D5. Understanding and analyzing the peaks (case study about concrete)
D5. Understanding and analyzing the peaks (case study about concrete)
The smoothing referred to in this chapter concerns the clipping of stresses or strains or the distribution of reinforcement, by averaging it, over a certain length (or width) of the calculated element.
D.5.1 Stress concentration and peak stress (forces)  Different types of stress visualization
D.5.1.1 Stress and strain peaks
Contrary to common ideas, stress and strain peaks did not appear with finite element software, but have always been part of the stress of structural engineers, they are inherent to the inclusion of point forces in the calculation of slabs.
Let us consider, for example, the study of an infinite isostatic slab subjected to a load concentrated in its center.
A possible approach to calculate the forces in the center of a slab is the use of Pücher’s charts.
The chart above (surfaces of influence) shows a peak of moments in the center: the line of influence increases rapidly as the load approaches the middle of the span: the values 3 / 4 / 5 then 6 and 7 narrow to an value which is infinite, but corresponds to a surface tending toward 0. Pücher arbitrarily truncated the representation to the value of 8.
Let us consider a slab with a span of 3 m articulated on its sides, subjected to a concentrated load of 10 KN in its center.
According to the chart, the theoretical moment at the load is infinite, which is not satisfactory for the engineer who has to dimension the reinforcement. In reality, point loads do not exist, especially since the common practice is to diffuse the loads to the layer of a slab. We are then led to calculate integrals to obtain the value of the moment in the slab (for more details, please refer to Pücher’s original publications).
By distributing the load over a 20×20 cm² square, the integration of the surfaces shows that the maximum moment is equal to 3.0 KN.m.
Another example of the use of Pücher’s charts is given in the example of modeling a Br wheel.
It should not be forgotten that the calculations are generally carried out with raw sections (formwork) and elastic materials (linear behavior). In reality, the cracking of reinforced concrete will lead to a redistribution of forces that tends to reduce these peak effects. This type of calculation is not (to date) current practice.
It is therefore necessary to know how to correct simply, often manually, the results of a linear calculation.
Examples of correcting the results of a linear calculation are given below.
D.5.1.2 Finite Element Peak Studies
What happens when this same slab is calculated using finite elements?
Slab of 3 m articulated span subjected to a concentrated load of 10 KN at its center (node stresses)
Same in 3D
We note the appearance of a peak whose maximum value is not infinite, but equal to 2.96 KN.m.
We saw that the software calculates the efforts at the elements' integration points, then extrapolates the results to the center and then to the nodes of the elements. A node is generally common to 4 elements, so there are 4 results per node (one for each element). What will the chosen result be? The maximum value? The average value? Indeed, finite element software does not propose a single result for each calculation, but several results depending on the options chosen by the engineer: the software is able to draw up maps of results from the forces on the nodes, or from the forces at the center of the elements, they can be smoothed, not smoothed, etc.
The engineer must choose the visualization options carefully, as the results vary greatly depending on the option chosen. This is what we propose to show with the example below.
The previous figure shows the moments calculated on the nodes of the elements whose maximum value is 2.96 KN.m, it is very close to the value of 3.0 KN.m calculated manually.
We take the same example by displaying the calculated moments in the center of the elements (instead of the nodes); the central peak is smaller: 1.92 KN.m for 2.96 KN.m previously. This result is also far from the manually calculated value of 3.0 KN.m.
Slab of 3 m articulated span subjected to a concentrated load of 10 KN distributed over 20×20 cm² in its center (forces in the center of the elements).
The representation below of the curve of moments on a section in the middle of the slab allows to understand these differences.

Node efforts
Below is the curve of Myy moments calculated by smoothing on the nodes.
The forces are calculated in the integration points of the elements and then extrapolated to the nodes.
The maximum moment is 2.96 KN.m in the center of the slab according to the manual calculation.

