Plane Thermoelastic Waves in Infinite Half-Space Caused FACTA UNIVERSITATIS Series: Mechanical Engineering Vol. 12, N o 3, 2014, pp. 261 - 276 1THE CRITICAL LOAD PARAMETER OF A TIMOSHENKO BEAM WITH ONE-STEP CHANGE IN CROSS SECTION UDC 534.1 Goran Janevski 1 , Marija Stamenković 2 , Mariana Seabra 3 1 University of Niš, Department of Mechanical Engineering, Serbia 2 Mathematical Institute of the SASA, Belgrade, Serbia 3 University of Porto, Faculty of Engineering – FEUP, Portugal Abstract. The paper analyzes the transverse vibration of a Timoshenko beam with one- step change in cross-section when subjected to an axial force. The axial force is equal in both of the beam portions. Three types of beam which occur commonly in engineering application are considered. The frequency equation of the Timoshenko beam with one- step change in cross-section is expressed as the fourth order determinant equated to zero. The critical compressive axial force is expressed as a function of the critical load parameter which is tabulated for four classical boundary conditions. Apart from the results presented in Tables, the paper also provides calculated values of the critical load parameter for other values of system parameters along with the graphic representation of their dependence on the step position parameter. Key Words: Timoshenko Beam, Frequency Equation, Critical Axial Load, Critical Load Parameter, One-step Beam 1. INTRODUCTION Mechanical and other engineering structures often contain beams with step changes in cross-section. Stepped beams are increasingly used in many fields of engineering and practically always as structural elements. Due to the frequent application, they can be very different structures, subjected to various types of loads. For this reason, their dynamic properties have been investigated by many authors. Among the first ones are Jang and Bert [1-2] who have given the first exact results for natural frequencies of a stepped beam. In Jang and Bert‟s paper [1] the exact solution for fundamental natural frequencies is compared with the results obtained by the use of the finite element method. In the next paper by the same authors [2] higher mode frequencies of a Received October 10, 2014 / Accepted November 19, 2014 Corresponding author: Goran Janevski University of Niš, Department of Mechanical Engineering, Serbia E-mail: gocky.jane@gmail.com Original scientific paper 262 G. JANEVSKI, M. STAMENKOVIĆ, M. SEABRA stepped beam are obtained, expressing the frequency equation as the fourth-order determinant equated to zero. The vibration of Euler-Bernoulli beam with step changes in cross-section and under axial forces is considered by Naguleswaran [2-7]. The author presents the first three frequencies and the first two critical axial forces for beams with several combinations of axial forces in two portions, and for 16 sets of boundary combinations and three types which occur commonly in engineering applications. Also, Naguleswaran [6] investigates the sensitivity of frequency parameters from the step location factor and the “active“ dimension factor. The sensitivity is presented for the selected system parameters. The vibration of an Euler- Bernoulli beam with three step changes and elastic end supports is investigated in [7]. The first three frequency parameters are tabulated for the selected sets of system parameters and classical end support and 35 types of elastic end supports. Vibration and stability of Euler- Bernoulli tie-bars are considered by Naguleswaran [3]. The first three frequency parameters and the first two buckling axial forces are tabulated for three type arrangements with different number of rings and various end supports. The frequencies, in graphical form, of a uniform Euler-Bernoulli beam under constant axial compressive and tensile force for the classical boundary conditions are presented by Bokaian [8-9]. By means of the Adomian decomposition method, Mao and Pietrzko [10] investigate the free vibrations of a stepped beam. The first four natural frequencies for classical end supports