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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 11, 1681-1697 (1977) STIFFNESS MATRIX FOR A BEAM ELEMENT INCLUDING TRANSVERSE SHEAR AND AXIAL FORCE EFFECTS ASHOK K. CHUGH* Rock Mass Behavior Group, DenverMiningResearchCenter, US. Bureau ofMines,Denver, Colorado, U.S.A. SUMMARY A simple and direct procedure is presented for the formulation of an element stiffness matrix on element co-ordinates for a beam member and a beam-column member including shear deflections. The resulting stiffness matrices are compared with those obtained using the alternative formulation in terms of member flexibilities: The relative effects of axial force and shear force on the stiffness coefficients are presented. The critical buckling loads, considering the effects of shear force, are computed and compared with those available in the literature. Only prismatic members are considered. INTRODUCTION In most civil engineering structures, the constituent members are either long and slender or short and stocky. In the former type, the effect of axial force on the response of the member is important, whereas in the latter type the effect of transverse shear force on the response of the member is very important. Accordingly, most of the developments reported in the literature account for the effect of one or the other, but not both, on the flexural behaviour of a member. If a structure is composed of both slender and stocky members, then one has to appropriately select the member load-displacement (stiffness or flexibility) characteristics. Since the slender or stocky classification of a member is relative, it would be convenient to utilize a single formulation of load-displacement response such that the specified dimensions and load mag- nitudes determine the influence of the so-called secondary effects on the flexural behaviour of the members and hence that of the structure. Furthermore, some engineering problems do not lend themselves to the solution methods devised for typical civil engineering problems, and hence more general and complete formulations are required. For example, a non-conventional structural problem provided the motivation for the development presented herein, which comprises a part of the U.S. Bureau of Mines investigation of the stability/collapse behaviour of a naturally jointed rock mass that spans an underground opening, where the loading is composed of both gravity and formation pressures. In the displacement matrix method of structural analysis one constructs the element stiffness matrix for each member in the structure and then synthesizes the element stiffness matrices to generate the structural stiffness matrix. The structural displacements, the element displace- ments, and hence the element forces can then be computed. * Research Structural Engineer. Received 30 June 1976 Revised 12 August 1976 @ 1977 by John Wiley & Sons, Ltd. 1681

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Page 1: Stiffness matrix for a beam element including transverse ...courses.ce.metu.edu.tr/ce483/wp-content/uploads/... · critical buckling loads, considering the effects of shear force,

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 11, 1681-1697 (1977)

STIFFNESS MATRIX FOR A BEAM ELEMENT INCLUDING TRANSVERSE SHEAR AND AXIAL

FORCE EFFECTS

ASHOK K. CHUGH* Rock Mass Behavior Group, DenverMining Research Center, US. Bureau ofMines, Denver, Colorado, U.S.A.

SUMMARY

A simple and direct procedure is presented for the formulation of an element stiffness matrix on element co-ordinates for a beam member and a beam-column member including shear deflections. The resulting stiffness matrices are compared with those obtained using the alternative formulation in terms of member flexibilities: The relative effects of axial force and shear force on the stiffness coefficients are presented. The critical buckling loads, considering the effects of shear force, are computed and compared with those available in the literature. Only prismatic members are considered.

INTRODUCTION

In most civil engineering structures, the constituent members are either long and slender or short and stocky. In the former type, the effect of axial force on the response of the member is important, whereas in the latter type the effect of transverse shear force on the response of the member is very important. Accordingly, most of the developments reported in the literature account for the effect of one or the other, but not both, on the flexural behaviour of a member.

If a structure is composed of both slender and stocky members, then one has to appropriately select the member load-displacement (stiffness or flexibility) characteristics. Since the slender or stocky classification of a member is relative, it would be convenient to utilize a single formulation of load-displacement response such that the specified dimensions and load mag- nitudes determine the influence of the so-called secondary effects on the flexural behaviour of the members and hence that of the structure. Furthermore, some engineering problems do not lend themselves to the solution methods devised for typical civil engineering problems, and hence more general and complete formulations are required. For example, a non-conventional structural problem provided the motivation for the development presented herein, which comprises a part of the U.S. Bureau of Mines investigation of the stability/collapse behaviour of a naturally jointed rock mass that spans an underground opening, where the loading is composed of both gravity and formation pressures.

