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Nonlinear finite difference analysis of composite beams with partial interaction
0
004s7949[90 s3.00 + 0.00 I990 Pergamon Press plc
NONLINEAR FINITE DIFFERENCE ANALYSIS OF COMPOSITE BEAMS WITH PARTIAL INTERACTION R. I. M. AL-AMIZRYand T. M. ROBERTS School of Engineering, University of Wales College of Cardiff, Newport Road, Cardiff CF2 IXH, U.K.
Abstract-A general formulation for the analysis of composite beams with partial interaction, including the influence of slip and separation at the interface between the two materials and nonlinear material and shear connector behaviour, is developed. Numerical solutions of the four basic equilibrium and compatibility equations are obtained by expressing the displacement derivatives in finite difference form and solving the resulting nonlinear algebraic equations iteratively.
NOTATION
; m P 4
r s t
u* w XJ
A E K
L
B
reinforced concrete slab, as shown in Fig. l(a). The steel beams are joined to the concrete slab by socalled ‘shear connectors’, which transfer shear and normal forces between the two components, thereby sustaining the composite action. The concrete slabs may be solid, as shown in Fig. l(b), or ribbed, as shown in Fig. l(c). Ribbed slabs are economical in the construction of buildings since the profiled steel sheeting used to form the ribs also acts as tensile reinforcement for the slab, spanning between the steel beams. For design purposes, a single steel beam is assumed to act compositely with an effective width of the concrete slab, which is limited by the influence of shear lag. Most modern Codes of Practice permit the ultimate moment of resistance of composite beams to be determined from assumed rectangular plastic stress blocks, and the required strength of the shear connectors to be determined from the horizontal force to be transmitted between sections of zero and maximum moment. Little or no account is taken explicitly of the slip at the interface between the steel and concrete, which can cause significant redistribution of strains and stresses, under both service and ultimate loading conditions. Analysis of the influence of slip in composite beams [l-5] has, in general, been based on an approach which has been attributed to Newmark et of. [I]. The equilibrium and compatibility equations for an element of the beam are reduced to a single second order differential equation in terms of either the resultant axial force in the concrete or the interface slip. Solutions for the axial force or interface slip are substituted back into the basic equilibrium and comILtibility equations which can then be solved to give displa~ments and strains throu~out the beam. This approach has been developed by Yam and Chapman [6,7] to incorporate nonlinear material and shear connector behaviour, the resulting nonlinear differential equations being solved iteratively. An alternative approach to the analysis of composite beams with partial interaction has been pre-
separation of local co-ordinate axes shear force bending moment normal force per unit length at interface shear force per unit length at interface representative displacement spacing of shear connectors axial force displacements in x and .z directions co-ordinate axes area material property relating stress to strain stiffness per unit length of shear connection length of composite beam representative load shear force per connector support reaction parameter defining load slip curve for shear connectors strain distributed load stress
Subsmpts
cu f i
materials a and b relative disnlacement of materials a and b construction strain concrete cube strength free strain interface between materials u and b
L
x=t
a, b ab c
;r P
s u X
Y
normal stiffness x=0 applied loading shear stiffness ultimate differentiation steel yield stress
Since the early 1950s composite steel and concrete beams have been used extensively in the construction of buildings and bridges. The most frequently encountered structural form consists of a number of steel I-section beams, on top of which is cast a 81
R. I. M. AL-AMERY
82 Shear
connectors
R-C slab
la) Solid
Ribbed
slob
slob
\ \ , ‘.: .;,.: “..A> :’ ‘,.*..~~;t:: ;‘:-.’ ‘,\I .,’: ‘2: ‘.I1 :. :: .;. . .; ? : *’ ?: . Steel beam
-
ICI
lb)
Fig. 1. Composite beam and slab construction. sented by Roberts [8], in which the basic equilibrium and compatibility equations are formulated in terms of the displacements of the concrete and steel. The resulting differential equations are then solved simultaneously by expressing the displacement derivatives in finite difference form. Herein, the development of this approach, to incorporate nonlinear material and shear connector behaviour, is described. The resulting nonlinear differential equations are expressed in finite difference form and solved iteratively. DISPLACEMENTS, STRAINS RESULTANTS
and T. M. ROBERT
Stresses 0 can now be related to strains via the material properties E, and Eb, which for linear elastic material behaviour are constants. However, for nonlinear elastic and elasto-plastic material behaviour, E, and Eb are functions of strain. The free strains due to shrinkage, temperature, etc. are denoted by cf, while the strains induced during the construction sequence are denoted by cc. Hence, if u and w are assumed to exclude the displacements corresponding to ce, the stresses in materials a and b are given by Q, = E,(u,, - =, w,s, + c,, - cfi )
(3)
Ob =
(4)
&&b.r
An element of a composite beam, length 6x, is shown in Fig. 2. The beam is made from two materials, a and b, joined by a medium of negligible thickness but which has finite normal and tangential stiffnesses. The two materials are subjected to moments m, shear forcesfand axial forces t, while q and p denote the shear and normal forces per unit length at the interface. Assuming that plane sections within each material remain plane, the axial strains c can be expressed in terms of displacements u and w relative to the local x and z axes, which are assumed to pass through the centroids of the two materials. Hence
=b wb,xx
+
ccb -
efb).
The axial forces t and moments m can now be obtained by integrating the stresses over the crosssection areas of materials a and b, denoted by A, and Ab. Hence t,=
a,dA,;
tb =
s
mo=-
AND STRESS
-
a,dA,
(5)
s
s
O,Z,dA,;
mb=
s
abzbdAb.
