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Stiffness matrices for non-prismatic members including transverse shear
Compulers & Slrwlures Vol. 40, No. 4. pp. 831-835. 1991 Printed in Great Britain.
STIFFNESS
MATRICES INCLUDING
@M-7949/91 s3.00 + 0.00 0 1991 Pcrgamon Press plc
FOR NON-PRISMATIC MEMBERS TRANSVERSE SHEAR
M. EISENBERGER Department of Civil Engineering, Technion-Israel Institute of Technology, Technion City 32000, Israel (Received 23 May 1990) Abstract-Explicit terms for the stiffness matrices, including the effect of transverse shear, of common non-prismatic members are derived. The stiffnesses are formulated from the flexibilities of the element. Comparison of the values derived by the suggested formulation with known results is made. The formulation is highly recommended for small in-house computers, and saves computer time and storage.
INTRODUCTION Structural engineers often analyze structures that involve non-prismatic members. Haunched or tapered elements are treated in a different manner depending on the analysis method. Analysis done by the classical methods will usually involve the use of tables or graphs for stiffness and carry-over factors for the different type of non-prismatic members [l-3]. When accurate results are needed the use of the tables and graphs will not suffice. A textbook [4] includes tables of flexibility coefficients for use in analysis by the flexibility method. More modern analysis, done on a digital computer, will involve the approximation of a typical non-prismatic member by a number of prismatic members. When this approximation is carried out, using a sufficient number of members, the results are adequate. The number of sub-divisions is dependent on the member geometry, and the desired accuracy. The use of this technique will increase significantly the effort needed for modeling the structure, the preparation of data for the computer analysis, the time needed for computer analysis, and in some cases will override the computer’s capabilities. Examples for the use of this technique, and accuracy analysis are given in [S]. The reference manual for ANSYS [6], suggests complicated formulas for equivalent cross-sectional area and moment of inertia for non-prismatic members. The exact stiffness matrix for several commonly used non-prismatic members were given in [S]. The recent publications by Banerjee and Williams [7,8] gave exact solutions for few cases of such members. The variation of the cross-sectional area, polar moment of inertia, and the moment of inertia of the area are not independent, and their variation is dependent on the choice of only two parameters. Karabalis and Beskos [13] gave exact static stiffness matrices for linear depth variation and constant width. For dynamics and stability they got good approximate solutions. Eisenberger and Reich [9] presented an 831
approximate method for general polynomial variation of width and/or depth for beams. The exact stiffness matrix for general polynomial variation of cross section properties was presented recently in [lo]. All the above, and many other earlier publications, do not consider the effect of shear deformations on the stiffness matrix. In this work, the exact terms of the stiffness for non-prismatic members including matrix shear deformations are derived. It is very often that non-prismatic members are short and thus the effect of transverse shear becomes significant. The accuracy of approximate analysis is studied compared to the exact solution. The exact terms can be used by designers who need very accurate results for deflections and rotations of their structures. STIFFNESSMATRIXDERIVATION The most convenient way to derive the stiffness matrix for non-prismatic members is by inversion of the corresponding flexibility matrix. For the member in Fig. 1 the flexibility matrix for the displacements at end k of the member is given as
(1)
The terms in this matrix are found using the unit load method [l 11. Inverting this matrix one can get the stiffness corresponding to the unit translations in the four and five directions, and unit rotation six. The matrix will be
%t=
A
0
0
0
B
C
[ OCD
1
(2)
M. EISENBERGER
832 2
hgpz5/G
5 t
t
I i-y /t 3
L 1
1
L
Fig. 1. General plane frame member.
Fig. 2. Linear height variation member.
and the terms are
(3)
ingly. The width of the beam is b, and L is the length. The terms of the flexibility matrix Fkk are F ,fr
+
lMj/h)
(9)
Ebh,-hk’
e,
12L3 Fss = Eb (h, - hk)’ x
(6)
[
(1 +g)lni+?-$J
/
6Lz
with
Fs
H = FssFh6 - F&.
