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Stability of columns subjected to an intermediate concentrated load with one end elastically restrained to move axially
004s7949/87
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PcrgamonJournals
STABILITY OF COLUMNS SUBJECTED TO AN INTERMEDIATE CONCENTRATED LOAD WITH ONE END ELASTICALLY RESTRAINED TO MOVE AXIALLY R. SUBFUMAN~ANand G. VENKATESWARA Structural
RAO
Engineering Group, Department of Space, Vikram Sarabhai Space Centre, Trivandrum 695022, India (Received 14 February 1986)
Abstract-The elastic stability of columns subjected to an intermediate concentrated load with one end elastically restrained to move axially is studied in this paper. The elastic axial restraint is represented by means of an axial spring. The finite element formulation including prebuckling load analysis and stability analysis is briefly described. Numerical results for several values of axial spring parameter and various positions of the intermediate concentrated load for four types of boundary conditions are obtained and are presented in the form of tables.
NOTATION as shown in Fig. I
area of cross-section of the column Young’s modulus moment of inertia element geometric stiffness matrix assembled geometric stiffness matrix axial spring stiffness element axial stiffness matrix element bending elastic stiffness matrix assembled axial stiffness matrix assembled bending elastic stiffness matrix element length length of the column intermediate concentrated load on the column element axial load critical load load vector axial displacement axial displacement vector lateral displacement axial coordinate =-
a
L
problem is the stability of columns subjected to an intermediate concentrated load. This problem has
been reinvestigated [3] with the effects of shear deformation included, which is a more realistic situation. If the support at one end of the column elastically restrains that end of the column to move in the axial direction, the problem discussed in [2,3] is to be remodified to include the effect of an axial elastic restraint. In the present paper the effect of an axial elastic restraint at one end of the column as shown in Fig. 1 is studied for various boundary conditions. In this study the effect of shear deformation is intentionally not included so that the effect of elastic axial restraint is explicitly brought out. Numerical results for the stability parameter of columns with four types of boundary condition are obtained using finite element method and are presented in the form of tables. The authors believe that this data will be readily useful for designers. In the next section, the finite element formulation of the problem under study is briefly presented.
eigen-vector axial spring stiffness parameter
stability parameter
9;
I P
(-pnL2> EI
1. INTRODUCTION
The elastic stability of structural components such as columns, circular and rectangular plates, and shells is well discussed in [ 1). A comprehensive compilation of elastic stability data covering many practical situations is given in [2]. One such interesting practical
Hinged-hinged
Fixed-hinged
,
Hinged-fixed
Fixed-fixed
Fig. I. Columns with one end elastically restrained to move axially, subjected to an intermediate concentrated load. I05
R.
106
SUBRAMASIANand
2. FOR.MULATION
2.1 Prebuckling load analysis This is done using a one-dimensional bar element with the axial displacement u as the degree of freedom and with two nodes per element. The strain energy U,, in axial extension of a bar element of length 1is given by
The second term in the strain energy expression is the strain energy of the spring and this term is used in eqn (1) only where the spring is attached to the particular element. Using a linear displacement distribution over the element the element stiffness matrix [k,] can be obtained [4]. The matrix equation after assembly of the element matrices is
(2)
It is to be noted here that for the present problem {R} contains all zeros except at the loaded point of the column with a value of P. By solving eqn (2) one can obtain {u} and finally the load in the element using strain-displacement and stress-strain relations. 2.2 Stability analysis The load distribution in the column is obtained following the procedure described in Sec. 2.1. For the stability analysis the bending elastic stiffness and the geometric stiffness matrices are used. The strain energy U, of an element in bending is given by El
LJ,= -
2
dx.
VENKATESWARARAO
the work done W by the load P, given by
Consider a column of length L subjected to an intermediate load P at C (Fig. 1). The load is directed towards D. At B the support offers an elastic restraint so that the point can not move freely. This is schematically represented by means of an axial spring of stiffness k,. Under this configuration the load distribution in the column from B to C is different from the load distribution from C to D. Before attempting to solve the stability problem, it is necessary to find the load distribution through a prebuckling load analysis.
Kl{u~ = {RI.
G.
(3)
A cubic displacement distribution is used to derive the element bending elastic stiffness matrix [KJ. The degrees of freedom are w and dw/dx with two nodes per element. Using the same displacement distribution, the element geometric stiffness matrix [G] is obtained from
(4)
where P is the axial load in an element as obtained from the prebuckhng load analysis. After assembly of the element matrices, the matrix equation governing the stability problem is [Kb]{d} - I.[G]{b} = 0.