_
EQUILIBRIUM AND COMPATIBILITY EQUATIONS
Since strains have been defined in terms of four independent displacement variables, four independent equations are required to obtain a solution. These four equations can be obtained by considering the equilibrium of an element of the composite beam and the compatibility at the interface between the two materials. For equilibrium of the element shown in Fig. 2, in the x-direction t,, + tb,x= 0.
t, = uP.X- z, W&XX ub.x
-
=b wb,x.r 9
in the z-direction
(2)
in which subscripts a and b denote the two materials, subscripts x denote differentiation and z is the distance from the origin of the co-ordinates.
f,.,
+.fb,,
=
P
=
Pa +
Pb +
P,J 1
(8)
in which p is the total distributed load per unit length (superimposed load p,, plus dead loads p, and Pb) since the strains induced during construction have been included in eqns (3) and (4). Loads due to removal of props used during construction should be considered as live loads, since their influence is not included in the evaluation of E,. Taking moments about the origin of co-ordinates in material a gives %., +
Fig. 2. Element of a composite beam.
(7)
(1) For equilibrium
Eb =
(6)
mb.x
=f,
+fb
+
tbJer
(9)
in which e is the separation of the coordinate axes in materials a and b. Combining eqns (8) and (9) gives
Analysis of composite beams with partial interaction
m
(I,xx
+
mb,x.r tb.xxe -
=
p.
(10)
The Slip, a.,$, at the interface between the two materials is defined as the relative displacement in the x-direction of initially adjacent particles. If ziOand z,~ denote the z co-ordinates of the interface in the two materials (zib being negative), ~4,~is given by hb
=
(%a -
zia wo,x)
-
(ub
-
zib wb,x ).
83
CONCRETE
Units
(11)
:
Newtons
If the shear stiffness of the joint per unit length is denoted by K,, then
/Can
yressio;
STEEL
q = KS urr6= t,, .
and millimetres
(12) 200 000
Hence, from eqns (11) and (12) t0.x-K,{(U,-Zi~W,,)-(Ub-Z,Wb.,)}=O.
(13)
The separation wbaat the interface between materials a and b is the relative displacement in the z-direction, i.e. w,,=w,-ww,.
(14)
If the normal stiffness of the joint per unit length is denoted by K,, then p=K,w,=K,(w,-w,).
For equilibrium z-direction
(15)
of an element of material a in the
f,,=P,+Pa+P.
For moment equilibrium about the interface
of an element of material u
ma.x+ ta.xzio Hence, combining
(16)
=_A,
.
(17)
eqns (16), (17) and (14)
mnsx+to.xrZio-Kn(Wb-Wo)=Ps+Po.
(18)
Equations (7), (lo), (13) and (18) are the four equilibrium and compatibility equations required for a complete solution, which can be expressed in terms of displacement derivatives, after substitution from eqns
(3x6).
MATERIAL
PROPERTIES
Integration of eqns (5) and (6) to determine the axial forces and moments in materials a and b, requires specification of the material properties E, and Eb in eqns (3) and (4). Solution of the four equilibrium and compatibility equations also requires specification of the shear and normal stiffnesses of the connectors, KS and K,, . In general, E,, Eb, KS and K,
Fig. 3. Stress-strain
curves for concrete and steel.
are all functions of strain or displacements and solutions have to be obtained iteratively. If materials a and b represent concrete and steel, respectively, the assumed uniaxial stress-strain curves, neglecting the influence of coexistent shear stress, are as shown in Fig. 3 [9], in which the appropriate units are Newtons and millimetres. The concrete is assumed to have no tensile strength and the ultimate compressive strain is limited to 0.0035. The curved portion of the stress-strain curve (CTvs 6) is defined by the equation d =5500&c
- 11.3 x 10662,
09)
in which 6, is the concrete cube strength. For simplicity, a bilinear stress-strain curve was assumed for the steel, with equal yield stresses, uY, in tension and compression. The assumed shear force Q vs slip u,,, curve for the shear connectors is as shown in Fig. 4, and can be represented by eqn [7].
Q = Quil -
exp(-au,)j9
in which Q. is the ultimate shear strength of a connector and a is a constant which can be determined from test results.
Shear force P Qu --------_-----______ Q
-__
k cab
Slip
uab
Fig. 4. Shear force vs slip curve for shear connectors.
R. I. M. AL-AMERY and T. M. ROBERTS
84
If, for example, the slip at load Q is equal to &, then from eqn (20)
ci (Eb)ji"b,x - czb)jwb,xx
tb =
(25)
(21)
m, = - 1 (Ea)j {ua,x - (za)j Wa,xx j
The secant values of the material properties E, used in the iterative analysis, are defined by the equation
Ed
tEco -Ef~)jl
(za6Ao)j
(26)
+
(%b -
(zb6Ab)j.
