Eng. K. N. KARTASHOV.

ALLOWING FOR THE DEGREE OF FIXITY OF BASE SHOES IN THE ANALYSIS OF RIGID FRAMES BY THE METHOD OF FOCAL RATIOS AND FOUR MOMENTS

Tomsk
1928

Eng. K. N. Kartashov.

Allowance for the Degree of Fixity of Base Shoes in the Analysis of Rigid Frames by the Method of Focal Ratios and Four Moments

General principles and analysis by the method of focal ratios

As is known, up to the present time, in the analysis of rigid frames—chiefly reinforced-concrete ones—only the elastic properties of the material selected for the construction of the structure, and the dimensions of the designed structure as determined by a preliminary approximate computation, have been taken into consideration; the properties of the foundation, which takes upon itself the vertical and horizontal reactions and the fixing moments at the supports, have been entirely disregarded.

Taking, as one of the most flexible methods of analysis, the analysis by the method of focal ratios,* we see that here the degree of fixity of the support sections of each member, depending on the dimensions of the entire structure, is fully characterized by the magnitude of the “focal ratio” for the focus nearest to the support under consideration. In the most common case of a rectilinear member with a constant section along its entire length, complete fixity corresponds to the focal ratio \(K = 0.5\), and a hinged support to \(K = 0\). Intermediate values of \(K\) give various degrees of elastic restraint of the support section.

The calculation of the left and right ratios for an arbitrary member \(AB\) (Fig. 1) is performed by the formulas:

\[ K^a = \frac{\beta}{\alpha + \varepsilon^a}; \qquad K^b = \frac{\beta}{\gamma + \varepsilon^b}; \]

in which the magnitudes of the angles \(\alpha\) and \(\beta\) depend only on the dimensions of the cross-section and the length of the member under consideration, while the quantities \(\varepsilon^a\) and \(\varepsilon^b\) represent the angle of rotation of the support under the influence of a moment equal to unity. All the angles entering into these formulas—\(\alpha\), \(\beta\), \(\gamma\), \(\varepsilon^a\), and \(\varepsilon^b\)—are determined from formulas known from theory, and from them the focal ratios are then found without difficulty.

In determining the lower focal ratios for the columns of frames resting on the ground through base shoes, difficulty is caused by finding the magnitude of the angle of rotation of the support section \(\varepsilon\). Therefore, in order to allow for the elastic restraint at the base of the columns, the designer, on the basis of his experience, in resolving the resulting “static ambiguity,” assigns the desired focal ratio, taking it, for the most common cases, as equal to from 0.2 to 0.4. Sometimes, however, in order to avoid indeterminacy in the solution of the problem, hinges are constructed at the bottoms of the columns, or an effort is made to create as nearly as possible complete fixity, by designing massive foundations running continuously under a row of columns, etc.

* E. Suter, “Die Methode der Festpunkte.”
A. Strassner, “Neuere Methode.”
Prof. G. V. Ulyaninsky, “Analysis of Frame Structures.”
Eng. K. N. Kartashov, “Analysis of Frame Structures by the Method of Focal Ratios.”

On the basis of the analysis of structures resting on an elastic foundation, we may use the well-known formula giving the relation between the soil pressure $\sigma$ and its deformation $y$:

\[ \sigma = Cy, \]

where the coefficient $C$ is called the “coefficient of subgrade reaction.”

To examine the nature of this coefficient, let us turn to the diagram showing the deformation of soil under the influence of an increasing load (Fig. 2). In this diagram, stresses are laid off along the axis of abscissas, to a certain scale, and the corresponding soil deformations downward along the axis of ordinates. From the curves presented (in the drawing, curve $\Pi$ shows the penetration of a circular loading plate into compacted sand, and curve $P$ into loose sand), we see that, as for any material, so also for soil, one can more or less precisely establish a limit of proportionality and a temporary resistance. At comparatively low loads there is approximate proportionality between stress and deformation. Within these limits the curvilinear portion of the diagram may be replaced by a straight line passing through the origin; the tangent of the angle which this line makes with the axis of abscissas is the required coefficient of subgrade reaction $C$ for the given soil, taken as constant at the permissible stresses, and representing nothing other than the soil pressure necessary to produce a deformation of 1 cm.