(5)
Equation (5) can be solved for obtaining eigenvalues (stability parameter) and eigenvectors using any standard algorithm. 3. NUMERICAL RESULTS AND DISCUSSION
Using the formulation described above the stability parameter 1 of columns with various boundary conditions is obtained for several of y and fi. The boundary conditions considered are: (1) Hinged-hinged (2) Fixed-hinged (3) Hinged-fixed (4) Fixed-fixed. The case of the cantilever column is not considered in the present study as an elastic axial restraint at the free end is not practically possible. Based on the convergence study a twenty-element idealization of the column is used for obtaining numerical results. The results are obtained up to the value of y = 1, where the orders of the axial stiffness of the column and the spring stiffness are equal. The numerical results for various boundary conditions are presented in Tables l-4. From these results the following observations can be made. (1) For y = 0.001, the axial elastic spring offers practically no resistance for axial motion and the present results agree very well with those of [2]. (2) For a given b as y increases, i. increases, and this increase in I. is more pronounced for y = 1.0. (3) For y =O.OOl and 0.01, as /I increases i. increases, and for y = 0.1 and 1.0, as /J increases i. increases up to some value of p and then decreases slightly with increase in 8. Further increase in p results in an increased value of 1. (4) The value 1 for a given 7 and /? for a hinged-fixed column is always higher than that of fixed-hinged column.
Elastic stability of columns
107
Table 1. Stability parameter 1 of a hinged-hinged
column
B 7
0.9
0.001
11.95 (11.90)t 12.07 13.22 24.95
0.01 0.1 1.0
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
14.20 16.42 18.09 18.68 18.74 19.59 23.13 36.95 (14.21) (16.40) (18.06) (18.66) (18.75) (19.62) (23.14) (36.97) 14.33 16.58 18.26 18.83 18.86 19.68 23.22 37.05 15.71 18.18 19.90 20.31 20.03 20.59 24.00 37.94 28.95 32.39 34.41 32.80 28.83 26.68 28.75 42.94
t Values in brackets are from [2]. Table 2. Stability parameter i. of a fixed-hinged column Y
0.9
0.001
20.44 (20.43)t 20.61 22.26 37.44
0.01 0.1 1.0
0.8
0.7
0.6
B 0.5
0.4
0.3
0.2
0.1
21.66 24.1 I 27.55 30.89 32.20 32.42 35.68 52.86 (21.62) (24.11) (27.56) (30.91) (32.15) (32.38) (35.64) (52.85) 21.83 24.28 27.74 31.09 32.37 32.56 35.78 52.96 23.45 25.99 29.59 33.03 34.08 33.83 36.75 53.90 37.45 39.89 43.96 47.56 46.24 42.05 42.45 59.09
t Values in brackets are from [2]. Table 3. Stability parameter I of a hinged-fixed column
B
Y
0.9
0.001
26.97 (26.94)t 27.25 30.03 58.71
0.01 0.1 I.0
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
34.67 42.62 47.02 47.39 50.87 63.99 105.4 318.7 (34.57) (42.64) (46.92) (47.33) (50.84) (64.00) (105.5) (318.3) 35.04 43.10 47.56 47.84 51.26 64.39 106.0 320.0 38.72 47.82 53.00 52.41 55.06 68.13 110.9 332.0 72.20 86.09 101.1 95.13 84.51 92.97 139.3 388.7
t Values in brackets are from [2]. Table 4. Stability parameter 1 of a fixed-fixed column Y
0.9
0.001
40.39 (40.32)t 40.72 44.03 74.47
0.01 0.1 1.0
0.8
0.7
0.6
B 0.5
0.4
0.3
0.2
0.1
44.79 53.39 65.78 74.47 75.37 85.23 126.4 347.2 (44.76) (53.29) (65.77) (74.48) (75.34) (85.19) (126.3) (346.7) 45.14 53.79 66.29 75.06 75.85 85.65 126.9 348.2 48.62 57.80 71.23 80.86 80.52 89.59 131.3 357.6 78.75 90.1 108.3 127.0 115.6 115.5 157.4 406.0
t Values in brackets are from [2].
4.
CONCLUDING REMARKS
REFERENCES
The stability of a column subjected to an intermediate concentrated load. with one end elastically restrained
to move axially
has been studied
in this
paper. The finite element method has been used to obtain the numerical results. It has been observed that for all practical purposes the axial spring does not offer any resistance to axial motion when the spring stiffness parameter is 0.001.
I. S. P. Timoshenko and J. M. Gere, Theory of Elastic Sfobiliry. McGraw-Hill. New York (I 961). 2. Column Research Committee of Japan, Hundbook of Structural Stability, pp. 1.6-1.7. Corona, Tokyo (1971). 3. B. P. Shastry and G. Venkateswara Rao, Stability of columns subjected to an intermediate concentrated
load. Cotnpu~ Svucr. (to appear). 4. 0. C. Zienkiewicz, Finite Element Method in Engineering Science. McGraw-Hill,
New York (1971).
f3.00 + 0.00 Lid.
