A Note on a Prestressed Bolted Connection

I. I. Bobarykov.

(with 2 tables)

If, during assembly of a connection, the bolt is installed with preliminary tightening, then it is called prestressed. The transverse dimension of the bolt is usually determined on the basis of the following considerations. Let a bolt with a rectangular thread, having outer and inner diameters respectively \(d\) and \(d_1\), be subjected to the action of a longitudinal tensile force \(P\). The latter is distributed uniformly over the contact surface of the threads of the nut and bolt. For convenience, one may imagine that this force is uniformly distributed along the mean helical line, whose diameter is

\[ d_0=\frac{d_1+d}{2}, \]

and whose lead angle is \(\angle \alpha\) *). During tightening, the external moment overcomes the resistance to displacement of the point of application of the force \(P\), the friction on the thread, and the friction on the lower end surface of the nut. Denoting the diameter of the circle inscribed in the hexagon by \(D\), and the coefficients of friction on the thread and on the end surface of the nut by \(f\) and \(f_1\), we find the greatest value of the moment:

\[ M=\frac{Pd_0}{2}(\operatorname{tg}\alpha+f)+\frac{Pd_0}{3}\frac{D^3-d_0^3}{D^2-d_0^2}f_1 \ . \ . \ . \ . \ . \ (1) \]

If we take the international thread scale (Table A) **), then we observe that within the limits of the most common bolt sizes from \(3/4''\) to \(2''\) (approximately \(d=20\text{–}52\ m/m\)) one may take, on average:

\[ d:d_1=1.2,\qquad d:d_0=1.1. \]

In addition, taking \(D=d\sqrt{\frac{3}{}}\) and \(f=f_1\), we find:

\[ M=(0.5\,\operatorname{tg}\alpha+1.19\,f)Pd_0 \ . \ . \ . \ . \ . \ (2) \]

*) The assumption made does not give an appreciable error; for uniformly distributed pressures on the contact surface of the threads, the resultant along the line will actually be farther from the axis, namely on a cylinder whose radius is

\[ x=\frac{d_0}{2}+\frac{d^2-d_0d_1}{6d_0}. \]

Taking on average \(d=1.1d_0\) and \(d_0=1.1d_1\), we have

\[ x=0.505d_0. \]

**) See at the end.

I. I. Esyarykov.

Of the two terms on the right-hand side, the second will have a much greater influence on the value of \(M\), as is seen from Table I.

Table I.

Value of \(M\).

Let us take, as an average value, \(f = 0.2\); then from (2) we have:

\[ M_{d=20\ \mathrm{mm}} = 0{,}253\,P d_0,\quad M_{d=52\ \mathrm{mm}} = 0{,}260\,P d \quad\text{and, on the average,} \]

\[ M = 0{,}256\,P d_0 \ldots \ldots \ldots \ldots \ldots (3) \]

Passing to a bolt with a triangular thread, in expression (2), instead of \(f\), one takes the quantity \(f' = f : \cos \xi\), where \(\xi\) is half the angle at the vertex of the generating triangle.

Consequently, in the international system, for the same limits we shall have, on the average:

\[ M = 0{,}30\,P d_0 \ldots \ldots \ldots \ldots \ldots (4) \]

A part of this quantity, namely, \(M_1 = 0.5(\operatorname{tg}\alpha + f')\), twists the bolt, producing tangential stresses in it. The average value of \(M_1\) for bolts within the limits \(d = 20\text{–}52\ \mathrm{mm}\) will be:

\[ M_1 = 0{,}135\,P d_0 \ldots \ldots \ldots \ldots \ldots (5) \]

Thus the body of the bolt is subjected to compound resistance; the tensile resultant force \(R\) will be

\[ R = 1.35\,P \ldots \ldots \ldots \ldots \ldots (6) \]

i.e., owing to tightening, the stress in the material is increased by \(35\%\), which is also taken into account in finding the safe dimensions of the bolt for a given force \(P\).

The conditions in which a bolt is found in stressed connections are by no means such as is assumed in the calculation method given.