In the displacement matrix method of structural analysis one constructs the element stiffness matrix for each member in the structure and then synthesizes the element stiffness matrices to generate the structural stiffness matrix. The structural displacements, the element displace- ments, and hence the element forces can then be computed.

* Research Structural Engineer.

Received 30 June 1976 Revised 12 August 1976

@ 1977 by John Wiley & Sons, Ltd.

1681

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1682 A. K. CHUGH

The stiffness matrix for a prismatic beam and a beam-column element can be derived in several ways.’’2 However, the derivation of the stiffness matrix for a prismatic beam which includes transverse shear deflection is not so straightforward. If shear deformations are not permitted in simple beam bending, then the requirement that plane sections remain plane can be used and this introduces internal When this restraint is removed to allow for shear deformation, there is an addition to the internal strain energy stored; to maintain the equality of internal and external work, the external work must be increased by the same amount. One possible approach to formulating the correct stiffness matrix is to form a flexibility matrix that accounts for shear deformation and then to obtain the stiffness matrix through the flexibility- stiffness tran~formation.~’~ In the technical literature there appears to be no formulation of a stiffness matrix for a beam-column, prismatic or non-prismatic, with shear deflections.

The objectives of the present paper are to present 1. a simple method to construct the stiffness matrix of a beam and a beam-column element of

constant cross-section, with bending in one principal plane, including shear deflections; 2. an alternative formulation for the element stiffness matrix of a beam-column of constant

cross-section, with bending in one principal plane, including shear deflection, in terms of member flexibilities; and

3. computation of critical buckling loads, including effects of transverse shear force, corres- ponding to any end conditions. Linear elasticity, small deformations, no warping, and no local cross-sectional buckling are assumed. Figure 1 shows the general form of the member and the element co-ordinate system.

The simple method presented (objective 1. in the preceding paragraph) helps one to describe the non-linear interaction among the axial, shear, and flexural effects intuitively. The alternative formulation in terms of member flexibilities presents a rigorous mathematical development which, in addition to being a contribution in itself, tends to confirm the validity of the simple

Y I v, v2

T

‘ P 4 d

Figure l(a). General layout of a beam element bending in one principal plane

Y

Figure l(b). Element co-ordinate system

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STIFFNESS MATRIX FOR A BEAM ELEMENT 1683

method. The computation of critical buckling loads is one of the areas of application of the element stiff neness matrix. These three items are described in the text in the order mentioned above.

STIFFNESS MATRIX FOR A BEAM ELEMENT INCLUDING SHEAR DEFLECTION

Translational stiffness

Without the shear deflection, the member forces for S translational displacement on element co-ordinate 3 , Figure 2(a), are

1 2 EIS k33 = -k43 =-

L3

See the Appendix for meaning of symbols. The additional displacement caused by transverse shear force k33 = 12EIS/L3 is

12EIS 61 =-

A,GL’

Total translational displacement on element co-ordinte 3, Figure 2(c), is

1 2 EIS A = S +S,=S+--S(I +a) A , G L ~ -

where

12EI a=- A,GL’

(4)

‘(5)

From equations ( l ) , (2), and (6), it follows that E13, Ez3, E33, and &43, the corresponding terms in the element stiffness matrix [El, including shear deflections, are given by

E - E - -6EI 1 13- 2 3 - 2 - L l + a

- - 12EI 1 k - - k 4 3 - L3 1+a 33 -

(7)

Similar expressions for member forces can be obtained for unit translational displacement on element co-ordinate 4, as shown in Figure 2(b).