PcrgamonJournals
STABILITY OF COLUMNS SUBJECTED TO AN INTERMEDIATE CONCENTRATED LOAD WITH ONE END ELASTICALLY RESTRAINED TO MOVE AXIALLY R. SUBFUMAN~ANand G. VENKATESWARA Structural
RAO
Engineering Group, Department of Space, Vikram Sarabhai Space Centre, Trivandrum 695022, India (Received 14 February 1986)
Abstract-The elastic stability of columns subjected to an intermediate concentrated load with one end elastically restrained to move axially is studied in this paper. The elastic axial restraint is represented by means of an axial spring. The finite element formulation including prebuckling load analysis and stability analysis is briefly described. Numerical results for several values of axial spring parameter and various positions of the intermediate concentrated load for four types of boundary conditions are obtained and are presented in the form of tables.
NOTATION as shown in Fig. I
area of cross-section of the column Young’s modulus moment of inertia element geometric stiffness matrix assembled geometric stiffness matrix axial spring stiffness element axial stiffness matrix element bending elastic stiffness matrix assembled axial stiffness matrix assembled bending elastic stiffness matrix element length length of the column intermediate concentrated load on the column element axial load critical load load vector axial displacement axial displacement vector lateral displacement axial coordinate =-
a
L
problem is the stability of columns subjected to an intermediate concentrated load. This problem has
been reinvestigated [3] with the effects of shear deformation included, which is a more realistic situation. If the support at one end of the column elastically restrains that end of the column to move in the axial direction, the problem discussed in [2,3] is to be remodified to include the effect of an axial elastic restraint. In the present paper the effect of an axial elastic restraint at one end of the column as shown in Fig. 1 is studied for various boundary conditions. In this study the effect of shear deformation is intentionally not included so that the effect of elastic axial restraint is explicitly brought out. Numerical results for the stability parameter of columns with four types of boundary condition are obtained using finite element method and are presented in the form of tables. The authors believe that this data will be readily useful for designers. In the next section, the finite element formulation of the problem under study is briefly presented.
eigen-vector axial spring stiffness parameter
stability parameter
9;
I P
(-pnL2> EI
1. INTRODUCTION
The elastic stability of structural components such as columns, circular and rectangular plates, and shells is well discussed in [ 1). A comprehensive compilation of elastic stability data covering many practical situations is given in [2]. One such interesting practical
Hinged-hinged
Fixed-hinged
,
Hinged-fixed
Fixed-fixed
Fig. I. Columns with one end elastically restrained to move axially, subjected to an intermediate concentrated load. I05
R.
106
SUBRAMASIANand
2. FOR.MULATION
2.1 Prebuckling load analysis This is done using a one-dimensional bar element with the axial displacement u as the degree of freedom and with two nodes per element. The strain energy U,, in axial extension of a bar element of length 1is given by
The second term in the strain energy expression is the strain energy of the spring and this term is used in eqn (1) only where the spring is attached to the particular element. Using a linear displacement distribution over the element the element stiffness matrix [k,] can be obtained [4]. The matrix equation after assembly of the element matrices is
(2)
It is to be noted here that for the present problem {R} contains all zeros except at the loaded point of the column with a value of P. By solving eqn (2) one can obtain {u} and finally the load in the element using strain-displacement and stress-strain relations. 2.2 Stability analysis The load distribution in the column is obtained following the procedure described in Sec. 2.1. For the stability analysis the bending elastic stiffness and the geometric stiffness matrices are used. The strain energy U, of an element in bending is given by El
LJ,= -
2
dx.
VENKATESWARARAO
the work done W by the load P, given by
Consider a column of length L subjected to an intermediate load P at C (Fig. 1). The load is directed towards D. At B the support offers an elastic restraint so that the point can not move freely. This is schematically represented by means of an axial spring of stiffness k,. Under this configuration the load distribution in the column from B to C is different from the load distribution from C to D. Before attempting to solve the stability problem, it is necessary to find the load distribution through a prebuckling load analysis.
Kl{u~ = {RI.
G.
(3)
A cubic displacement distribution is used to derive the element bending elastic stiffness matrix [KJ. The degrees of freedom are w and dw/dx with two nodes per element. Using the same displacement distribution, the element geometric stiffness matrix [G] is obtained from
(4)
where P is the axial load in an element as obtained from the prebuckhng load analysis. After assembly of the element matrices, the matrix equation governing the stability problem is [Kb]{d} - I.[G]{b} = 0.