Usually a bolted connection intended to resist a given force \(Q\) is assembled at a time when the force \(Q\) is not yet act-

...exists. When the nut is tightened by the moment \(M\), a preliminary elongation is produced by the force \(P\), which appears as the result of compression, by the bolt, of the parts being joined together. This force \(P\), called the initial tightening force, together with the twisting moment \(M_1\), causes normal and tangential stresses in the material of the bolt. If now a force \(Q\) is applied to the bolt, then, owing to its additional elongation, the stress in the parts clamped by the bolt decreases; the initial tightening force falls, and the bolt begins to be stretched by a certain force \(P_1\), intermediate between \(Q\) and \(P+Q\). It is easy to see that the value of this force depends both on the initial tightening \(P\) and on the elastic properties of the materials of the bolt and of the parts joined by it.

Let a bolt with outside diameter \(d\) (Fig. 1) tighten, with force \(P\) (initial tightening), a hollow prism \(A\), situated between the head \(C\) and the nut \(B\); the height of the prism is \(h\). Let the cross section of the bolt and of the prism be denoted by \(\omega\) and \(\Omega\), and their moduli of elasticity by \(E\) and \(E_1\). The prism is compressed, evidently, by the force \(P\); if now two forces \(Q\) and \(Q_1\) are applied to the bolt along its axis, then the force stretching the bolt will be \(P_1>P\), and the force compressing the prism \(T<P_1\), and there will always be the relation:

\[ Q+T=P_1 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (7) \]

Let the absolute elongation of the bolt shank over the length \(h\) be denoted by \(\lambda\). Its additional elongation \(\lambda'\) is proportional to the increase in the tensile force, i.e., to the difference \(P_1-P\). The decrease in the deformation of the prism, \(\lambda''\), is evidently equal to:

\[ \lambda''=\lambda--\lambda \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (8) \]

On the basis of the adopted notation, one may write instead of (8)

\[ \frac{T}{E_1\Omega}=\frac{P}{E_1\Omega}-\frac{P_1-P}{\omega E}\ldots \ldots \ldots \ldots (10) \]

From (7) and (9) we have

\[ P_1=P+\frac{E\omega}{E\omega+E_1\Omega}\,Q=P+bQ\ldots \ldots (10) \]

\[ T=P-\frac{E_1\Omega}{E\omega+E_1\Omega}\,Q=P-(1-b)Q\ldots \ldots (11) \]

Relation (10) holds as long as the value of \(Q\) does not exceed the limit...

\[ Q < \frac{P}{1-b} \ldots \ldots \ldots \ldots \ldots \ldots \ldots (12) \]

Thus the normal stress \(k_z\) in a tightened bolt under the action of the tensile force \(Q\) will be less than the sum of the forces \(P+Q\), but may be greater than the force \(1.35\,Q\). The resultant stress \(k\) is found by combining the stress \(k_z\) and the tangential stress \(k_s\), determined from the twisting moment \(M_1\) (Eq. 5), i.e., referring the required stress to the smallest cross-section \(\omega_1=\dfrac{\pi d_1^2}{4}\), we shall have:

\[ k = 0.35\,\frac{P_1}{\omega_1} + 0.65\sqrt{ \left(\frac{P_1}{\omega_1}\right)^2 +4\left(\frac{0.135\,P\,d_0}{\dfrac{\pi d_1^3}{16}}\right)^2 }. \]

Whence the resultant tensile force \(R\):

\[ R=\omega_1 k =0.35\,(P+Qb)+0.65\sqrt{(P+Qb)^2+1.4\,P^2} \ldots \ldots (13) \]

Equation (13) takes into account all the circumstances, such as the tightening \(P\), the dimensions of the connected parts, and their elastic properties; but, unfortunately, it cannot have immediate practical significance, since the magnitude of the initial tightening is unknown. If it is assumed that the length of the wrench is equal to 15 diameters, then, denoting by \(S\) the force of the worker’s hand, we shall have:

\[ 15\,d\,S = 0.30\,P d_0 \ldots \ldots \ldots \ldots (14) \]

Taking \(S=20\text{--}40\ \mathrm{kg}\), we find

\[ P = 1100\text{--}2000\ \mathrm{kg}. \]

