Handles pressure deformations of wood

Tilt resistance of dowel connections

content

Preliminary remarks

Assignments

Equations

Tables

in this work

Roman numerals

Roman numerals

from DIN 1052: 2004-08

Arabic numbers> 100

starting with letters

from DIN 1052-2: 1996-10

Arabic numbers <9

Arabic numbers

Table I: Assignments of equations and tables of work and DIN's

thanksgiving

After the article on the tilting load capacity of a dowel connection was published, a very fruitful discussion with a specialist colleague took place. In evaluating this conversation, the article has been expanded to include the following sections: Problem and effective number of anchors (effective number of anchors).

Problem

As a rule, the restraint of a column is stressed in two axes (x and y direction). For stability analysis, the restraint in the two mutually perpendicular (orthogonal) axes must be examined jointly and / or separately. The main stress is often a tilting stress and the fastening is done with dowels. The dowels can be subjected to normal tensile or compressive stress in one of the tilting axes (normal case of DIN 1052), while the dowel connections are subjected to tilting in the axis perpendicular to this. This stress is the subject of the present study. Because of the low permissible moments, it can also be seen why this tilting load should never occur. This is probably why the tipping load is not included in DIN 1052: This load should never play a role for static safety.

There is another reason for the fact that this stress must never occur. Connections are often a weak point and therefore connections may only be considered as existing if they are standardized. However, there is no standard for this stress on the connection, i.e. this connection must not be used. If, however, no connection is made, a support in this axis is to be treated as a pendulum support.

If one believes that the connection is to be assumed to be completely rigid (cantilever support), it must be proven that the deformation under the forces acting is less than 1/1000 (DIN 1052 part 1 point 10.2.5) of the length. Without this verification, assuming a column as a cantilever is a statics error.

For a moment in the direction of the anchor axes (i.e. tilting perpendicular to the anchor axes), at least two anchor groups must be present. The permissible moment for this load is obtained as the product of the permissible tensile or compressive force of each anchor group (according to DIN 1052) times the distance between the two anchor groups. If someone calculates a moment for the perpendicular direction (moment perpendicular to the anchor axes) analogously (permissible tensile or compressive force of a group of anchors times anchor length) and describes the resulting moment as permissible, then he is making a big mistake, because the permissible one The moment is at least 6 times smaller, as will be described in the following work.

Two examples are given to illustrate the problem.

The base point restraint of a column is subject to tilting stress when a horizontal force (e.g. due to wind) acts on the column head (or along the column). This horizontal force must be derived. If there are no other conductors (e.g. a bracket on the column head), this is done by clamping the base. This derivation must be proven and usually does not cause any problems and is therefore not considered further in this article. In contrast to the relatively problem-free derivation of the H-force, deriving the following moment is more difficult. This moment arises because the H-force and its derivation have different points of application. This moment must also be derived from the base point restraint. The moment load on a connection is unusual in wooden structures, as wooden structures are usually stiffened. On the other hand, unreinforced steel structures are more likely, but bracing is better arranged for these as well.

A second example is a flagpole. The flag causes a horizontal force to act at the upper end of the mast. The resulting forces and moments have to be diverted back into the ground through the mast clamping. How should the mast bracket be measured? The dimensioning of the mast itself is not a problem.

The purpose of the article is to consider the derivative of moments. Can a given construction fulfill this task under all normal loads or not?

In the case of the tensile-compressive load of a connecting structure, the permissible load of a group of anchors is less than the product of the permissible load of the individual anchor times the number of anchors. This reduction is treated in the standard as the effective number of dowels and treated by equation (210) in DIN 1052: 2004-08. What is the effective number of dowels for tilting loads?

Note 1: If the base point restraint is not just a short piece of sheet steel, but a long, rigid restraint, the moment is not transferred via the dowels, but directly to the restraint. This case (often found on flagpoles) is not the subject of the investigation.

Note 2: In the case of flagpoles, the mast clamping usually acts as a quiver (or sleeve), the screws that hold both halves of the quiver together only have the task of holding the two halves together. In order for the "quiver" to do its job, the quiver must be sufficiently stable. That is why they are often double-T profiles or thick reinforced concrete elements. In the perpendicular (orthogonal) direction, the screws that hold the case halves together act as normal bolt connections that can be calculated in accordance with DIN 1052.

