Abstract
An axial follower force acting on the free end of a beamcolumn is known to remain tangential to its elastica at that point. Elastic beamcolumns exhibit infinitely high buckling resistance to static compressive follower load. Loss of their dynamic stability is known to occur at critical follower loads, by flutter characterized by vanishing lateral displacement and infinitely high natural frequency. Classical theory deals with physically linear nonconservative beamcolumns. Physical nonlinearity exhibited by concrete beamcolumns under service loads is caused by the closing and reopening of the extant transverse cracks. In this Paper, analytical expressions for the lateral displacement and lateral stiffness of such concrete beamcolumns are derived. Using these expressions, the stability of physically nonlinear elastic flanged concrete beamcolumns under the action of a follower compressive axial force and a lateral force is investigated. The significance of the analytical approach and the theoretical predictions is discussed.
Concrete beamcolumns; tangential follower load; flutter; divergence; snapthrough
Stability of concrete beamcolumns under follower forces
Mamta R. Sharma^{II}; Arbind K. Singh^{I}; Gurmail S. BenipalI,^{*}* Author email: gurmail@civil.iitd.ernet.in
^{I}Associate Professor, Department of Civil Engineering, I IT Guwahati
^{II}Research Scholar, Department of Civil Engineering, I IT Delhi
ABSTRACT
An axial follower force acting on the free end of a beamcolumn is known to remain tangential to its elastica at that point. Elastic beamcolumns exhibit infinitely high buckling resistance to static compressive follower load. Loss of their dynamic stability is known to occur at critical follower loads, by flutter characterized by vanishing lateral displacement and infinitely high natural frequency. Classical theory deals with physically linear nonconservative beamcolumns. Physical nonlinearity exhibited by concrete beamcolumns under service loads is caused by the closing and reopening of the extant transverse cracks. In this Paper, analytical expressions for the lateral displacement and lateral stiffness of such concrete beamcolumns are derived. Using these expressions, the stability of physically nonlinear elastic flanged concrete beamcolumns under the action of a follower compressive axial force and a lateral force is investigated. The significance of the analytical approach and the theoretical predictions is discussed.
Keywords: Concrete beamcolumns, tangential follower load, flutter, divergence, snapthrough
1 INTRODUCTION
Like dead loads, many types of live or imposed loads acting on the structures are gravitational forces. During the deformation of the structures, the points of action of such loads do not shift and their directions also remain same. The stiffness matrix of structures under such loads is symmetric. The energy is unique pathindependent reversible additive function of the state variables and is conserved during closed load cycles. Such conservative forces are also called gradient or potential forces because they can be derived as gradients of a potential in the space of state variables. Energy theorems are valid for these conservative or selfadjoint structures.
In contrast, the fluidstructure interactions impose some nonconservative forces on the structures. Either the point of action of these forces shifts on the structures during their deformation or motion, or their direction of action does not remain invariant. For example, the line of action of a follower force remains parallel to the deformed centroidal axis of a beamcolumn. Viscous fluids flowing past a surface of a deformable structure impose a tangential drag force on it. Examples of follower forces include a fluidconveying cantilevered pipe, rocket thrust caused by solid propellant fuel, powered jets, rockets or missiles, etc. Aeroplane wing flutter caused by aeroelastic follower forces is well known in aeronautical engineering (Bazant and Cedolin, 2010; Bolotin, 1964; Chen, 1976; Thomsen, 2003; Leipholz, 1978; Sugiyama, et al, 1999). Follower forces are also encountered, though as yet not recognized as such, even in the machine foundations and structures supporting rotating or reciprocating machinery (Srinivasulu and Vaidyanathan, 1976). A cantilever type beamcolumn supporting a rotating machine at the top end can also be considered to be under the action of pulsating tangential and transverse follower forces. As a special case, two motors with opposite direction of motion result only in sinusoidal tangential follower force. Long span cablestayed and suspension bridges are shown to be vulnerable to flutter instability at critical wind velocities (Kreis and Andre, 2005). The extensive literature available till that time on the theoretical and experimental studies on the dynamic stability of columns subjected to follower forces has been evaluated (Langthjem and Sugiyama, 2000).