The value of $C$ varies within wide limits, from 0 for a liquid to $\infty$ for an absolutely rigid body. To determine actual values of this coefficient for various types of soil, a number of experimenters have carried out tests, the results of which are given in Table 1 below.

Unfortunately, these investigations are not sufficiently numerous, and the soils encountered in practice are highly varied; for this reason the data presented can serve only as a first approximation. Moreover, experiments have shown that even for one and the same soil this coefficient is not constant, but varies depending on the depth of embedment of the foundation and on the size and shape of the loading plate; nevertheless, with comparatively small error, at stresses which of course do not exceed the limit of proportionality, one may take $C = \mathrm{const}$.

Having determined, in one way or another, the quantities $C$ and, on the basis of preliminary estimates, chosen the sections of the frame members and the dimensions of the shoes, we take one of them and, cutting it off from the overlying structure, apply to it the axial force $P$, the transverse force $Q$, and the bending (overturning) moment $M$ acting in the lower section of the column.

If, in determining the internal forces acting in the frame and the support reactions, the length of the columns was taken from the bottom of the shoes, then all these three external forces $(P, Q,$ and $M)$ will be applied at point $d$ (Fig. 3); if, however, the length of the columns was measured from the top of the shoe, then, in transferring these forces from point $b$ to point $d$, we obtain a new overturning moment:

\[ M' = M + Qh. \]

In general, in order to simplify the exposition, it is more convenient to adopt the first method of determining the column length, taking the moment of inertia of its cross-section over the height of the shoe $h$ as equal to infi-

Table I.

[[unclear: beginning of word]]ness. In the second method, we introduce into the calculation the quantity \(Q\), which is a function not only of the lower but also of the upper restraining moments of the column, as well as of the external load along its length.

In ordinary cases the force \(Q\) is fully taken up by the frictional forces developing in the plane of the sole of the shoe under the action of the compressive force \(P\); the normal stresses acting over this same surface may be divided into two parts: the stress

\(\sigma_0\), uniformly distributed over the entire area of the shoe and caused by the central force \(P\), and the stress \(\sigma_u\), caused by the overturning moment. The former stresses, although they often have a predominant value and therefore also produce large deformations of the soil, will not be included in our consideration, on the grounds that these stresses are more or less uniform at all the shoes of the structure, and only when it is partially loaded by a temporary load is some nonuniformity in the settlement of individual supports to be expected; in that case one would have to introduce into the calculation not the absolute magnitude of the resulting displacements, but only their difference, whereas the deformations caused by the bending moment and expressed in the rotation of the shoe must be introduced in their full amount.

In view of the fact that the height of the bent section of a reinforced-concrete shoe is usually taken not from the conditions of permissible bending stresses, but on the basis of a calculation of shearing and oblique tensile stresses, which entails a large increase in its height, we shall not make a large error if we assume that its deflection deformations are so small in comparison with the deformations of the soil that they may be neglected. In that case the foundation stresses caused by the overturning moment may be determined from the formula:

\[ \sigma_u = \frac{M}{J} x, \]

to which there corresponds the soil deformation

\[ y = \frac{\sigma_u}{C} \]

and the greatest deformation, occurring at the edge of the shoe,

\[ y_{\max} = \frac{1}{C}\cdot \sigma_{u\max} = \frac{1}{C}\cdot \frac{M}{J} a \]

Under the influence of these deformations the shoe will rotate through a certain angle, whose tangent is written as follows:

\[ \operatorname{tg}\alpha = \frac{y_{\max}}{a} = \frac{1}{C}\cdot \frac{M}{J}, \]

but in view of the fact that the angle \(\alpha\) is very small, it may be considered that the tangent of the angle is equal to the angle itself, or

\[ \alpha = \operatorname{tg}\alpha = \frac{M}{JC}. \]

