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Deflection analysis of expanded open-web steel beams
Computerr & Structuriw. Vol. 4. pp. 327-336. Rrgamon
DEFLECTION
Press iid.
+ckNbd in &eat
Britain
ANALYSIS OF EXPANDED STEEL BEAMS?
OPEN-WEB
M. U. HO~AIN:, W. K. CHENGQ and V. V. NEDI/ Department of Civil Engineering, University of Saskatchewan, Saskatoon S7N OWO, Canada Abstract-TheGnite element method is very appropriate for calculating stresses at isolated areas of expanded open-web steel beams. For deflection analysis, however, the entire beam, or one4alf the beam if the system is symmetrica1, must be included in the idealization and therefore it is not practical to obtain deBeetions by a direct application of the Iinite element method. In this paper, the authors present a method of de&etion analysis which treats the castellated beam as an assemblage of typical segments and utilizes the finite element method to form the stiffness matrix for the typical segment. The beam deflections at the panel points am then computed by the conventional stiiTneas method. Two different idealizations were tried for the finite ekment analysis of a typical segment. These idealizations resulted in a 13 x 13 and a 7 x 7 stiffness matrix. De5ection vahtes calculated by using the 7 x 7 stiffness matrix showed close agreement with those obtained experimentally.
1. ~RODU~ON THE FINITEelement method has recently been utilized by several researchers to calculate the stresses [l, 2, 31 and deflections [3] in castellated steel beams. Since stress analysis is normally required only at isolated areas, where stresses are critical, the finite element method is very appropriate for stress analysis and may be applied to a segment of a castellated beam of any size. For deflection analysis, however, the entire beam or one-half the beam, if the system is symmetrical, must be included in the idealization. From a consideration of the number of su~i~sions required for the finite element method; available storage in the computer; and the round-off error associated with the solution of an excessively large number of simultaneous equations; it is seen that the idealization of the entire beam is possible only for short beams [3]. For long castellated beams, it is not practical to obtain de3eetions by a direct apportion of the finite element method. Since castellated steel beams are most effective only in long span lengths, there is a definite need to develop a method for predicting deflections in such beams. Referring to Fig. 1, the hatched area represents a typical segment (panel) of the castellated steel beam shown. The beam can be considered as an assemblage of such typical segments if the non-existing half-panel (shown by dotted line) is included as part of each end segment. The error involved should be insignilicant since the extra material occurs only at the free ends of the beam. The method presented in this paper utilizes the finite element method to form the stiffness matrix for the typical segment and then computes the beam deflection at the panel points by the conventional stiffness method. t Presentedat the National Symposium on Computer&d Structural Analysis and Design at the School of Eng&uing and Appbed !Mence. George W~hin~on University, Washington, D.C., 27-29 March (1972). $ Associate Professor. $ Design Engineer, Read, Jones and Christoffersen Ltd., Vancouver, Canada. )1Prof~r* 327
328
M. U. Hcxur~, W. K. CHJZNG and V. V. NEIS Typacoi
-cmbet
FIG. 1. Typical member of a castellated bwn
2, FEWTE llUMENT
ANUYSIS
OF THE TYPIC& IWWJHW
Twodimeusional ilnite element analysis is employed in the present work. Trkngular elements fit the geometry of the web of the castellated beam p&e&y. However, the ordinary‘constantstraintriangIe’is not satisfactory for problems with high stmss gradients. For. this reason, a mixture of rectangular and triangular elements is used in the ldealixation of the web with the use of triangular elements kept to a minimum. The flange of the stuzl beam, which presents a three-dimensional problem, is replaced by one-dimensional bar element having an area equivalent to that of the flange. Treating the flang;eas elements in the plane of the web with a much greater thickness or with a magnilled modulus of elasticity is not satisfactoryas the shear stiffness of the flange is apparently exaggerated. Again, if the shear lag effect is not sign&ant, it is not necessary to treat the flange as a separate plate with properly prescribed displacements relating to the web. If symmetry about the neutral axis of the beam is utilized, only half of the typical segment, as shown in Fig 2, needs to be considered. The number of nodes, and thus the degrees of freedom, associated with this member depends on the idealization used in the finite element model of the typical segment. The foliowing criteria, however, must be satisfied : (a) Full continuity between members at common boundary. (b) A node at mid-panel (point C in Fig. 2) to facilitate the application of load on the beam aad provide support conditions. (c) The resulting degrees-of-freedom of the member are to be kept to a minimum. (d) The terms of the resulting stiffness matrix shall not have a great difference in magnitude to avoid excessive computational round-off error. Three Merent idealizations were tried. Idealization 1, shown in Fig. 3, was first used to idealize the typical member. Edge MN is constrained in the xdirection. Edges RE and FJ are the edges connecting the adjacent