(27)
E’
in which Q is the stress on the assumed stress-strain curve corresponding to strain 6. Similarly, the secant value of the shear stiffness per unit length of the connection K,, is defined as KS
+
=e %b’
(23)
in which Q is the shear force on the assumed shear force vs slip curve, corresponding to slip uobr and s is the spacing of the connectors. The normal stiffness per unit length of the connectors K,, can be defined in a similar manner to KS. However, for most composite construction, separation of the two materials is negligible since K, is relatively large. Therefore, for simplicity, a relatively large constant value of K,, is assumed. NUMERICALINTEGRATIONOF FORCE-DISPLACEMENT EQUATIONS When the material properties E are constants, eqns (5) and (6) can be integrated analytically to give the axial forces t and moments m in terms of displacements u and w. Alternatively, if the material properties are nonlinear functions of strain, eqns (5) and (6) can be evaluated numerically. This can be achieved by dividing the cross-section area of each material into a number of elemental strips having area (6A)j at distance (z), from the origin of co-ordinates, as shown in Fig. 5, and replacing the integrals by a summation over the appropriate area. Hence, from eqns (3H6),
The appropriate values of(E), for the strips (aA), are the secant values determined from the assumed stress-strain curves, as discussed in the previous section, corresponding to the total strains (L), in strips (&4,). Therefore, for material a, (E,), corresponds to a strain (E,,)~,which is given by
FINITE DIFFERENCEANALYSIS Substituting eqns (24H27) into eqns (7), (lo), (13) and (18) provides a set of four simultaneous differential equations in terms of the displacements u,, w,, ub and wb . These equations can be solved by expressing the displacement derivatives in finite difference form and solving the resulting set of algebraic equations. The governing differential equations contain derivatives of third order in u and fourth order in w, which can be expressed in central difference form using five node points as shown in Fig. 6(a). For node n, the derivatives of w, for example, can be expressed as [lo] W WxC
-W,-1
“+I
(cCcl - Ej2),}CaAo)j
(24)
(29)
2Ax W n+,-2W,-tW,-I Wxx =
(30)
Ax2 +2w,_,-W”_2 2Ax’
W n+2-2Wn+I
W.XXX =
dX
+
cfb)j}
n-2
_
bx
n-1
Ax "
._____;o______. EA"Ol nodes
Boundary node
4x n*1
(31)
_ n*2
\ I"tA nodes
Ia)
lb)
Id
& Ix.01
Fig. 5. Subdivision of cross-section into elemental areas.
2
Fig. 6. (a) and (b) Finite difference nodes. (c) Composite beam-support reactions.
Analysis of composite beams with partial interaction &+*-4w”+,
w,X.LY =
+6w,-4w,_,+w,_,
, (32)
Ax4
in which w, denotes the value of w at node n, etc., and Ax is the spacing of the nodes. Solution of the resulting set of algebraic equations requires the specification of six natural boundary conditions at each boundary, i.e. two each for w, and wb and one each for u, and at,. In general, therefore, two external nodes, as shown in Fig. 6(b), are required to specify the boundary conditions, which may involve derivatives of up to third order in w and second order in u. However, if each external node is assigned four degrees of freedom (u,, w,, ub and wg), to be consistent with the internal nodes, eight boundary conditions are required to obtain a solution. The two additional boundary conditions can be provided by setting the fourth derivatives of u, and u, equal to zero, which does not influence the solution, since fourth derivatives of u do not appear in the governing differential equations. To illustrate the foregoing discussion, the boundary conditions for a simply supported, composite beam of length L, shown in Fig. 6(c), are Atx =0
85
r Idisplacement I Fig. 7. Graphical illustration of iterative procedure.
For the first stage of the solution, the material properties are assumed constant and a set of nodal displacements corresponding to a specified applied loading is determined. From these displacements, slip at the interface and strains throughout the composite beam are determined, which are used to define the secant values of the material properties for the second stage of the solution. The process is repeated until the calculated displacements have converged, according to a prescribed criterion. For subsequent values of the applied loading, the iterative procedure is commenced with secant values of the material properties corresponding to the previously converged solution, which reduces the number of iterations required.
Atx=L ILLUSTRATIVE
% =o
u0.X=o
EXAMPLE
(33)
To illustrate the application of the theory presented herein, a study was made of the behaviour of a 9-m span, simply supported, composite steel and concrete u, = 0 (35) beam, having the cross-section dimensions shown in ub,x --0 Fig. 8. For simplicity, the beam was assumed to be =o ub,,x ub,xxxx =o (36) supported during construction, so that the construction strains E, were zero, and subjected to a uniformly W a.xx =o (37) distributed load. The free strains c, were also assumed WWX=o zero. w,=o w,=o (38) The cube strength of the concrete c,, and yield stress of the steel uYwere taken as 30 and 280 N/mm*, W b.xx --0 W b,xx --0 (39) respectively. The connection between the concrete slab and steel beam was assumed to be produced by pairs of 19-mm diameter, lOO-mm long, headed studs, (40) f,+fb=Rl f,+fb=%. with a spacing s of 240mm. The ultimate shear Equations (40) specify that the sum of the shear strength of a single stud, QU, was taken as 100 kN, forces in materials a and b are equal to the support while the slip tiab corresponding to a shear force reactions R. and RL. 0 = 62 kN (see Fig. 4) was taken as 0.5 mm. This Considering the moment equilibrium of elements gave an initial tangent value of KS= 1.61 kN/mm*. of materials a and b (see Fig. 2) gives The normal stiffness of the connection K, was assumed constant and equal to lo6 kN/mm*. For numerical integration of eqns (24)-(27), the f, = mo,~+ zi* tw (41) concrete slab was divided into 10 equal strips. Each fb = mb.x + zib tb.x ’ (42) %X*X= 0
u(IJXXX = 0
(34)
Substituting from eqns (3)-(6) or from eqns (24)-(27),
f,and fb, and hence eqns (40), can be expressed in terms of displacement derivatives. After introducing the boundary conditions, the nonlinear algebraic equations can be solved iteratively, as illustrated in Fig. 7, in which P and r denote a representative load and displacement respectively.