Note 3: If the brackets are too soft for the bolts, the brackets cannot act as a quiver. If, for example, flat steel 10 mm x 100 mm and a height of 60 cm is used in the example calculated at the end, the bracket would bend by 19 cm (calculated as a cantilever arm) under the effect of the forces permissible for the dowel connection. This shows that a thin holder has no quiver effect. The quiver effect only sets in when the bending of the bracket is less than the dowel obstructing the tilting of the support. At the same time, a distinction is made between when there is a thick or thin restraint.

Note 4: If the restraint acts as a quiver, the sufficient strength of the quiver must of course be proven.

Effective number of dowels

A base point restraint with several dowels, which are held by steel sheets, is considered here.

Figure I: base point restraint

Under the action of a moment, the restraint is deformed, as shown in Figures II and III below. The resulting deformation figures are shown in red. There are two extreme cases of deformation, whereby the changes in length of the steel sheets are always small compared to the deformations of wood, so that the changes in length of the steel sheets can be neglected:

1. If the steel sheets are thin, the vertical steel sheets will follow the tilting movement of the support. The pivot point of the tilting movement is approximately at the level of the sheet steel fastening. The dowel / sheet steel connection points remain at the same height and all dowel connections basically have the same deformations. The total moment for a given deformation is the product of the individual moment times the effective number of anchors (which is less than the actual number of anchors). In other words: the product of the individual anchor load times the number of anchors must be reduced by the ratio of the effective number of anchors to the actual number of anchors. This type of deformation is shown in Figure II.

2. The pivot point is in the middle of the anchor group. With each dowel, the wood is deformed differently depending on its distance from the center of rotation (closer distances less). The contribution of each dowel to the total moment depends on its distance from the center of rotation, so the most distant dowels make the largest contribution, closer ones less. The closer dowels deform the wood less. This type of deformation is shown in Figure III.

Fig. II: Deformation in thin steel sheets Figure III: other deformation

The deformations of the wood are the differences between the rest position and the deformed position. This deformation is shown in green in Figures II and III.

With all stresses, the permissible stress is limited by the maximum permissible wood deformations. In the case of deformations Figure II, the maximum deformations occur on all shear surfaces. In the case of deformations in Figure III, the maximum deformations only occur on the shear surfaces of the outermost dowels.

One more remark: Often a column stands on a plate of the base point restraint. In principle, this "connection" also contributes to the strength of the connection - but the contribution is so insignificant that it can be ignored. In the rarest of cases there will be a perfectly fitting connection that could contribute something.

In the case of deformation in Fig. II, it is sufficient to examine a dowel connection. Even with deformation as shown in Figure III, the total moment can be traced back to the individual dowel connection. This happens in the following:

A single dowel connection is considered. It is only considered here that the dowel does not deform. The wood deformation (green) is the difference between the black rest position and the red deformed position - initially only when rotating in the center of the dowel (Fig. IV) and then when rotating around the center of rotation with the distance e between the center of rotation and the center of the dowel.

Figure IV: Deformation at the distance between the center of rotation and the dowel

Usually the maximum deformation of the wood is yMax the limit of resilience. If the pivot point is in the center of the dowel and if the dowel is rotated until the wood is deformed, the maximum angle of rotation is reached:

or

The linear relationships between wood deformation and the force occurring are considered. With a constant K (not further investigated here), the moment when rotating through an angle β (x is the coordinate in the direction of the anchor - zero point at the point of rotation, y perpendicular to this is the direction of deformation:

The following consideration is simplified when working with trigonometric functions. For this purpose, the following substitution (with any length e) is agreed:

According to the rules of integration calculation, it becomes

and

It is used:

The integral is not solved any further because the value has already been calculated.

Now the moment is to be determined when the column rotates around a center of rotation. The definition of the moment remains of course unchanged. An angle ψ is substituted here instead of φ. This becomes (here e is the distance between the center of rotation and the center of the dowel):

Now x and y (wood deformation) have to be determined as a function of ψ. ψ determines a point on the rest axis of the anchor and the distance from the pivot point. The y-coordinate is independent of the angle and is always e. The following applies to the x coordinate:

For the distance l* applies:

A point on the rotated axis has different coordinates. The point of intersection of the connecting line pivot point with the point under consideration on the rest position of the anchor axis with the deformed anchor axis is determined. The angle between the twisted dowel center and this connecting line should be called φ. The following relationship then exists between an angle of rotation α of the anchor and the angles ψ and φ:

ψ = α + φ

The following relationship exists for the length l between the point of intersection and the pivot point:

The following coordinates of the intersection result from the radiation equations:

The y-coordinate y results analogouslyk:

The deformation of the wood results from the difference between the y-coordinates of untwisted and twisted wood:

The values ​​can now be used, resulting in:

Now ψ is expressed by φ:

With the addition theorems:

The denominator has to be examined differently:

Assuming that tan φ tan α <1 (i.e. the maximum deflection is smaller than the distance - see later note), the denominator can be expanded into a geometric series:

In practice tan φ tan α << 1, so the series converges quickly. The integral is:

Note: If e becomes so small that the series expansion is no longer permissible, then the difference between M (if e> 0) and M is0 (at e = 0) so small that with M0 can be expected.

The following is now reversed:

According to the rules of integration calculation, it becomes

and

The integral is:

After re-sorting (and extra representation of the expression in brackets [Kla]):

With the expression in brackets:

The integration can now be carried out:

Only terms for which the exponent of x is odd make contributions to the integral. Even exponent terms add up to zero. With i = 2j or i = 2j +1:

Now a part of the t-potencies is extracted:

The maximum deformation of the wood results:

Or multiplied:

Because of the additional term, only smaller angles α are permitted and intended for a given maximum deformation. The equation shows: the larger e is, the smaller is α. For all e> 0 with the same deformation, the permissible angle α is always smaller than the angle β.

Note: This reduction in the deformation angle, which reduces the disruptive deformation under load, is also the cause of a lower permissible load.

Further note: From the above equation follows the above-mentioned assumption that yMax

By agreement, t / (2e) is equal to tan φG. This will:

The following is changed from this:

As long as yMax

If the dowel goes through the center of rotation, the formula for M does not apply, since then the maximum deflection is not less than the distance, which was the prerequisite for the derivation. Then the formula derived above is to be used:

The distance between the dowel and the center of rotation results in a different moment than when the dowel goes through the center of rotation. This becomes for the ratio:

This is transformed into:

Since the angle α is always smaller than the angle β and the distances ei the dowels i are different for different dowels, the contribution of the individual dowels to the total moment is different. If the total moment of all anchors is related to the maximum permissible moment of an individual anchor, this quotient can be expressed as the effective number of anchors nd describe. With the above equation, the effective number of dowels becomes:

The effective number of dowels nd is calculated under the assumption that each individual wood deformation has no effect on any other wood deformation. This assumption is not correct, so that the effective number of dowels has to be reduced further:

This further reduction has the same causes as in the case of tension-compression loading: wood deformation under the action of a dowel is not limited to the immediate vicinity of the dowel, but also spreads further in the wood. Another dowel, which is located in the deformation area of ​​this dowel, will achieve the maximum permissible wood deformation even with a lower load than without this pre-deformation. This effect is made possible by the effective number of dowels nef of equation (210) in DIN 1052: 2004-08.

Since the effects nd and nef are independent of each other and act at the same time, both reduction factors must be treated multiplicatively.

Since it depends on many factors in individual cases whether a deformation according to Fig. II or Fig. III (or any transition in between) occurs, the maximum permissible load after the deformation according to Fig. III must be taken into account for safety reasonsd has led. If the deformation is also to be taken into account, the deformation according to Figure II must be taken into account as a possible deformation and not the smaller deformation according to Figure III, because the deformation also depends on many factors in individual cases.

If the distance (unavoidable in the case of deformations according to Figure III) between the wood and the sheet steel is very large, the dowel itself deforms outside the wood, so that the wood deformation is less. As a result, lower forces are transmitted from the dowel to the support, so that the effective number of dowels must be further reduced. This reduction can be up to 0.5.

DIN 1052not standardized (note other direction of force)

Force directions according to DIN 1052 and in this work

Actually, the tilting resistance (tilting safety) of a dowel connection should not be of interest, because it should be insignificant for the verification of the stability of a construction.Quotation from a script of the timber construction institute of the Technical University of Munich http://www.fgh.bv.tum.de/Daten/Skript/WS0506%20-%20Kapitel%208.pdf: "Skeletal systems are like houses of cards: They have to be stiffened."