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1684 A. K. CHUGH

k24

Figure 2. Translational stiffness of a beam element

Rotational stiffness

co-ordinate 1 , Figure 3(a), are Without the shear deflection, the member forces for 8 rotational displacement on element

The translational displacement caused by the transverse shear force k31 = 6E18/L2, without any change in the rotational displacement, if permitted, would be Figure 3(c),

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STIFFNESS MATRIX FOR A BEAM ELEMENT 1685

-I

4+a 2-a -6 6 L L L2 L2

2-cu 4 + ( ~ -6 6 . -6 -6 12 -12

- - - -

- - - - E' L L L2 L2

[El =-

The force required to cause 6' = 0, without causing any change in rotational displacement would be, Figure 3(d),

12EI 1 aL0 6EI a9 k' 31- - -k' 4 1 - 3 - - = - - - L l+a 2 L2 l+a

If, in Figure 3(e), gl1, k;', &I, and l 4 1 are the corresponding terms in the element stiffness matrix [&I including shear deflections, then from equations (9)-(14)

- - - -12 l 2 I I L2 L2 L3 L3

6

STIFFNESS MATRIX FOR A BEAM-COLUMN ELEMENT INCLUDING SHEAR DEFLECTIONS

Translational stifness

Without shear deflection, the member forces for 8' translational displacement on element co-ordinate 3, Figure 4(a), arelS2

-u2(1 -c)EI8' B'3 = B23 = L2(2 - 2c -us)

u 3sEIS' B 3 3 = - B 4 3 =

L3(2-2c-us)

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1686 A. K. CHUGH

Y A

F@re 3. Rotational stiffness of a beam element

Ignoring the presence of the axial force P, the transverse shear caused by BI3 and B23 is

2u2(1 -c)EIs’ L3(2-2c-us)

Bk3 = -B;3 =

The additional displacement caused by the transverse shear force B53 = [2uz(1 -c)EIs’]/ [L’(~-~c-us)] is

2u2(l -C)EI8’ s; = L2A,G(2-2c-u~)

Total translational displacement on element co-ordinate 3, Figure 4(c), is

2u2(1 - c)p I [ (2-2c-us) A =s‘+s; = 6’ 1 +

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STIFFNESS MATRIX FOR A BEAM ELEMENT 1687

where = EI1L’A.G .. (2 - 2c - u s w 6’ =

[2 - 2c - us + 2u2(1 - C ) P ]

The axial force P acting through the translational displacement A’ causes the equilibrating shear force of magnitude PA’IL, Figure 4(d). From equations (20), (22), (25) and the equilibrating shear force with the total translational displacement A’ set equal to unity the corresponding terms in the element stiffness matrix, [B] , including shear deflections, Figure 4(e), are

- - -u’EI(l -c) B13 = B23 = L2[2-2c -us +2uZ(1 - C ) P ]

(27) 2u2(1 -c)EI --= P u’EI[us -2u2(1 -c)P]

8 3 3 = - B 4 3 = 3 L [2-2c -us +2u2(1 - C ) P ] L L3[2 -2c -us +2u2(1 - C ) P ]

Figure 4. Translational stiffness of a beam-column element

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1688 A. K. CHUGH

Similar expressions for member forces can be obtained for unit translational displacement on element co-ordinate 4, as shown in Figure 4(b).

Rotational stifiess

co-ordinate 1, Figure 5(a), are Without the shear deflection, the member forces for 0‘ rotational displacement on element

Ignoring the presence of the axial force P, the translational displacement caused by the transverse shear force &I = [u2(l -c)E10’]/[L2(2-2c -us)], without any change in the rota- tional displacement, if permitted, would be, Figure 5(c)

The axial force P acting through the translation1 displacement 6’; causes an equilibrating shear force of magnitude PS’;/L, Figure 5(d). The force required to restore 6’; = 0, without causing any change in the rotational displacement, would be, Figure 5(e),

- -u2EI(1 - c ) u2(i -c)Ew B’l = B’l = B136’; =L2[2-2c -us +2u2(1 -c)P] LASG(2-2c -us)