1800 ;,‘T
I%
Fig. 8. Composite beam-cross-section dimensions.
R. I. M. AL-AMERY and T. M. ROBERTS
86
p =GO
loo_ l/L
(Nhml,
cIj
span
l/L
span
mid span
O~Ocil76 jgyjY
p’
r
0.0021
0.0026
0.0059
0.0063
I; I::-;f{
100 200 Centraldeflection (mml 100
p =70N/mm
N/mm mid
0.0006
pUz76 N/mm ___o__---Q __o_____------
P
span
r pU=7GN/mm
P INlllWll
,rO__ ;
. %
260
_________--------0
IStud
2.80
260
260
Fig. 10. Strain and stress profiles.
failure
50-f (bl f I .I I I S
10 Maximum
slip
(mm]
Fig. 9. Applied load vs central deflection and maximum slip. flange of the steel beam was divided into four equal strips and the web of the steel beam was divided into 10 equal strips. The following results were obtained using 25 nodes along the length of the beam, including the four external nodes required to specify the boundary conditions. Solutions were considered to have converged when the change in the maximum slip at the ends of the beam was less than 0.02mm. The variation of the central deflection wrnaxwith the applied uniformly distributed load p is shown graphically in Fig. 9(a). The numerical results appear to be converging to a value py = 76 N/mm, which is the uniformly distributed load corresponding to the ultimate flexural strength of the composite beam, based on assumed rectangular plastic stress blocks. The variation of the maximum slip at the ends of the beam with the applied load is shown in Fig. 9(b). If failure of the shear connectors is assumed to occur at a slip of 4 mm, the maximum load that the beam can sustain is approx. 0.9 pU.
Strain and stress profiles throughout the depth of the composite beam, corresponding to applied uniformly distributed loads of 60 and 70 N/mm, are shown in Fig. 10. The discontinuous strain profiles indicate the existence of slip at the interface between the concrete and steel, while the stress profiles indicate the spread of plasticity. To assess the convergence of the nonlinear (iterative) finite difference solution, results were obtained for different numbers of nodes along the beam (including the four external nodes required to specify the boundary conditions) and different convergence limits on the maximum slip at the ends of the beam. These results are summarised in Table 1. Assuming the results for 45 nodes and a convergence limit &Sm = 10e6 mm to be correct, the results obtained using 15 nodes and a convergence limit 6Sm = 0.02mm differ by less than 5% for applied loads up to 60 N/mm, i.e. 80% of the ultimate load. It is also apparent that the number of iterations required for convergence increases very significantly as the ultimate load is approached, due to the large changes taking place in the secant material properties. DISCUSSION AND CONCLUSIONS
A general formulation for the analysis of composite beams with partial interaction, which incorpo-
Table 1. Convereence of iterative solution No. nodes
45
25
15
6Sm
1O-6
0.02
0.02
Load (N/mm)
It
Sm
Wm
It
Sm
Wm
It
Sm
Wm
40 50 60 70
11 20 41 37
0.447 0.657 1.347 11.027
30.2 39.7 59.4 172.7
3 4 9 43
0.440 0.628 1.288 11.953
30.3 39.4 59.2 191.7
3 5 10 15
0.432 0.632 1.317 11.366
30.8 40.7 62.0 195.6
Wm:
central deflection
It: number of iterations; Sm: maximum slip at support (mm); (mm); 6Sm: convergence limit-change in Sm per cycle (mm).
Analysis of composite beams with partial interaction rates the influence of slip and separation at the interface between the two materials and nonlinear material and shear connector behaviour, has been developed. Solutions of the four basic equilibrium and compatibility equations can be obtained by expressing the displacement derivatives in finite difference form and solving the resulting nonlinear algebraic equations iteratively. A convergence study indicates that reasonably accurate results can be obtained, for the entire range of loading up to failure, with only 15 nodes for a single span (60 degrees of freedom), including the
four external nodes required to specify the boundary conditions. The number of iterations required for convergence increases rapidly as the ultimate load is approached, due to the large strains and corresponding changes in the material properties. Accelerated convergence techniques are desirable, therefore, in order to increase the efficiency of the solution. REFERENCES 1. N. M. Newmark, C. P. Siess and I. M. Vies& Tests
and analysis of composite
beams with incomplete
87
interaction. Proc. Sot. exp. Stress Anal. 9, 75-92 (1951). 2. R. P. Johnson, Composite Structures of Steel and Concrete. Crosby-Lockwood-Staples, London (1975). 3. L. C. P. Yam, Design of Composite Steel-Concrete Structures. Surrey University Press, London (1981). 4. R. P. Johnson and I. M. May, Partial interaction design of composite beams. Struct. Engr 53, 305-311 (1975). 5. R. P. Johnson, Loss of interaction in short span composite beams and plates. J. Construct. Steel Res. 1, 11-16 (1981). 6. L. C. P. Yam and J. C. Chapman, The inelastic behaviour of simply supported composite beams of steel and concrete. Proc. Inst. ciu. Engrs 41, 651-683 (1968). 7. L. C. P. Yam and J. C. Chapman, The inelastic behaviour of continuous composite beams of steel and concrete. Proc. Inst. civ. Enars 53. 487-501 (1972). 8. T. M. Roberts, Finite difference analysis of composite beams with partial interaction. Comput. Struct. 21, 469473 (1985). 9. BS5400: Part 5: Code of Practice for Design of Composite Bridges. British Standards Institution, U.K. (1979). 10. F. B. Hildebrand, Finite-Difference Equations and Simulations. Prentice-Hall, Englewood Cliffs, NJ (1968).