are tabulated. Also, the authors obtain the frequencies for the stepped beam with a translational and rotational spring at one and both ends. Their results are compared with those obtained in [7]. Mao [11] has extended the study in [10] to multiple-stepped beams and compared them with those obtained in [3]. By using the continuous-mass transfer matrix method, Wu and Chang [12] have investigated free vibration of the axial loaded multi-step Timoshenko non-uniform beam carrying any number of concentrated elements. In this paper the authors take into consideration the effects of shear deformation, rotary inertia and their joint action term, and thus determine the exact natural frequencies. Zhang at al. [13] have developed an analytic method to study transverse vibrations of double-beam systems, in which two parallel Timoshenko beams are connected by discrete springs and coupled with various discontinuities. By dividing the entire structure into a series of distinct components, and then systematically organizing compatibility and boundary conditions with matrix formulations, closed-form expressions for the exact natural frequencies, mode shapes and frequency response functions can be determined. Parametric studies are performed for a practical example to illustrate the influences of the parametric variabilities on the dynamic behaviors. In the present paper, the emphasis is on the critical axial force and natural frequencies of a Timoshenko stepped beam. The three types of frequent use in engineering applications are considered. The frequency equations for four combinations of the classical boundary conditions are expressed as the second order determinants equated to zero. The critical axial force is presented in the function of the dimensionless parameter. The influence of a flexural rigidity ratio, slenderness ratio and type of beam are considered. The obtained results are displayed graphically for a set of combinations of dimension factors and location parameters. Critical Load Parameter of a Timoshenko Beam with One-Step Change in Cross Section 263 2. PROBLEM FORMULATION The basic differential equations of motion for the analysis will be reduced by considering the Timoshenko-beam of length L, subjected to axial compressive force F. This will be applied on the basis of the following assumptions:  The behavior of the beam material is linear elastic,  The cross-section is rigid and constant throughout the length of the beam and has one plane of symmetry,  Shear deformations of the cross-section of the beam are taken into account while the elastic axial deformations are ignored,  The equations are derived bearing in mind the geometric axial deformations, and,  Axial forces F acting on the ends of the beam do not change with time. Fig. 1 The coordinate system and notation for the beam: a) Timoshenko-beam subjected to an axial compressive force F ; b) Deflected differential beam element of length dx A beams element of length dx between two cross-sections taken normal to the deflected axis of the beam is shown in Fig. 1b. Since the slope of the beam is small, the normal forces acting on the sides of the element can be taken as equal to axial compressive force F. Shearing force FT is related to the following relationship:           x v kGAF T (1) where v = v(x,t) is the displacement of the cross-section in y-direction, v/x is the global rotation of the cross section,  =  (x,t) is the bending rotation, G is the shear modulus, A is the area of the beam cross-section, and k is the shear correction factor of cross-section. Analogously, the relationship between bending moment M and bending angles  is given by: z M EI x     (2) where E is the Young‟s modulus and Iz is the second moment of the area of the cross- section. Finally, forces and moments of inertia are given by: 2 2 2 2 , z w f A J I t t           , (3) 264 G. JANEVSKI, M. STAMENKOVIĆ, M. SEABRA respectively, where  is the mass density. The dynamic-forces equilibrium conditions of these forces are given by the following equations: 2 2 2 2 2 2 0 v v v m kGA F