Equating the moment overturning the shoe to unity, we obtain the required angle of rotation of the lower support of the column \(\varepsilon\), entering into the formula for determining the lower focal ratio:

\[ \varepsilon = \frac{1}{J\cdot C}. \]

In the case where the pressure of the structure is transmitted to the soil through a pile foundation, as occurs with weak soil, the determination of the magnitude of the angle of rotation of the shoe \(\varepsilon\) and of the lower focal ratio of the column remains in principle the same; only in calculating the moment of inertia of the shoe one must take into account not its full ...

area, but only the cross-sectional area of the piles, and instead of the bed coefficient \(C\) the pile-settlement coefficient \(C_1\) is introduced.

If in many cases the bed coefficient is a quantity not yet established, then, as regards the coefficient \(C_1\), there are even fewer experiments.

Below, as approximate data, Table II gives certain values of \(C_1\), depending on the quality of the soil and the length of the piles.

Table II*)

In conclusion it should be noted that, with an increase in the length of the driven pile, its settlement coefficient increases rapidly.

To clarify all that has been said, below we give an example of the analysis of a frame resting on soil.

Example. Let us be given a two-bay symmetrical frame with the dimensions indicated in Fig. 4, resting on soil possessing a bed coefficient \(C = 9\ \text{kg}\cdot\text{cm}^{-3}\), and loaded on the left girder with a uniformly distributed load \(p = 3\) tons per running meter of projection. Suppose that, on the basis of preliminary estimates, the sections of the columns \(0.4 \times 0.6\) m and of the girders \(0.35 \times 0.9\) m were selected, which gives moments of inertia:

\[ J_1 = J_4 = J_5 = 0.4 \frac{0.6^3}{12} = 0.0072\ \text{m}^4 \]

\[ J_2 = J_3 = 0.35 \cdot \frac{0.9^3}{12} = 0.0212\ \text{m}^4 \]

and shoes with an area of \(1.5 \times 1.5\) m, whence the moment of inertia of their bearing surface:

\[ J_0 = J_{\mathrm{IV}} = J_{\mathrm{V}} = \frac{1.5^4}{12} = 0.422\ \text{m}^4 = 4.22 \cdot 10^7\ \text{cm}^4 \]

*) Tables I and II are borrowed from Terzaghi’s book Erdbaumechanik auf bodenphysikalischer Grundlage.

Hence the angle of rotation of the shoe under the action of the overturning moment \(M=1\) ton·m \(=10^5\) kg·cm is equal to

\[ \varepsilon_0=\varepsilon_{\mathrm{IV}}=\varepsilon_{\mathrm{V}}=\varepsilon=\frac{M}{CJ} =\frac{10^5}{9\cdot 4.22\cdot 10^7} =2.625\cdot 10^{-4}. \]

Proceeding to the determination of the focal ratios, we introduce, in computing all angles, the constant multiplier

\[ EJ_2=2.1\cdot 10^6\cdot 0.0212=4.46\cdot 10^4\ \text{ton·m}^2 \]

In that case:

\[ EJ_2\varepsilon=4.46\cdot 10^4\cdot 2.625\cdot 10^{-4}=11.71; \]

\[ EJ_2\alpha_1=EJ_2\alpha_4=\frac{l_1}{3}\cdot \frac{J_2}{J_1} =\frac{5}{3}\cdot \frac{0.0212}{0.0072}=4.92; \]

\[ EJ_2\beta_1=0.5\,EJ_2\alpha_1=0.5\cdot 4.92=2.46; \]

\[ EJ_2\alpha_5=\frac{l_5}{3}\cdot \frac{0.0212}{0.0072}=6.9; \]

\[ EJ_2\beta_5=0.5\,EJ_2\alpha_5=0.5\cdot 6.9=3.45; \]

\[ EJ_2\alpha_2=\frac{l_2}{3}=\frac{10.2}{3}=3.4;\qquad EJ_2\beta_2=0.5\,EJ_2\alpha_2=1.7. \]