004s7949[90 s3.00 + 0.00 I990 Pergamon Press plc
NONLINEAR FINITE DIFFERENCE ANALYSIS OF COMPOSITE BEAMS WITH PARTIAL INTERACTION R. I. M. AL-AMIZRYand T. M. ROBERTS School of Engineering, University of Wales College of Cardiff, Newport Road, Cardiff CF2 IXH, U.K.
Abstract-A general formulation for the analysis of composite beams with partial interaction, including the influence of slip and separation at the interface between the two materials and nonlinear material and shear connector behaviour, is developed. Numerical solutions of the four basic equilibrium and compatibility equations are obtained by expressing the displacement derivatives in finite difference form and solving the resulting nonlinear algebraic equations iteratively.
NOTATION
; m P 4
r s t
u* w XJ
A E K
L
B
reinforced concrete slab, as shown in Fig. l(a). The steel beams are joined to the concrete slab by socalled ‘shear connectors’, which transfer shear and normal forces between the two components, thereby sustaining the composite action. The concrete slabs may be solid, as shown in Fig. l(b), or ribbed, as shown in Fig. l(c). Ribbed slabs are economical in the construction of buildings since the profiled steel sheeting used to form the ribs also acts as tensile reinforcement for the slab, spanning between the steel beams. For design purposes, a single steel beam is assumed to act compositely with an effective width of the concrete slab, which is limited by the influence of shear lag. Most modern Codes of Practice permit the ultimate moment of resistance of composite beams to be determined from assumed rectangular plastic stress blocks, and the required strength of the shear connectors to be determined from the horizontal force to be transmitted between sections of zero and maximum moment. Little or no account is taken explicitly of the slip at the interface between the steel and concrete, which can cause significant redistribution of strains and stresses, under both service and ultimate loading conditions. Analysis of the influence of slip in composite beams [l-5] has, in general, been based on an approach which has been attributed to Newmark et of. [I]. The equilibrium and compatibility equations for an element of the beam are reduced to a single second order differential equation in terms of either the resultant axial force in the concrete or the interface slip. Solutions for the axial force or interface slip are substituted back into the basic equilibrium and comILtibility equations which can then be solved to give displa~ments and strains throu~out the beam. This approach has been developed by Yam and Chapman [6,7] to incorporate nonlinear material and shear connector behaviour, the resulting nonlinear differential equations being solved iteratively. An alternative approach to the analysis of composite beams with partial interaction has been pre-
separation of local co-ordinate axes shear force bending moment normal force per unit length at interface shear force per unit length at interface representative displacement spacing of shear connectors axial force displacements in x and .z directions co-ordinate axes area material property relating stress to strain stiffness per unit length of shear connection length of composite beam representative load shear force per connector support reaction parameter defining load slip curve for shear connectors strain distributed load stress
Subsmpts
cu f i
materials a and b relative disnlacement of materials a and b construction strain concrete cube strength free strain interface between materials u and b
L
x=t
a, b ab c
;r P
s u X
Y
normal stiffness x=0 applied loading shear stiffness ultimate differentiation steel yield stress
Since the early 1950s composite steel and concrete beams have been used extensively in the construction of buildings and bridges. The most frequently encountered structural form consists of a number of steel I-section beams, on top of which is cast a 81
R. I. M. AL-AMERY
82 Shear
connectors
R-C slab
la) Solid
Ribbed
slob
slob
\ \ , ‘.: .;,.: “..A> :’ ‘,.*..~~;t:: ;‘:-.’ ‘,\I .,’: ‘2: ‘.I1 :. :: .;. . .; ? : *’ ?: . Steel beam
-
ICI
lb)
Fig. 1. Composite beam and slab construction. sented by Roberts [8], in which the basic equilibrium and compatibility equations are formulated in terms of the displacements of the concrete and steel. The resulting differential equations are then solved simultaneously by expressing the displacement derivatives in finite difference form. Herein, the development of this approach, to incorporate nonlinear material and shear connector behaviour, is described. The resulting nonlinear differential equations are expressed in finite difference form and solved iteratively. DISPLACEMENTS, STRAINS RESULTANTS
and T. M. ROBERT
Stresses 0 can now be related to strains via the material properties E, and Eb, which for linear elastic material behaviour are constants. However, for nonlinear elastic and elasto-plastic material behaviour, E, and Eb are functions of strain. The free strains due to shrinkage, temperature, etc. are denoted by cf, while the strains induced during the construction sequence are denoted by cc. Hence, if u and w are assumed to exclude the displacements corresponding to ce, the stresses in materials a and b are given by Q, = E,(u,, - =, w,s, + c,, - cfi )
(3)
Ob =
(4)
&&b.r
An element of a composite beam, length 6x, is shown in Fig. 2. The beam is made from two materials, a and b, joined by a medium of negligible thickness but which has finite normal and tangential stiffnesses. The two materials are subjected to moments m, shear forcesfand axial forces t, while q and p denote the shear and normal forces per unit length at the interface. Assuming that plane sections within each material remain plane, the axial strains c can be expressed in terms of displacements u and w relative to the local x and z axes, which are assumed to pass through the centroids of the two materials. Hence
=b wb,xx
+
ccb -
efb).
The axial forces t and moments m can now be obtained by integrating the stresses over the crosssection areas of materials a and b, denoted by A, and Ab. Hence t,=
a,dA,;
tb =
s
mo=-
AND STRESS
-
a,dA,
(5)
s
s
O,Z,dA,;
mb=
s
abzbdAb.