t x x x                (4) 2 2 2 2 0 v I kGA K t x x                 (5) where m = A is the mass per unit length and K = EIz is the flexural rigidity. The equations on motion (4-5), which are coupled together, are reduced by standard procedure, eliminating  , to the following fourth-order partial differential equation: 4 2 4 2 4 4 2 2 2 2 4 1 1 0z z IF v v F E v v v K F I m kGA x x kGA kG x t t kG t                             (6) 3. THE FREQUENCY EQUATION OF THE STEPPED BEAM In this section of the paper a general theory for the determination of the natural frequencies and the critical buckling load of a beam with step change in cross-section is given. Ends A and B are on classical clamped (cl), pinned (pn) and free (fr) supports. Each beam portion is made of some material with Young‟s modulus E and mass density , and has a cross-section with a uniform cross-section of area Ai and moment inertia Ii. The flexural rigidity, mass per unit length and the length of the beam portion are Ki, mi and Li. The axial force in beam portion is Fi. The coordinate systems with origin at A and B are in opposite directions. Fig. 2 The coordinate system and notation for the stepped beam The dynamics of each beam portion are treated separately. If we apply the above- mentioned procedure to a differential element of each beam portion, the following set of coupled differential equations will be obtained: 4 2 4 2 4 1 1 1 1 1 1 1 1 14 2 2 2 42 1 11 1 1 1 1 0 v v v v I vF F E EI F I m kGA kGA kG t kG tx x x t                            (7) 4 2 4 2 4 2 2 2 2 2 2 2 2 24 2 2 2 42 2 22 2 2 1 1 0 v v v v I vF F E EI F I m kGA kGA kG t kG tx x x t                            (8) The standard approach to solving Eqs. (7, 8) is by separating the variables, and the same procedure may be applied here. Thus, assuming that: Critical Load Parameter of a Timoshenko Beam with One-Step Change in Cross Section 265 ( , ) ( ) ( ), 1, 2, i i i i v x t y x T t i  (9) where yi(xi), are the known mode shape functions which will be determined on the basis of the type of beam supports. Assuming time harmonic motion, the unknown time function can be assumed to have the following form: ( ) , 1 , j t T t e j     (10) where  denotes the circular natural frequency of the system. To express the set of Eqs. (7-8) in the dimensionless form one defines dimensionless abscissas Xi, amplitude Yi and step position parameter Ri as follows: , , , 1, 2,i i i i i i x y L X Y R i L L L     (11) Also, adopt the “reference beam” with the following characteristics: area of the beam cross-section AR, length L, mass per unit mR and flexural rigidity EIR. Then we can define dimensionless flexural rigidity ratio i, mass per unit ratio i and dimensionless axial force  in the form: 2 , , , 1, 2.i i i i R R R EI m FL i EI m EI       (12) Introducing the general solutions (9) into Eqs. (7-8), considering dimensionless values, one gets the system of dimensionless differential equations: 4 2 4 4 2 4 2 4 ( ) ( ) 1 1 1 1 ( ) 0 , i i i i i R R Ri R i i i i i i R R R i i i d Y X d Y X k dX k dX Y X k                                                  (13) where 6.2 G E . In Eq. (13) R is the natural frequency parameter and R is the slenderness ratio of the beam: 24 4 2 2 , ,R R R R R R R m L I i EI A L L             (14) where iR is the radius of gyration of the reference beam. The solutions of Eq. (13) are assumed to be:   ( ) , 1, 2.i i u X i i i Y X A e i  (15) For the non-trivial solution, we can determine ui as the solution of the characteristic equation in the form: 2 2 2 4 0 4 4 , 1, 2, 2 i i i i i i u i           (16) where: 266 G. JANEVSKI, M. STAMENKOVIĆ, M. SEABRA 4 4 2 4 2 4 0 1 , 1 1 , 1 . i i R i R R i R i i i i i R R R i k k k                                                  . (17) With the use of Euler‟s formula, solution (15) can be written as: 1, 2, 3, 4, ( ) cosh( ) sinh( ) cosh( ) sinh( ), i i i i i i i i i i i i i i Y X C a X C a X C b X C b X    (18) where Cj,i (i = 1,2, j = 1,2,3,4) are eight constants to be determined from the initial conditions and: 2 2 2 2 4 0 2 2 4 