members of the beam. There are 5 joints, and therefore 10 degrees-of-freedom, on each of the edges AE and FJ. However, if the same y-displacement is prescrii to all joints on the same edge, the degree-of-freedom is reduced to 6 for each common edge as shown in Fig. 3. One more dtgree-of-fieuiom has to be introdu& at joint X to allow for the application of load and also to allow for the presence of a support at that point. Therefore, Id~zation 1 will result in a 13 x 13 sti&ess matrix for the typical member. However, there are two drawbacks for adopting this idealization. Firstly, the resiiltlng member stiffness matrix is rather large (13x 13). Sccqndly, if a unit x-displacement is introduced at joint H, large forces will result at joints’1 and G while very small forces will result in all other joints. Hence very large and very small terms
Deflection Analysis of
=====
WpAMMB#&W& Steel &ms
329
* ---------
-
-7
Neutral
\
axis
\
\
\
\ \
-_-.-
FIG. 2. Half-beam idealization Typical
bar
to replace
element flonqe
FIG.3. Idealization 1 for the typical member
will be present in the member stiffness matrix, and this is highly undesirable. For these reasons, Idealization 1 was modified to Idealization 2. In this idealization, just one element is used along the common edge, and this element is linked to the fine mesh by triangular elements as shown in Fig. 4. This idealization then results in a 7 x 7 member stiffness matrix. The co-ordinate number corresponding to each degree of freedom is also shown Typical
bar element
FIG.4. Idealization 2 for the typical member
&c
330
M.
U. Hw,
W. K. CHENOand V. V. NEIS
on the figure. The rather coarse mesh used at the common edge should not jeopardize the accuracy on the whole for two reasons. Firstly, the stress gradient is not high at the common edge. Secondly, it can be shown that a fine mesh in the y-direction is of minor importance for the type of rectangular element used [4$ The mod&d idealization was therefore adopted for the analysis. A third idealization, representing the full segment and involving 12degreesof-freedom is now being investigated. Adoption of Idealization 3 is expected to improve the final results further. 3. FORMATION OF MEMBER SMATRIX With the shape of the typical member chosen and the associated degrees-of-freedom determined, the member stiffness matrix can be obtained from the first principles, i.e. by obtaining the resulting forces when 8 unit d~p~~rnent is introduced, one by one, according to the co-ordinate number of the member. Because of Maxwell’s Reciprocal Theorem, only four such analyses 8re required to form the 7 x 7 stiffness matrix. An indirect approach, consisting of the following steps, WBSadopted by the authors. (1) A unit force is applied along one of the seven co-ordinate directions, direction (4) for example, while constraints 8re imposed in the other six directions. Using 8 plane stress finite element program, the model shown in Fig. 4 is analysed to give all joint displacements, (2) The nodal forces for the elements 1, 2, 3 and 4 and forces in the relevant Lange elements, i.e. bar elements 205 and 212 in Fig. 6, are computed next utilizing the displacements obtained in step (1). (3) By summing the appropriate nodal forces, the seven forces corresponding to the seven co-ordinate directions can be obtained. These are the resulting forces when a unit force is applied along co-ordinate direction (4). (4) The seven forces obtained in step (3), when divided by the displacement in direction (4), yield the seven forces assockted with 8 unit displacement in direction (4) and therefore represent the 4th column of the stiffness matrix of the typical member. The complete
matrix may be formed by repeating the analysis with the unit force applied along a different co-ordinate each time. The stiffness matrix of the typical member formed in section 3 can be used to compute deflections in castellated beams of any length having the same typical member. A separate computer program based on the conventional stiffness method was written for this purpose. The program which generates most of the information within itself requires the minimum input information. 5. APPLICATLQN AND RESULTS The method of analysis is ante by application to two series of castellated steel beams which were tested elsewhere [5]. It was.necessary to form the stiffness matrices for only two typical members, one for each series, since all test specimens in any one series were composed of the same typic8l segment. A computer program, written specifical1yfor the analysis of castellated beams [4], WBSused for the finite element an8lysis of the typical segments. As a numerical example, the analysis of specimen F-l is briefiy discussed below. 5.1 Anaiysis of specimen F- 1 (a) Id~~~~~~~~. Figure 5 shows a typical id~tion used for the deflection analysis by the conventional stiffness method. The solid line in Fig. 5b represents the physical
Deflection Analysis of Expa&?d @mi-Web %cI Beams
331
shape of the indeterminate structure being analysed. The solid line together with the dotted line show the actual system the idealization represents. The simulated system implies the appro~~tely correct caption [4] that shear is divided equally between the upper and lower tee sections. Constraints were imposed on direction 4 of members 1 and 8 to represent the supports of the real beam. No constraint in the x-direction was specified as the neutral axis of the beam is already constrained in the x-direction.