The stiffness matrix of the nonconservative structures being asymmetric, the work done by these forces is not a unique function of the state variables. Instead, it is pathdependent resulting in energy deficit or excess in the closed load cycles. Invalidity of the principle of conservation of energy implies the invalidity of energy theorems for structural analysis. To be specific, theorems of Maxwell, Betti and Castigliano as well as principle of minimum potential energy are invalid. Euler's method of buckling of elastic structures is equivalent to the energybased methods. Hence, it is valid only for investigating the stability of conservative structures. For example, the critical buckling load of a column under a follower force is predicted by the Euler method to be infinitely high. This implies that column under a follower force is always stable. However, Euler method can only determine states of static equilibrium adjacent to the given equilibrium state. For determining the alternative states of dynamic equilibrium, the methods of dynamic stability analysis have to be employed. Indeed, the elastic columns have been observed to lose their dynamic stability by flutter. Such stability analysis has proved to be useful in the design of many types of structures like aeroplanes (Bazant and Cedolin, 2010; Bolotin, 1964; Chen, 1976; Thomsen, 2003; Leipholz, 1976; Sugiyama, et al, 1999; Langthjem and Sugiyama, 2000). Interestingly, even in the presence of abundant literature dealing with the stability of structures under follower forces published in the past few decades, the admissibility of the concept of static follower force has recently been questioned on operational grounds (Elishakoff, 2005; Sugiyama, et al, 1999).
With increase in applied loads, concrete structures exhibit cracking of concrete, yielding and debonding of reinforcement, crushing of concrete, etc., prior to failure. Being designoriented, the popular methods of concrete structural analysis attempt to incorporate the effect of all the above factors (Chopra, 2007). Such is also the case with stability analysis of concrete columns and beam columns subjected to conservative loads. To the best of Authors' knowledge, stability of nonconservative concrete structures has not yet been analysed. Exceptionally, AristizabalOchoa has emphasized the significance of investigating the behaviour of beamcolumn connections undergoing finite rotations under the action of tangential follower forces (AristizabalOchoa, 2004, 2005). Need has been felt for carrying out more accurate analysis of concrete structures including the effects of material and geometric nonlinearity (Nogueira et al, 2013). Stiffness degradation caused by fire (Caldas et al, 2014) can render such concrete structures vulnerable to static and dynamic instabilities. Concrete structures are cracked in flexural tension even under service loads. Fully cracked concrete structures undergoing small deformations exhibit physically nonlinear elastic behaviour. Conventional stability analysis generally deals with physically linear structures. Static and dynamic stability of fully cracked concrete columns and beamcolumns subjected to conservative loads has been earlier investigated by the Authors. Two values of the critical axial and lateral loads are identified for the concrete beamcolumns in the form of cantilever. Concrete beam columns are predicted to fail by brittle mode at peak axial loads and small lateral displacements. For constant lateral load, quite complex axial loadlateral displacement behaviour is exhibited. For lesser lateral loads, two equilibrium displacements are possible, out of which the one with lesser displacement is dynamically stable while the other is dynamically unstable. In this latter case, the loss of dynamic stability occurs by divergence. At two critical axial loads, the loss of stability occurs by vanishing natural frequency. The load space and the phase plane of initial conditions are partitioned into stable and unstable regions (Sharma et al. 2010; 2012a; 2012b).
The objective of the present Paper is to investigate the elastic stability of such beamcolumns under the action of a conservative lateral force and a tangential follower force. Expressions for the lateral displacement and lateral stiffness are derived and the nonlinear differential equation of motion is formulated. The theoretical predictions using the analyticalcomputational methodology are discussed.
2 THEORETICAL FORMULATION
Consider a fully cracked flanged concrete beamcolumn in the form of a vertical cantilever under the action of an tangential follower force P and a conservative lateral force Q. The cross section has equal reinforced flanges with concentrated areas of concrete and reinforcement as shown in Fig. 1. For some load combinations, the upper segment BC of the beamcolumn can have existing cracks in the closed state, while the tension flange of the lower segment AC has cracks in the open state.