_
EQUILIBRIUM AND COMPATIBILITY EQUATIONS
Since strains have been defined in terms of four independent displacement variables, four independent equations are required to obtain a solution. These four equations can be obtained by considering the equilibrium of an element of the composite beam and the compatibility at the interface between the two materials. For equilibrium of the element shown in Fig. 2, in the x-direction t,, + tb,x= 0.
t, = uP.X- z, W&XX ub.x
-
=b wb,x.r 9
in the z-direction
(2)
in which subscripts a and b denote the two materials, subscripts x denote differentiation and z is the distance from the origin of the co-ordinates.
f,.,
+.fb,,
=
P
=
Pa +
Pb +
P,J 1
(8)
in which p is the total distributed load per unit length (superimposed load p,, plus dead loads p, and Pb) since the strains induced during construction have been included in eqns (3) and (4). Loads due to removal of props used during construction should be considered as live loads, since their influence is not included in the evaluation of E,. Taking moments about the origin of co-ordinates in material a gives %., +
Fig. 2. Element of a composite beam.
(7)
(1) For equilibrium
Eb =
(6)
mb.x
=f,
+fb
+
tbJer
(9)
in which e is the separation of the coordinate axes in materials a and b. Combining eqns (8) and (9) gives
Analysis of composite beams with partial interaction
m
(I,xx
+
mb,x.r tb.xxe -
=
p.
(10)
The Slip, a.,$, at the interface between the two materials is defined as the relative displacement in the x-direction of initially adjacent particles. If ziOand z,~ denote the z co-ordinates of the interface in the two materials (zib being negative), ~4,~is given by hb
=
(%a -
zia wo,x)
-
(ub
-
zib wb,x ).
83
CONCRETE
Units
(11)
:
Newtons
If the shear stiffness of the joint per unit length is denoted by K,, then
/Can
yressio;
STEEL
q = KS urr6= t,, .
and millimetres
(12) 200 000
Hence, from eqns (11) and (12) t0.x-K,{(U,-Zi~W,,)-(Ub-Z,Wb.,)}=O.
(13)
The separation wbaat the interface between materials a and b is the relative displacement in the z-direction, i.e. w,,=w,-ww,.
(14)
If the normal stiffness of the joint per unit length is denoted by K,, then p=K,w,=K,(w,-w,).
For equilibrium z-direction
(15)
of an element of material a in the
f,,=P,+Pa+P.
For moment equilibrium about the interface
of an element of material u
ma.x+ ta.xzio Hence, combining
(16)
=_A,
.
(17)
eqns (16), (17) and (14)
mnsx+to.xrZio-Kn(Wb-Wo)=Ps+Po.
(18)
Equations (7), (lo), (13) and (18) are the four equilibrium and compatibility equations required for a complete solution, which can be expressed in terms of displacement derivatives, after substitution from eqns
(3x6).
MATERIAL
PROPERTIES
Integration of eqns (5) and (6) to determine the axial forces and moments in materials a and b, requires specification of the material properties E, and Eb in eqns (3) and (4). Solution of the four equilibrium and compatibility equations also requires specification of the shear and normal stiffnesses of the connectors, KS and K,, . In general, E,, Eb, KS and K,
Fig. 3. Stress-strain
curves for concrete and steel.
are all functions of strain or displacements and solutions have to be obtained iteratively. If materials a and b represent concrete and steel, respectively, the assumed uniaxial stress-strain curves, neglecting the influence of coexistent shear stress, are as shown in Fig. 3 [9], in which the appropriate units are Newtons and millimetres. The concrete is assumed to have no tensile strength and the ultimate compressive strain is limited to 0.0035. The curved portion of the stress-strain curve (CTvs 6) is defined by the equation d =5500&c
- 11.3 x 10662,
09)
in which 6, is the concrete cube strength. For simplicity, a bilinear stress-strain curve was assumed for the steel, with equal yield stresses, uY, in tension and compression. The assumed shear force Q vs slip u,,, curve for the shear connectors is as shown in Fig. 4, and can be represented by eqn [7].
Q = Quil -
exp(-au,)j9
in which Q. is the ultimate shear strength of a connector and a is a constant which can be determined from test results.
Shear force P Qu --------_-----______ Q
-__
k cab
Slip
uab
Fig. 4. Shear force vs slip curve for shear connectors.
R. I. M. AL-AMERY and T. M. ROBERTS
84
If, for example, the slip at load Q is equal to &, then from eqn (20)
ci (Eb)ji"b,x - czb)jwb,xx
tb =
(25)
(21)
m, = - 1 (Ea)j {ua,x - (za)j Wa,xx j
The secant values of the material properties E, used in the iterative analysis, are defined by the equation
Ed
tEco -Ef~)jl
(za6Ao)j
(26)
+
(%b -
(zb6Ab)j.