support I
support
Support
support (bl
Idcolized
brom
FIO.5. Real and idealized beam: Specimen F-1
(b) Stilgnessmatrix of a t,ypicalmember. Figure 6 shows the dimensions and the coordinate directions of the typical member for thespecimens of seriesF[4]. The forcesf l-$7, corresponding to the seven co-ordinate directions, caused by a unit displacement along one of the co-ordinate directions, were computed according to the procedure outlined in Section 3. The results corresponding to a unit displacement along directions 4, 5, 6 and 7 respectively are shown in Fig. 7. These forces represent the columns 4,5, 6 and 7 respectively of the member stiffness matrix. Because of symmetry, no independent analysis was necessary for columns 1,2 and 3. The resulting matrix, which is very nearly symmetrical, is shown in Fig. 8a. The agreement between corresponding terms is quite good. For example, the terms k,, and kt, are - 939.285 and -939.2964 respectively. The computation round off errors seem to be acceptable. Checking for CFv==Oalso indicates a good agreement. In each column of the matrix the 3rd, 4th and 7th terms should add up to zero. The result for the 1st column shows an upward force of 1000.1822 compared to a downward force of 999.5585 kipsfin. This represents an error of about 0.063 per cent. The matrix was adjusted slightly to form a symmetrical matrix with WV=0 for all cohnnns. The adjusted matrix is shown in Fig. 8b. Such adjustment shall become unnecessary if double precision (i.e. working with approximately 16 figures instead of 7) is used in the computation. For the above calculation, the Poisson’s ratio was assumed to be 0.3 and the modulus of elasticity was taken as 29,000 ksi.
M. W. HOMIN,W. K. CHEIWand V. V. NEIS
332
221 $5 I64
Joints Tr~ar(luiar
e.ements
Rectangular
elements
26 Ear e’ements
FIG. 6. Idealization 2 for the typical member: Series F
5.2 Results and diwussion. The computed vertical deflections for specimens F-l, F-2 and F-3, along with the observed vahaea, are presented in Fig. 9. The computed and observed values for specimens F-l and F-2 show good agreement although the computed values
are always on the unsafe side. This is expected since a finite element idealization is - 668,441s
lS4S.SOO2
(al
Unit
dirplacrmrnt
in direction
4
(b)
Unit
33OS”f
Unit
displacement
FIG.7.
in direction
- 624.169
423.3469
(c)
displacement
in direction
6
,
(d) Unit
1
displacement
Nodal forces in typical member: Series F
, lOOO$6596
ir direction7
5
Deflection Analysis of Expan*
(0)
(b)
As
:opUr-WebSteel Beams
obtained
Adjusted
Fro. 8. Stiffness matrix for the typical member: Series F
deflection
: I Symmetrical about
Experimentally deflectmn
=O.O69in.
observed =O.O56in.
T-
-
Experlmenlolly deflection Symmetrical about FIG.
observed =0,2in.