Unsmoothed efforts on the elements
The forces are calculated in the integration points and then averaged to obtain the force at the center of the element.
The maximum value is 1.92 kN.m , average of the central elements.
We do not find here the manually calculated value of 3.0 kN.m, but a "smoothed" value on the elements surrounding the peak. From this example, we will note that the forces at the nodes give results that are consistent with those calculated manually, which is not the case for the forces calculated at the center of the elements.

Smoothed efforts on the elements
The forces are calculated in the center of the elements and then smoothed between them.
This curve gives the impression of a curve which is extrapolated on the nodes, whereas the extrapolations are carried out only on the results at the centers of the elements, the resulting curve is without physical meaning and therefore "false".
On the other hand, in another case, this option of smoothing the efforts at the center could be useful if we want to know the efforts in the plane of the walls.
To be valid, the width of the mesh should be equal to the thickness of the wall.
The visualizations of the finite element results at the peaks give very different results depending on the options chosen by the engineer. These results cannot be taken as they are, but must be analyzed and interpreted by the engineer.
The figure below shows the values obtained with a point force of 10 kN at the center of the slab.
The moment value increases to 5.2 kN.m and the pace of the moment curve shows a clear peak.
Later in this chapter, we will see that point forces (which have no physical meaning) lead to effort peaks and that it is better to avoid using them in order to obtain accurate local results.
D.5.2 Peak analysis method
The reinforcement specifications often show peaks of steel which have very important consequences in the dimensioning of the reinforcement.
The user is often unaware of these peaks: should they be taken into account by considering that they are structural, or should they be ignored by assimilating them to numerical calculation problems?
Example: the peaks shown above at the fixed end of this cantilever beam are of course structural.
The answer to this problem lies in understanding the functioning of the structure and the path of forces at the peak level, an analysis that is indispensable to solve peaks (limit the maximum demandclipping, linearization of reinforcement).
This can be difficult in the case of complex models, but is always essential.
Three types of analyses are possible:
Analyses 
Objective 
1: Geometrical analysis 
Identifying the singularities of the modeling at the peak to determine its geometric origin. 
2: Steel sections analysis Axi, Axs, Ayi, Ays 
Making a first distinction between membrane, shear and bending forces 
3: Analysis of the efforts generating the peak 
Detecting the "faulty" component(s) and quantifying efforts 
D.5.2.1 First analysis: geometrical
Experience shows that 90% of the peaks are located at the level of geometric singularities (columns, supports, wall/wall or wall/slab intersections, etc.). This is due to the fact that the modeling, carried out from middle planes or middle fibers, does not represent the elements with their real volumetric geometry (for example, a slab is represented by a plane element and the columns on which it rests, by wire elements). In a pictorial way, this leads, as explained in D.5.1, to taking into account forces applied to null surfaces, thus inevitably leading to numerical problems. It is therefore essential to identify these singularities on the model.

Examples of peaks related to geometry
These peaks must be interpreted in detail, it is possible to reduce the values of the dimensioning moments by limiting the maximum demandclipping at the beams.
Example of a slab on a grid n array of columns and beams
Visualization of bending moments in the slab
Peaks appears; they can be limited clipped at the beams:
In the above case, conventional clipping is performed in the plane of the edges of the beams or column in the case of a flat slab on headed columns.
Some types of software allow to define supports with plan dimensions to directly obtain the efforts in the plane. Example of a 10×10 m² slab, 30 cm thick, subjected to its selfweight, supported linearly on an edge and on two point supports in line with two posts 1 m from each corner. The left support is a classic point support, the right support is a column type support of 50×50 cm².
The representation of efforts with and without reduction of effort shows a significant difference in values.
Note: It is always advisable to check the methodology used by the software and to ensure that it is compatible with the regulatory justifications to be carried out.