0 4 4 4 4 , 1, 2 2 2 i i i i i i i i i i i i a b i                  . (19) The need for Eq. (18) to satisfy the boundary conditions at A and B may be used to eliminate four of the constants. The mode shape of the beam portions may be expressed as: 1 1 2 2 ( ) ( ) ( ), 1, 2. i i i i i i i i Y X A U X A U X i   (20) where Ai1 and Ai2 are unknown constants and functions Ui1(Xi) and Ui2(Xi) for the classical boundary conditions are: For clamped, 1 2 ( ) cosh( ) cosh( ), ( ) sinh( ) sinh( ). i i i i i i i i i i i i i i U X a X b X a U X a X b X b     For pinned, 1 2 ( ) sinh( ), ( ) sinh( ). i i i i i i i i U X a X U X b X  (21) (21) for free, 2 1 3 2 3 ( ) cosh( ) cosh( ), ( ) sinh( ) sinh( ). i i i i i i i i i i i i i i i i i i i i a U X a X b X b a a U X a X b X b b                Taking into account the opposite direction coordinate axes at A and B, the need to satisfy the continuity of deflection and the slope and compatibility of bending moment and shearing force at C will result in the following equations: 1 1 2 2 1 1 1 2 1 2 ( ) ( ) ( ) ( ), , dY R dY R Y R Y R dX dX    (22) 2 2 3 3 1 1 2 2 1 1 1 1 2 2 2 2 1 2 1 22 2 3 3 1 2 1 1 2 2 ( ) ( ) ( ) ( ) ( ) ( ) , . d Y X d Y X d Y X dY R d Y X dY R dX dX dX dX dX dX                 Substituting the solution (20) into Eq. (22) and rewriting in the matrix form, one obtains the following homogenous set of four algebraic equations: Critical Load Parameter of a Timoshenko Beam with One-Step Change in Cross Section 267 11 1 12 1 21 22 11 1 12 1 21 22 11 1 1 2 2 12 2 2 2 2 2111 1 12 1 21 22 1 1 2 22 2 2 2 221 1 2 2 11 1 12 1 21 2 2 2 2 2 2 2 222 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) u R u R u R u R du R du R du R du R A dX dX dX dX A Ad u R d u R d u R d u R AdX dX dX dX B R B R B R B R                            0 0 0 0                         , (23) where:   3 3 ( ) ( ) .mn m mn m mn m m m m d u R du R B R dX dX    (24) For the non-trivial solution, the coefficient matrix must be singular, one gets the frequency equation: 11 1 12 1 21 2 22 2 11 1 12 1 21 2 22 2 1 1 2 2 2 2 2 2 11 1 12 1 21 2 22 2 1 1 2 22 2 2 2 1 1 2 2 11 1 12 1 21 2 22 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) u R u R u R u R du R du R du R du R dX dX dX dX d u R d u R d u R d u R dX dX dX dX B R B R B R B R          (25) 4. CRITICAL AXIAL FORCE, CRITICAL LOAD PARAMETER Of all the modes of failure, buckling is probably the most common and most catastrophic one. For this reason, the critical buckling load is an important characteristic of a mechanical structure. As it is known, the critical buckling load for uniform beam is: , 2 ,           r ucr k (26) where kr depends on the boundary conditions and one has  kr = 1 for pinned/pinned beam  kr = 1/2 for clamped/clamped beam  kr = 2 for clamped/free beam  kr = 0.69915566 for clamped/pinned beam. In order to verify the proposed method for the analysis of the vibration of the stepped beam, several numerical examples with different boundary conditions, step positions, slenderness ratio and moment of inertia parameter will be discussed in this section. Assuming that the stepped beam has a uniform Young‟s modulus E and density , the present paper considers three of the types which occur commonly in engineering application. The first two are beams of a rectangular cross-section with step changes in breadth and depth and constant depth and breadth, respectively. The third type of beam has a circular cross- section with the step change in diameter. 268 G. JANEVSKI, M. STAMENKOVIĆ, M. SEABRA Fig. 3 Three types of beam Three representative types are shown in Fig. 3, where „active‟ dimensions for Type 1, 2 and 3 are breadth, depth and diameter, so that: For Type 1, 21 , , , , , 12 i R i i i i i R R b h b L             For Type 2, 3 21 , , , , , 12 i R i i i i i R R h h b L             (27) For Type 3, 2 4 21 , , , , , 1 , 2. 