Or point 8.8.1 (9) of DIN 1052: 2004-08: "... the torsional stiffness of all connections is taken into account ...". Point 8.8.3 (6) of DIN 1052: 2004-08: "... shocks may be assumed to be torsionally rigid if ... at 3 times the rated value of the moments ..."

When a structure is stiffened, the individual connections are not subjected to tilting, but rather to tension, pressure and shearing. Because of the inexpediency of the stress on tilting, there are also no regulations for determining the tilting strength.

Also, no manufacturer of structural analysis software supports the calculation of this tilting strength, because no responsible structural engineer calculates a house of cards.

If, for whatever reason, a structure is still not stiffened or cannot be stiffened, it can happen that connections are subject to tilting stress. While the deformations almost never play a role in the case of tension or compression (evidence of buckling, etc. is of course required), the deformation is usually significant in the case of tipping. In the case of supports, the flexibility of the connection increases the tilting length dramatically (see e.g. Heimeshoff, B .: Dimensioning of wooden supports with flexible foot connection. Holzbau-Statik-aktuell, volume 3 May 1979, ISSN 0720-9568) - again in contrast to this if the tilting loads dowels lying apart, with each individual dowel (each individual row of dowels) only being subjected to tension or pressure with the corresponding flexibility. The fact that the anchor is also subjected to shearing loads does not matter in the present context.

The strengths that can be expected are given below. Then it is deduced how these formulas are created. Two-shear dowel connections are considered.

Comparison of the new DIN 1052: 2004-08 with the old DIN 1052-2: 1996-10

The old and new DIN equations provide similar values ​​- but sometimes deviations of up to 50% (see examples at the end). It should also be taken into account that the new DIN mainly provides the force data on a shear joint, while the old DIN gives the total force of the dowel, i.e. for 2 shear joints. It follows that in equations for moment, etc., the Nst, b included (according to the old DIN) Nst, b by 2 rowsk to be replaced.

Note: Equation (199) of DIN 1052: 2004-08 corresponds to equation (4) of DIN 1052-2: 1996-10 - almost only with the difference that the exponent is not 2, but approx. 1.8. Paragraph 12.2.3 (7) gives Eq. (199) modified so that it becomes equation (3) - the exponent of the modified equation (199) being almost 1, in contrast to Eq. (3) where it is exactly 1.

permissible loads

In the case of chains of inequalities, the left value is the limit value for short anchors, the right value the limit value for long anchors.

If the permissible torque is inserted into the equation for the torsion spring, the dependence on the dowel length becomes clear (with the third power, since Nst, b depends on a.). If the reduction and / or the length dependency are not taken into account, this can have dramatic effects.

or

According to DIN with additions, this means:

perm Nst, b: permissible load according to section 5 of DIN 1052-2: 1996-10
perm M.st, b: permissible load torque (additionally defined)
ν: shift at perm Nst, b according to table 13 of DIN 1052-2: 1996-10
C: Displacement module according to Table 13 of DIN 1052-2: 1996-10
C.d: Torsional stiffness (additionally defined - analogous to Heimeshoff literature)
a: Wood thickness according to Section 5.8 of DIN 1052-2: 1996-10 = dowel length
dst, b: Diameter of the anchor according to Section 5.8 of DIN 1052-2: 1996-10

Except for the allowable moment perm M.st, b When tilting, the deformation and the increase in the buckling length must be taken into account.

Mathematical proof of the formulas for the load limit

Since there are no standards for dimensioning the tilting of fasteners, existing standards are also to be applied to the case of tilting of fasteners.

What are these starting points ?:

  • The loads (forces) along the anchor axis are not constant [see e.g. images in table G.3 of DIN 1052: 2004-08) and the formula (199)].
  • Even if there are small partial forces in any section of the anchor, the partial forces must not be increased in other sections of the anchor.
  • In each sub-area, the partial force is proportional to the deformation of the wood.
  • The dowel is deformed by the forces from the wood.

These assumptions provide sufficient prerequisites to prove the tilting. But they also show differences:

The proportionality factor between partial force and deformation is not entirely proportional to the deformed area. Otherwise, e.g. no reduction as in equation (203) of DIN 1052: 2004-08 would be necessary. This can be explained: If the wood is pressed by the dowel, not only is the wood directly under the dowel deformed, but also the wood to the side of it. A deformation trough is created that is much wider than the dowel. As a result, with the same maximum wood deformation due to this lateral expansion, the specific force is smaller with thicker dowels (or the hollow is deeper with the same specific force).