Let us denote the ratios \(E_1:E\) and \(\Omega:\omega\) by

\[ \xi = E_1:E \quad \text{and} \quad \eta=\Omega:\omega \]

then we shall have

\[ b=\frac{1}{1+\eta\xi} \ldots \ldots \ldots \ldots (15) \]

Assigning various values to \(\eta\) and \(\xi\), for example:

\[ \begin{aligned} \xi &= 2\ --\ 1\ --\ 0.5\ --\ 0.1\ --\ 0.05,\\ \eta &= 1\ --\ 3\ --\ 5\ --\ 10\ --\ 20 \end{aligned} \]

we find for the coefficient \(b\) the following values (Table II):

Table II.

For clarity, in Fig. 2 this table is represented by curves constructed for $b$ as the coefficient $\xi = E_{1}:E$ varies. Since expressions (10) and (11) are valid for such values of the force $Q$ that satisfy condition (12), it is of interest to see how the limit for the force $Q$ varies for different combinations of the coefficients $\eta$ and $\xi$. In Fig. 3 curves of the value $Q$ are plotted for the values of $b$ taken from Table II. The magnitude of the initial tightening, $P$, is taken as unity.

Consideration of the curves in Fig. 2 leads to the conclusion that the influence of the additional load is weaker the larger the value of $\xi$. Moreover, this influence is the more noticeable the smaller the ratio $\eta = Q:\omega$. Consequently, for example, at one and the same value of the coefficient $\eta$, an iron bolt tightening wooden parts $(\xi \approx 0.05)$ is in less favorable conditions than a yellow-copper bolt $(\xi \approx 0.1)$.

Figure 3 presents curves indicating the dependence between the limiting value of the force $Q$ (according to Eq. 12) and the coefficients $\eta$ and $\xi$. The ordinates give the magnitudes $Q$, the abscissas the values of $\eta$. From Fig. 3 it is evident that the influence of the ratio between $P$ and $Q$ is the more significant the smaller $\xi$ and $\eta$ are. Therefore this value is more substantial for small moduli of elasticity of the parts being connected and for a small cross section of them.

For example, let it be given: an iron bolt has an internal diameter

$d_{1} = 16 \ m/m,\quad P = 1000 \ klg,\quad Q = 1000 \ klg.$

Let us take four cases:

1) $\xi = 0.05 \quad \eta = 3 \quad b = 0.87.$

2) $\xi = 0.05 \quad \eta = 20 \quad b = 0.50$

I. I. Bobarykov.

\[ \begin{aligned} 3)&\quad \xi = 1 \qquad \eta = 3 \qquad b = 0.25,\\ 4)&\quad \xi = 1 \qquad \eta = 20 \qquad b = 0.0475. \end{aligned} \]

From eq. (13) we find:

\[ \begin{aligned} 1)&\quad R = 2100\ \mathrm{kg}, \quad k = 10.4\ \mathrm{kg/mm^2}\\ 2)&\quad R = 1770\ \mathrm{kg}, \quad k = 8.8\ \mathrm{kg/mm^2}\\ 3)&\quad R = 1560\ \mathrm{kg}, \quad k = 7.75\ \mathrm{kg/mm^2}\\ 4)&\quad R = 1400\ \mathrm{kg}, \quad k = 6.95\ \mathrm{kg/mm^2} \end{aligned} \]

In order to verify to what extent expression (10), obtained on the basis of theoretical considerations, gives results agreeing with reality, I carried out experiments in the Mechanical Laboratory of the Institute on the stretching of a stressed tightening bolt, approximately according to the scheme in Fig. 1.

Six cylindrical rods with ends cut with triangular thread were subjected to investigation.

The dimensions of the rods are as follows, Table III.

Table III.

The prisms that were compressed between the nuts of the rod were made in the form of tubes, six in number, of the following dimensions:

*) Rod No. 3 served only for determining the relation between the twisting moment and the tightening force.

Table IV.

Tube No. 5 was made of dry birch; in cross-section it had the external outline of a square with beveled corners; the latter had to be done so that the measuring instrument could fit.

