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Structural Integrity

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Procedia Structural Integrity 6 (2017) 216-223

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XXVII International Conference "Mathematical and Computer Simulations in Mechanics of Solids and Structures". Fundamentals of Static and Dynamic Fracture (MCM 2017)

Development of the applied approach to analysis of beam-columns

L.M. Kagan-Ro senzweig*

Saint-Petersburg State University of Architecture end Civ— Engineer—g, Russia

Abstract

The known approximate formula for calculating bending moment in statically determinate elastic rod under bending and compression is generalized to the case of a statically indeterminate one loaded with a compressive force distributed along its length. The accuracy of the proposed formula corresponds with the accuracy of engineering calculations. Presented examples show that for the statically determinate rod of constant cross-section compreired by the load amounting to 990%% oi a critical one the moment error is about 3%; if the rod is statically indeterminate it is about: 6%.

CopyrigVt © 2017 The Authors. Pub lished byElsevier B. V. Peer-review under responsibility of the MCM 2017 organizers.

Keywords: ton^udmaRransverse bendmg bendmg moment beam-cotamns.

1 h—trs—d uctkin

In beantscokmns bendmg moment AU is often cakcidaEed u smg approxmiate tormdat leaving an— engrnee—tng de-grtt of accuracy. For tlie tceam-cohmn on two hinged supportE, comjsressed by at force p of constant direction on the end suclt formula iM = M0 +-P-(1)

1-P / Pc

In tltis formula P— is ttie Ituler's critical force. Here and ttektw the superscript "'zero" means computation m the absence of compression.

* Corresponding author. E-maü address: Kagan_R@maU.ru

2452-3216 Copyright © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the MCM 2017 organizers. 10.1016/j.prostr.2017.11.033

Formulas of the type Eq. (1) are known only for statically determinate rods. For statically indeterminate rods, less precise relations are used in engineering calculations.

The presented paper generalizes the simple result Eq. (1), firstly, to the case of a statically indeterminate rod, secondly, to the case of compressive load varying along the rod. This paper is the part of investigation that exploits the potentialities of the rarely used differential bending equation

M" + —M =-q . (2)

This equation is valid for elastic rod of variable cross-section compressed by force P at one end. EI is the flexural rigidity, q is the load intensity. To author's knowledge, A. R. Rzhanitsyn was the first who applied this equation to rod of variable cross-section, Rzhanitsyn (1955). Other results of the mentioned above investigation can be found in papers Kagan-Rozenzweig (2015), (2016), (2017).

2. The rod compressed by a force at the end

The rod in Fig. 1 has a variable cross-section, S is a moment at support, H is a horizontal reaction (it does not coincide with the shearing force). Subscripts "zero" and "one" indicate cross-sections at the origin and at the opposite end, respectively.

Fig. 1. Beam-column under consideration

Approximate formula is constructed right for a moment M omitting deflection w calculation. The starting point is the differential Eq. (2), supplemented by the bending equation in traditional notation

M = — EIw

and the corresponding boundary conditions.

The moment under investigation is written as the sum of two terms:

M = M 0 + AM

Term M0 is the solution of transverse bending problem, satisfies the equation

M "0 = —q

Term AM is the correction due to the force P . Combining Eq. (3) - (5), we get:

AM" + — AM = ——M 0. (6)

The effect of the equilibrium conditions is the following exact equation

AM = P[(w- w0) + a + bx], (7)

in which the term P(a + bx) accounts for a change in support reactions H0, S0 evoked by deflection w0; a, b are the constants.

The deflection w is also divided into two terms: w = w0 + Aw . Constants a, b coupled with deflection w0 are denoted as a0, b0, the first-order correction is introduced:

A1M = P(Aw0 + a0 + b0x), Aw0 = w0 - w00. (8)

The total correction AM is written as the sum

AM = A1M + A 2M . (9)

As Eq. (3) is also valid for the moment M0, according to Eq. (3), (8) we have A1M = -(P/EI)M0, and Eq. (6) results in

A 2 M +— A 2 M =--A1M . (10)

2 EI 2 EI 1

The last equation is exact, but it is solved approximately. Let Mc be the form of the moment at the stability loss, satisfying the homogeneous differential equation

Mc + P Mc = 0 (11)

with the corresponding boundary conditions. The correction A2M is considered proportional to Mc:

A 2 M = CM c. (12)

According to Eq. (11), (12)

A2M =—^ A2M ,

therefore approximately

" P P P

A2M + — A2M = (—^ + —)A2M .