I 1
9. Calculated and observed deflections: Series F
333
334
M. U. H-IN.
W. K.
CHENG
and V. V. NEIS
inherently stiffer than the actual structure it represents. The present analysis shows that the deflected shape of a castellated beam resembles that of a truss instead of a beam from which some material has been removed. This confirms the view expressed to this effect by Halleux [6.] In order to check the statical equilibrium of the beams in the vertical direction, the resulting seven nodal forces for all the members were calculated for specimens Fl and F-2. The results are presented in Table 1. The vertical forces associated with the support reactions, i.e.f, for members 1 and 8, are nearly identical with the corresponding simple beam reactions. These nodal forces may be conveniently used to compute the stress distributions in an isolated area of the castellated beam. Member
fl
1 2 3 4
0.0 3.5214 11.8365 20.4716 21.4550 14.8056 8.8056 2.6166
: 7 8
f3
f2
0.0 1.3806 2.5551 3.3311 3 -5443 3.1727 1.9417 1.0322
f4
09 6.8569 6.8572 6.8576 -5.1422 -5.1418 -5.1415 -5.1414
f6
J-5
6.8566 09001 OM)ol -119999 OXXXl1 0.0 0.0 5.1412
-3.5214 -11.8366 -20.4716 -21.4550 -14.8056 -8.8056 -2.6165 0.0
f7
-1 a3806 -6.8568 -2.5551 -6.8571 -3.3311 -6.8574 -3.5443 5.1423 -3.1726 5.1418 5.1415 -19416 -1.0322 5.1414 0.0 0.0
(a) Beam F-I Member
fl
1
f2
0.0 3.1079 10.6886 14.1635 13.8348 14.1636 10.6888 3.1080
3 4 i 7 8
f3
0.0 1.1874 1.7911 2.5215 2.9598 2.5216 1.7911 1.1874
f4
0.0 5.9993 5.9996 -0.ooo3 09001 0.0005 -59995 -5.9993
f6
fs
59990 0.0 -6.0 oQOo1 OGOOl -6.0 E391
-3.1079 -10.6887 -14.1636 -13.8349 -14.1636 -106888 -3.1079 0.0
-1.1873 -1.7911 -2.5215 -2.9598 -2.5215 -1.7911 -1.1874 0.0
(b) BeamF-2 TABLE1. Calculatednodalforcesfor member:BeamsF-l AND F-2
3%4.55)8 -1c66.7005
1715.465
Symmetrical
- 945.3812
455.6597
682.9443
711.1957
-391.0468
-594.9011
1189.8022
-1551.l419
-404.4353
-234.1855
-710.3922
- 604.4363
- %.5573
64.6U9
234.I.855
- 64.6109
- 68.0432
390.0116 -594.9011
3%4.5538 -1a6.7005 944..57%
1715.465 - 454.6235
Adjusted
Fio.
10. Stiffnessmatrixfor the typicalmember:!kries E
682.9443
f7
-59993 -59995 -0.0003 0.0 -0.0004 59995 59993 0.0
Deflection Analysis of’ Fxpan+&Open-Web
Steel Beams
335
FIG. 11. Calculated and observed deflections: Series E
The member stiffness matrix for the specimens of series E is shown in Fig. 10. The ideali~tion used was exactly similar to the oneusedinconnection withthe F-seriesspecimens, the only difference being the overall member dimensions. The computed and observed deflections for specimens El, E-2 and E-3 are presented in Fig. 1I * The agreement between experimenti and calculated values is not consistant. There is considerable difference between the computed and observed values for specimens F-3 and E-3. There is no apparent explanation for the disagreement. Further experimental verification of the proposed method of analysis is therefore planned. Adoptation of IIdealizatian 3, mentioned in Section 2, is expected to improve the fkal results in general, investigation was carried out with financial supportfrom the National Research CounciI of Canada. The anth6rs am indebted to Mr_ Akx Kozlow for his assistance in the preparation of the Sgures and to Miss. Darlene Waite who typed the manuscript.
~~~~~~~~~~r~-T~is
336
M. U. Hoeurr~. W. K. Giaffi and V. V. Nars
PI A. T. Hu~~rraar and V. K. SUNLEY.Finite dsmsnt analysis of an expanded I-section beam and an axisymmetric flanged cylinder. CorJsmcc
Racy&, A&an&
fasnccrofStrew Atutlysf. Vol. 3, No. 14 (1968).
Stress Analysis, 3-14 Joint British Con-
(31 W. K. CHlmo, M. U. Hosu~ and V. V. NEIS,Analysis of castellated steel beams by the finite element method. Proc&&gs of the Speckdty Confmnce ONFfnlte E/went Method in Civil Engineering, Montreal, Canada, 1-2 June (1972). i41 W. K. CLIBNO,M. U. HOSAINand V. V. NE& Application of finite element method to expanded openwd, I-sections. Structural Enlphcerino Report No. 4, Civil Engineering Department, University of SMkaMewa& Saskatoo~ CanaQ, September (1971). [s] M. U. H~MIN and W. 0. Spuns, Failure of castellated beams due to rupture of welded joints. AciwSruhf-&eel, No. 1 (1971). [6] P. HALLBUX, Limit analysis of castellated steel beams. Acier-St&Steel, No. 3 (1967). (Received 29 February 1972)
DEFLECTION
Press iid.