First of all, the moduli of elasticity \(E\) and \(E_1\) were determined for the materials of the rods and tubes, the former in tension, the latter in compression. The following data were obtained:

Table V.

Rod No. 1 — \(E=21345\ kg/mm^2\). Tube No. 1 — \(E_1=20200\ kg/mm^2\)

» No. 2 — \(E=21727\) »        » No. 2 — \(E_1=22191\) »

» No. 3 — \(E_1=9428\) »

» No. 4 — \(E=10030\) »        » No. 4 — \(E_1=7001\) »

» No. 5 — \(E=20671\) »        » No. 5 — \(E_1=1376\) »

» No. 6 — \(E_1=7550\) »

In view of the fact that in the first experiments, when tightening the tubes with nuts on the rod, twisting of the tubes was observed, i.e. a circumstance not provided for by relation (10), it was necessary to see to it that the twisting moment was not transmitted to the tube. For this purpose the following device was made (Fig. 4): a cast-iron crosspiece \(AB\) has in the middle a hole for passing the rod \(E\). At its ends two posts \(a, a\) are fastened (screwed in), serving as guides for a similar crosspiece \(CD\), which can move along the rod and is set depending on the length of the clamped…

of my tube \(F\). Under such conditions twisting of the tube becomes possible only to the extent permitted by the deflection of the guides \(a-a\), which is very small.

In measuring tensile and compressive deformations, a Martens–Kennedy apparatus with a gauge length of \(150\ m/m\) and equipped with a vernier was used. The apparatus permits measurements to be made with an accuracy of up to \(1:300\ m/m\), if the reading is limited to tenths of a millimeter.

The experiment was carried out as follows. The tube was clamped between the crossheads \(AB\) and \(CD\) of the apparatus (Fig. 4) and tightened with nuts by means of a wrench. From the reading on the measuring apparatus the compressive force \(P\), i.e., the initial tightening, was determined. Then the rod \(EE\) was subjected to longitudinal tension *); after each load a reading was taken of the deformation of the tube, which, obviously, was being unloaded. Upon reaching the greatest load, the latter was gradually reduced to zero; the deformation of the tube went back, i.e., increased. The coincidence of the readings of the apparatus at the beginning and at the end of the experiment was proof that the stress in the material did not pass beyond the elastic limit. From the readings obtained on the apparatus for the deformation of the tube, it was possible to establish, for each load of the machine, the magnitude of the corresponding force \(T\) compressing the tube.

Consequently, the measure of the load for the tube was its deformation; therefore, in finding the values of the moduli of elasticity \(E_1\), the readings of the apparatus corresponding to a load of \(500\) or \(1000\ klg.\) (depending on the material) were also determined. Compression was carried out on an Amsler press, and the displacements of the apparatus were read every \(1000\) (or \(500\)) \(klg\). After repeating the loading several times, the tube was turned through \(90^\circ\) and again loaded in the same order.

The average reading from the figures obtained in this way differed little from the extreme values, by no more than \(5\%\). The results are given in Table VI, containing the readings \(m\).

*) The tension was carried out on a vertical Amsler machine with a capacity of \(50t\). Fig. 5 shows the setup for the experiment with tube No. 5. Fig. 5a shows the apparatus (Fig. 4) on rod No. 2 with tube No. 6.

Table VI.

Table VII.

Rod No. 1. Tube No. 2.

Ratios: \(\eta = 0.79,\ \xi = 1.04,\ b = 0.52.\)

Reading \(m = 0.57\) per 1000 kg.

Limiting value \(Q = 8400\) kg.

Table VIII.

Rod No. 1. Tube No. 3.

Ratios: \(\eta=0.825.\quad \xi=0.440\quad l=0.73.\)
Reading \(m=1.28\ m/m\) per \(1000\ kg.\)
Limiting value \(Q=9650\ kg\)

Table IX.

Rod No. 1. Tube No. 6

Ratios: \(\eta=2.10;\ \xi=0.353;\ l=0.575.\)
Reading \(m=0.625\ m/m\) per \(1000\ kg.\)
Limiting value \(Q=8750\ kg.\)

Table X.