Here, these upper 'uncracked' and lower 'cracked' segments and their section properties, like moments of inertia and the location of the centroids, are denoted by subscripts 1 and 2 respectively. As in the earlier derivations (Bolotin, 1964 ; Bazant and Cedolin, 2010), the tangential follower force P is resolved into vertical and horizontal components. Assuming the slope θ of the elastica at the free end to be small, these components turn out to be P and (Pθ) respectively.
In this Paper, the moments of inertia of the uncracked as well as cracked sections are determined in reference to their respective elastic centroids. Also, the same centroidal axes play the role of the reference axes for measuring the eccentricity of vertical load component for determining the flexural moment introduced by it. The flexural moment at any section in the uncracked segment BC yields the following second order differential equation:
Or equivalently,
Here, the prime denotes differentiation with respect to space coordinate x. Also, , , , . Similarly, y(L) = w and y'(L) = θ. The general solution for the above differential equation (2c) is of the form:
Using equations (2), the boundary conditions at the free end (y"(L) = 0 and y"'(L) = ) yields the following expressions:
Similarly, the differential equation of flexural equilibrium in the cracked segment AC is stated as
Or equivalently,
The general solution is obtained as
The boundary conditions at the fixed end (y(0) = 0 and y'(0) = 0) yields the following expressions
The following two more equations are obtained by using continuity conditions for displacement and slope at C (x=g):
Elementary structural mechanics requires that, for nil strain in the 'tension' flange, the moment at C in CB and CA segments must amount to Pr_{1} and Pr_{2} respectively. In view of this, equations (2a) and (6a) respectively yield the following expressions:
Similarly, the equations (2b) and (6b) respectively yield the following expressions
One obtains the equality
Using equations (4), (8), (9), (11) and (13), the unknown coefficients are determined as
C_{1} = , C_{2} = , C_{3} = , C_{4} =X_{1} = (B cos k_{2}g + 1 B) sin k_{1} (L g) + sin k_{2}g cos k_{1} (L g)
X_{1} = cos k_{2}g cos k_{1} (L g) sin k_{2}g sin k_{1} (L g)
X_{1} = sin k_{2}g cos k_{1} (L g) + cos k_{2}g sin k_{1} (L g)
Thus, the expression for the lateral displacement w, rotation θ and the lateral stiffness K are determined as follows:
Another useful expression in terms of loads is obtained from equation (10a) for determining the length 'g' of the cracked segment AC as
When the cracks at all the sections are closed, the expressions for the lateral displacement and lateral stiffness for such uncracked concrete beamcolumns are obtained as
These expressions derived for physically linear beamcolumns are identical to the wellknown expressions for the physically linear columns under the action of the axial follower force (Bolotin, 1963; Bazant and Cedolin, 2010). Clearly, the lateral stiffness of the physical linear beamcolumns is independent of the magnitude of the lateral force. The flutter loads for the fully uncracked beamcolumns can be obtained using the conventional theory as 20.19, while for the fully uncracked state, the corresponding flutter loads are B times lower.
For this SDOF nonlinear dynamical system, the stiffness and damping coefficients are vibration amplitudedependent. The instantaneous values of the natural frequency ω_{n} and the damping coefficient C depend upon the instantaneous stiffness coefficient K and the constant mass M lumped at the free end as
where ξ is the damping ratio.
Thus, the second order nonlinear differential equation of motion of the SDOF partlycracked beamcolumn subjected to forcing function F(t) is stated as
The above analysis is intended for cracked concrete flanged beamcolumns is also valid for flanged beamcolumns composed of bimodular solids. Concrete or ferrocement with distributed reinforcement can be modelled as bimodular solids. Let the bimodular ratio c be defined as where E_{t} and E_{c} denote the values of Young's moduli of elasticity in tension and compression respectively. This bimodular ratio c is related to the parameter B as B(1 + c) = 2c.