(7)
The member stiffness matrix S,,,, is found from equilibrium considerations in the deflected shapes that correspond to the unit displacements in the stiffness method, using the terms in Skk, is given as
-A
0
0
B
C+BL D+2CL+BL*
s,=
F
=
1, (10)
1.5
k
(11)
2
Ebh,h,’
=6L(hj+hk) 66 Ebh?h* I k
(12)
’
where g =fE(hj - hk)* 12GL*
-A
0
0
0
-B
-c
0
-C-BL
A
0
-D-CL
(8)
,
0
B
Symmetric
(13)
c D
where L is the member length. NON-PRISMATIC
MEMBER FLEXIBILITIES
Explicit terms of the flexibility matrix for two most commonly used non-prismatic members are given in this section. The effect of transverse shear is included in the appropriate terms. For each cross-section shape, one has to know also the form factor for shear f WI. Linear height variation
Members with linear height variation and constant width are commonly used in frames. A typical such member is shown in Fig. 2. We denote the height at ends j and k of member, as hi and hk, correspond-
Parabolic height variation
Another type of commonly used non-prismatic member is a member with parabolic height variation, shown in Fig. 3. Using the same notation as defined for case (a) one gets
F
_
6(
L
arctan JP
Ebhk
Jp
L’ F,‘=sEI,
1
5+3p (1----
(14)
’ 2
1
l+P
P
(15)
Stiffness matrices for non-prismatic members
Fig. 3. Parabolic height variation member. 2
L2
3
4
6
7
8
Number of elements
2+p
(16)
F56 = 4EI, (1 +py + 3 arctan Jp JP
1’
Fig. 4. Example 1: relative errors in stiffness.
(17)
where p=+ h/
Sheardefom~rtionrnot inducedlO.W7l
(18)
k
lkJ!!i,
I
I
I
I
I
I
2
3
4
6
8
7
8
Number of elements Fig. 5. Example 2: convergence of tip deflection.
2fEh; g=m. NUMERICAL
‘1
(1%
(20)
by including the shear deformations (as discussed in [l l]), denoted by Sss, S56, S,. The importance of including the shear deformations is evident: even when one divides the member into a large number of segments, if shear deformations are not included convergence will be to values with relative errors of W-100%.
RESULTS
Example 1
The exact expressions for stiffness that were derived using the outlined procedure are compared in this section, with values that were computed by dividing the member into n sections. For members with linear height variation, a member for which the height varies from 0.75 to 0.3 m (a = hj/hk = 2.5), and L = 100 cm, was tested. For all the examples E was taken as 1, G as 0.4, and the cross-section was rectangular (S = 1.2). The member was divided into 1, 2, 3, 4, 5, 6 and 8 sections with constant stiffness, equal to that of the middle of the element. The results for the relative error are given in Fig. 4. The comparison is made to the stiffness that are found by neglecting shear deformations (in the constant cross-section members), denoted by ST,, Sf,, S&, and those found
Example 2
A cantehver with linearly varying height was analyzed using the proposed formulation and compared to results from finite element analysis. The beam dimensions are: b = 0.2 m, hj = 0.75 m, hk = 0.3 m, (a = 2.5). The load was P = 1. The finite element solution was calculated twice: with and without consideration of shear deformations. For each member the average height was taken. The results are shown in Fig. 5. The convergence is to the values calculated using the formulation in this work, and to the
P
Fig. 6. Example 3: variable cross-section beams with a = 2, 3 and y = 3.
834
M .EISENBERGER solution from [5] for the case where shear deformation is not included. The relative error of the converged solution when shear is not considered is 25% * Example 3
In this example the effect of shear deformation as a function of the member aspect ratio and tapering ratio is presented. The deflection under the load in the three beams in Fig. 6 are given in Table 1. In the table, results are given for L = 1,2, 3,4 and 5 m. The aspect ratio for the beam, L, is calculated as follows: for the linear height variation cases as the ratio of L over the average height, and for the parabolic variation as the ratio of L over the height of constant cross-section beam with equal material volume (in this case 6 = 0.5 m). In the table both the deflections including shear deformations, dsh, and neglecting them, S,, are given. The relative error, e, is given for each case. Also given is 2 which is the relative error between the deflections of a constant cross sections member with the same aspect ratio and material volume, when shear deformations are included and not included. Several conclusions can be drawn from the table: (a) for larger d the relative errors e and 2 is smaller. (b) The relative errors e are significantly higher than e. (c) for larger taper ratio a and y (not given in this example) the relative errors increase.
CONCLUSIONS
As was seen in the examples the effect of shear deformations is much more important for tapered members. Neglecting them will result in significant errors. The effect is larger for high values of a and y and for relatively short beams, because in these cases part of the beam undergoes large shear deformations, which effect the overall stiffness and the size of the deflections significantly. The use of proposed method of analysis also has the following advantages (that were presented in [51): (a) The use of one member instead of five members, on the average, will reduce the number of linear equations by twelve for these members. For an average frame with five non-prismatic members this results in savings of about one third of the core memory of in-house computers, thus increasing the capacity of the program. (b) The reduction in the number of equations that should be solved will result in big savings in computer time. For one member, computation time is reduced by a factor of 20 where the number of equations drops from 18 to 6. (c) Addition of other types of non-prismatic members is easy. One needs only the terms of the flexibility matrix for the member. The rest of the procedure is identical.