+ckNbd in &eat
Britain
ANALYSIS OF EXPANDED STEEL BEAMS?
OPEN-WEB
M. U. HO~AIN:, W. K. CHENGQ and V. V. NEDI/ Department of Civil Engineering, University of Saskatchewan, Saskatoon S7N OWO, Canada Abstract-TheGnite element method is very appropriate for calculating stresses at isolated areas of expanded open-web steel beams. For deflection analysis, however, the entire beam, or one4alf the beam if the system is symmetrica1, must be included in the idealization and therefore it is not practical to obtain deBeetions by a direct application of the Iinite element method. In this paper, the authors present a method of de&etion analysis which treats the castellated beam as an assemblage of typical segments and utilizes the finite element method to form the stiffness matrix for the typical segment. The beam deflections at the panel points am then computed by the conventional stiiTneas method. Two different idealizations were tried for the finite ekment analysis of a typical segment. These idealizations resulted in a 13 x 13 and a 7 x 7 stiffness matrix. De5ection vahtes calculated by using the 7 x 7 stiffness matrix showed close agreement with those obtained experimentally.
1. ~RODU~ON THE FINITEelement method has recently been utilized by several researchers to calculate the stresses [l, 2, 31 and deflections [3] in castellated steel beams. Since stress analysis is normally required only at isolated areas, where stresses are critical, the finite element method is very appropriate for stress analysis and may be applied to a segment of a castellated beam of any size. For deflection analysis, however, the entire beam or one-half the beam, if the system is symmetrical, must be included in the idealization. From a consideration of the number of su~i~sions required for the finite element method; available storage in the computer; and the round-off error associated with the solution of an excessively large number of simultaneous equations; it is seen that the idealization of the entire beam is possible only for short beams [3]. For long castellated beams, it is not practical to obtain de3eetions by a direct apportion of the finite element method. Since castellated steel beams are most effective only in long span lengths, there is a definite need to develop a method for predicting deflections in such beams. Referring to Fig. 1, the hatched area represents a typical segment (panel) of the castellated steel beam shown. The beam can be considered as an assemblage of such typical segments if the non-existing half-panel (shown by dotted line) is included as part of each end segment. The error involved should be insignilicant since the extra material occurs only at the free ends of the beam. The method presented in this paper utilizes the finite element method to form the stiffness matrix for the typical segment and then computes the beam deflection at the panel points by the conventional stiffness method. t Presentedat the National Symposium on Computer&d Structural Analysis and Design at the School of Eng&uing and Appbed !Mence. George W~hin~on University, Washington, D.C., 27-29 March (1972). $ Associate Professor. $ Design Engineer, Read, Jones and Christoffersen Ltd., Vancouver, Canada. )1Prof~r* 327
328
M. U. Hcxur~, W. K. CHJZNG and V. V. NEIS Typacoi
-cmbet
FIG. 1. Typical member of a castellated bwn
2, FEWTE llUMENT
ANUYSIS
OF THE TYPIC& IWWJHW
Twodimeusional ilnite element analysis is employed in the present work. Trkngular elements fit the geometry of the web of the castellated beam p&e&y. However, the ordinary‘constantstraintriangIe’is not satisfactory for problems with high stmss gradients. For. this reason, a mixture of rectangular and triangular elements is used in the ldealixation of the web with the use of triangular elements kept to a minimum. The flange of the stuzl beam, which presents a three-dimensional problem, is replaced by one-dimensional bar element having an area equivalent to that of the flange. Treating the flang;eas elements in the plane of the web with a much greater thickness or with a magnilled modulus of elasticity is not satisfactoryas the shear stiffness of the flange is apparently exaggerated. Again, if the shear lag effect is not sign&ant, it is not necessary to treat the flange as a separate plate with properly prescribed displacements relating to the web. If symmetry about the neutral axis of the beam is utilized, only half of the typical segment, as shown in Fig 2, needs to be considered. The number of nodes, and thus the degrees of freedom, associated with this member depends on the idealization used in the finite element model of the typical segment. The foliowing criteria, however, must be satisfied : (a) Full continuity between members at common boundary. (b) A node at mid-panel (point C in Fig. 2) to facilitate the application of load on the beam aad provide support conditions. (c) The resulting degrees-of-freedom of the member are to be kept to a minimum. (d) The terms of the resulting stiffness matrix shall not have a great difference in magnitude to avoid excessive computational round-off error. Three Merent idealizations were tried. Idealization 1, shown in Fig. 3, was first used to idealize the typical member. Edge MN is constrained in the xdirection. Edges RE and FJ are the edges connecting the adjacent members of the beam. There are 5 joints, and therefore 10 degrees-of-freedom, on each of the edges AE and FJ. However, if the same y-displacement is prescrii to all joints on the same edge, the degree-of-freedom is reduced to 6 for each common edge as shown in Fig. 3. One more dtgree-of-fieuiom has to be introdu& at joint X to allow for the application of load and also to allow for the presence of a support at that point. Therefore, Id~zation 1 will result in a 13 x 13 sti&ess matrix for the typical member. However, there are two drawbacks for adopting this idealization. Firstly, the resiiltlng member stiffness matrix is rather large (13x 13). Sccqndly, if a unit x-displacement is introduced at joint H, large forces will result at joints’1 and G while very small forces will result in all other joints. Hence very large and very small terms