3 STABILITY ANALYSIS
The theoretical predictions presented below pertain to a particular concrete beamcolumn with following numerical details:
L = 20m d = 4.45m B = 0.18006 M = 450,000 kg ξ = 0.03The results apply to a bimodular ratio of 0.099. The adopted computational algorithm is based upon average acceleration NewmarkBeta method. When only the lateral force Q is acting, the concrete beamcolumns are cracked all through their length. Application of an axial follower force P results in the closure of extant cracks in the upper segment. The lateral force remaining constant, the typical axialload lateral displacement (P w) and axial loadend rotation (P θ) curves are shown in Fig. 2. With increase in axial force, the lateral displacement initially decreases. With further increase in axial load, quite complex response is predicted. Segment AB of the (P w) curve is associated with an increase in axial load upto P_{max} and a decrease in the length 'g' of the cracked segment of the concrete beamcolumn from its length L to a certain value. To be specific, cracks at all the sections do not close up at P_{max}. Further decrease in 'g' along segment BC of (P w) curve is accompanied by a decrease in axial load from P_{max} to P_{c}. Here, P_{c} signifies the magnitude of axial load corresponding to vanishing 'g'as per Eq. (17). The segment CD of the P w curve pertains to uncracked beamcolumns.
The variation of end rotation with P resembles that of the lateral displacement. The variation of P_{max} and P_{c} with Q is shown in Fig.3. Equation (17) gives an expression for P_{c} in terms of Q, while P_{max} has been determined by numerical computations. It can be observed that P_{max} exists only for lateral load exceeding Q_{1}. For this particular case the magnitude of Q_{1} turns out to be 0.437x10^{9} N with the corresponding value of P_{c} as 2.352x10^{9} N. Systematically varying the magnitude of the lateral force, the variation of P_{max} and P_{c} with their corresponding lateral displacements is plotted in Fig.4. With increase in Q, both P_{max} and P_{c} are predicted to increase. Similarly, P_{c} increases monotonically with w_{c}. The displacement w_{m} corresponding to the peak load P_{max} first increases, but reaches a maximum value at Q = 1.34x10^{9}N before decreasing with further increase in Q. The lateral displacement vanishes when Q attains the magnitude Q_{2} given by 3.014 10^{9}N. For higher values of Q the lateral displacements can be negative as discussed later.
From the enlarged view of the P w curve in Fig.2 presented in Fig.5, it can observed that for P < P_{c} and for P > P_{max}, the concrete beamcolumns are predicted to exhibit unique response. However, for P_{c}< P < P_{max}, three equilibrium states with different lateral displacements are possible. Corresponding to the segments AB, BC and CD of the P w curve discussed in Fig.2 above, these three displacements are denoted as w_{1}, w_{2} and w_{3} respectively. For a particular load set (P = 6.2Sx10^{9}N, Q = 2x10^{9}N) the above three equilibrium displacements turn out to be 3793 mm, 5433.5 mm and 6646.7 mm respectively. Further, the dynamic stability of these three alternative equilibrium states is studied. This is achieved by analysing the phase plane representations of the dynamic response of concrete beamcolumns taking the above equilibrium displacements as the initial displacements in different cases. In all these investigations, vanishing initial velocity is assumed. Phase plane plots shown in Fig.6 reveal that, irrespective of the particular initial displacement for each case, the lateral displacement asymptotically attains the same value w_{2}. It can be observed that the trajectories originating from the other equilibrium displacements w_{1} and w_{3} diverge, i.e., are repelled away. It implies that the equilibrium states corresponding to displacements w_{1} and w_{3} are dynamically unstable. Such a dynamic instability is called divergence. It has been verified, but not presented here, that the above conclusion deduced for a particular case is valid for other magnitudes of the axial and lateral loads as well.
It is worth mentioning here that purely static structural response is a theoretical construct. In fact, under 'static' load variations, structures exhibit quasistatic behaviour. Real structures located in the natural environment as well as the structural models in the laboratory are subjected to small perturbations. Obviously, the dynamically unstable equilibrium states are de facto inaccessible to measurement and the structures are observed to exhibit only dynamically stable 'static' response. In view of this argument, the P w curve discussed above can be reinterpreted. Fig.7 depicts the observed static response of a concrete beamcolumn subjected to a constant lateral force and a monotonically increasing axial follower force. Here, the observed P w curve is identified as AcCBbD. As the axial load is increased upto P_{c}, the segment Ac of the curve depicts the stable structural response. When the axial load slightly exceeds P_{c}, a sudden increase cC in the lateral displacement is predicted. For P_{c} < P < P_{max}, only the segment CB of the curve is relevant. Another sudden increase Bb in lateral displacement occurs as the axial load crosses P_{max}.