We compute the value of the focal ratios. For the end columns

\[ K_1^{0}\,K_1^{\mathrm{IV}} =\frac{\beta_1}{\alpha_1+\varepsilon} =\frac{2.46}{4.92+11.71} =0.1475. \]

For the middle column

\[ K_5^{\mathrm{V}} =\frac{\beta_5}{\alpha_5+\varepsilon} =\frac{3.45}{6.90+11.71} =0.1852. \]

Hence we see that, for one and the same size of shoe, for columns of greater length, i.e. having less stiffness, the fixity has proved to be greater, as was to be expected. Finding the end focal ratios for the cross-bars, we have:

\[ EJ_2\varepsilon_2^{\mathrm{I}}=EJ_2\varepsilon_2^{\mathrm{I}} =\alpha_1-K_0^{1}\beta_1 =4.92-0.1475\cdot 2.46=4.657, \]

whence

\[ K_2^{\mathrm{I}}=K_3^{\mathrm{III}} =\frac{\beta_2}{\alpha_2+\varepsilon_2^{\mathrm{I}}} =\frac{1.7}{3.4+4.657} =0.211 \]

and for the middle ratios (near the ridge):

\[ EJ_2\tau_3^{\mathrm{II}}=EJ_2\tau_2^{\mathrm{II}} =\alpha_2-K_2^{\mathrm{I}}\beta_2 =3.4-0.211\cdot 1.7=3.14, \]

\[ EJ_2\tau_5^{\mathrm{II}} =\alpha_5-K_5^{\mathrm{V}}\beta_5 =6.9-3.45\cdot 0.1852=6.26, \]

\[ EJ_2\varepsilon_2^{\mathrm{II}} =\frac{\tau_3^{\mathrm{II}}\tau_5^{\mathrm{II}}}{\tau_3^{\mathrm{II}}+\tau_5^{\mathrm{II}}} =\frac{3.13\cdot 6.261}{3.14+6.261} =2.09, \]

whence

\[ K^{II}_{2}=K^{II}_{3}=\frac{\beta_{2}}{\alpha_{2}+\varepsilon^{II}_{2}} =\frac{1.7}{3.4+2.09}=0.31. \]

To find the upper focal ratio of the middle column, we have:

\[ EJ_{2}\,\varepsilon^{II}_{5} =\frac{\tau^{II}_{3}\tau^{II}_{2}}{\tau^{II}_{3}+\tau^{II}_{2}} =\frac{3.14^{2}}{2\cdot 3.14}=1.57, \]

whence,

\[ K^{II}_{5}=\frac{\beta_{5}}{\alpha_{5}+\varepsilon^{II}_{5}} =\frac{3.45}{6.9+1.57}=0.407 \]

and for the upper ratios of the end columns

\[ EJ_{2}\,\varepsilon^{I}_{1}=EJ_{2}\tau^{I}_{2} =\alpha_{2}-K^{II}_{2}\beta_{2}=3.4-0.31\cdot 1.7=2.874, \]

whence,

\[ K^{I}_{1}=K^{III}_{4} =\frac{\beta_{1}}{\alpha_{1}+\varepsilon^{I}_{1}} =\frac{2.46}{4.92+2.874}=0.315. \]

After all the focal ratios have been determined, the further work of determining the transition coefficients and bending moments, for fixed and displacing joints, contains nothing special; therefore, omitting it, we give directly in Table III the computed support bending moments.

Table III.

Bending support moments in ton-meters for various degrees of fixity of the bottoms of the columns.