Deflection Analysis of
=====
WpAMMB#&W& Steel &ms
329
* ---------
-
-7
Neutral
\
axis
\
\
\
\ \
-_-.-
FIG. 2. Half-beam idealization Typical
bar
to replace
element flonqe
FIG.3. Idealization 1 for the typical member
will be present in the member stiffness matrix, and this is highly undesirable. For these reasons, Idealization 1 was modified to Idealization 2. In this idealization, just one element is used along the common edge, and this element is linked to the fine mesh by triangular elements as shown in Fig. 4. This idealization then results in a 7 x 7 member stiffness matrix. The co-ordinate number corresponding to each degree of freedom is also shown Typical
bar element
FIG.4. Idealization 2 for the typical member
&c
330
M.
U. Hw,
W. K. CHENOand V. V. NEIS
on the figure. The rather coarse mesh used at the common edge should not jeopardize the accuracy on the whole for two reasons. Firstly, the stress gradient is not high at the common edge. Secondly, it can be shown that a fine mesh in the y-direction is of minor importance for the type of rectangular element used [4$ The mod&d idealization was therefore adopted for the analysis. A third idealization, representing the full segment and involving 12degreesof-freedom is now being investigated. Adoption of Idealization 3 is expected to improve the final results further. 3. FORMATION OF MEMBER SMATRIX With the shape of the typical member chosen and the associated degrees-of-freedom determined, the member stiffness matrix can be obtained from the first principles, i.e. by obtaining the resulting forces when 8 unit d~p~~rnent is introduced, one by one, according to the co-ordinate number of the member. Because of Maxwell’s Reciprocal Theorem, only four such analyses 8re required to form the 7 x 7 stiffness matrix. An indirect approach, consisting of the following steps, WBSadopted by the authors. (1) A unit force is applied along one of the seven co-ordinate directions, direction (4) for example, while constraints 8re imposed in the other six directions. Using 8 plane stress finite element program, the model shown in Fig. 4 is analysed to give all joint displacements, (2) The nodal forces for the elements 1, 2, 3 and 4 and forces in the relevant Lange elements, i.e. bar elements 205 and 212 in Fig. 6, are computed next utilizing the displacements obtained in step (1). (3) By summing the appropriate nodal forces, the seven forces corresponding to the seven co-ordinate directions can be obtained. These are the resulting forces when a unit force is applied along co-ordinate direction (4). (4) The seven forces obtained in step (3), when divided by the displacement in direction (4), yield the seven forces assockted with 8 unit displacement in direction (4) and therefore represent the 4th column of the stiffness matrix of the typical member. The complete
matrix may be formed by repeating the analysis with the unit force applied along a different co-ordinate each time. The stiffness matrix of the typical member formed in section 3 can be used to compute deflections in castellated beams of any length having the same typical member. A separate computer program based on the conventional stiffness method was written for this purpose. The program which generates most of the information within itself requires the minimum input information. 5. APPLICATLQN AND RESULTS The method of analysis is ante by application to two series of castellated steel beams which were tested elsewhere [5]. It was.necessary to form the stiffness matrices for only two typical members, one for each series, since all test specimens in any one series were composed of the same typic8l segment. A computer program, written specifical1yfor the analysis of castellated beams [4], WBSused for the finite element an8lysis of the typical segments. As a numerical example, the analysis of specimen F-l is briefiy discussed below. 5.1 Anaiysis of specimen F- 1 (a) Id~~~~~~~~. Figure 5 shows a typical id~tion used for the deflection analysis by the conventional stiffness method. The solid line in Fig. 5b represents the physical
Deflection Analysis of Expa&?d @mi-Web %cI Beams
331
shape of the indeterminate structure being analysed. The solid line together with the dotted line show the actual system the idealization represents. The simulated system implies the appro~~tely correct caption [4] that shear is divided equally between the upper and lower tee sections. Constraints were imposed on direction 4 of members 1 and 8 to represent the supports of the real beam. No constraint in the x-direction was specified as the neutral axis of the beam is already constrained in the x-direction.