For P > P_{max}, the structure again exhibits unique mechanical response. Such also happens to be the case when the constant lateral force exceeds a value (3.0134x 10^{9}N). For lower magnitudes of constant lateral force, slightly different P w curve is obtained. As shown in Fig.8, jump phenomenon in the form of sudden changes in the response at P_{c} and P_{max} are predicted in this case as well. Here, again the observed P w curve is AcCBbD. However, in this case, there is a sudden decrease cC in larteral displacement occurs at P_{c}. It can be observed from Fig. 2 that the sudden increase/decrease in the lateral displacement is accompanied by the sudden increase/decrease in the rotation at the free end of the beamcolumn.
Fig.9 shows the P w curves for the special case of the fully uncracked concrete beamcolumn subjected to different lateral forces. The magnitudes of the axial force corresponding to vanishing lateral displacements are denoted by P_{1}_{uc} < P_{2}_{uc} < P_{3uc}, etc. For fully uncracked beamcolumns the values of P_{1}_{uc} and P_{2}_{uc} are respectively obtained as 20.036x10^{9}N and 60.189x10^{9}N. The corresponding values of the follower forces for the fully cracked beamcolumn turn out to be B times the values for the uncracked beamcolumns. These values of axial follower force are independent of the magnitude of the constant lateral force. In the classical theory of columns under the action of an axial follower force, vanishing lateral displacement is interpreted to imply infinitely high lateral stiffness and so infinitely high natural frequency. Such a dynamic instability is called flutter. For P < P_{luc}, the positive natural frequency implies dynamic stability of the system. When P_{1}_{uc} < P < P_{2uc}, the lateral stiffness assumes negative values implying dynamic instability. This system again exhibits dynamically stable behaviour when axial load P is in the range P_{2}_{uc} P_{3uc}, and so on.
To recapitulate, for P < P_{max}, the lateral displacement of partly cracked concrete beam columns do not vanish at lower magnitudes of the constant lateral force. When the lateral force exceeds a particular value Q_{2} = 3.014x 10^{9}N, displacement vanishes at two values P_{1} and P_{2} of the axial follower force. Fig.10 shows the Pw curve for a particular case when P_{1} and P_{2} are obtained as 4.987x 10^{9} N and 9.888x 10^{9} N respectively. It can be observed that these values of the flutter loads for the partly cracked concrete beamcolumns lie between their corresponding values for the two limiting cases of fully uncracked and fully cracked concrete beamcolumns discussed above. When the axial load is increased beyond P_{max}, the lateral displacement also vanishes as and when the flutter loads (P_{3}, etc.) for the fully uncracked concrete beamcolumns are reached.
The corresponding variation of the lateral stiffness of concrete beamcolumns with the axial follower force is plotted in Fig. 11. Here, the flutter loads P_{3} is the same as the lowest flutter load of the uncracked beamcolumns. For P < P_{1}, the stiffness increases with increase in the axial force. As the axial force reaches P_{1}, the stiffness, as expected, attains infinitely high values. When axial force crosses P_{1} or P_{3}, the stiffness is expected to change discontinuously form infinitely high positive values to infinitely high negative values. Such indeed happens to be the case here as well. However, the stiffness of concrete beamcolumns is predicted to vary in complex manner when the axial force varies in the range P_{1} P_{3}.