Peaks caused by point forces
The peak treatment at concentrated loads is similar to the peaks caused by point links. (A force or torsor can represent the effects of point support; there is a strict equivalence).
Example:
We take the trivial example of a load arriving via a column on a wall (30 cm thick wall, 55×30 cm² column).
Depending on what we want to calculate, the approach will not be the same. If we are looking for a global vertical load calculation, the approaches on the left, either via a point load or via a wire bar are perfectly suitable. However, if we are interested in local effects, it is absolutely necessary to use a distributed load to minimize the stress peak, which does not facilitate automated calculations (and does not anticipate other manual calculations to be performed: diffusion, punching ...).
A concentrated load which is perpendicular to a slab generates a moment peak (see D.5.1). This peak must be processed to calculate the reinforcement. It can be reduced by replacing the concentrated load by a loading block that takes into account the diffusion of the load in the slab (diffusion of a wheel on a bridge slab).
Example:
The longitudinal moment for the same 100 kN load is compared using four approaches:

a point force in the center of a mesh;

a point force on a node;

a pressure corresponding to a force of 100 kN on a surface of 0.25×0.25 m² (the mesh);

the force of 100 kN/4 = 25 kN distributed to the 4 nodes framing a mesh of 0.25×0.25 m².
100 kN force in the center of a mesh → Mx=26.62 kN.m/m
100 kN force on a node → Mx=48.12 kN.m/m
Distributed pressure on a mesh → Mx=26.62 kN.m/m
1/4 force on 4 nodes → Mx=26.62 kN.m/m
In conclusion, we realize that modeling a distributed load by a point force can be very detrimental, especially if this load is applied to a node of the mesh: it is better to use pressure (knowing that the software distributes the loads to the nodes), or alternatively to split the force into several loads to avoid the potential peak force.
Tip: Avoid modeling distributed forces by a point resultant for the calculation of forces in slabs or oneway joist slab!
Make sure that the mesh size is in correct proportions with both the thickness of the plate and the impact area of the smallest load.
In addition, to illustrate the different results that can be obtained for an extremely simple case, the Working Group calculated the forces generated by a Br wheel of installment 62 Title II, using several software programs and several smoothing approaches. The study is available in part 3 or directly below: example of modeling a Br wheel.
Punching
It has already been evoked, several times, the fact that finite elements did not deal with certain subjects such as the shifting of moment curves, the limiting of maximum demandclipping of shear forces close to the supports or the punching checks.
Except in special cases, therefore, the checks related to punching still have to be done manually. We will illustrate this on the example of the load on the oneway joist slab bridge shown above.
The modeler might be tempted to reason about the shear stresses averaged over the punched surface from the FE calculation. We refer to the Limit state design (LSD), also known as Load And Resistance Factor Design (LRFD), which has the advantage of being simple. The comparison stress is simply the load divided by the perimeter of the impact which is diffused at the average layer and by the thickness of the slab.
Let τ=100 kN/(4*0.25 m)/0.25 m=400 kPa (the slab is 25 cm thick).
The FE calculation, on the other hand, results in smoothed stresses (here in the most unfavorable zone, not strictly on the load diffusion perimeter!) of the order of:

τ=5.33 kN (integral over 0.125 m, read below)/0.125 m/0.25 m= 171 KPa;

τ=5.42 kN (integral over 0.125 m, read below)/0.125 m/0.25 m= 173 KPa.
We are far from the 400 KPa that the regulatory approach to the BAEL gives  taking into account FE values would not be safe. But these are two different approaches: shear in the slab is used to calculate the shear reinforcement and punching is another type of verification.
For the calculation of reinforced concrete structures, it should be noted that the peaks are caused by point forces or supports.
3. Peaks caused by mesh size problems
The geometrical marking of the peak often makes it possible to detect the causes.
Example of an inconsistency of results related to modeling: a graphical construction can make us think that an edge is common to two shells, whereas in reality there is an extremely small shift that leads to aberrant results. To illustrate this case, we extend the slab of an example above by another rectangular element that we consider to be strictly in the same plane as the previous one. The results should alert us that some peaks appear:
By zooming, we can see, on the right side, that the connection is only made on a few points (corresponding to a geometrical tolerance of the software) while on the left side there a very small shift (and which is caused in this case!) but which prevents the connection.
You can see in the figure below that not all the nodes on the common edge are connected: this should alert us.
The observation of the deformation should also alert us:
In case of difficulties, the user can himself or herself create beforehand the nodes of the joined edge, if necessary with bar elements that will be deleted once the mesh is reliable and fixed. Although this approach may seem tedious at first glance, the time saved can be well worth it.
We must remain cautious about the use of automatic corrections proposed by some software (in this case the linking of nodes through kinematic links) which can lead to local stress peaks. In this case, it is better to erase some shells and start again some parts of the model. See also § C.3.7.
D.5.2.2 Second Analysis: Study of steel sections perpendicular to the peak.
To help us locate the origin of the peak, we can analyze the steels details on each face, in both reinforcement directions. Indeed, the analysis of the steel sections of Axi, Axs, Ayi and Ays perpendicular to the peak quickly provides important information:

high NXX and NYY membrane stresses are detected by large and equal steel sections on both sides. If all 4 steel sections are equal, the element works in shear in the NXY plane;

high bending moments MXX and MYY can be detected by important steel sections on one fiber and very weak ones on the other one.
To illustrate this, the table below summarizes the consequences on the reinforcement of each stress component:
The code X indicates an important value of the steel section, the code 0 a low value.
From there, the analysis described below is carried out.
D.5.2.3 Third Analysis: Effort analysis perpendicular to the peak
If the two previous analyses are not enough to explain the peak, it is then necessary to study the effort components in detail to determine their intensities and identify cases of dimensional loads.
This analysis, which is often long and delicate in the case of complex modeling, can be greatly simplified for simple modeling by carrying out effort mapping or local cuts.
D.5.3 Peak Resolution: determination of final reinforcement
After having understood the functioning of the structure and mastered the effort paths, the engineer has all the elements to solve the peak and deduce the strictly necessary reinforcement.
D.5.3.1 Cases where maximum limitingclipping or smoothing are not possible
In some cases, the study of the effort path shows that the peaks are real and cannot be limitedclipped. This is the case of lintels which show very important peaks at their ends, which is logical because they are fixed end beams that must be calculated according to the mechanics of materials rules and reinforced concrete (fixed end beam subjected to constant shear).
D.5.3.2 Limiting Clipping moments on supports
Article 9.5.3.2.2 of EC21 allows to limitingclipping of the moments in the plane of the supports (walls).

Beam resting on a column  example 1
Consider the example of a slab with 2 spans of 6 m which are uniformly loaded by 25 kN/m².
Theoretical moments are 11.2 kN.m on supports and 6.3 kN.m in span.
The FE software shows a peak of 12.1 kN.m on the support, which is real.
Limiting Clipping at the plane of the wall:
To obtain the moment at the plane of the wall (thickness = 20 cm), the user has several solutions:

either making cuts at the support level;

or not visualizing the support area (but this requires an adaptation of the mesh);
Moment Clipping at the plane of the supports = 10.6 kN.m

or adapting the mesh in order to make the width of the supported mesh equal to the thickness of the wall, as shown below.
The moment in the center of the mesh will then represent the moment at the plane of the support.
The same type of reflection and approach is to be carried out in order to reduce the transverse shear effort, if necessary.