2 EI 2 EI EI 2

Taking Eq. (10) into account, we have the proportionality of A1M , A2M :

A2M =-A,M . (13)

2 Pc - P

Combining the above equations, we get the required formula:

M = M0 +-P-(Aw0 + a0 + b0x), Aw0 = w0 - w00. (14)

1 - P / Pc

In what follows the lower support is assumed to be immovable, that is w00 = 0 and Aw0 = w0. Constants a0, b0 of this formula are calculated from the boundary conditions, have the meaning of specific (divided by the force P ) support reactions at the lower end, arising due to deflection w0.

For the cantilever rod it turns out that a0 = -wf , b0 = 0.

For the hinged rod a0 = 0, b0 = 0.

In these two cases Eq. (14) converts into known approximate formulas.

When the rod is fixed at the bottom and hinged at its upper end, constants a0, b0 provide the zero moment and deflection at the upper end. Applying the Maxwell-More formula, we obtain

b0 = fw°( x)(l - X) dx / f(l - X)2 dx , a0 = - lb0. (15)

0 EI /i EI

For the rod with fixed both ends, linear and angular deflections of the lower end with fixed upper one are equal zero, constants a0, b0 are the solution of the system

l i l l 0r \ l l 2 l 0 / \

i±dx • a0 + fXdx • b0 + fw^)dx = 0, f-^dx • a0 + f^dx • b0 + f^^dx = 0. (16)

0 EI 0 EI 0 EI 0 EI 0 EI 0 EI

3. The rod compressed by a system of forces

The system of concentrated and distributed compressive forces is specified by the load parameter K . Concentrated forces Pi = Kpi, i = 1,2...,n are applied in sections with coordinates x = ; R(x) = Kr(x) is distributed compressive force. P0 = Kp0 is the vertical reaction at lower support. Kc is the critical value of K .

Let w0 be the deflection in the absence of compressive load. We divide the rod into two pieces and calculate the moment m = Kf (w0) in the section of curved rod due to vertical load only. Such calculation of m does not provide equilibrium, so the function f (w0) can be written in two different forms, depending we consider the upper or the lower part of the rod:

f (w0) = P0(w0 - w0) - XPi(w0 - w0)| - ir(y)[w0(x) - w0(y)]dy (17.1)

i=1 x>xi 0

f (w0) = £ Pi (w0 - w0)|x< + i r(3-Xw0(x) - w0(y)]dy (17.2)

i =1 x<x' x

Eq. (17.1) and (17.2) differ linearly with respect to x, so that the choice of particular expression for f (w0) affects

only a0, b0 values, but does not affect the result. In the equations above, if condition x > xi (x < xi) written below the line is violated, the corresponding term is to be omitted.

When solving a particular problem, we choose a convenient expression for f (o0), then write the first-order correction

A1M = L[ f (o0) + a0 + b0 x]. (18)

Here, constants a0, b0 keep the meaning of the specific support reactions at the rod's end (at the bottom or top end

depending on the f (o0) choice), they are found by formulas for the rod with a force at its top, replacing in these

formulas deflection Ao0 with function f (o0). Eq. (14) takes the form

M = M0 +-L-[ f (o0) + a0 + b0 x]. (19)

1 - L / Lc

Eq. (19) is deduced the same way as the Eq. (14). Now, however, the following more complicated equation is used instead of Eq. (2):

M" + NM - o'R = -q . EI

In it N is the projection of forces on one side of the section onto the axis of the stainless rod. Note that both Eq. (14), (19) relate to the rod of variable cross-section.

Below are examples of beam-columns, bent by uniformly distributed transverse load q , illustrating the accuracy of approximate formulas.

4. Examples

4.1. Cantilever rod

L(1-m)

Fig. 2. Cantilever rod

The rod of length l is loaded by transverse load q and by three different systems of vertical forces (Fig. 2).

M0 = ql2 / 2 is the moment's module at support in the absence of compression, deflection w00 = 0 . Constants are

as follows: a0 = -w0, b0 = 0.

For the load in Fig. 2, a), Eq. (14) and Eq. (19) coincide, move to a known result

M = M0 +-P-(w0 - w,0).

1 - P / Pc 1

When compressive force amounts to 90% of a critical one, exact and approximate moments at support differ in 3.5%.

The load in Fig. 2, b) is not covered by the traditional Eq. (20), we must apply Eq. (19). Let's take the sum of compressive forces as the load parameter K , parameter m sets the forces ratio. In Eq. (17.1) n = 2 , p0 = 1, Pl = m , p2 = 1 - m, xj = al, X2 = l, / = 0 . The error of the approximate solution and the critical K value Kc depend on m and a. When K amounts to 90% of Kc, the error of the moment calculation is 1-3%.