16 i R i i i i i R R d d i d L              Without the loss of generality, the beam with length L and the characteristic of the first beam portion, i.e., EIR = EI1 and mR = m1, is chosen as the „reference‟ beam. This means that in all the examples 1 = 1, 1 = 1 and 1 = 1. Taking the above into account and the fact that R2 = 1  R1, the system dimensionless parameters are 2,  and R1. Clearly, the critical load is a function of this parameter. Our analysis shows that the critical force can be represented in the form: , 2 , k cr cr u    (28) where k = k (type of beam, 2,  , R1) is a dimensionless critical load parameter (CLP). For the selected set of 2,  and R1, to calculate k one writes R = 0 in the frequency equation (25). The roots of the frequency equation (25) are determined by an iterative procedure based on linear interpolation and proposed in [4]. This procedure is used to calculate k for 2 = 0.6, 0.7 & 0.8 R1 = 0.1, 0.2,...,1and  = 1/20 & 1/100. Also, for the sake of comparison, we have calculated the CLP force values for  = 0 which correspond to the case of Euler- Bernoulli beam. The CLT is tabulated in Table 1 for pinned-pinned beam. It can be observed that the values of CLP for the uniform Euler-Bernoulli beam 0 or 1, i.e. the critical force is: 1 2 , 1 , ( 0) , ( 1) cr cr u cr cr u R R        , (29) The previous expression applies to all the types of supports. The CLP values for Timoshenko-beam are higher which means that the critical force has less value. In order to show the trend of the change in the CLP value depending on step position R1, the CLT values for other values of R1 with the step of 0.001 are calculated and graphically displayed. Critical Load Parameter of a Timoshenko Beam with One-Step Change in Cross Section 269 The Fig. 4 shows the CLP dependence on changes R1 for values 2 = 0.8 and  = 1/20 & 1/100 considering the influence of the beam type. It can be noticed that the CLP values are maximal for beam type 3, while for beam type 1 are minimal. For larger values of δ (greater influence of shear) in the case when the step position is near the supports, the CLP values are maximal for type 1 and minimal for type 3. The CLP dependence on changes R1 for types 1 and 3 and value  = 1/20, considering the influence of relations 2, is shown in Fig. 5. For smaller values 2, the CLP has higher values except in the case when the step position is near the supports; then the CLP has higher values for greater values 2. Table 1 CLP for pn-pn beam Table 2 CLP for cl-cl beam 1R type  2 1R type  2 0.6 0.7 0.8 0.8 0.7 0.8 0 1 20 1.0125184 1.0179287 1.0286575 0 1 20 1.0496005 1.0710373 1.1135468 100 1.0005022 1.0007193 1.0011497 100 1.0020083 1.0028763 1.0045974 EB 1 1 1 EB 1 1 1 2 20 1.0015053 1.0029331 1.0061206 2 20 1.0060004 1.0116777 1.0243335 100 1.0000603 1.0001175 1.0002453 100 1.0002411 1.0004699 1.000981 EB 1 1 1 EB 1 1 1 3 20 1.0007843 1.0015284 1.0031898 3 20 1.0031296 1.0060937 1.0127052 100 1.0000314 1.0000612 1.0001277 100 1.0001256 1.0002448 1.000511 EB 1 1 1 EB 1 1 1 0.3 1 20 0.9004332 0.8963132 0.8978066 0.3 1 20 0.790734 0.817013 0.8632652 100 0.8881282 0.8788707 0.8701333 100 0.7347316 0.7407416 0.7464902 EB 0.887614 0.8781417 0.8689767 EB 0.7323622 0.7375177 0.7415579 2 20 0.931096 0.9173096 0.9027403 2 20 0.7203619 0.7308869 0.7564571 100 0.9295037 0.9142647 0.8965305 100 0.7112445 0.7154073 0.7280566 EB 0.9294372 0.9141376 0.8962712 EB 0.7108618 0.7147567 0.7268615 3 20 0.942686 0.9280114 0.910252 3 20 0.7310421 0.7198191 0.736254 100 0.9418468 0.9264071 0.9069895 100 0.7257155 0.7107592 0.7202147 EB 0.9418118 0.9263402 0.9068534 EB 0.7254923 0.7103792 0.7195413 0.5 1 20 0.6063826 0.5844888 0.570687 0.5 1 20 0.6444095 0.6400219 0.6578166 100 0.5934776 0.5666266 0.5427583 100 0.5907593 0.5673238 0.5458978 EB 0.5929383 0.5658802 0.541591 EB 0.5884946 0.5642562 0.5411757 2 20 0.7353931 0.6824913 0.6268342 2 20 0.6820122 0.6666843 0.6408012 100 0.7333133 0.6787341 0.6196697 100 0.6724187 0.650626 0.6114626 EB 0.7332265 0.6785773 0.6193705 EB 0.6720159 0.649951 0.6102279 3 20 0.7803152 0.7240699 0.6577481 3 20 0.6807354 0.6787212 0.6520273 100 0.7791806 0.7220238 0.6539021 100 0.6746779 0.6692922 0.635663 EB 0.7791332 0.7219384 0.6537418 EB 0.6744238 0.6688966 0.6349761 0.7 1 20 0.2156296 