In this same table, for comparison, are given the moments obtained with absolute fixity of the lower ends of the columns \((J_0=\infty)\) and with hinges arranged at the bottom \((J_0=0)\). We see that the difference in some moments is so substantial that in both of the latter cases of calculation we would obtain results far from reality. Depending on the fixity of the posts, not only the distribution of moments changes, but also the magnitude and character of the deformation. Thus, to the three adopted cases of calculation there correspond displacements of the upper joints of the posts to the right by the amounts: 2 mm, 0.682 mm, and 2.83 mm. In Fig. 5, on an enlarged scale, the general form of deformation of the structure under the action of the external load is shown schematically.

Example II. Let the same frame have base shoes with an area of \(2.0 \times 2.0\) m. In that case

\[ J_0=J_{\mathrm{IV}}=J_{\mathrm{V}}=4^4\cdot\frac{1}{12}=1.333\ \mathrm{m}^4=1.333\cdot10^8\ \mathrm{cm}^4 \]

whence

\[ EJ_2 \varepsilon=4.46\cdot10^4\cdot\frac{10^5}{9\cdot1.333\cdot10^8}=3.71, \]

which gives the lower focal ratios for the posts:

\[ K_1^0=\frac{2.46}{4.92+3.71}=0.285, \]

\[ K_5^v=\frac{3.45}{6.9+3.71}=0.326. \]

Example III. Suppose that the preceding frame rests on a pile foundation, with a group of nine piles of diameter 20 cm under each post (Fig. 6), and with coefficient of pile settlement

\[ C_1=62\ \mathrm{kg}\cdot\mathrm{cm}^{-3}. \]

Hence the area of each pile is

\[ f=\frac{\pi d^2}{4}=314\ \mathrm{cm}^2, \]

and the moment of inertia with respect to a diameter is

\[ J_0^1=\frac{\pi d^4}{64}=7854\ \mathrm{cm}^4, \]

whence the moment of inertia of the section of the pile group is

\[ J_0=9J_0^1+6\cdot f\cdot60^2=9\cdot7854+6\cdot314\cdot60^2=6850000\ \mathrm{cm}^4 \]

\[ EJ_2 \varepsilon=4.46\cdot10^4\cdot\frac{10^5}{6.85\cdot10^6\cdot62}=10.5, \]

to which there correspond:

\[ K_1^0=0.161;\qquad K_5^v=0.1985. \]

Calculation by the method of four moments

In the case of calculating a frame structure by the “method of four moments,” it is possible in exactly the same way to take into account the degree of restraint of the lower sections of the posts. To simplify the understanding of the derivation, we shall use the same frame that was presented in the preceding example, and shall solve it in general form, i.e., without substituting numerical values—

*) 1. F. Bleich, “Die Berechnung statisch unbestimmter Tragwerke nach der Methode des Uebermomentensatzes”.

1. Prof. G. V. Ulyaninsky, “Calculation of Frame Structures.”

...of individual quantities. As is known from the theory[^1], the equation of four moments has the following form:

\[ M_1^0 l_1' + 2M_1^1 l_1' + 2M_2^1 l_2' + 2M_2^{\mathrm{II}} l_2' = 6EJ_0(\vartheta_1 - \vartheta_2) + C_{1-2} \]

First of all, let us establish the relation between the angles of rotation of the individual members \(\vartheta\) (Fig. 7). Under the action of the external load, the upper joints of the frame may receive certain lateral displacements by an amount \(\rho\), which, when the angles \(\vartheta\) are small, may be regarded as perpendicular to the initial position of the axes of the members. Owing to the fact that it is usually possible to neglect the elongations of the members, and in the presence of only columns parallel to one another (which is the case in our particular example), the displacements of all the upper ends of the columns will be equal to one another, and both girders will move parallel to themselves. Thus we may write:

\[ \vartheta_1 = \vartheta_4 = \frac{\rho}{l_1}; \qquad \vartheta_5 = \frac{\rho}{l_5} = \frac{\vartheta}{l_1}\cdot \frac{l_1}{l_5} = \vartheta_1 \frac{l_1}{l_5}; \qquad \vartheta_2 = \vartheta_3 = 0. \]