If Bll, Bzl, B31, and 841 are the corresponding terms in the element stiffness matrix, [B], including shear deflections, then from equations (28)-(33) with 0‘set equal to unity, Figure 5(f), one gets

u(s -uc)EI u2EI(1 - c ) u2(1 -c)EI Bll= Bll +Bi l = L(2-2~ -us)-L2[2-2c -US +2u2(1 -c )P] LA,G(2 - 2 ~ -US)

- u(s -uc +PU2S)EI - L[2 - 2c -us +2u2(1 - C ) P ]

uEI(u - S -Pu’s) B - 21 - B21+BL =L[2-2c -us +2u2(1 -c )@]

(34)

(35)

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STIFFNESS MATRIX FOR A BEAM ELEMENT 1689

Figure 5. Rotational stiffness of a beam-column element

-u2(1 -c)EI + [ u2EI{us -2u2P(1 -c ) } u2(1-c)EI LASG(2-2c -US) 8 3 1 =B31 +B;1= 2 L (2 - 2~ -US) L3{2 - 2~ -US + 2u2(1 -c )P}

- -u2(1 -c)EI - L2[2 - 2c -us + 2u2(1 -c )P ]

Similar expressions for member forces can be obtained for unit rotational displacement on element co-ordinate 2, as shown in Figure 5(b).

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1690 A. K. CHUGH

The element stiffness matrix for a beam-column element including shear deflection, for unit displacements, can conveniently be expressed as

X - uEI

= L[2 - 2c -us +2u2(1 -C)P] r

-u(l-c) L (s - uc + u zs/3) (u - s - u 2sp) I u(1-c)

L -u(l -c) u(1-c)

L L (u - s - U Z S / 3 ) (s - uc + U 2 S P )

-u(l -c) -u(l-c) u2[s-2u(1-c)p] -U2[S-2U(l-C)/3] I L L LZ LZ

u(1 -c) u(1 -c) - U Z [ S - 2u(l -c)/3] u2[s - 2 4 1 - C)S] 1 7 L LZ L2

AN ALTERNATIVE FORMULATION

An alternative formulation for the stiffness matrix of a beam element considering shear deformation is to construct a flexibility matrix that accounts for shear deformation and then obtain the stiffness matrix. This approach is well documented''2 and is not repeated here. A parallel approach for a beam-column element, with shear deflections, is to form the flexibility matrix for a simply supported beam subjected to flexure plus shear, with only flexure accounting for the axial load and to then use the flexibility-stiffness transf~rmation.~ This approach is presented below:

The flexibility matrix, [f l , for a simply supported beam, Figure 6(a), (b), subjected to flexure plus shear is

L(s-uc) 1 -L(U-s)

-L(u-s) 1 L(s-uc) 1

+- (39)

+- +- u'sEI LA,G u2sEI LA,G

The transformation from a flexibility matrix to a stiffness matrix, without the axial force P, can be accomplished by3

where matrix [R], defined as {F,} = [R] {Ff}, is derived from the static equilibrium requirement of the element. {Ff} and {F,} are the external and support forces respectively. From Figure 6(a), (b)

[R]=L[-l -11 L 1 1

The triple product [R][fl-'[RIT represents the stiffness coefficients corresponding to the unit translational displacement on element co-ordinates 3 and 4. Since for a beam-column member, the axial force P moves through a unit translational displacement, Figure 6(c), (d), the triple product [R][fl-'[R]' is modified to include the equilibrating force due to P acting through the

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STIFFNESS MATRIX FOR A BEAM ELEMENT

Y

t

1691

- Pfl-??

I L

P T -

L P I

Figure 6. Beam-column element with simple support

unit distance, to become

Thus the transformation from a flexibility matrix to a stiffness matrix,.including an axial force P, is accomplished by

Knowing equations (39) and (41) and solving for [n in equation (42), one obtains exactly the same stiffness coefficients as given by equation (38), that is, Zj =Bij for all i, j . These stiffness coefficients will be designated Bij hereafter.