support I
support
Support
support (bl
Idcolized
brom
FIO.5. Real and idealized beam: Specimen F-1
(b) Stilgnessmatrix of a t,ypicalmember. Figure 6 shows the dimensions and the coordinate directions of the typical member for thespecimens of seriesF[4]. The forcesf l-$7, corresponding to the seven co-ordinate directions, caused by a unit displacement along one of the co-ordinate directions, were computed according to the procedure outlined in Section 3. The results corresponding to a unit displacement along directions 4, 5, 6 and 7 respectively are shown in Fig. 7. These forces represent the columns 4,5, 6 and 7 respectively of the member stiffness matrix. Because of symmetry, no independent analysis was necessary for columns 1,2 and 3. The resulting matrix, which is very nearly symmetrical, is shown in Fig. 8a. The agreement between corresponding terms is quite good. For example, the terms k,, and kt, are - 939.285 and -939.2964 respectively. The computation round off errors seem to be acceptable. Checking for CFv==Oalso indicates a good agreement. In each column of the matrix the 3rd, 4th and 7th terms should add up to zero. The result for the 1st column shows an upward force of 1000.1822 compared to a downward force of 999.5585 kipsfin. This represents an error of about 0.063 per cent. The matrix was adjusted slightly to form a symmetrical matrix with WV=0 for all cohnnns. The adjusted matrix is shown in Fig. 8b. Such adjustment shall become unnecessary if double precision (i.e. working with approximately 16 figures instead of 7) is used in the computation. For the above calculation, the Poisson’s ratio was assumed to be 0.3 and the modulus of elasticity was taken as 29,000 ksi.
M. W. HOMIN,W. K. CHEIWand V. V. NEIS
332
221 $5 I64
Joints Tr~ar(luiar
e.ements
Rectangular
elements
26 Ear e’ements
FIG. 6. Idealization 2 for the typical member: Series F
5.2 Results and diwussion. The computed vertical deflections for specimens F-l, F-2 and F-3, along with the observed vahaea, are presented in Fig. 9. The computed and observed values for specimens F-l and F-2 show good agreement although the computed values
are always on the unsafe side. This is expected since a finite element idealization is - 668,441s
lS4S.SOO2
(al
Unit
dirplacrmrnt
in direction
4
(b)
Unit
33OS”f
Unit
displacement
FIG.7.
in direction
- 624.169
423.3469
(c)
displacement
in direction
6
,
(d) Unit
1
displacement
Nodal forces in typical member: Series F
, lOOO$6596
ir direction7
5
Deflection Analysis of Expan*
(0)
(b)
As
:opUr-WebSteel Beams
obtained
Adjusted
Fro. 8. Stiffness matrix for the typical member: Series F
deflection
: I Symmetrical about
Experimentally deflectmn
=O.O69in.
observed =O.O56in.
T-
-
Experlmenlolly deflection Symmetrical about FIG.
observed =0,2in.