Stiffness matrices .for non-prismatic members REFERENCES 1.
H. Cross and N. D. Morgan, Continuous Frames of Concrete. John Wiley (1951). R. Agent, Methoda Approximatilor Successive. Editura de Stat Pentru Arhitectura Si Constructii, Bukarest (1965). R. J. Roark and W. C. Young, Formulasfor Stress and Strain, 5th edn. McGraw-Hill (1975). W. Pytowsky, Obliczenia Statycze, Pretow 0 Xmiennej Sztywnosk, Metoda A. Winokura, Arkady, Warsaw (1951). M. Eisenberger, Explicit stiffness matrices for nonprismatic members. Comput. Struct. 20,715-720 (1985). P. C. Kohnke, ANSYS Theoretical Manual, Swanson Analysis Systems (1977). J. R. Banerjee and F. W. Williams, Exact BernoulliEuler dynamic stiffness matrix for a range of tapered Reinforced
2. 3. 4.
5. 6. 7.
835
beams. Int. J. Numer. Meth. Engng 21, 2289-2302 (1985). 8. J. R. Banerjee and F. W. Williams, Exact BemoulliEuler static stiffness matrix for a range of tapered beams. Int. J. Numer. Meth. Engng 23, 1615-1628 (1986).
9. M. Eisenberger and Y. Reich, Static, vibration, and stability analysis of non-uniform beams. Comput. Struct. 31, 567-573 (1989).
10. M. Eisenberger, Exact static and dynamic stiffness matrices for variable cross section members. AIAA J128, 1105-l 109 (1990). 11. W. Weaver Jr and J. M. Gere, Matrix Analysis of Framed Structures. Van Nostrand (1980). 12. S. P. Timoshenko and J. M. Gere, Mechanics of Materials. Van Nostrand (1972). 13. D. L. Karabalis and D. E. Beskos, Static, dynamic and stability of structures composed of tapered beams. Comput. Struct. 16, 731-748 (1983).
STIFFNESS
MATRICES INCLUDING
@M-7949/91 s3.00 + 0.00 0 1991 Pcrgamon Press plc
FOR NON-PRISMATIC MEMBERS TRANSVERSE SHEAR
M. EISENBERGER Department of Civil Engineering, Technion-Israel Institute of Technology, Technion City 32000, Israel (Received 23 May 1990) Abstract-Explicit terms for the stiffness matrices, including the effect of transverse shear, of common non-prismatic members are derived. The stiffnesses are formulated from the flexibilities of the element. Comparison of the values derived by the suggested formulation with known results is made. The formulation is highly recommended for small in-house computers, and saves computer time and storage.
INTRODUCTION Structural engineers often analyze structures that involve non-prismatic members. Haunched or tapered elements are treated in a different manner depending on the analysis method. Analysis done by the classical methods will usually involve the use of tables or graphs for stiffness and carry-over factors for the different type of non-prismatic members [l-3]. When accurate results are needed the use of the tables and graphs will not suffice. A textbook [4] includes tables of flexibility coefficients for use in analysis by the flexibility method. More modern analysis, done on a digital computer, will involve the approximation of a typical non-prismatic member by a number of prismatic members. When this approximation is carried out, using a sufficient number of members, the results are adequate. The number of sub-divisions is dependent on the member geometry, and the desired accuracy. The use of this technique will increase significantly the effort needed for modeling the structure, the preparation of data for the computer analysis, the time needed for computer analysis, and in some cases will override the computer’s capabilities. Examples for the use of this technique, and accuracy analysis are given in [S]. The reference manual for ANSYS [6], suggests complicated formulas for equivalent cross-sectional area and moment of inertia for non-prismatic members. The exact stiffness matrix for several commonly used non-prismatic members were given in [S]. The recent publications by Banerjee and Williams [7,8] gave exact solutions for few cases of such members. The variation of the cross-sectional area, polar moment of inertia, and the moment of inertia of the area are not independent, and their variation is dependent on the choice of only two parameters. Karabalis and Beskos [13] gave exact static stiffness matrices for linear depth variation and constant width. For dynamics and stability they got good approximate solutions. Eisenberger and Reich [9] presented an 831
approximate method for general polynomial variation of width and/or depth for beams. The exact stiffness matrix for general polynomial variation of cross section properties was presented recently in [lo]. All the above, and many other earlier publications, do not consider the effect of shear deformations on the stiffness matrix. In this work, the exact terms of the stiffness for non-prismatic members including matrix shear deformations are derived. It is very often that non-prismatic members are short and thus the effect of transverse shear becomes significant. The accuracy of approximate analysis is studied compared to the exact solution. The exact terms can be used by designers who need very accurate results for deflections and rotations of their structures. STIFFNESSMATRIXDERIVATION The most convenient way to derive the stiffness matrix for non-prismatic members is by inversion of the corresponding flexibility matrix. For the member in Fig. 1 the flexibility matrix for the displacements at end k of the member is given as
(1)
The terms in this matrix are found using the unit load method [l 11. Inverting this matrix one can get the stiffness corresponding to the unit translations in the four and five directions, and unit rotation six. The matrix will be
%t=
A
0
0
0
B
C
[ OCD
1
(2)
M. EISENBERGER
832 2
hgpz5/G
5 t
t
I i-y /t 3
L 1
1
L
Fig. 1. General plane frame member.