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1692 A. K. CHUGH

CRITICAL BUCKLING LOAD CONSIDERING EFFECT OF SHEARING FORCE

The critical buckling load, including effect of shearing force and corresponding to any end conditions, can be derived from the stiffness matrix in equation (38). For example, the critical buckling load for a member with one end hinged and the other end fixed, i.e., with D1 free to occur while D2 = D3 = D4 = 0, is obtained from Bll = 0, Figure 7(a). The Euler buckling load, P,, for this case is lr2EI/0.49L2. The numeric factor p = EI/L2A,G can be written in the form; p = (0-49/lr2)(nP,/AG). Figure 7(b) shows the relationship between nP,/AG and PJP,.

The critical buckling load for a cantilever post, Figure 7(c), including transverse shear effects is obtained by solving

U[S-UC+U’SP]EI -u2(1 -c)EI L[(2 - 2c - us)+2u2(1 - C ) P ] L2[(2 - 2c -us)+ 2u2(1 - c ) p ]

-u2(1 -c)EI u ~ [ s - ~ u ( ~ - c ) ~ ] E I L2[(2 -2c -us)+ 2u2(1 - c )@] L3[(2 - 2c -us)+ 2u2(1 - c ) p ]

Expansion of equation (43) leads to the transcendental equation:

c-usp=o

= O (43)

(44)

The Euler critical load for this case is P, = (7r2/4)(EI/L2). The numeric factor p, in this case, can be written as p = (4/lr2)(nP,/AG). The solution of the equation (44) is graphed in Figure 7(d), where it can be compared with a similar graph of equation (2-57) of Reference 6.

It may be mentioned that the zeros of the equation Bll = 0 and equation (44) were approximated by the one-step linear interpolation f~r rnula ,~ x3 = ( x l y 2 -x2y1)/(y2 - y l ) where x1 to x2 is the range in which the solution lies. This range was obtained numerically by incrementing u through 0.1.

Taking the Taylor series expansion of sin u and cos u in equation (44), retaining only the first two terms in each series, and ignoring a term containing (EIp in the denominator, one obtains

Equation (2-57) in Reference 6 for the critical load of a cantilever post, including the effect of transverse shear force, is P,, = Pe/[ 1 + (nP,/AG)].

General comments

Several problems were analyzed to study the relative effects of axial force and shear force on the elements of the stiffness matrix in equation (38). Specifically, the parameter p was varied from 0 to 0.3 in steps of 0.01, and for each value of p the parameter u was varied from 0.1 to 6-0 in steps of 0.1. Figure 8 shows the relative effect of p, and hence the shearing force, for varying magnitudes of u, the axial load parameter, on four independent stiffness coefficients. Figure 9 represents the converse. In Figure 8, the value of u corresponding to the stiffness element Bij = 0 is the critical buckling load for the associated end conditions. Figure 8 also indicates the decay in rotational and translational stiffnesses of a member with increasing axial and shear forces. For a

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STIFFNESS MATRIX FOR A BEAM ELEMENT 1693

L I 1

30 1'. aa 2'. 00 n P.

0 Solution of Eq. (44) to f i rs t

X Eq.(2-57) in R e f . 6 approximation

0 0

0.40 0.80 n b -

( d ) AG

Figure 7. Effect of shearing force on the critical load

given value of p, the carryover moments increase with increasing axial loads. However, the carryover moments decrease for increasing value of p.

Illustration

Figure 10 shows the classical problem of a cantilever beam carrying an axial load P and a concentrated tip load Q. If this cantilever is represented by a single element, the fixed end condition is enforced by removal of the rows and columns from equation (38) which correspond to the rotational and translational degrees-of-freedom at this end, i.e., by deleting row 1, column 1 and row 3, column 3 in equation (38). Inverting the resulting 2 X 2 matrix and performing the

, the vertical deflection and matrix multiplication of the inverse with the load vector

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1694 A. K. CHUGH

?- 4.00 8 . 0 0 0 . 0 0

( a 1 U

U U

W

( C ) U

a CI

c

'G. 00 4.00 8 . 00

( b ) U

Figure 8. Sample analyses-(Cal(=omp plots)

rotational displacement, respectively, of the free end of the cantilever, are

- L 3 ~ s - u c + u 2 s s m v -

U 3 ( C - U S / 3 ) E I

~ ~ ( 1 -C)Q

u2(c -usp)Er 8 = (47)