(5)
Equation (5) can be solved for obtaining eigenvalues (stability parameter) and eigenvectors using any standard algorithm. 3. NUMERICAL RESULTS AND DISCUSSION
Using the formulation described above the stability parameter 1 of columns with various boundary conditions is obtained for several of y and fi. The boundary conditions considered are: (1) Hinged-hinged (2) Fixed-hinged (3) Hinged-fixed (4) Fixed-fixed. The case of the cantilever column is not considered in the present study as an elastic axial restraint at the free end is not practically possible. Based on the convergence study a twenty-element idealization of the column is used for obtaining numerical results. The results are obtained up to the value of y = 1, where the orders of the axial stiffness of the column and the spring stiffness are equal. The numerical results for various boundary conditions are presented in Tables l-4. From these results the following observations can be made. (1) For y = 0.001, the axial elastic spring offers practically no resistance for axial motion and the present results agree very well with those of [2]. (2) For a given b as y increases, i. increases, and this increase in I. is more pronounced for y = 1.0. (3) For y =O.OOl and 0.01, as /I increases i. increases, and for y = 0.1 and 1.0, as /J increases i. increases up to some value of p and then decreases slightly with increase in 8. Further increase in p results in an increased value of 1. (4) The value 1 for a given 7 and /? for a hinged-fixed column is always higher than that of fixed-hinged column.
Elastic stability of columns
107
Table 1. Stability parameter 1 of a hinged-hinged
column
B 7
0.9
0.001
11.95 (11.90)t 12.07 13.22 24.95
0.01 0.1 1.0
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
14.20 16.42 18.09 18.68 18.74 19.59 23.13 36.95 (14.21) (16.40) (18.06) (18.66) (18.75) (19.62) (23.14) (36.97) 14.33 16.58 18.26 18.83 18.86 19.68 23.22 37.05 15.71 18.18 19.90 20.31 20.03 20.59 24.00 37.94 28.95 32.39 34.41 32.80 28.83 26.68 28.75 42.94
t Values in brackets are from [2]. Table 2. Stability parameter i. of a fixed-hinged column Y
0.9
0.001
20.44 (20.43)t 20.61 22.26 37.44
0.01 0.1 1.0
0.8
0.7
0.6
B 0.5
0.4
0.3
0.2
0.1
21.66 24.1 I 27.55 30.89 32.20 32.42 35.68 52.86 (21.62) (24.11) (27.56) (30.91) (32.15) (32.38) (35.64) (52.85) 21.83 24.28 27.74 31.09 32.37 32.56 35.78 52.96 23.45 25.99 29.59 33.03 34.08 33.83 36.75 53.90 37.45 39.89 43.96 47.56 46.24 42.05 42.45 59.09
t Values in brackets are from [2]. Table 3. Stability parameter I of a hinged-fixed column
B
Y
0.9
0.001
26.97 (26.94)t 27.25 30.03 58.71
0.01 0.1 I.0
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
34.67 42.62 47.02 47.39 50.87 63.99 105.4 318.7 (34.57) (42.64) (46.92) (47.33) (50.84) (64.00) (105.5) (318.3) 35.04 43.10 47.56 47.84 51.26 64.39 106.0 320.0 38.72 47.82 53.00 52.41 55.06 68.13 110.9 332.0 72.20 86.09 101.1 95.13 84.51 92.97 139.3 388.7
t Values in brackets are from [2]. Table 4. Stability parameter 1 of a fixed-fixed column Y
0.9
0.001
40.39 (40.32)t 40.72 44.03 74.47
0.01 0.1 1.0
0.8
0.7
0.6
B 0.5
0.4
0.3
0.2
0.1
44.79 53.39 65.78 74.47 75.37 85.23 126.4 347.2 (44.76) (53.29) (65.77) (74.48) (75.34) (85.19) (126.3) (346.7) 45.14 53.79 66.29 75.06 75.85 85.65 126.9 348.2 48.62 57.80 71.23 80.86 80.52 89.59 131.3 357.6 78.75 90.1 108.3 127.0 115.6 115.5 157.4 406.0
t Values in brackets are from [2].
4.
CONCLUDING REMARKS
REFERENCES
The stability of a column subjected to an intermediate concentrated load. with one end elastically restrained
to move axially
has been studied
in this
paper. The finite element method has been used to obtain the numerical results. It has been observed that for all practical purposes the axial spring does not offer any resistance to axial motion when the spring stiffness parameter is 0.001.
I. S. P. Timoshenko and J. M. Gere, Theory of Elastic Sfobiliry. McGraw-Hill. New York (I 961). 2. Column Research Committee of Japan, Hundbook of Structural Stability, pp. 1.6-1.7. Corona, Tokyo (1971). 3. B. P. Shastry and G. Venkateswara Rao, Stability of columns subjected to an intermediate concentrated
load. Cotnpu~ Svucr. (to appear). 4. 0. C. Zienkiewicz, Finite Element Method in Engineering Science. McGraw-Hill,
New York (1971).