Fig. 2. Linear height variation member.
and the terms are
(3)
ingly. The width of the beam is b, and L is the length. The terms of the flexibility matrix Fkk are F ,fr
+
lMj/h)
(9)
Ebh,-hk’
e,
12L3 Fss = Eb (h, - hk)’ x
(6)
[
(1 +g)lni+?-$J
/
6Lz
with
Fs
H = FssFh6 - F&.
(7)
The member stiffness matrix S,,,, is found from equilibrium considerations in the deflected shapes that correspond to the unit displacements in the stiffness method, using the terms in Skk, is given as
-A
0
0
B
C+BL D+2CL+BL*
s,=
F
=
1, (10)
1.5
k
(11)
2
Ebh,h,’
=6L(hj+hk) 66 Ebh?h* I k
(12)
’
where g =fE(hj - hk)* 12GL*
-A
0
0
0
-B
-c
0
-C-BL
A
0
-D-CL
(8)
,
0
B
Symmetric
(13)
c D
where L is the member length. NON-PRISMATIC
MEMBER FLEXIBILITIES
Explicit terms of the flexibility matrix for two most commonly used non-prismatic members are given in this section. The effect of transverse shear is included in the appropriate terms. For each cross-section shape, one has to know also the form factor for shear f WI. Linear height variation
Members with linear height variation and constant width are commonly used in frames. A typical such member is shown in Fig. 2. We denote the height at ends j and k of member, as hi and hk, correspond-
Parabolic height variation
Another type of commonly used non-prismatic member is a member with parabolic height variation, shown in Fig. 3. Using the same notation as defined for case (a) one gets
F
_
6(
L
arctan JP
Ebhk
Jp
L’ F,‘=sEI,
1
5+3p (1----
(14)
’ 2
1
l+P
P
(15)
Stiffness matrices for non-prismatic members
Fig. 3. Parabolic height variation member. 2
L2
3
4
6
7
8
Number of elements
2+p
(16)
F56 = 4EI, (1 +py + 3 arctan Jp JP
1’
Fig. 4. Example 1: relative errors in stiffness.
(17)
where p=+ h/
Sheardefom~rtionrnot inducedlO.W7l
(18)
k
lkJ!!i,
I
I
I
I
I
I
2
3
4
6
8
7
8
Number of elements Fig. 5. Example 2: convergence of tip deflection.
2fEh; g=m. NUMERICAL
‘1
(1%
(20)
by including the shear deformations (as discussed in [l l]), denoted by Sss, S56, S,. The importance of including the shear deformations is evident: even when one divides the member into a large number of segments, if shear deformations are not included convergence will be to values with relative errors of W-100%.