An enlarged view of Fig. 11 around P_{2} is presented in Fig.12. For proper interpretation of this plot, it must be remembered that the points D and G marked on the K P curve correspond to axial forces P_{max} and P_{c} respectively. Thus, the flutter load P_{2} lies within the range P_{c} P_{max}. In this range, there are three values of stiffness corresponding to the three equilibrium lateral displacements. As the axial force is increased from P_{1} to P_{max} along CD, the lateral stiffness increases from infinitely high negative value to a certain negative but finite value. From this finite negative value, the stiffness decreases along DE again to infinitely high negative value as the axial load decreases from P_{max} to P_{2}. When the decreasing load crosses P_{2}, the stiffness increases from infinitely high negative values to an infinitely high positive value. On decreasing the axial force further blow P_{2} upto P_{c} along FG, the stiffness decreases from infinitely high positive value to a certain positive finite value at P_{c}. When axial force is increased along GH from P_{c} to P_{3}, the stiffness again increases to infinitely high positive value. Thus, contrary to expectations, as axial force increases across P_{2}, the stiffness varies from infinitely high positive value to infinitely high negative value.
Further, the dynamic stability of concrete beamcolumns is investigated in reference to magnitude of the axial follower force relative to the flutter loads P_{1}, P_{2}, P_{3}, etc. This is achieved by analysing their dynamic response on being perturbed from their equilibrium state. The forcing function F(t) has been assumed to equal constant lateral force Q. When the axial force lies outside the range P_{c} P_{max}, there is only one equilibrium state. However, within this range of axial force, there exist three equilibrium lateral displacements. The segments ABCD, DEFG and GHIJK of the KP curve pertain respectively to the lateral displacements w_{1}, w_{2} and w_{3}.
For appreciating the following dynamic stability analysis, the relative magnitudes of follower loads for this particular case should be noted: P_{1} < P_{c} < P_{2} < P_{max} < P_{3}. From the analysis of dynamic stability of equilibrium states, the following conclusions can be drawn: For P < P_{1}, the unique equilibrium state (w = w_{1}) corresponds to the positive stiffness and hence is stable. This equilibrium state is rendered dynamically unstable because of negative lateral stiffness in the range P_{1} P_{c}. It has been discussed earlier that, of the three equilibrium displacements within the range P_{c} P_{max}, only that corresponding to displacement w_{2} is stable. Such happens to be the case here as well as in range P_{c} P_{2} because of the corresponding positive stiffness. Negative stiffness in the range P_{2} P_{max} even destabilizes this one otherwise stable equilibrium state. It implies that in the ranges P_{1} P_{c} and P_{2} P_{max} of the tangential follower force, there are no dynamically stable equilibrium states. Of course, P_{max} P_{3}, the unique equilibrium state (w = w_{3}) continues to be stable. As expected, there exist other dynamically unstable and stable states of the uncracked beamcolumns at higher follower loads. It can be concluded that cracking lowers the magnitude of the flutter loads for the concrete beamcolumns.
Based upon the above results, the load space is shown to be partitioned into dynamically stable and unstable regions in Fig.13. Here, the variation of various flutter loads, P_{c} and P_{max} with Q is plotted. While the flutter loads P_{1}_{uc} = P_{3} for the uncracked structures remains constant, the flutter loads for the partly cracked structures vary with Q > Q_{2}. Specifically, the lower flutter load P_{1} decreases with Q to asymptotically reach the flutter load BP_{1}_{uc} for the fully cracked structures. The higher flutter load P_{2} increases with Q while the flutter load P_{3} equals that P_{1}_{uc}for the uncracked structure. For Q < Q_{2}, no flutter occurs for P < P_{1}. Based upon their vulnerability to flutter instability, the ranges P_{1} P_{c} and P_{2} P_{max} of the follower force corresponds to dynamically unstable equilibrium states. In contrast, the ranges P_{c} P_{2} as well as P_{max} P_{3} represent dynamically stable behaviour. Within the range Q_{1} Q_{2} of the lateral force, P_{max} exceeds P_{c}. In this range, out of the three equilibrium states, only one w_{2} is dynamically stable. As discussed earlier, snapthroughlike discontinuities in the P w curves are predicted in this range as and when P equals either P_{c} or P_{max}. Thus, based upon the vulnerability to divergence instability, in the load space bounded by P_{c} < P < P_{max} and Q_{1} < Q < Q_{2} is also dynamically unstable. It can be observed that for Q ≳ Q_{1}, Pc and P_{max} are below even the lowest flutter load BP_{1}_{uc} for the fully cracked concrete beamcolumns.