0.202338 0.198505 0.7 1 20 0.2989858 0.3163953 0.3576915 100 0.2028625 0.1846704 0.1707437 100 0.2562517 0.2540968 0.2552395 EB 0.2023291 0.1839321 0.1695835 EB 0.2544512 0.2514707 0.2509195 2 20 0.3712844 0.2909344 0.2322638 2 20 0.3812923 0.3147333 0.2932055 100 0.3680207 0.2858599 0.22374 100 0.368565 0.2960827 0.2620313 EB 0.3678844 0.2856479 0.2233838 EB 0.3680302 0.2952983 0.2607184 3 20 0.4593584 0.3517904 0.2614351 3 20 0.4648444 0.3611526 0.2934004 100 0.4573784 0.3488207 0.2567818 100 0.4566664 0.3498693 0.2770316 EB 0.4572958 0.3486968 0.2565878 EB 0.4563232 0.3493967 0.276345 1 1 20 0.0125184 0.0179287 0.0286575 1 1 20 0.0496005 0.0710373 0.1135468 100 0.0005022 0.0007193 0.0011497 100 0.0020083 0.0028763 0.0045974 EB 0 0 0 EB 0 0 0 2 20 0.0041728 0.0059762 0.0095525 2 20 0.0165335 0.0236791 0.0378489 100 0.0001674 0.0002398 0.0003832 100 0.0006694 0.0009588 0.0015325 EB 0 0 0 EB 0 0 0 3 20 0.0021755 0.0031157 0.0049801 3 20 0.0086444 0.0123804 0.019789 100 0.000087 0.0001249 0.0001996 100 0.0003487 0.0004994 0.0007983 EB 0 0 0 EB 0 0 0 270 G. JANEVSKI, M. STAMENKOVIĆ, M. SEABRA Fig 4 Variation of CLP with R1 for pn-pn beam and 2 0.8  : a) 1 20   , b) 1 100   Fig 5 Variation of CLP with R1 for pn-pn beam and 1 20   : a) type 1, b) type 3 Fig. 6 Variation of CLP with R1 for pn-pn beam and 2 0.8  : a) type 1, b) type 3 The CLP dependence on changes R1 for types 1 and 3 and value 2 = 0.8 with influence of change values  are shown in Fig. 6. It can be noticed that the CLP value is higher for greater values of . The influence of δ is higher for type 1 than for type 3. Critical Load Parameter of a Timoshenko Beam with One-Step Change in Cross Section 271 Fig. 7 Variation of CLP with R1 for cl-cl beam and 2 0.8  : a) 1 20   , b) 1 100   Fig. 8 Variation of CLP with R1 for cl-cl beam and 1 20   : a) type 1, b) type 3 Fig. 9 Variation of CLP with R1 for cl-cl beam and 2 0.8  : a) type 1, b) type 3 The CLT is tabulated as shown in Table 2 for clamped-clamped beam. In Figs.7-9, the CLP dependence on change R1 is presented while the beam type influence, dimension ratio 2 and slenderness ratio  are taken into consideration. The influence of the beam type is higher for lower values , as well as that of dimension ratio 2 for beam type 1 in relation to beam type 3. An especially large influence of slenderness ratio  is observed for beam type 1, while for beam type 3 is considerably smaller. 272 G. JANEVSKI, M. STAMENKOVIĆ, M. SEABRA Table 3 CLP for cl-fr beam Table 4 CLP for cl-pn beam 1R type  2 1R type  2 0.6 0.7 0.8 0.6 0.7 0.8 0 1 20 0.999139839 0.998768088 0.998030899 0 1 20 1.025524562 1.036555975 1.058431446 100 0.999965631 0.999950777 0.999921322 100 1.001027398 1.001471427 1.002351944 EB 1 1 1 EB 1 1 1 2 20 0.999896942 0.999799043 0.999580305 2 20 1.003075743 1.005990663 1.012494595 100 0.999995865 0.999991949 0.999983202 100 1.000123308 1.000240365 1.000501796 EB 1 1 1 EB 1 1 1 3 20 0.999946335 0.99989537 0.999781511 3 20 1.001603102 1.00312319 1.006515908 100 0.999997843 0.999995801 0.999991243 100 1.000064225 1.000125195 1.000261365 EB 1 1 1 EB 1 1 1 0.3 1 20 0.516563665 0.493505096 0.47300968 0.3 1 20 0.746833293 0.762347877 0.788574418 100 0.51772272 0.495023672 0.475235116 100 0.717997588 0.723030128 0.728359103 EB 0.517770844 0.49508674 0.475327561 EB 0.716786788 0.721379896 0.72583265 2 20 0.652268724 0.595615643 0.5401181 2 20 0.719999623 0.716830331 0.728191711 100 0.652459409 0.595956048 0.540735108 100 0.715396893 0.708900445 0.713536209 EB 0.652467341 0.595970201 0.540760754 EB 0.71520442 0.708568591 0.712922479 3 20 0.704485327 0.640117804 0.571758088 3 20 0.737832143 0.71734048 0.717983769 100 0.704588851 0.640308298 0.57210797 100 0.735173101 0.712769566 0.709764639 EB 0.704593159 0.640316223 0.572122522 EB 0.735062002 0.712578473 0.709420883 0.5 1 20 0.231981741 0.214571151 0.200117438 0.5 1 20 0.704941873 0.707837666 0.720463214 100 0.233370034 0.216264853 0.202474946 100 0.676340365 0.6689679 0.660927761 EB 0.233427646 0.216335175 0.202572866 EB 0.675139695 0.667336845 0.658430238 2 