas velocidades de paso de la rueda de ardilla. Estas mediciones se tomaron a lo largo de todo el experimento. Se tomaron radiografias a intervales regulares para evaluar la estabilidad y mediante el marcado secuencial en distintos colores y las microrradiografias se investigb el proceso de consolidation. Se estudiaron las posibles relaciones y/o interacciones entre la tension de la placa y la consolidation osea. La carga de1 implante bajo deformation por flexion se redujo al maxim0 para la combination de placa sobredoblada con tornillo de traction. La insertion de un tornillo de traction reduce la deformation superficial en la placa, tanto si se trata de placas perfiladas coma de placas sobredobladas. Las fuerzas de flexion y de torsion son mayores si se utiliza una placa recta solamente y no se aplica el metodo de placas con banda de tension. La consolidation osea directa se observe solo en el grupo con placa perfilada y tornillo de traction. El sobredoblado de la placa en combination con el uso de tornillo de traction result6 en una fijacion relativamente inestable. La existencia de un espacio residual inmediatamente debajo de la placa permite la c