Structural mechanics

THEME 12. CALCULATING STATICALLY

INDETERMINATE TRUSSES

12.1. Types of Statically Indeterminate Trusses

This chapter discusses the features of calculating trusses as articulat- ed-rod systems with extra connections (with redundant links). It is essential to remember that the nodal joints of the hinge-rod systems are ideal hinges without friction.

The degree of static indeterminacy of the hinge-rod system is determined by the formula

B L 2N ,

where B is the number of rods making up the truss; L – the number of support rods of the truss; N – the number of truss nodes.

Examples of several types of statically indeterminate trusses are shown below (Figures 12.1. and 12.2, a).

A three-span continuous beam truss with parallel chords and a triangular lattice (Figure 12.1, a) is twice externally statically indeterminate. After being detached from the supports, this truss has a geometrically unchangeable statically determinate structure.

Figure 12.1

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The seven-panel beam truss with a crossed lattice (Figure 12.1, b) contains seven extra links. This truss is internally statically indeterminate. Externally, it is statically determinate: the reactions of its supports can be found from the equilibrium equations, as in a simple beam.

A beam truss with parallel chords, with a triangular lattice and additional struts, but strengthened with a polygonal tie (Figure 12.1, c), is also internally statically indeterminate once.

A thrusting two-hinged truss with an additional brace in the central panel (Figure 12.2, a) is statically indeterminable both externally and internally.

12.2. Features of Calculating Statically Indeterminate Trusses

The calculation of statically indeterminate trusses is performed, as a rule, by the force method. The primary system of the force method is selected by cutting the truss rods, or by removing the support links (Figure 12.2, b), which are not absolutely necessary.

Figure 12.2

The canonical equations of the force method have standard form

where the index n means the number of primary unknowns of the forces method.

With a nodal load in the rods of statically indeterminate trusses, as well as other hinged-rod systems, only longitudinal internal forces will

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arise. Therefore, the displacements in the trusses will depend only on the longitudinal deformations of their rods, and the one-term Maxwell formula should be used to calculate the displacements.

Consequently

where the summation sign ∑ extends to all the truss rods;

Ni , Nk , NF – accordingly, the efforts in the rods of the primary

system of the method of forces due to unit values of the primary unknowns ( Xi 1, Xk 1) and the given load F;

s and EA – length and tensile-compression rigidity of the corresponding truss rod.

The final effort in the rods of statically indeterminate truss is calculated by the formula

n

N NF Ni Xi .

i 1

All calculations are conveniently carried out in a tabular form. For a truss with two primary unknowns (Figure 12.2), such a table can have the following form (Table 12.1).

Table 12.1

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The first column of the table shows the rod numbers in a selected order. The second column contains the deformability of the rods, i.e. the ratio of the lengths of the rods to their longitudinal rigidity. The third, fourth and fifth columns contain the internal forces in the truss rods, calculated in the primary system of the force method due to the given load and unit values of the primary unknowns.

In the next five columns, the actual calculations are performed, the meanings of which are indicated in the header of the table. The sums of the elements of the columns 6,...,10 give the values of the coefficients at primary unknowns and the values of the free terms of the canonical equations of the force method, i. e. the displacements from unit unknowns and the displacements from given load in the primary system.

After the values of the basic unknowns are determined from the solution of the system of canonical equations, columns 11 and 12 are filled in. In other words, the actual values of the efforts in the primary system are calculated by the found values of the primary unknowns.

Finally, summing columns 3, 11 and 12, the final values of the internal forces in the rods of a statically indeterminate truss are obtained. If necessary, additional columns can be added to the table for intermediate and final kinematic checks in accordance with the force method.

Calculation of trusses by the displacement method leads to a significantly larger number of primary unknowns. As a rule, the displacement method is used in the automated calculation of trusses using computers.

12.3. Constructing Influence Lines for Efforts in Truss Rods

The influence lines are used to calculate trusses under the action of a moving load to determine its most unfavorable location. Based on the theorem on the reciprocity of reactions and displacements (the kinematic method of constructing influence lines), the influence line for the internal force in any rod (link) of a statically indeterminate truss coincides with the deflection line of the loaded chord of the truss caused by the action of a unit displacement in the direction of this internal force (in the direction of the corresponding link).

The process of constructing the influence line for the effort in a certain rod (link) of a statically indeterminate truss can be carried out somewhat differently, based on the theorem on reciprocity of displacements. To build the influence line for the effort in a truss rod, it is neces-

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sary to cut this rod (remove the corresponding link). The degree of static indeterminacy of the truss is reduced by one. A truss with a removed link can be considered as the primary system of the force method, in the general case, statically indeterminate. The primary unknown, the reaction in the removed constraint, depends on the point of application of the mobile force equaled to one. The law of change of this primary unknown determines the desired line of influence.

