Strand7: Web notes: Stiffened plates

Modelling stiffened plates

Following is a short discussion on some of the problems that can arise when modelling stiffened plates and some recommended modelling techniques.

Stiffened plates are common in all types of engineering design and examples include concrete slabs with beam stiffeners and aluminium aircraft structures that comprise a thin skin stiffened by stringers. Examples of both of these structures are shown in the figure below. There are many methods that can be used to model stiffened plates. The one that is chosen will depend on the type of structure, the information required from the model, etc. Following is a brief look at some of the more common methods and a note about their advantages, disadvantages and problems that can occur.

Brick Model - In the case of the concrete slab example above, the slab and the beam could be modelled entirely from brick elements (see figure below). This will give a very good representation of the structure and good results for the stresses and deflections. The problems are that in order to build a model representative of any real structure the model will rapidly become very large.

Plate Model - In the case of the aircraft structure example above the entire structure including the stringer could be modelled from plates (see figure below). This can give good results for the stress and deflections. This model also has the advantage that it can be used to perform a local buckling analysis on the stiffeners. As for the brick model, the main disadvantages are that the model can become very large and thus this technique is best suited to detailed modelling of some local part of the structure.

Orthotropic plates - When a very regular structure is involved - that is a panel with evenly spaced constant section stiffeners - and you are more interested in the overall behaviour of the structure as opposed to the detailed stresses in stringers etc., then a good representation of the panel behaviour can be obtained by using orthotropic plate elements.

Orthotropic structures can have different stiffness in each of the two principal directions and the membrane and bending stiffness can be independent of one another. To specify the orthotropic characteristics for a stiffened panel the [C] and [D] matrices must be calculated.

Plates with Stiffeners Modelled by Offset Beams. Probably the most practical way to model stiffened plates in the majority of situations is to use a combination of plate elements and offset beams. It is this method that we receive the most questions about. The method of modelling and the points to watch when using this method are best explained by an example.

Example - Consider the centrally loaded, simply supported beam shown in figure below.

This will be modelled with a plate and offset beam combination and various aspects of the modelling and results will be discussed. Following is an analysis of this beam by (a) a conventional manual method and (b) by Strand7 using a combination of plates and offset beams.

Hand Calculation - Theoretical Results

1.	Bottom of Beam
	S1 = (2500*0.32875)/0.010630953	= 77309 Pa
2.	Centroid of Beam
	S2 = (2500*(0.32875-0.25))/0.010630953	= 18519 Pa
3.	Top of Beam/Bottom of Plate
	S3 = (2500*(0.5-0.32875))/0.010630953	=-40271 Pa
4.	Centroid of Plate
	S4 = (2500*(0.525-0.32875))/0.010630953	=-46150 Pa
5.	Top of Plate
	S5 = (2500*(0.55-0.32875))/0.010630953	=-52029 Pa
	The Deflection at the centre of the beam is:
	y = PL3/48EI
	= (1000103)/(481.0E7*0.010630953)
	= 0.195968634m

Finite Element Results

Displacements - The vertical deflection of the node to which the force is applied is 0.2016 (a difference of 2.87%)

	Position	Stress (Pa)	Difference (%)
1.	Bottom of Beam	82950	7.39
2.	Centroid of Beam	15992	13.65
3.	Top of Beam	-50960	26.54
	Bottom of Plate	-37872	5.96
4.	Centroid of Plate	-44503	3.57
5.	Top of Plate	-51135	1.72

Shear Force and Bending Moments - In both cases the bending moments are plotted in the graph above; in the first case (a) for the complete beam by the consideration of equilibrium of external and internal forces; in the second case (b) the moments in the offset beam are plotted. Some Strand7 users have assumed that this will represent the bending moment of the beam as a whole. Clearly this is not the case. The peak bending moment obtained from the hand analysis is 2500 Nm compared with only 1395 Nm for the beam element in the finite element model. In this case the beam and the plate elements work together to carry the moment since the finite element model must be in equilibrium at any point with the external applied loads.

The forces acting on the elements at any section through the beam are as follows:

The sum of the forces and moments at this section must equal the forces and moments due to the externally applied loads. For convenience the sum of the moments can be taken about the mid plane of the plate; this effectively removes the plate membrane stress from the calculation. So the sum of the moments is as follows:

SM_AA=0=(F_b*Offset)-M_b-M_p=M_section

Obviously the signs of the forces, moments etc. will depend on each individual problem and the orientation of the plates and beams.

In many such problems the moment in the plate is small and can be neglected. However it is suggested that users intending to do modelling of this type carry out their own tests on a model representative of their structure to verify this assumption.