Rod No. 2, tube No. 1.

Ratios: \(\eta=0.82;\ \xi=0.932;\ b=0.568.\)

Reading: \(m=0.60\ m/m\) per \(1000\ kg\).

Limiting value \(Q=9360\ kg\).

Table XI.

Rod No. 2, tube No. 2.

Ratios: \(\eta=0.79;\ \xi=1.02;\ b=0.553.\)

Reading \(m=0.57\ m/m\) per \(1000\ kg\).

Limiting value \(Q=9450\ kg\).

Table XII.

Rod No. 2, Tube No. 3.

Ratios: $\eta=0,825;\ \xi=0,435;\ b=0.735.$

Readings: $m=1,28\ m/m$ per $1000\ kg.$

Limiting value: $Q=14000\ kg.$

Table XIII.

Rod No. 2, Tube No. 4.

Ratios: $\eta=0,82;\ \xi=0,323;\ b=0{,}79.$

Readings: $m=0,87\ m/m$ per $500\ kg.$

Limiting value: $Q=17000\ kg.$

Table XIV.

Rod No. 2, Tube No. 5.

Ratios: \(\eta=3,62;\ \xi=0,0635;\ b=0,807.\)

Reading \(m=1\ m/m\), per \(500\ kg\).

Limiting value \(Q=17600\ kg\).

Table XV.

Rod No. 2, Tube No. 6.

Ratios: \(\eta=2,10;\ \xi=0,348;\ b=0,58.\)

Reading \(m=0,625\ m/m\) per \(1000\ kg\).

Limiting value \(Q=5700\ kg\).

Table XVI

Rod No. 4, tube No. 1.

Ratios: \(\eta = 0{,}83;\ \xi = 2{,}00;\ b = 0{,}76.\)

Reading \(m = 0{,}6\ m/m\) per \(1000\ kg\).

Limiting value \(Q = 4100\ kg\).

Table XVII.

Rod No. 4, tube No. 3.

Ratios: \(\eta = 0{,}83;\ \xi = 0{,}942;\ b = 0{,}56.\)

Reading \(m = 1{,}28\ m/m\) per \(1000\ kg\).

Limiting value \(Q = 5400\ kg\).

A Note on a Prestressed Bolted Connection

Table XVIII

Rod No. 5, tube No. 1.

Ratios: \(\eta=1.16;\ \xi=0.975;\ b=0.462.\)

Reading \(m=0.6\) mm per 1000 kg.

Limiting value \(Q=5650\) kg.

Table XIX

Rod No. 5, tube No. 3.

Ratios: \(\eta=1.2;\ \xi=0.455;\ b=0.63.\)

Reading \(m=1.28\) mm per 1000 kg.

Limiting value \(Q=6650\) kg.

Table XX.

Rod No. 5, tube No. 4.

Ratios: \(\eta=1,19;\ \xi=0,338;\ b=0,715.\)

Reading \(m=0,87\ \mathrm{m}/\mathrm{m}\) per \(500\ \mathrm{kg}\).

Limiting value \(Q=9750\ \mathrm{kg}\).

Table XXI.

Rod No. 5; tube No. 5.

Ratios: \(\eta=5,25;\ \xi=0,065;\ b=0,74.\)

Reading \(m=1,00\ \mathrm{m}/\mathrm{m}\) per \(500\ \mathrm{kg}\).

Limiting value \(Q=8300\ \mathrm{kg}\).

Table XXII

Rod No. 5; tube No. 6.

Ratios: $c_1 = 3.03$; $\xi = 0.365$, $b = 0.475$.

Readings: $m = 0.625\ m/m$ per 1000 kg.

Limiting value $Q = 4700$ kg.

In Tables VII—XXII are given the results of observations*) for obtaining the values of the force $P_1$, for various combinations of rods and tubes. In the fourth column the values of $P_1$ were found from equation (7), in the fifth from equation (10). As is seen from the tables, complete agreement between the observed and calculated values for $P_1$ is not obtained. But in the majority of cases the difference between these quantities is small and varies within the limits of $\pm 3\%$. In individual tests, however, this difference is larger and even reaches 10% (Table XXII). The reason for such a deviation may lie in the lagging of the readings of the Kennedy instrument during unloading of the tube, which was observed, for example, in tubes Nos. 4 and 6 when determining the modulus of elasticity. In addition, for cast-iron tubes the modulus $E_1$ was calculated from mean readings, which, of course, already introduces a certain error.