From the corresponding canonical equation it may be found that:

Inf .Line of X1 1F (x) F1(x)

11 11

where 11 is the displacement in the primary system in the direction of

the removed constraint from the unit value of the force in this constraint, that is constant quantity;

1F (x) is the displacement in the primary system in the direction

of removed constraint from the unit force and is the function of argument x that is the abscissa of the point of application of the mobile unit force;

F1(x) is a function of the same argument x, but which expresses

displacements in the direction of the mobile force from the unit value of the immobile primary unknown X1 1 , i.e. is a diagram of displace-

ments (is a diagram of deflections of the chord that will be loaded) in the truss with the removed link due to unit value of the force in this link.

Thus, in order to construct a line of influence of a certain effort in a statically indeterminate truss, it is necessary to remove the member perceiving this effort. Then a unit force is applied to the truss with the removed link in the direction of this link. The applied unit force causes the deflections of all nodes of the chord that will be loaded by vertical mobile unit force. The diagram of the displacements of this chord should be

constructed (deflection line). The displacement 11 in the direction of

the removed link should also be calculated. Usually last displacement is non-zero and positive. Consequently, the ordinates of the deflection line,

reduced by 11 of time, are the ordinates of the desired line of influence.

When using computer technology, the influence line for any effort can be built by its direct definition, as a result of the multiple calculation of this effort from the action of a single vertical force equaled to one and

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applied alternately to each nodes of the chord on which the unit force will move.

If, in one way or another, the influence lines for forces are constructed in all the redundant links of a statically indeterminate truss

(Inf . Line Xk , (k 1,..., n)),

then the line of the influence for effort in any other rod (Inf . Line N j ) can be constructed using a simple formula:

n

Inf . Line N j Inf .LineN0j Nkj (Inf .LineX k ) ,

k 1

where Inf .Line N0j is the influence line of the force in question in the primary system of the force method with n removed links;

Nkj is the force in the considered rod in the primary system of the force method from a unit unknown X k 1 .

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THEME 13. CALCULATING STATICALLY INDETERMINATE ARCHES, SUSPENSION AND COMBINED SYSTEMS

13.1. Kinds of Statically Indeterminate Arches

The following types of statically indeterminate arches are most commonly used in construction practice: two-hinged arches, single-hinged arches and hingeless arches.

The two-hinged arch (Figure 13.1, a) is characterized by two immovable hinged supports. The single-hinged arch (Figure 13.1, b) contains, as a rule, one hinge in the middle of arch span. The hingeless arch (Figure 13.1, c) represents a continuous curved bar, absolutely rigidly supported at the ends.

From the point of view of static indeterminacy, a two-hinged arch has one “extra” link, a single-hinged arch is twice statically indeterminate, and a hingeless arch is three times statically indeterminate.

Figure 13.1

As a rule, according to the outline, the arches are symmetrical. Depending on the nature of the load, the axis of the arch can be outlined along a square parabola, along an arc of a circle or other curve, and can be polyline. The cross section of the arch can be either constant or variable along the length of the arch.

All types of arches are thrusting systems, i. e. when a vertical load is applied to an arch, horizontal reactions also arise in its supports.

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Therefore, arches require the creation of powerful supporting devices. Arches with ties are used in order not to transmit significant horizontal forces to the underlying structures. Typically, ties of different designs are arranged in two-hinged arches (Figure 13.2). A two-hinged arch with a tie retains the properties of thrust systems, has supports as a simple beam. Therefore, it transfers only vertical forces from a vertical load to the underlying supporting structures and can be located on tall columns or walls without the use of special buttresses

Figure 13.2

The features of calculating once statically indeterminate arches will be considered using the example of calculating a two-hinged arch with a tie.

13.2. Calculation of a Two-Hinged Arch with a Tie

The two-hinged arch with a tie is outwardly non-thrusting. The thrust is perceived by a tie, and should be considered as an internal tensile force in the tie. A two-hinged arch with a tie has only one redundant link and may be easy calculated by the force method.

Let us consider a two-hinged arch with a straight tie located at the level of the supports. The arch has a cross sectional area that may be variable along the span. The arch is loaded with a vertical load (Figure 13.3, a). The primary system of the force method can be obtained by dissecting the tie (more precisely, removing from the tie a link that perceives longitudinal force). The primary unknown of the force method will be

the internal tensile force in the tie Ntie X1 (Figure 13.3, b). The canonical equation of the force method has the form

11X1 1F 0 ,

where 11 is the mutual displacement of the ends of the tie in the cut, caused by the unit effort in the tie X1 1, 1F the mutual displacement of the ends of the cut tie from external loads.

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line of the axis of the arch (Figure 13.3, d). The unit primary unknown causes the constant tensile longitudinal force Ntie 1 in the tie (Figu-

re 13.3, e).

The main feature of calculating arches is that the Mohr integrals for calculating displacements in arches must be taken along the length of the axis of the arch, i.e. they are curvilinear integrals. The free term of the canonical equation is found by the one-term Mohr formula

The coefficient at the primary unknown (the unit displacement), calculated taking into account the longitudinal deformation of the tie, is found by the two-term formula

To go to the integration over the span, i.e., over the abscissa x, in these curvilinear integrals, it is necessary to introduce the replacement

where (х) is the angle of inclination to the horizontal of the tangent to

the axis of the arch in cross section with the abscissa x. As a result, the following formulas are obtained to calculate the coefficient and the free term of the canonical equation of the force method:

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