In order to take into account the angle of rotation of the shoes under the action of the fixing moments, let us suppose that below the support joints there are additional members of arbitrary length \(l\) with an infinitely large moment of inertia \(J=\infty\), which gives their length, reduced to some finite moment of inertia \(J_2\) common to the whole frame:

\[ l' = l \frac{J_2}{J} = l \frac{J_2}{\infty} = 0. \]

At the same time, under the action of the moments transmitted to these fictitious members from the frame at points O, IV, and V and equal to the support fixing moments of the columns—\(M_1^0\), \(M_4^{\mathrm{IV}}\), and \(M_5^{\mathrm{V}}\)—these members will rotate through certain angles \(\vartheta_0\), \(\vartheta_{\mathrm{IV}}\), and \(\vartheta_{\mathrm{V}}\). Taking into account the sign convention usually adopted in this method of analysis, whereby a moment bending an element convexly to the right, counting in the direction of the numbering of the joints (shown by arrows in drawing 8), is taken as positive, we have:

\[ \vartheta_1 = - M_1^0 \varepsilon; \qquad \vartheta_{\mathrm{IV}} = M_4^{\mathrm{IV}} \varepsilon; \qquad \vartheta_{\mathrm{V}} = M_5^{\mathrm{V}} \varepsilon, \]

where the quantity \(\varepsilon\) has the same meaning as in the analysis by the method of focal ratios. The signs adopted in determining the values of the angles of rotation are explained as follows: for a column in which the numbering proceeds from the shoe upward, in the case of a positive fixing moment at the bottom, the moment transmitted to the fictitious element (Fig. 8) has a counterclockwise rotation; consequently, in this direction the angle of rotation \(\vartheta\) will be laid off, whereas, according to the sign convention adopted in deriving the basic formulas, this direction is taken as negative. For members in which the numbering of the joints goes from top to bottom, in the presence of a positive support moment in the column, the moment rotating the fictitious member will act clockwise, i.e., we obtain a positive angle \(\vartheta\).

Passing to the determination of the bending moments under the action of a one-sided distributed load \(p\) on the left girder, we take two types of loading (Fig. 9): the first type consisting of a symmetrical load \(\frac{p}{2}\) distributed over both spans, and the second type consisting of a mirror-symmetrical load of the same magnitude.

By summing the bending moments from the two accepted cases of loading, we obtain the desired quantities.

In the first part of the calculation, in view of the fact that under the action of a symmetrical load the section of the girder above the column will not rotate, we may consider a system consisting only of two members \(0—1—II\) with an absolute fixity at point \(II\), owing to which the middle column will not be subjected to bending at all. According to the number of unknown support moments, we set up three equations of four moments:

\[ 2 M_a^0 l_1^1 + M_a^I l_1^1 = 6 E J_2 \vartheta_0 = -6 E J_2 M_a^0 \varepsilon ; \]

\[ M_a^0 l_1^1 + 2 M_a^I \left(l_1^1 + l_2^1\right) + M_a^{II} l_2^1 = + C_{1-2}; \]

\[ M_a^I l_2^1 + 2 M_a^{II} l_2^1 = + C . \]

In these equations the lengths of all members have been reduced to the length of a member of the same stiffness, but with a moment of inertia equal to the moment of inertia of the girder \(J_2\), i.e.

\[ l_1^1 = l_4^1 = l_1 \frac{J_2}{J_1}; \qquad l_2^1 = l_3 = l_2 \frac{J_2}{J_2} = l_2 . \]