I 1
9. Calculated and observed deflections: Series F
333
334
M. U. H-IN.
W. K.
CHENG
and V. V. NEIS
inherently stiffer than the actual structure it represents. The present analysis shows that the deflected shape of a castellated beam resembles that of a truss instead of a beam from which some material has been removed. This confirms the view expressed to this effect by Halleux [6.] In order to check the statical equilibrium of the beams in the vertical direction, the resulting seven nodal forces for all the members were calculated for specimens Fl and F-2. The results are presented in Table 1. The vertical forces associated with the support reactions, i.e.f, for members 1 and 8, are nearly identical with the corresponding simple beam reactions. These nodal forces may be conveniently used to compute the stress distributions in an isolated area of the castellated beam. Member
fl
1 2 3 4
0.0 3.5214 11.8365 20.4716 21.4550 14.8056 8.8056 2.6166
: 7 8
f3
f2
0.0 1.3806 2.5551 3.3311 3 -5443 3.1727 1.9417 1.0322
f4
09 6.8569 6.8572 6.8576 -5.1422 -5.1418 -5.1415 -5.1414
f6
J-5
6.8566 09001 OM)ol -119999 OXXXl1 0.0 0.0 5.1412
-3.5214 -11.8366 -20.4716 -21.4550 -14.8056 -8.8056 -2.6165 0.0
f7
-1 a3806 -6.8568 -2.5551 -6.8571 -3.3311 -6.8574 -3.5443 5.1423 -3.1726 5.1418 5.1415 -19416 -1.0322 5.1414 0.0 0.0
(a) Beam F-I Member
fl
1
f2
0.0 3.1079 10.6886 14.1635 13.8348 14.1636 10.6888 3.1080
3 4 i 7 8
f3
0.0 1.1874 1.7911 2.5215 2.9598 2.5216 1.7911 1.1874
f4
0.0 5.9993 5.9996 -0.ooo3 09001 0.0005 -59995 -5.9993
f6
fs
59990 0.0 -6.0 oQOo1 OGOOl -6.0 E391
-3.1079 -10.6887 -14.1636 -13.8349 -14.1636 -106888 -3.1079 0.0
-1.1873 -1.7911 -2.5215 -2.9598 -2.5215 -1.7911 -1.1874 0.0
(b) BeamF-2 TABLE1. Calculatednodalforcesfor member:BeamsF-l AND F-2
3%4.55)8 -1c66.7005
1715.465
Symmetrical
- 945.3812
455.6597
682.9443
711.1957
-391.0468
-594.9011
1189.8022
-1551.l419
-404.4353
-234.1855
-710.3922
- 604.4363
- %.5573
64.6U9
234.I.855
- 64.6109
- 68.0432
390.0116 -594.9011
3%4.5538 -1a6.7005 944..57%
1715.465 - 454.6235
Adjusted
Fio.
10. Stiffnessmatrixfor the typicalmember:!kries E
682.9443
f7
-59993 -59995 -0.0003 0.0 -0.0004 59995 59993 0.0
Deflection Analysis of’ Fxpan+&Open-Web
Steel Beams
335
FIG. 11. Calculated and observed deflections: Series E
The member stiffness matrix for the specimens of series E is shown in Fig. 10. The ideali~tion used was exactly similar to the oneusedinconnection withthe F-seriesspecimens, the only difference being the overall member dimensions. The computed and observed deflections for specimens El, E-2 and E-3 are presented in Fig. 1I * The agreement between experimenti and calculated values is not consistant. There is considerable difference between the computed and observed values for specimens F-3 and E-3. There is no apparent explanation for the disagreement. Further experimental verification of the proposed method of analysis is therefore planned. Adoptation of IIdealizatian 3, mentioned in Section 2, is expected to improve the fkal results in general, investigation was carried out with financial supportfrom the National Research CounciI of Canada. The anth6rs am indebted to Mr_ Akx Kozlow for his assistance in the preparation of the Sgures and to Miss. Darlene Waite who typed the manuscript.
~~~~~~~~~~r~-T~is
336
M. U. Hoeurr~. W. K. Giaffi and V. V. Nars
PI A. T. Hu~~rraar and V. K. SUNLEY.Finite dsmsnt analysis of an expanded I-section beam and an axisymmetric flanged cylinder. CorJsmcc
Racy&, A&an&
fasnccrofStrew Atutlysf. Vol. 3, No. 14 (1968).
Stress Analysis, 3-14 Joint British Con-
(31 W. K. CHlmo, M. U. Hosu~ and V. V. NEIS,Analysis of castellated steel beams by the finite element method. Proc&&gs of the Speckdty Confmnce ONFfnlte E/went Method in Civil Engineering, Montreal, Canada, 1-2 June (1972). i41 W. K. CLIBNO,M. U. HOSAINand V. V. NE& Application of finite element method to expanded openwd, I-sections. Structural Enlphcerino Report No. 4, Civil Engineering Department, University of SMkaMewa& Saskatoo~ CanaQ, September (1971). [s] M. U. H~MIN and W. 0. Spuns, Failure of castellated beams due to rupture of welded joints. AciwSruhf-&eel, No. 1 (1971). [6] P. HALLBUX, Limit analysis of castellated steel beams. Acier-St&Steel, No. 3 (1967). (Received 29 February 1972)