Load Transfer in Series and Parallel Paths by Kotur Raghavan
Recent articles were devoted for understanding of load paths in structures. The topic was discussed in the context of human body ( https://www.fembestpractices.com/2020/10/human-body-further-structural-insights.html ) and bicycle ( https://www.fembestpractices.com/2020/10/load-paths-in-structures-3-bicycle.html ). These two simple examples bring home what is basically common to all structures. They are load application points and reaction or support points. In a mathematical sense, they form the extremities of any given structure which are connected to one another through the load path. Structural analysis is concerned with study of the components which occupy the load path.
Of special significance are the members which are in series and those which are in parallel. By definition, the same load passes through contiguous members which are in series. On the other hand, members in parallel share load coming from a common contiguous member. In the human body, the two legs share the load coming from the torso and hence they are structural items in parallel. If the load is carried in the right hand the same load passes through the palm, wrist, forearm and the arm. Those parts are members in series. A weight placed on the head passes through the skull, neck joint and the spinal column (consisting of 33 vertebrae located one upon other and interlinked) on to the pelvic. Thus skull, neck joint and the spinal column are in series. In a bicycle the two wheels can be considered to be members in parallel. They share the loads coming from the central frame. The frame acts as a single integral unit and does not contain clearly identifiable serial or parallel members.
Schematically, structural elements in series and parallel are represented using linear springs as shown in the figure below.
In the series arrangement, the same load will pass through K1 and K2 and each will stretch by lengths inversely proportional to stiffness. The deflection at the load application point is (F/K1+F/K2). In the parallel arrangement both K1 and K2 will stretch by equal lengths. The deflection at the load application point is (F/(K1+K2)). The force in K1 and K2 are (F*K1/(K1+K2)) and (F*K2/(K1+K2)) respectively. It is obvious that flexibilities get added in the case of series arrangement and stiffnesses get added in the case of parallel arrangement. For the two cases, the equivalent stiffness values are as stated below.
It can be recognised that the equivalent stiffness is governed more by the lower individual value in the series arrangement and by higher individual value in the parallel arrangement. Suppose that K1 is much greater than K2. Then the equivalent stiffnesses will be nearly equal to K2 and K1 in series and parallel arrangements respectively. The phrase ‘much greater than’ has certain amount of subjectivity. However, in engineering analysis a factor of 1000 or more can be deemed to be ‘much greater’.
The foregoing arguments are of significance and can be exploited beneficially in developing optimum simulation models. A few illustrations will help understand the approach.
1. Cantilever Beam with Spring Support
The cantilever beam and the spring are structural elements in parallel. These are just symbolic for the purpose of illustration and may represent something more complex. The total stiffness at the tip can be seen to be (K + (3*E * I)/ L**3). If the spring stiffness is equal to the beam stiffness the load will be equally shared between the beam and the spring. As the spring stiffness becomes less and less its influence keeps on decreasing and the behaviour approaches that of simple cantilever beam. The spring becomes practically ineffective when the spring ratio is 0.001 or less. On the other hand, if the spring stiffness is increased gradually it will dominate the beam. If the ratio is of the order of 1000, the spring effectively becomes the reaction point and the beam will hardly bend.
2. Rotor Bearing systems
In rotor dynamics studies, modelling of the rotor-bearing systems normally pose some challenge. The models are needed for critical speed and unbalance response evaluations. A typical rotor-bearing system is shown in the figure below.
Analysis models typically are as shown in the figure below.
Three structural members are represented by stiffnesses KR, KB and KH represent the rotor, bearing and housing (pedestal) respectively. The rotor is in series with the support systems. The two support systems, by themselves, are in parallel. MH is the housing mass. Relative values of these stiffnesses are to be used as guideline for arriving at the right model. Generally the rotor happens to be the most flexible of the three. Bearing stiffnesses are generally of comparable value in the case of oil film bearings and much higher in the case of anti-friction bearings. Housing or pedestal stiffnesses are normally high. As a result, frequently it happens that the analysis model shown below will be adequate.
Models like this are very useful for initial evaluation. They are also quite reliable for the first critical speed which is dominated by flexure of the rotor.
3. Slab Column Assemblies
In the figure below are shown typical constructional features in buildings.
The slabs are in flexure and the columns are in axial compression. The slab is in series with the columns and columns, by themselves, are in parallel. As flexural stiffness is normally low, quite often it will suffice to analyse only the slab by applying constraints at column locations.
1. It is a good practice to identify structural members which are in series and those which are in parallel.
2. If the members are in series, flexibilities add. In the case of parallel arrangement stiffness values add.
3. The value of ‘equivalent stiffness’ is governed by smaller value when they are in series. Higher values govern equivalent stiffness when the members are in parallel.
A prior knowledge of relative stiffness values will be helpful for arriving at simpler simulation models particularly in cases wherein stiffness ratios are of the order of 1000 or more.