To make it easier to understand the tipping load capacity, the relationships between tension and compression are considered first.

With the above-mentioned prerequisites, a differential equation for the dowel and wood deformation must be set up and solved. The type of dowel holder is included in the solution. If the holding steel sheets are thin, the connection between the bracket and the dowel is relatively "loose" since the dowel can rotate with its axis in the bracket or the sheet metal follows a movement of the dowel. If the holding steel sheet is thick, the dowel is "firmly" held in the holder in its initial direction and the direction of the dowel axis cannot rotate.

In the new DIN 1052, a distinction is therefore made between thin and thick (or loose and firm). The solutions for the two-shear connections are (the proportionality constant for tension or pressure and the proportionality constant times the dowel length have to be added - the angle functions used are shown in abbreviated form):

C: coshyp (x)
S: sinhyp (x)
c: cos (x)
s: sin (x)

Table II

In Fig. V and Fig. VI these functions are shown for the same maximum wood deformation - as a function of the relative length. The reference length is half of treq with fixed bracket. Figure V also shows the limit values ​​for the load-bearing capacity according to equation (197) or (199) of DIN 1052: 2004-08 - including requirement 12.2.3 (6). The break point of the limit curve (transition from the linearly increasing part to the constant value) is the minimum timber thickness treq.

Fig. V: Tensile (compressive) force as a function of the dowel length with the same maximum deformation

Fig. VI: Tilting moment as a function of the anchor length with the same maximum deformation

Figure VII: Quotient of tilting moment and tensile force as a function of the dowel length with the same maximum deformation

In the old DIN 1052-2 equations (3) and (4) only the "fixed" curve is used. The transition from equation (3) to (4) takes place at a certain wood thickness areqthat have the same meaning as treq the new DIN has. For areq applies with the designations of the old DIN:

The deformation of the wood along the dowel is shown in Figure IIX. In the case of long dowels, the deformation is practically only present in the edge area. This edge deformation as an excerpt from Figure IIX is shown in Figure IX. With short dowels, the decrease in wood deformation (due to the increase in dowel deformation) does not yet play a role - therefore the load capacity increases with the length of the dowel, with long dowels the dowel deforms in the middle area so that the dowel follows the movement of the wood. This means that there is no deformation of the wood and therefore no forces act on the wood in this area of ​​the dowel.

Fig. IIX: Wood deformation as a result of tension (pressure) in dowels as a function of the total length

Fig. IX: Deformation of wood due to tension (pressure) with long dowels in the edge area

The load limit for a long dowel length and a loose bracket is greater than half the load limit for a fixed bracket (that would be the value "1") and, according to the standard, has the value "Root (2)". Although picture V suggests the value "1", the larger value is still correct. It is about the failure mechanism and negative wood deformations do not play a role - see Fig. IX that the wood deformations (and only these are responsible for the force effects) shows in the edge area. The deformations of the wood are much less with a loose bracket and also have greater negative proportions.

The load limit of a dowel connection is therefore determined by the maximum permissible deformation of the wood and it is irrelevant how this comes about. In the old DIN (Table 13), the maximum wood deformation was specified regardless of the dowel thickness. In the new DIN (Table G.1) it depends on the dowel thickness - approximately with d0,3. The exponent results from the quotient from Table G.1 (1.5) and the dependence on R.k with about 1.8.

The formulas in Table II already show the factors for tilting loads. The deformation of wood with dowels of different lengths under tilting load is shown in Figure X. With short dowels, the deformation of wood is linear and concentrates more and more at the two dowel ends with increasing dowel length. In the case of long dowels, the course of the edge deformation is shown as an extract from Figure X in Figure XI. This course is almost the same as with tensile loading. Note: In the case of tensile load, the same tensile or compressive load is present at both ends of the dowel, so that the total load capacity = 2 Rk (DIN 1052: 2004-08) is or equal to Nst, b (DIN 1052-2: 1996-10). Under this condition (very long dowel), the permissible load moment must be determined directly from the moment definition (force couple times distance between the force couple).

allow M = Rk × t

Image X: Deformation of wood due to tilting of dowels depending on the total length

Fig. XI: Wood deformation due to tilting with long dowels in the edge area

With shorter dowels, the calculation of moments is wrong with this equation, since there is no force couple, but approximately a triangular line load - the shorter the dowel, the more triangular. For such a triangular line load, the forces that the dowel exerts on the bracket may not have the same value as the forces under tensile load - despite the same maximum wood deformation, because the wood deformation and the forces of the wood on the dowel are completely different. The forces are considerably lower - only 1/3 with short dowels. The course of this reduction factor AF is shown in Fig. VII, the values ​​result from the quotient of the factors of tension and tilting in Table II. The value AF starts with 1/3 for short anchors and has an end value of 1 (or 0.5 for "loose" Dowels) for long dowels.