Beam resting on a column  example 2
The example below shows two ways of modeling a simple columnbeam structure to highlight the effects of the real dimensions of the structures.
The principle is as follows, for an outofplane width of 1 m:
Firstly, a wireframe model is used to calculate the moments at the central support (truncated view):
The moment on the central support is 82.09 kN.m, giving an extreme upper and lower fiber stress of 6M/(bh²)=(6*82.09)/(1.0*0.5²) = 1970 kPa = 1.97 MPa. The stress in the plane is (6*70.00)/(1.0*0.5²) = 1680 kPa = 1.68 MPa.
Below is the result (in stresses) of the same calculation performed on a model with plates, locked in outofplane displacement. This model allows to represent the behavior in section, with the real thickness of the elements. (truncated view).
The stress obtained in the upper fiber is 1.58 MPa. (The lower fiber stress is not representative because it is calculated in a singular area that constitutes the right angle).
This model highlights the legitimacy of limiting clipping in practice.
D.5.4 Reinforcement smoothing
The following method, derived from a common practice in the field of nuclear civil engineering, provides simple rules for smoothing the reinforcement.
Its application is nevertheless subject to the engineer's judgment. In particular, in the case of floors of ordinary buildings or bridge slabs, these values are probably too favorable and it seems possible to reduce the size of the smoothing to half of the values below, therefore limiting this size to 2h (h being the thickness of the slab).
In this context, the smoothing of the longitudinal and shear reinforcement sections must be carried out:

between adjacent elements (and not successive in relation to the reinforcement direction); the smoothing is done perpendicularly to the reinforcement direction;

over a reasonable distance (engineer's judgement) and less than a value that is correlated to the plate thickness and the plate span.
The current common practice is to average the results of three elements: the element for which the maximum is observed and the two adjacent elements, limiting the width over which redistribution takes place to 4 times the slab thickness (see table below).
E.g.: if the size of the elements is 1 m, averaging over three elements is the same as averaging over a width of 3 m. For a 0.5 m slab, this width is limited to 4 times the thickness of the slab (2m) which leads to averaging over only two elements.
The distribution width should also be limited according to the slab span or the wall height, because the smaller the span (or the height), the smaller the distribution width should be.
Proposed rules for smoothing the longitudinal reinforcement peaks of slabs (resulting from N, M forces)
They are presented in the following table. They are valid for a slab subjected essentially to membrane forces and bending moments due to outofplane distributed loads, and with a sufficiently fine mesh size that has:

an odd number of elements in both span directions;

at least 5 elements according to the small span (7 elements if possible);

a mesh size equal to the thickness of the loadbearing elements.
Width over which longitudinal and transverse reinforcement sections can be smoothened 
Limitation of the distribution width as a function of the plate thickness h 
Limitation of the distribution width according to the plate span L 
Zone where efforts can be redistributed in both directions 
4h 
0.5L 
Zone where the redistribution of efforts can only be done in one direction (at the edge of the shaft) 
2h 
0.25L 
For outofplane concentrated loads, the distance of the load from the support and its diffusion must also be taken into account.
Reinforcement peaks are frequently located at the edge of the shaft, in which case the redistribution of forces can only be in one direction and therefore over limited widths:

if additional reinforcement is required, it is always better to place it as close as possible to the shaft edges.

if, after smoothing the reinforcement as specified above, the current reinforcement is sufficient, it is however recommended to place additional reinforcement at the edge of the shaft if more than one current reinforcement is cut by the shaft in one of the two directions.
In the case of small openings (sleeves in particular) that fit into the reinforcement mesh or cause the interruption of a single reinforcement, it is possible to dispense with additional reinforcement.
Proposed rule for smoothing longitudinal reinforcement peaks in membrane elements (walls)
For elements subjected to tensile membrane stresses, redistribution can only take place in one direction.
Width over which longitudinal and transverse reinforcement sections can be smoothened 
Limitation of the distribution width according to the plate thickness h 
Limitation of the distribution width according to the plate span L 
Effort can only be redistributed in one direction (at the shaft edge). 
2 h 
0.25 L 
In bracing walls subjected to an axial bending moment perpendicular to the wall plane, the tensile membrane stress varies linearly.
When smoothing the reinforcement peaks, the smoothed reinforcement must be extended over a length large enough to maintain the bending capacity: F1 x d1 > F0 x d0, where:

F1 = Resulting stress taken up by the reinforcement after smoothing over the length L;

d1 = distance between the resultant F1 and the zero moment point;

above parameters with index 0 = before smoothing.
For elements subjected to membrane shear, it is possible to transfer part of the required section AX to section AY, if it is overabundant and vice versa. The Capra Maury method, which optimizes the sum of the reinforcement sections AX + AY, is used in common software.
The AY crosssection can sometimes be determined by the minimum reinforcement condition and the expected AY crosssection is then bigger than the required AY crosssection from the strength calculation, thus allowing a redistribution of reinforcement crosssections. The strength of the section should then be checked with the new reinforcement sections.
Conclusion concerning the smoothing of longitudinal reinforcement peaks
In all cases, it is necessary to take into account the origin of the steel requirements by analyzing the stresses (Nxx, Nyy, Nxy, Mxx, Myy, Mxy) and to interpret the results casebycase, with a concrete approach.
In general, the wall reinforcement comes mainly from membrane forces (Nxx,Nyy,Nxy), the bending forces (Mxx,Myy,Mxy) being then negligible. On the other hand, floor reinforcement is mainly due to bending forces (Mxx,Myy,Mxy), and in some cases by membrane forces (Nxx,Nyy,Nxy) when the building is subjected to horizontal forces (wind, earthquake) or irregularities (wall beams).
A wall beam is a good example of an irregularity producing horizontal forces in floors: the following example is based on the wall beam resting on 2 columns studied in chapter D.4.5, but with a lower floor; the tensile stresses in red show that the bottom tie is formed not only at the base of the wall but also in the lower floor:
A wall beam is a good example of an irregularity producing horizontal forces in floors: the following example is based on the wall beam resting on 2 columns studied in chapter D.4.5, but with a lower floor; the tensile stresses shown in red show that the bottom tie rod is formed not only at the base of the wall but also in the lower floor:
Tractions are even more important (here +50%) in the case of openings at the wall base:
Proposed rule for smoothing shear reinforcement peaks (the shear being perpendicular to the elements)
Generally, peaks in shear forces which are perpendicular to the elements occur at the intersection of several plates.
Reinforcements shear stress peaks often occur when shear stress is concomitant with high traction.
As a reminder, for the justification of reinforced concrete, the concomitance of a shear force with a traction requires particular attention because it means that there is no compression strut formation in the concrete and therefore a risk of breakage.
The resolution of this type of peak requires to look back at the stresses, to average the shear and normal stresses and to recalculate the reinforcement.
Approach illustrations:
Example 1:
Example 2:
The reinforcement peaks read on the charts are smoothed according to the following principle:
Smoothing principle of the local peaks of longitudinal reinforcement
Smoothing principle of the local peaks of longitudinal reinforcement
Smoothing principle of the local peaks of transverse reinforcement
D.5.5 Stress distribution in beams and slabs
It is important to remember that, with few exceptions, the calculation of the internal forces in the elements is a linear elastic calculation with a constant concrete modulus.
Peak stresses often determine a cracking of the reinforced concrete section and thus a local reduction of the peak stresses and a redistribution of the stresses.
It is sometimes useful  and it is accepted  to redistribute bending moments. A distinction must be made between:

the SLS (limited redistribution possibility, taking into account the weakening of the section’s stiffness due to concrete cracking in the highly stressed area of the peak);

the ULS (possibility of wider redistribution: same phenomenon as the one taken into account for the SLS with the addition of the wider redistribution possibilities indicated in Eurocode 2 in paragraphs 5.5 "Linearelastic analysis with limited redistribution of moments" and 5.6 "Plastic analysis".
However, it should be noted that the professional recommendations for the application of standard NF EN 199211 (NF P 187111) authorize, for buildings, to use in the SLS, a moment redistribution with the same redistribution coefficients as in the ULS.
Within the Eurocodes, the ratio δ of the moment after redistribution to the elastic bending moment must be greater than or equal to 0.7 for class A longitudinal reinforcement and 0.8 for class B or C.
This redistribution of moments in a continuous beam is possible if the beam does not participate in the bracing. It is more delicate if the beam belongs to a portal frame (beware of the elastic moments coming from the portal frame effect).
We must not forget to take into account the impact of the redistribution of bending moments on shear forces.
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