(27)
E’
in which Q is the stress on the assumed stress-strain curve corresponding to strain 6. Similarly, the secant value of the shear stiffness per unit length of the connection K,, is defined as KS
+
=e %b’
(23)
in which Q is the shear force on the assumed shear force vs slip curve, corresponding to slip uobr and s is the spacing of the connectors. The normal stiffness per unit length of the connectors K,, can be defined in a similar manner to KS. However, for most composite construction, separation of the two materials is negligible since K, is relatively large. Therefore, for simplicity, a relatively large constant value of K,, is assumed. NUMERICALINTEGRATIONOF FORCE-DISPLACEMENT EQUATIONS When the material properties E are constants, eqns (5) and (6) can be integrated analytically to give the axial forces t and moments m in terms of displacements u and w. Alternatively, if the material properties are nonlinear functions of strain, eqns (5) and (6) can be evaluated numerically. This can be achieved by dividing the cross-section area of each material into a number of elemental strips having area (6A)j at distance (z), from the origin of co-ordinates, as shown in Fig. 5, and replacing the integrals by a summation over the appropriate area. Hence, from eqns (3H6),
The appropriate values of(E), for the strips (aA), are the secant values determined from the assumed stress-strain curves, as discussed in the previous section, corresponding to the total strains (L), in strips (&4,). Therefore, for material a, (E,), corresponds to a strain (E,,)~,which is given by
FINITE DIFFERENCEANALYSIS Substituting eqns (24H27) into eqns (7), (lo), (13) and (18) provides a set of four simultaneous differential equations in terms of the displacements u,, w,, ub and wb . These equations can be solved by expressing the displacement derivatives in finite difference form and solving the resulting set of algebraic equations. The governing differential equations contain derivatives of third order in u and fourth order in w, which can be expressed in central difference form using five node points as shown in Fig. 6(a). For node n, the derivatives of w, for example, can be expressed as [lo] W WxC
-W,-1
“+I
(cCcl - Ej2),}CaAo)j
(24)
(29)
2Ax W n+,-2W,-tW,-I Wxx =
(30)
Ax2 +2w,_,-W”_2 2Ax’
W n+2-2Wn+I
W.XXX =
dX
+
cfb)j}
n-2
_
bx
n-1
Ax "
._____;o______. EA"Ol nodes
Boundary node
4x n*1
(31)
_ n*2
\ I"tA nodes
Ia)
lb)
Id
& Ix.01
Fig. 5. Subdivision of cross-section into elemental areas.
2
Fig. 6. (a) and (b) Finite difference nodes. (c) Composite beam-support reactions.
Analysis of composite beams with partial interaction &+*-4w”+,
w,X.LY =
+6w,-4w,_,+w,_,
, (32)
Ax4
in which w, denotes the value of w at node n, etc., and Ax is the spacing of the nodes. Solution of the resulting set of algebraic equations requires the specification of six natural boundary conditions at each boundary, i.e. two each for w, and wb and one each for u, and at,. In general, therefore, two external nodes, as shown in Fig. 6(b), are required to specify the boundary conditions, which may involve derivatives of up to third order in w and second order in u. However, if each external node is assigned four degrees of freedom (u,, w,, ub and wg), to be consistent with the internal nodes, eight boundary conditions are required to obtain a solution. The two additional boundary conditions can be provided by setting the fourth derivatives of u, and u, equal to zero, which does not influence the solution, since fourth derivatives of u do not appear in the governing differential equations. To illustrate the foregoing discussion, the boundary conditions for a simply supported, composite beam of length L, shown in Fig. 6(c), are Atx =0
85
r Idisplacement I Fig. 7. Graphical illustration of iterative procedure.
For the first stage of the solution, the material properties are assumed constant and a set of nodal displacements corresponding to a specified applied loading is determined. From these displacements, slip at the interface and strains throughout the composite beam are determined, which are used to define the secant values of the material properties for the second stage of the solution. The process is repeated until the calculated displacements have converged, according to a prescribed criterion. For subsequent values of the applied loading, the iterative procedure is commenced with secant values of the material properties corresponding to the previously converged solution, which reduces the number of iterations required.
Atx=L ILLUSTRATIVE
% =o
u0.X=o
EXAMPLE
(33)
To illustrate the application of the theory presented herein, a study was made of the behaviour of a 9-m span, simply supported, composite steel and concrete u, = 0 (35) beam, having the cross-section dimensions shown in ub,x --0 Fig. 8. For simplicity, the beam was assumed to be =o ub,,x ub,xxxx =o (36) supported during construction, so that the construction strains E, were zero, and subjected to a uniformly W a.xx =o (37) distributed load. The free strains c, were also assumed WWX=o zero. w,=o w,=o (38) The cube strength of the concrete c,, and yield stress of the steel uYwere taken as 30 and 280 N/mm*, W b.xx --0 W b,xx --0 (39) respectively. The connection between the concrete slab and steel beam was assumed to be produced by pairs of 19-mm diameter, lOO-mm long, headed studs, (40) f,+fb=Rl f,+fb=%. with a spacing s of 240mm. The ultimate shear Equations (40) specify that the sum of the shear strength of a single stud, QU, was taken as 100 kN, forces in materials a and b are equal to the support while the slip tiab corresponding to a shear force reactions R. and RL. 0 = 62 kN (see Fig. 4) was taken as 0.5 mm. This Considering the moment equilibrium of elements gave an initial tangent value of KS= 1.61 kN/mm*. of materials a and b (see Fig. 2) gives The normal stiffness of the connection K, was assumed constant and equal to lo6 kN/mm*. For numerical integration of eqns (24)-(27), the f, = mo,~+ zi* tw (41) concrete slab was divided into 10 equal strips. Each fb = mb.x + zib tb.x ’ (42) %X*X= 0
u(IJXXX = 0
(34)
Substituting from eqns (3)-(6) or from eqns (24)-(27),
f,and fb, and hence eqns (40), can be expressed in terms of displacement derivatives. After introducing the boundary conditions, the nonlinear algebraic equations can be solved iteratively, as illustrated in Fig. 7, in which P and r denote a representative load and displacement respectively.
1800 ;,‘T
I%
Fig. 8. Composite beam-cross-section dimensions.