Nevertheless, the results obtained permit one to suppose that relation (10) is correct; for a final conclusion it is desirable to carry out observations on a larger scale, which is intended to be done subsequently in experiments on the distribution of stresses in a flanged joint, which have already been begun in the laboratory.

*) Table VII is given in full, i.e., with the readings of the Kennedy instrument (left and right readings) from the beginning of the test to the greatest load $Q$ and then the gradual unloading to 0 and until the nut was unscrewed. Since these figures were needed only to prove that during the test the phenomenon remained within the limits of proportionality, in the remaining tables the indicated figures have been omitted.

From Tables X–XI it can be seen that the initial tightening reaches values of \(P = 4000\ klg\) and higher. It would be very interesting to establish at least an approximate dependence between the bolt diameter \(d\) and the magnitude \(P_1\), although theoretically (eq. 14) the force \(P\) does not depend on \(d\). Unfortunately, such a dependence cannot be found in general, since it is affected by the value of the coefficient of friction \(f\), the quality of workmanship, etc.

Using the apparatus shown in Fig. 4, I determined, for rods Nos. 3, 4, and 5, the twisting moment for various tightening forces. The apparatus with the rod and tube was placed on a horizontal torsion machine (Fig. 6); one nut was fastened in clamp \(A\), connected with the counterweight. Onto the other was fitted a double wrench \(B\), turned by the driver \(C\), mounted on sector \(D\) of the front clamp \(E\). By tightening the nut with a known, preassigned moment, the deformation of the tube was read by means of the Martens-Kenedy apparatus, and consequently the corresponding tightening force \(P\) was also read.

In Tables XXIII–XXV are given the results of tightening tubes Nos. 2 and 3 on rods 3, 4, and 5. These figures show that, within the limits of the observations made, the magnitude of the tightening does not depend on the diameter of the rod or on its material. On average it may be taken that \(1\ klg\) of twisting moment produces \(p = 180\ klg\) of longitudinal tensile force.*)

Table XXIII.

Rod No. 3. Tube No. 2.

Table XXIV.

Rod No. 3. Tube No. 3.

*) It was indicated above that the magnitude \(P\) depends on the coefficient of friction. How great this dependence is may be seen from an experiment made in parallel just now …

Table XXV.

If it is assumed that the force \(S\) of the hand on the wrench will be \(S = 40—30—20\ kg\), then we obtain the following values of tightening, taking the torque moment to be

\[ M = 15\, d.S, \]

Table XXVI.

i.e., the tightening, under identical conditions, increases with increasing diameter. For diameters of \(20\ m/m\) and \(24\ m/m\), one may take as mean values \(P = 1500\ kg\).

To show how sharply the tightening affects the resulting force \(R\), let us take a numerical example. Let it be given: \(\xi = 0.05\), \(\eta = 5\), and \(P = 1500\ kg\); then \(b = 0.8\).

of that described, on the same rod No. 3 with tube No. 2, but the washer was replaced by two rings with grooves, in which balls \(5\ mm\) in diameter were placed. With the same screwing moments as in Table XXIII, the tightening force doubled.

If \(Q=1000\ klg\), then by (13) \(R=2720\ klg\).

If \(Q=3000\ klg\), then by (13) \(R=3170\ klg\).

i.e., in the first case \(R\) exceeds \(Q\) by more than 2.5 times; in the second, \(R\) differs hardly at all from \(Q\).

After the observations were completed, the moduli of elasticity \(E\) and \(E_1\) for the rods and tubes were again determined; the following was found.

Table XXVII.

i.e., the deviations, in comparison with Table V, are negligible.

Table A.

International system of thread.

Table I
for the article by
V. V. Bobarykov

Fig. 2

Fig. 1

Fig. 4.

Fig. 5A.

Fig. 5.

Fig. 6.