Passing to the second part of the calculation, namely to the determination of the effect of the mirror-symmetrical load, we may consider not the whole frame, but only its half up to the axis of symmetry, taking the moment of inertia of the fifth column and of its shoe reduced by half (Fig. 10). In this case the reduced length of the middle column will be:

\[ l_5^1 = l_5 \frac{2 J_2}{J_5} \]

and we obtain the system of equations:

\[ 2 M_b^0 l_1^1 + M_b^I l_1^1 = 6 E l_2 \left(-M_b^0 \varepsilon - \vartheta_1\right); \]

\[ M_b^0 l_1^1 + 2 M_b^I \left(l_1^1 + l_2^1\right) + M_b^{II} l_2^1 = 6 E l_2 \vartheta_1 + C_{1-2}; \]

\[ M_b^I l_2^1 + 2 M_b^{II} \left(l_2^1 + l_5^1\right) + M_b^V l_5^1 = -6 E J_2 \vartheta_1 \frac{l_1}{l_5} + C_{2-5}; \]

\[ M_b^{II} l_5^1 + 2 M_b^V l_5^1 = 6 E J_2 \left(\vartheta_1 \frac{l_1}{l_5} - M_b^V \varepsilon\right). \]

Since here four unknown moments and also the angle of rotation of the first column \(\vartheta_1\) enter into the equations, we need one more equation; to set it up we equate to zero the projection of all forces on the horizontal axis, which gives us:

\[ \Sigma X = H_1 - H_5 = O \]

or

\[ \frac{M_b^I - M_b^0}{l_1} - \frac{M_b^I - M_b^V}{l_5} = O . \]

Solving the two systems of equations that have been set up and then summing their results, we obtain all the required bending moments:

\[ M^0 = M_a^0 + M_b^0; \qquad M^{IV} = M_a^0 - M_b^0; \]

\[ M^I = M_a^I + M_b^I; \qquad M^{III} = M_a^I - M_b^I; \]

\[ M_2^{II} = M_a^{II} + M_b^{II}; \qquad M_3^{II} = M_a^{II} - M_b^{II}; \]

\[ M_5^{II} = 2 M_b^{II}; \qquad M^V = 2 M_b^V; \]

Solving the problem in numerical values, we shall of course arrive at the same results as were obtained by the method of focal ratios.

Tomsk. February 1928.

To the article by Eng. K. N. Kartashov

Fig. 1.

Visible labels: \(A\), \(B\), \(M=1\), \(\alpha\), \(\beta\), \(\gamma\), \(M_a=1\), \(M_b=1\), \(\varepsilon_a\), \(\varepsilon_b\).

Fig. 3.

Visible labels: \(2a\), \(b\), \(P\), \(M\), \(d\), \(Q\), \(x\), \(h\), \(\delta\), \(\delta'\), \(\delta''\), \(\delta_n^{\max}\).

Fig. 2.

Visible labels: \(0.0\), \(1\), \(2\), \(3\ \mathrm{kg/cm^2}\), \(0.2\), \(0.4\), \(0.6\ \mathrm{cm}\), \(P\), II.

Fig. 6.

Visible labels: \(0.20\), \(1.60\), \(1.20\).

Fig. 4.

Visible labels: \(10.20\), \(10.00\), \(5.0\), \(7.0\), \(0\), I, II, III, IV, V, 1, 2, 3, 4, 5.

To the article by K. N. Kantashev

Fig. 7

Visible labels and annotations: \(l_2\), \(l_3\), \(\beta\), I, II, IV, V, \(p_1\), \(p_3\), \(p_4\), \(v_2=0\), \(v_3=0\), \(v_1\), \(v_5\), \(v_4\), \(v_0\), \(v_V\), \(v_{IV}\), \(e\).

Fig. 8

Visible labels and annotations: \(n+1\), \((-)\), \((-)\), \(m\), \(n\), \((+)\), \((+)\), \(m+1\), \(v_n^{(-)}\), \(v_{m+1}^{(+)}\).

Fig. 9

Visible labels and annotations: I, II, 0.

Fig. 10

Visible labels and annotations: I, II, 0, V, \(J_1\), \(J_2\), \(J_3\), \(J_0\), \(0.5J_V\), \(0.5J_3\).

Fig. 5

Visible labels and annotations: \(2.83\), \(2.0\,\mathrm{m}\), \(0.682\,\mathrm{m}\), \(J_0=0\), \(J_0=0.422\,\mathrm{m}^4\), \(J_0=\infty\).