20 0.373238269 0.305011992 0.251479256 2 20 0.681825858 0.69320834 0.696071668 100 0.373617013 0.305577286 0.25231405 100 0.676884599 0.685081612 0.681244199 EB 0.373632747 0.305600765 0.252348727 EB 0.676677903 0.684741481 0.680623243 3 20 0.44925751 0.357372974 0.280582333 3 20 0.663081249 0.684467961 0.693158972 100 0.449483015 0.357736179 0.281108551 100 0.659887918 0.679616675 0.684818598 EB 0.449492388 0.357751272 0.281130422 EB 0.659754399 0.679413805 0.684469755 0.7 1 20 0.055186907 0.050063457 0.045640544 0.7 1 20 0.379309729 0.361963181 0.360261413 100 0.056450765 0.051615789 0.047851137 100 0.35135486 0.324284617 0.302297179 EB 0.056503226 0.051680255 0.047942968 EB 0.350182851 0.322704721 0.29986644 2 20 0.108655097 0.079229345 0.061213511 2 20 0.524759271 0.460060425 0.396540649 100 0.109305745 0.079925259 0.062080655 100 0.518851854 0.450425138 0.379786092 EB 0.109332728 0.079954145 0.062116669 EB 0.518604613 0.450021642 0.37908408 3 20 0.156066172 0.101028811 0.07073479 3 20 0.57511257 0.510509096 0.429191083 100 0.156648268 0.1015835 0.071306225 100 0.571486178 0.504829574 0.419799388 EB 0.156672393 0.101606523 0.071329969 EB 0.57133452 0.504592024 0.419406539 1 1 20 -0.00086016 -0.00123191 -0.00196910 1 1 20 0.025524562 0.036555975 0.058431446 100 -0.00003 -0.00005 -0.00008 100 0.001027398 0.001471427 0.002351944 EB 0 0 0 EB 0 0 0 2 20 -0.00028672 -0.00041063 -0.00065636 2 20 0.008508185 0.012185324 0.019477148 100 -0.00001 -0.00002 -0.00003 100 0.000342464 0.000490475 0.000783981 EB 0 0 0 EB 0 0 0 3 20 -0.00014906 -0.00021370 -0.00034159 3 20 0.004440154 0.006359143 0.010164521 100 -0.000006 -0.000009 -0.000014 100 0.000178377 0.000255475 0.000408356 EB 0 0 0 EB 0 0 0 The CLT is tabulated as shown in Table 3 for clamped-free beam. Figs. 10-12 show the CLP dependence on changes R1 while the beam type influence, dimension ratio 2 and slenderness ratio  are taken into consideration. Here, the influences of parameters on the CLP values are totally clear: in the above Figs. it can be noticed that the CLP values are higher for lower values of dimension ratio 2, maximum for beam type 3, less for beam type 2 and minimum for beam type 1. The influence of slenderness ratio  on the CLP values is insignificant, since the influence of shear in this manner is very small. Critical Load Parameter of a Timoshenko Beam with One-Step Change in Cross Section 273 Fig. 10 Variation of CLP with R1 for cl-fr beam and 2 0.8  : a) 1 20   , b) 1 100   Fig. 11 Variation of CLP with R1 for cl-fr beam and 1 20   : a) type 1, b) type 3 Fig. 12 Variation of CLP with R1 for cl-fr beam and 2 0.8  : a) type 1, b) type 3 274 G. JANEVSKI, M. STAMENKOVIĆ, M. SEABRA Fig. 13 Variation of CLP with R1 for cl-pn beam and 2 0.8  : a) 1 20   , b) 1 100   Fig. 14 Variation of CLP with R1 for cl-pn beam and 1 20   : a) type 1, b) type 3 Fig. 15 Variation of CLP with R1 for cl-pn beam and 2 0.8  : a) type 1, b) type 3 The CLP is tabulated as shown in Table 4 for clamped-pinned beam. In Figs.13-15, CLP dependence on change R1 is presented while the beam type influence, dimension ratio 2 and slenderness ratio  are taken into consideration. The conclusions presented in the previous cases are here confirmed, especially in the part when the step position is near the supports. Critical Load Parameter of a Timoshenko Beam with One-Step Change in Cross Section 275 5. CONCLUSIONS The frequency equations of one-step Timoshenko beams under compressive axial forces and four combinations of classical boundary conditions are expressed as the fourth order determinant equated to zero. The critical axial forces are expressed as a function of the critical load parameter and the critical load for uniform beam. The critical load parameters are tabulated for the beams with the classical boundary conditions. To determine the trend of change and sensibility, the dependence of the critical load parameter is displayed graphically for three types of beam in the function of step position, flexural rigidity ratio and slenderness ratio. Less sensitive are the beams