For the load in Fig. 2, c), the load parameter is K = Rl. In Eq. (17.2) it is necessary to put / = 1/1, pt = 0 . For K = 0.9Kc, the difference between exact and approximate moments at support is 1.8%.

Fig. 3. Pinned-fixed rod

4.2. Statically indeterminate rod of constant cross-section

Eq. of type (1) are not applicable to the rod in Fig. 3. We are to use Eq. (14), (19). In Eq. (19) function f (w0) is taken in the form Eq. (17.2); Aw = w . In the absence of compression, M 0 = ql2/8 is the moment's module at the support. Constants a0, b0 are calculated according to Eq. (15). For load in Fig. 3, a), we obtain

b0 = J w0(x)(l - x)dxt

3 240EI

If P = 0.9Pc, the moment at support increases in 6.5 times, but its exact and approximate values differ by 6.7%. If P = 0.8Pc, the difference is reduced to 4.8%. Pay attention that calculations are made for a very high level of compressive load not found in structural design.

For the load in Fig. 3, b), load parameter is taken as K = Rl, so that pt = 0 , / = 1/1. Eq. (17.2) takes the form

f ( x) = - J [o0( x) - o0( y)]dy .

For this rod of constant cross-section, factor b0 is

b0 = J f (x)(l- x)dx / J(l- x)2dx = --

2240EI

If K = 0.9Kc, the error in moment at support and the maximum span moment calculation is less than 1%.

4.3. Statically indeterminate rod of variable cross-section

The rigidity of the rod loaded according to Fig. 3, a, is given in the form

EI = EI 0(1 -ax)4,

where parameter a < 1. In the absence of compressive load, the horizontal reaction in the upper support is

H 0 = q J dJ J V-lt dx ,

J 2EI /i EI

and the bending moment and deflection are

M 0 = H 0 - q i-xL. o» = J (Ç-x)

0 (Ç)

Parameters a0 , b0 are calculated by Eq. (15). Eq. (14) with o° = 0 gives:

M = M0 +

0 ' P (o0 + a0 + b0x). 1 - P / P„

0 0.25 0.5 0.75 „ 1 x/l

0.4 0.2 0

1 2 i i a = 0.8

0 0.25 0.5 0.75 „ 1 x/l

Fig. 4. Moments in the rod of variable cross-section The critical force and the exact solution needed to evaluate the proposed results, both depend on parameter a

are found numerically. Exact and approximate solutions are compared in Fig. 4. If a = 0.5, the rigidity at the bottom and top end differs by 16 times, Pcl2 /EI0 = 5.0476 , the exact and approximate moments at support differ by 7.8%. For a = 0.8, the rigidity differs by 625 times, Pcl2/ EI0 = 0.80762 , moments at support differ by 11%. P is taken as P = 0.8Pc. Curve 0 - no compression, 1 - exact solution, 2 - approximate solution.

As the degree of variability of cross-section increases, the accuracy of approximate solution decreases, but remains acceptable for technical calculations. Again, note that presented accuracy estimations consider a very high level of rod's compression.

5. Conclusion

The traditional approximate formula for bending moment in statically determinate beam-column is generalized to the case of the statically indeterminate one, compressed by a system of concentrated and distributed forces. The result of this generalization consists in the Eq. (14), (19). They have engineering accuracy and are useful for design calculations.

6. References

Rzhanitsyn A. R. 1955. Stability of the equilibrium state of elastic systems. Gostehizdat, Moscow, 475 p. [in Russian]

Kagan-Rozenzweig L. M. 2015. Calculation of natural frequencies of compressed rods with variable cross-section. Simplified bending equation

(I). Bulletin of civil engineers 6, 84-88. [in Russian] Kagan-Rozenzweig L. M. 2016. Method for internal forces calculation at transverse-longitudinal bending of elastic rods with variable cross section. Simplified bending equation (II). Bulletin of civil engineers 1, 75-82. [in Russian] Kagan-Rozenzweig L. M. 2016. Method to calculate the critical load in elastic rods with variable cross section. Simplified bending equation (III).

Bulletin of civil engineers 2, 61-67. [in Russian] Kagan-Rozenzweig L. M. 2017. Development of the applied approach to analysis of bars subjected to bending and compression/ Bulletin of civil engineers 4, 130-134. [in Russian]