When the effect of transverse shear force is ignored, i.e., /3 = 0 in equations (46) and (47), one obtains the usual expressions for deflection and rotation of the free end of a cantilever with axial

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STIFFNESS MATRIX FOR A BEAM ELEMENT 1695

8 ( a )

4

VI

‘a . ao 0.20 a. uo

(C) P P

( d )

LEGEND 0 u.o.1 8 ---- u= 1.0 t - - - - u.2.0 x - - - - ~ ~ 3 . 0

Figure 9. Sample analyses-(CalComp plots)

Figure 10. Example problem

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1696 A. K. CHUGH

load P and transverse load Q applied at the tip. It is interesting to note that the effect of transverse shear force on the response of a beam-column member is not simply additive.

CONCLUSION

An apparently simple formulation is presented for the development of the stiffness matrix of a prismatic beam element considering shear deflections. The same ideas have been extended to apply to a beam-column element of uniform cross-section. An alternative formulation for the stiffness matrix of a beam-column, including shear effects, in terms of member flexibilities is also included. Any structural synthesis scheme can be used to synthesize the element stiffness matrices. Although the apparent advantage of the algorithm presented lies in the use of the displacement method of matrix analysis, the information developed can also be used in the classical methods of structural analysis. The critical buckling loads of struts with various end conditions can be easily obtained. The results of several analyses are presented in which the relative effects of axial force and shear force on the stiffness coefficients were investigated. The effect of transverse shear force on the response of a beam-column is not simply additive. The element stiffness matrix has been used with success in the analysis of many structural problems.

ACKNOWLEDGEMENTS

The author would like to express his sincere appreciation to Dr. Louis A. Panek for providing the incentive for the work reported in this paper, and to Professors Sherrill Biggers, Richard Gallagher, Hans Gesund and William Weaver, Jr., for their prompt and helpful comments on the general subject matter of this paper in response to the author’s personal correspondence.

APPENDIX

List of symbols

A = cross-sectional area A, = shear area = A/n Bii = stiffness coefficient of a beam-column Bii = i, jth beam-column stiffness including shear deflection

Di = displacement degree-of-freedom on element co-ordinate i E = modulus of elasticity [A= flexibility matrix G = shear modulus I = planar moment of inertia

kij = stiffness coefficient of a beam Gi = i, jth beam stiffness including shear deflection L = length of the member n = numeric factor depending upon the shape of the member cross-section P = applied axial force

P,,= buckling load for given member considering shear deflections P, = Euler buckling load Q = concentrated load

c =cos u

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STIFFNESS MATRIX FOR A BEAM ELEMENT 1697

[ R ] =static equilibrium matrix s = sin u Ti = i, jth beam-column stiffness including shear deflection, used for an alternative formulation

J G u = numeric factor = L

12EI a = numeric factor = ~

A , G L ~ EI p = numeric factor = ~

L ~ A , G S = translational displacement

a1 = shear deflection 0 = rotational displacement

REFERENCES

1 . A. Ghali and A. M. Neville, Structural analysis, Intext Educational Publishers, 1972, pp. 364-402. 2. J. M. Gere and W. Weaver, Analysis of Framed Structures, Van Nostrand, 1965, pp. 428-430. 3. R. H. Gallagher, Finite Element Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1975, pp. 37-50 and 361-367. 4. R. T. Severn, ‘Inclusion of shear deflections in the stiffness matrix for a beam element’, J. Strain Analysis, 5,239-241

5. M. G. Salvadori and M. L. Baron, NumericalMethods in Engineering, Prentice-Hall, Englewood Cliffs, N.J., 1961,

6. S. P. Timoshenko and J. M. Gere, Theory of Elastic Stabiliry, McGraw-Hill, 1961, pp. 132-135.

(1970).

pp. 18-20.