with a rectangular cross-section (type 1 and type 2), while the most sensitive is the beam with a circular cross-section (type 3). Sensitivity is minor for small values of slenderness ratio due to the small influence of shear. Influence of slenderness ratio is almost nonexistent for a clamped-free beam. As far as the type of end support is concerned, there are fields with very small changes of CLP in the beams in which one of the support is clamped. Regarding a near-supports position, sensitivity is minor for pinned and free end supports and larger for a clamped end support. Also, in the beams where one of the supports is clamped there exists a field where sensitivity is very small. The method is applicable for any type of the boundary conditions and other system parameters. REFERENCES 1. Jang, S.K., Bert, C.W., 1989, Free vibration of stepped beams: exact and numerical solutions, Journal of sound and vibration, 130, pp. 342-346. 2. Jang, S.K., Bert, C.W., 1989, Free vibration of stepped beams: higher mode frequencies and effects of steps on frequency, Journal of sound and vibration, 132, pp. 164-168. 3. Naguleswaran, A., 2006, Vibration and stability of ring-stiffened Euler-Bernoulli tie-bars, Applied mathematical modelling, 30, pp. 261-277. 4. Naguleswaran, A., 2004, Transverse vibration and stability of an Euler-Bernoulli beam with step change in cross-section and in axial force, Journal of sound and vibration, 270, pp.1045-1055. 5. Naguleswaran, A., 2003, Vibration and stability of an Euler-Bernoulli beam with up to three-step changes in cross-section and in axial force, International journal of mechanical sciences, 45, pp.1563-157 9. 6. Naguleswaran, A., 2002, Natural frequencies, sensitivity and mode shape details of an Euler-Bernoulli beam with one-step change in cross-section and with endes on classical supports, Journal of sound and vibration, 252, pp.1045-1055. 7. Naguleswaran, A., 2002, Vibration of an Euler-Bernoulli beam on elastic end supports and with up to three step changes in cross-section, International journal of mechanical sciences, 44 pp.2541-2555. 8. Bokaian, A., 1990, Natural frequncies of beams under tensile axial loads, Journal of sound and vibration, 142, pp. 481-498. 9. Bokaian, A., 1988, Natural frequncies of beams under compressive axial loads, Journal of sound and vibration, 126, pp. 49-65. 10. Mao, Q., 2011, Free vibration analysis of ultiple-stepped beams by using Adomian decomposition method, Mathematical and computer modelling, 54, pp.756-764. 11. Mao, Q., Pietrzko, S., 2010, Free vibration analysis of stepped beams by using Adomian decomposition method, Applied Mathematics and computation, 217, pp. 3429-3441. 12. Wu, J-S., Chang, B-H., 2013, Free vibration of axial-loaded multi-step Timoshenko beam carrying arbitrary concentrated elements using continuous-mass transfer matrix method, European Journal of Mechanics A/Solids, 38, pp. 20-37. 13. Zhang, Z., Huang, X., Zhang, Z., Hua, H., 2014, On the transverse vibration of Timoshenko double-beam systems coupled with various discontinuities, International Journal of Mechanical Sciences, 89, pp. 222-241. 276 G. JANEVSKI, M. STAMENKOVIĆ, M. SEABRA KRITIĈNA OPTEREĆENJE TIMOŠENKOVE STEPENASTE GREDE SA JEDNOM PROMENOM POPREĈNOG PRESEKA U radu se analiziraju transferzalne oscilacije Timošenkove grede sa jednom promenom poprečnog preseka i pritisnute aksijalnim silama koje su konstantne duž grede. Razmatrane su tri tipa grede koje se često koriste u inžinjerskoj praksi. Frekventna jednačina Timošenkove grede je prikazana u obliku determinante četvrtog reda. Kritično opterećenje je izraženo uvodjenjem parametra kritičnog opterećenja čije su vrednost iza gredu sa standardnim graničnim uslovima prikazane u tabelama. Na osnovu tih rezultata i drugih vrednosti koje su sračunate za odredjene parametre sistema, zavisnost parametra kritičnog opterećenja je prikazana grafički u funkciji položaja promene poprečnog preseka. Ključne reči: Timošenkova greda, frekventna jednačina, kritična sila, parametar kritičnog opterećenja, stepenasta greda.