Stiffness of the connection

The torsional stiffness Cd of the connections results from the quotient of the load torque and the angle that is established in the process. With small angles of rotation, the tangent of the arc angle is almost equal to the arc length. Since the connection rotates around the longitudinal center point of the dowel, the angle is equal to the edge displacement through half the dowel length.

Values ​​that belong together must always be considered for M and φ. If M is increased with point 12.3 (8) (equation (209)), φ also increases to the same extent, i.e. the torsional stiffness is independent of the load at which it is determined - as long as the linearity range is not exceeded. According to the compliance load curve, the rigidity becomes lower when the load exceeds the linearity range.

Inserting the values ​​results from the new DIN with Kser from table G.1:

Enlargement of the buckling length

If the base point is flexibly connected, the buckling length s increasesk the column with the height h.

With Heimeshoff we get:

After solving the equation, the buckling length can be determined with β:

For a cantilever column, Cd = ∞ and accordingly β = π / 2 and therefore sk = 2 h.

Column head displacement

If the prop tilts a little under load, the prop head will shift. The shift results:

Column head displacement = h φ

Invoice for short dowels

With the short dowel, the deformation of the wood is almost exactly triangular (Fig. IX) and so the maximum permissible load can be calculated without a differential equation. But integrations are required.

Why is the final maximum permissible moment equal to the maximum permissible force times the dowel length impermissible? In order to make this clear, the elastic conditions under tension and compression will first be explained, which lead to equations (3) and (4) of DIN 1052-2.

The forces transmitted to the wood of the column deform the wood. In the linear area of ​​deformation, force and deformation are proportional. With a tensile (compressive) load, the deformation of the wood is constant over the entire length of the dowel with short dowels and therefore every partial force along a dowel section is the same, so that the total force as the sum of all partial forces is proportional to the dowel length a [equation (3)]. In terms of formula, the force N with a spring constant K and the displacement ν is to be represented as follows (with regard to the later calculation of the torque, the left and right dowel halves are named separately):

With larger dowel lengths, the forces also deform the dowel itself in the direction of the load, so that the deformation of the wood is less and thus the force on the wood (and thus the deformation of the wood) is less in the area of ​​the deformation of the dowel and ultimately becomes zero. Only in the edge area is the wood (and not the dowel) deformed. As a result, only the edge area contributes to the permissible load and the permissible load becomes independent of the dowel length (provided the dowel is sufficiently long!). The size of this edge area depends on the dowel diameter dst, b from and thus the permissible load is correctly described by equation (4) of DIN 1052-2. If the resulting deformation of the anchor is solved as the like and thus the maximum permissible load on the anchor connection is determined as a function of the anchor length, a closed solution (with hyperbolic functions) is obtained, which, as borderline cases for short (or long) anchors, equations (3 ) or (4) as borderline cases and in the transition area from short to long dowels only deviates from equations (3) and (4) by approx. 8%.

After this preliminary remark, the maximum permissible moment load of an anchor. When tilting, the deformation of the wood along the dowel is not constant and, with short dowels, changes linearly from compression on one side to zero in the middle to elongation on the other side (Fig. IX). The wood deformation is greatest at the edge - but must not exceed the maximum permitted wood deformation there either. Since the wood is destroyed if the permitted wood deformation is exceeded, the maximum value of the wood deformation under tensile load (pressure load) and tilting is the same. The following applies to the deformation of the wood:

Now the corresponding partial moment of the force couple (= force times distance) has to be determined for 2 partial sections on the left and right at the same distance from the center:

To integrate all partial moments to the total moment (a further 2 does not occur, since the left and right halves are already taken into account in the force couple):

From the comparison with N = K v a it follows:

Example - the following table

The example shows that the interpretation of the product of perm N * dowel length as perm M is inadmissible.

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