RESULTS
Example 1
The exact expressions for stiffness that were derived using the outlined procedure are compared in this section, with values that were computed by dividing the member into n sections. For members with linear height variation, a member for which the height varies from 0.75 to 0.3 m (a = hj/hk = 2.5), and L = 100 cm, was tested. For all the examples E was taken as 1, G as 0.4, and the cross-section was rectangular (S = 1.2). The member was divided into 1, 2, 3, 4, 5, 6 and 8 sections with constant stiffness, equal to that of the middle of the element. The results for the relative error are given in Fig. 4. The comparison is made to the stiffness that are found by neglecting shear deformations (in the constant cross-section members), denoted by ST,, Sf,, S&, and those found
Example 2
A cantehver with linearly varying height was analyzed using the proposed formulation and compared to results from finite element analysis. The beam dimensions are: b = 0.2 m, hj = 0.75 m, hk = 0.3 m, (a = 2.5). The load was P = 1. The finite element solution was calculated twice: with and without consideration of shear deformations. For each member the average height was taken. The results are shown in Fig. 5. The convergence is to the values calculated using the formulation in this work, and to the
P
Fig. 6. Example 3: variable cross-section beams with a = 2, 3 and y = 3.
834
M .EISENBERGER solution from [5] for the case where shear deformation is not included. The relative error of the converged solution when shear is not considered is 25% * Example 3
In this example the effect of shear deformation as a function of the member aspect ratio and tapering ratio is presented. The deflection under the load in the three beams in Fig. 6 are given in Table 1. In the table, results are given for L = 1,2, 3,4 and 5 m. The aspect ratio for the beam, L, is calculated as follows: for the linear height variation cases as the ratio of L over the average height, and for the parabolic variation as the ratio of L over the height of constant cross-section beam with equal material volume (in this case 6 = 0.5 m). In the table both the deflections including shear deformations, dsh, and neglecting them, S,, are given. The relative error, e, is given for each case. Also given is 2 which is the relative error between the deflections of a constant cross sections member with the same aspect ratio and material volume, when shear deformations are included and not included. Several conclusions can be drawn from the table: (a) for larger d the relative errors e and 2 is smaller. (b) The relative errors e are significantly higher than e. (c) for larger taper ratio a and y (not given in this example) the relative errors increase.
CONCLUSIONS
As was seen in the examples the effect of shear deformations is much more important for tapered members. Neglecting them will result in significant errors. The effect is larger for high values of a and y and for relatively short beams, because in these cases part of the beam undergoes large shear deformations, which effect the overall stiffness and the size of the deflections significantly. The use of proposed method of analysis also has the following advantages (that were presented in [51): (a) The use of one member instead of five members, on the average, will reduce the number of linear equations by twelve for these members. For an average frame with five non-prismatic members this results in savings of about one third of the core memory of in-house computers, thus increasing the capacity of the program. (b) The reduction in the number of equations that should be solved will result in big savings in computer time. For one member, computation time is reduced by a factor of 20 where the number of equations drops from 18 to 6. (c) Addition of other types of non-prismatic members is easy. One needs only the terms of the flexibility matrix for the member. The rest of the procedure is identical.
Stiffness matrices .for non-prismatic members REFERENCES 1.
H. Cross and N. D. Morgan, Continuous Frames of Concrete. John Wiley (1951). R. Agent, Methoda Approximatilor Successive. Editura de Stat Pentru Arhitectura Si Constructii, Bukarest (1965). R. J. Roark and W. C. Young, Formulasfor Stress and Strain, 5th edn. McGraw-Hill (1975). W. Pytowsky, Obliczenia Statycze, Pretow 0 Xmiennej Sztywnosk, Metoda A. Winokura, Arkady, Warsaw (1951). M. Eisenberger, Explicit stiffness matrices for nonprismatic members. Comput. Struct. 20,715-720 (1985). P. C. Kohnke, ANSYS Theoretical Manual, Swanson Analysis Systems (1977). J. R. Banerjee and F. W. Williams, Exact BernoulliEuler dynamic stiffness matrix for a range of tapered Reinforced
2. 3. 4.
5. 6. 7.
835
beams. Int. J. Numer. Meth. Engng 21, 2289-2302 (1985). 8. J. R. Banerjee and F. W. Williams, Exact BemoulliEuler static stiffness matrix for a range of tapered beams. Int. J. Numer. Meth. Engng 23, 1615-1628 (1986).
9. M. Eisenberger and Y. Reich, Static, vibration, and stability analysis of non-uniform beams. Comput. Struct. 31, 567-573 (1989).
10. M. Eisenberger, Exact static and dynamic stiffness matrices for variable cross section members. AIAA J128, 1105-l 109 (1990). 11. W. Weaver Jr and J. M. Gere, Matrix Analysis of Framed Structures. Van Nostrand (1980). 12. S. P. Timoshenko and J. M. Gere, Mechanics of Materials. Van Nostrand (1972). 13. D. L. Karabalis and D. E. Beskos, Static, dynamic and stability of structures composed of tapered beams. Comput. Struct. 16, 731-748 (1983).