From this summation of moments it is clear that the moment due to the externally applied forces is balanced by the bending moment in the beam element plus the couple due to the axial force in the beam times the offset. Any beam design will have to consider this total moment. There are two distinct types of structures that we have to consider:

In the case of a structure where the plate sits on but is not rigidly connected to a beam then the beam will have to be designed to carry the total moment. It should be noted that the moment obtained from the finite element model will be independent of the section of the beam included in the model. The offset beam approach should not really be used to model structures of this type since this method assumes that the beams are rigidly connected to the plates.
In the second case where a stiffener is rigidly attached to the plate as in the case of a concrete slab with an integral stiffener then the design would be approached differently. This problem is similar to the one shown in the previous example. For this case it is inappropriate to attempt to design the plate and the beam individually. It will be necessary to assume that the stiffener beam acts together with an effective section of the plate to resist the total bending moment. How much of the plate is assumed effective in resisting bending will depend on the particular application, the engineering discipline and in some cases the codes in use. The most conservative assumption would be to take only the section of plate to which the beam connects as being effective. In other not so conservative applications such as aeronautical applications it is common to assume that a considerable section of the plate either side of the stringer is also effective in resisting the bending. Whatever section is chosen it will be necessary to extract the total moment on the section using the above method and check the bending of the beam by manual calculation. This may sound time consuming but in most cases only a few sections will have to be considered. Since many beams are essentially constant in cross section, only the locations of peak moment will need to be checked. In any case the stresses in the finite element model will be approximately correct and these can be used as a guide to the locations that need detailed investigation.

It is evident from looking at the bending moment diagram for the beam element in the previous example that this is not continuous - it has a pronounced saw tooth pattern. This does not represent the physical situation as the total bending moments must be continuous. To gain an understanding of the reasons behind this difference it is necessary to study the forces in the various elements.

In any given beam element the axial force is constant whilst the bending moment varies linearly. Additional loads can only be transferred to the beam elements at the nodal points. Now the total bending moment varies linearly and according to the above equation for the summation of moments, if the total bending moment is to be continuous across adjacent elements then the beam axial force must also vary linearly. Since the axial force in the beam elements is a constant stepwise approximation to the linear variation, it follows that the bending moment in the beam elements must also be discontinuous.

Nodal equilibrium is one of the most basic of all conditions that must be met in a finite element analysis. The fundamental point to be understood is that irrespective of the distribution of the forces and moments within the beam elements and plates the summation of the total moment at any section is correct. This can be verified for any section of the example beam by taking the beam axial force and moment and substituting this into the moment equilibrium equation and comparing with the calculated bending moments.

The implications of these stepwise approximations to the linearly varying bending moments and axial forces are that the distribution of stress in the beam and plate elements will not be continuous. This can be seen by comparing the stresses at various points on the cross section of the beam as obtained from the FE model to the theoretical values.

Point	Theoretical	Strand7	% Difference
1	77309	82950	7.29
2	18519	15992	13.65
3	-40271	-50960 (Beam) -37872 (Plate)	26.54 5.96
4	-46150	-44503	3.57
5	-52029	-51135	1.72

The maximum difference in the stress is 26.54% however this is not as bad as it would seem since this refers to some internal point of the beam about which we have little interest. The maximum difference in extreme fibre stress is 7.29%.

The accuracy of the solution can be improved by refining the mesh. As the mesh is refined the piecewise approximation of the beams to the axial force will approach the correct continuous distribution and hence the moment distribution will also improve.

Plates with Stiffeners Modelled by Beams in the Plane of the Plates - We often see models that have stiffeners modelled with beams in the plane of the plate (i.e. the beams have no offsets). Unless special care is taken this can lead to erroneous results. If the beam is given the cross section of the beam attached to the plate, then the bending stiffness will be too low since the effect of the offset of the beam on the bending moment of inertia has been neglected. In this case the deflections will be wrong.

If this method is to be used an effective moment of inertia must be calculated for the beam. This should be equal to the total moment of inertia of the beam section being modelled about its centroid (0.010630953 m4 in the above example) minus the moment of inertia of the effective plate width (negligible in many cases). In the above example the equivalent beam section required is 0.597345205m square.

If the beam in the above example is modelled using this technique then the deflections are exact. The moment in the beam element should approximately equal the total moment on the section. The plate will carry some moment but since it lies on the neutral plane, its contribution will be insignificant. The peak moment in the beam elements obtained for the above model was 2495 Nm compared with 2500 Nm from theory. The bending moments in the beam will be continuous since the beam elements lie on the mid plane and do not carry any axial force. The stresses in this model will be wrong in both the beam and the plate elements as the section will have the wrong y values in the S=My/I calculation. Obtaining the correct stresses would be difficult since a section would have to be devised with stress calculation points in the correct location whilst preserving the required moment of inertia. This method may be of some value to those designing with codes that require bending moments and where stresses are not required. The main advantage of this method is that the bending moments can be obtained directly and do not suffer from the discontinuity problems of offset beams discussed above.

In both cases above it may be necessary to modify either the I or E of the beams and/or plate to make allowance for certain requirements in design codes such as AS3600 for concrete slabs. In this particular case there are factors to reduce the stiffness to make allowance for the cracking of the concrete.

Rigid Link Modelling Offsets - An alternative technique is to connect a series of rigid links between the plate and the main beam as shown in the following figure.