4 DISCUSSION
To recapitulate, the classical theory deals with physically linear beamcolumns subjected to tangential follower force only. In contrast, the present Paper deals with the physically nonlinear beamcolumns subjected to lateral force in addition to the follower force. Concrete beamcolumns cracked a priori at all the sections exhibit such physical nonlinearity. Beamcolumns composed of bimodular solids with Young's modulus of elasticity in tension being smaller than that in compression constitute another example. Thus, in this Paper, the conventional theory of beam columns is extended by incorporating the effect of physical nonlinearity. In this respect, this Paper can claim to constitute a definite significant contribution to structural stability theory. However, the hypothesis of small end rotations at the basis of the classical theories as well as the proposed theory does not seem to hold specifically at lower magnitudes of the follower force.
The predicted behaviour of concrete beamcolumns turns out to be quite interesting as discussed below:
Conventional beamcolumns subjected to tangential follower force only are known to exhibit infinitely high resistance to static buckling. In contrast, when subjected to lateral forces exceeding Q_{1}, concrete beam columns exhibit finite resistance to quasistatic follower force. The jump phenomenon resembles static instability called snapthrough and occurs at follower forces lower than the lowest flutter load for the fully cracked beamcolumns.
When the concrete beamcolumns are either uncracked or cracked all through their length, their axial loadlateral displacement curves are qualitatively similar to those of the conventional beamcolumns. However, in the latter case, the magnitudes of the flutter loads are not affected by the magnitude of the lateral force, even through the Pw curves depend upon it. Generally, the concrete beamcolumns are partly cracked. When Q < Q_{2}, even these physically nonlinear beam columns have the same flutter loads as the fully uncracked beamcolumns. Otherwise, the two more values of the flutter loads are predicted. These values lie between the lowest values of the flutter loads for fully uncracked and fully cracked concrete beamcolumns. These flutter loads are in addition to those for the fully uncracked case. In other words, cracking reduces the magnitudes of the flutter loads for concrete beamcolumns under sufficiently high lateral loads. From the design point of view, when the lateral force is lesser than Q_{2}, the lowest flutter load equals P_{1}_{uc}while, for higher lateral forces, it reduces to BP_{1uc}. However, snapthrough like jump phenomenon can occur even at lower loads for the lateral force in the range Q_{1} Q_{2}.
As discussed above, concrete beamcolumns supporting machine foundations are subjected to sinusoidal tangential follower forces. The scope of the present Paper is restricted to constant follower forces. Also, the theoretical predictions have not yet been empirically validated. Typical qualitative predictions of the concrete beamcolumns subjected to a tangential follower force and a lateral force are presented in this Paper. The theory of concrete beamcolumns presented here has to be extended further and validated before it can be of any relevance to structural design.
5 CONCLUSION
In this Paper, motivated by cracked concrete beamcolumns, the stability of physically nonlinear elastic beamcolumns subjected to a tangential follower force and a conservative lateral force is investigated. Analytical expressions for the lateral displacement and stiffness are derived. For certain range of follower loads lesser than the lowest flutter load for the uncracked structures, the beamcolumns are predicted to exhibit three equilibrium states out of which only one is dynamically stable. The loss of dynamic stability of the remaining two equilibrium states is of divergence type. Consequently, snapthrough type behaviour of static instability is predicted under quasistatic increase of the follower axial force. Such instabilities are predicted to occur even below the lowest flutter load for the fully cracked concrete beamcolumns. Quite complex flutter instability response is exhibited. For higher values of the lateral force, the flutter loads turn out to be lesser than those for the uncracked beamcolumns. The load space is partitioned into flutterbased dynamically stable and unstable regions. The proposed theory is shown to be valid for bimodular beamcolumns as well.
Received in 02 Mar 2013
In revised form 11 Jul 2013
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Publication Dates

Publication in this collection
03 Feb 2014 
Date of issue
Oct 2014
History

Received
02 Mar 2013 
Accepted
11 July 2013