R. I. M. AL-AMERY and T. M. ROBERTS
86
p =GO
loo_ l/L
(Nhml,
cIj
span
l/L
span
mid span
O~Ocil76 jgyjY
p’
r
0.0021
0.0026
0.0059
0.0063
I; I::-;f{
100 200 Centraldeflection (mml 100
p =70N/mm
N/mm mid
0.0006
pUz76 N/mm ___o__---Q __o_____------
P
span
r pU=7GN/mm
P INlllWll
,rO__ ;
. %
260
_________--------0
IStud
2.80
260
260
Fig. 10. Strain and stress profiles.
failure
50-f (bl f I .I I I S
10 Maximum
slip
(mm]
Fig. 9. Applied load vs central deflection and maximum slip. flange of the steel beam was divided into four equal strips and the web of the steel beam was divided into 10 equal strips. The following results were obtained using 25 nodes along the length of the beam, including the four external nodes required to specify the boundary conditions. Solutions were considered to have converged when the change in the maximum slip at the ends of the beam was less than 0.02mm. The variation of the central deflection wrnaxwith the applied uniformly distributed load p is shown graphically in Fig. 9(a). The numerical results appear to be converging to a value py = 76 N/mm, which is the uniformly distributed load corresponding to the ultimate flexural strength of the composite beam, based on assumed rectangular plastic stress blocks. The variation of the maximum slip at the ends of the beam with the applied load is shown in Fig. 9(b). If failure of the shear connectors is assumed to occur at a slip of 4 mm, the maximum load that the beam can sustain is approx. 0.9 pU.
Strain and stress profiles throughout the depth of the composite beam, corresponding to applied uniformly distributed loads of 60 and 70 N/mm, are shown in Fig. 10. The discontinuous strain profiles indicate the existence of slip at the interface between the concrete and steel, while the stress profiles indicate the spread of plasticity. To assess the convergence of the nonlinear (iterative) finite difference solution, results were obtained for different numbers of nodes along the beam (including the four external nodes required to specify the boundary conditions) and different convergence limits on the maximum slip at the ends of the beam. These results are summarised in Table 1. Assuming the results for 45 nodes and a convergence limit &Sm = 10e6 mm to be correct, the results obtained using 15 nodes and a convergence limit 6Sm = 0.02mm differ by less than 5% for applied loads up to 60 N/mm, i.e. 80% of the ultimate load. It is also apparent that the number of iterations required for convergence increases very significantly as the ultimate load is approached, due to the large changes taking place in the secant material properties. DISCUSSION AND CONCLUSIONS
A general formulation for the analysis of composite beams with partial interaction, which incorpo-
Table 1. Convereence of iterative solution No. nodes
45
25
15
6Sm
1O-6
0.02
0.02
Load (N/mm)
It
Sm
Wm
It
Sm
Wm
It
Sm
Wm
40 50 60 70
11 20 41 37
0.447 0.657 1.347 11.027
30.2 39.7 59.4 172.7
3 4 9 43
0.440 0.628 1.288 11.953
30.3 39.4 59.2 191.7
3 5 10 15
0.432 0.632 1.317 11.366
30.8 40.7 62.0 195.6
Wm:
central deflection
It: number of iterations; Sm: maximum slip at support (mm); (mm); 6Sm: convergence limit-change in Sm per cycle (mm).
Analysis of composite beams with partial interaction rates the influence of slip and separation at the interface between the two materials and nonlinear material and shear connector behaviour, has been developed. Solutions of the four basic equilibrium and compatibility equations can be obtained by expressing the displacement derivatives in finite difference form and solving the resulting nonlinear algebraic equations iteratively. A convergence study indicates that reasonably accurate results can be obtained, for the entire range of loading up to failure, with only 15 nodes for a single span (60 degrees of freedom), including the
four external nodes required to specify the boundary conditions. The number of iterations required for convergence increases rapidly as the ultimate load is approached, due to the large strains and corresponding changes in the material properties. Accelerated convergence techniques are desirable, therefore, in order to increase the efficiency of the solution. REFERENCES 1. N. M. Newmark, C. P. Siess and I. M. Vies& Tests
and analysis of composite
beams with incomplete
87
interaction. Proc. Sot. exp. Stress Anal. 9, 75-92 (1951). 2. R. P. Johnson, Composite Structures of Steel and Concrete. Crosby-Lockwood-Staples, London (1975). 3. L. C. P. Yam, Design of Composite Steel-Concrete Structures. Surrey University Press, London (1981). 4. R. P. Johnson and I. M. May, Partial interaction design of composite beams. Struct. Engr 53, 305-311 (1975). 5. R. P. Johnson, Loss of interaction in short span composite beams and plates. J. Construct. Steel Res. 1, 11-16 (1981). 6. L. C. P. Yam and J. C. Chapman, The inelastic behaviour of simply supported composite beams of steel and concrete. Proc. Inst. ciu. Engrs 41, 651-683 (1968). 7. L. C. P. Yam and J. C. Chapman, The inelastic behaviour of continuous composite beams of steel and concrete. Proc. Inst. civ. Enars 53. 487-501 (1972). 8. T. M. Roberts, Finite difference analysis of composite beams with partial interaction. Comput. Struct. 21, 469473 (1985). 9. BS5400: Part 5: Code of Practice for Design of Composite Bridges. British Standards Institution, U.K. (1979). 10. F. B. Hildebrand, Finite-Difference Equations and Simulations. Prentice-Hall, Englewood Cliffs, NJ (1968).