# Analysis

Version 19.1 by Fredrik Lagerström on 2020/03/27 10:43

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Depending on the current FEM-Design module you can do different calculations: displacement, internal forces, stresses, stability, imperfections, stability analysis, eigenfrequencies and/or seismic analysis. Some extra settings such as cracked-section analysis, non-linear behaviour etc. are also available for certain modules.

 Analysis type/settings Analysis for load cases Analysis for load combinations Analysis for maximum of  load groups Imperfections Second order analysis Stability analysis Eigenfrequencies Seismic analysis Non-linear behavior Cracked-section analysis Peak-smoothing algorithm

Table: Analysis features by FEM-Design Modules

Analysis can be done independently from any design calculations by entering to  tabmenu and clicking  Calculate command, or together with designs (RC, Steel or Timber) with the same command.

Figure: Analysis calculations

Analysis settings contain general and calculation-dependent settings. This chapter summarizes these settings and their effect on the result. Clicking OK runs Analysis according to the settings and selected calculation types. Other chapters introduce the display of results and their documentation (such as listing results in tables).

### General Analysis Settings

#### Finite Element Types

In the 3D modules, you can choose between “standard” and “accurate” 2D element types. With standard elements you can run 4-times faster but less accurate analysis than with the fine elements.

#### Peak Smoothing

To solve singularity problem in analysis results (internal forces), it is not enough to create peak smoothing regions in the finite element mesh. The use of the peak smoothing algorithm in the calculations have to be allowed. Without that permission, peak smoothing regions cause only mesh refinements (densifications) around objects.

Figure: Peak smoothing algorithm for Analysis

#### Setup calculation by load combinations

The calculation of the load combinations can be run with different options. They can be set in Calculations dialog by selecting the Load combination items and clicking   on Setup by load combinations.

The User has the opportunity to choose

• Which load combination should be calculated (Calc)
• Non-linear elastic calculation (NLE),
• Plastic analysis (PL),
• Non-linear soil (NLS),
• Cracked-section analysis (Cr.),
• Second order analysis (2nd),
• Imperfection calculation (Im., the selected shape will be taken into account in Second order analysis)

For example, in practice it can be useful to set 2nd order analysis only for the ULS and Cracked-section analysis only for the SLS combinations.

The above mentioned calculation types are described in details below.

Non-Linear Behavior

Non-linear behavior of supports (e.g. uplift), connections and truss members (e.g. tension-only) can be considered in analysis calculations (for load-combinations, imperfections and stability) by ticking NL checkbox at Calculations > Analysis > Load combinations > Setup load combinations.

Uplift” can be modeled both in 2D and 3D design modules by defining compression-only support/connection (tension = 0 (free)) and by selecting non-linear calculation for a load combination.

There is a possibility for the user to set the maximum iteration number of nonlinear calculation in Non-linear calculations tab in Setup load combination calculation dialog.

Plastic Analysis

In FEM-Design 3D Structure there is a plastic calculation option by the setup of load combinations.

Plastic calculation is available for trusses, supports and connections and edge connections of all shell elements (Plane plate and wall, Profiled plate and wall, Timber plate and wall, Fictitious shell).

The options above are considered only for load combinations calculated as non-linear elastic. Plastic behaviour is considered for load combinations calculated as non-linear elastic + plastic. See more details in the next chapter.

For further information check the documentation.

Cracked-Section Analysis

Cracked-section analysis means that the displacement of RC plates, walls, columns and beams can be calculated based on their cracked state and designed reinforcement.

Figure: Iteration steps of cracked-section analysis

Cracked section analysis for load combinations is available by ticking the Cr. checkbox at Analysis > Calculations > Load combinations > Setup load combinations > By load combinations

The iteration process settings available at Analysis > Calculations > Load combinations > Non-linear calculations:

For example, 20% means 5 load steps (= 100/20[%]). Less percentage generates more steps and more running time.

• Maximum iteration number:

The value must be in range 1 and 100.

• Allowed displacement error [%]:

Iteration ends, when the relative displacement error becomes less than the allowed value.

Second Order Analysis

In the   and   modules, 2nd order theory can be applied for load combination calculations of 3D structures. The 2nd order analysis considers the placement of the loads that changes together with the displacement, so it results additional moments derived from the new load positions.

To allow the 2nd order analysis for load combinations, just tick the 2ND checkbox at Calculations > Analysis > Load combinations > Setup load combinations > By load combinations

The 2nd order analysis is recommended to be done together with imperfection calculation. In Setup load combinations dialog, choose load combinations which you would like to apply the 2nd order theory for, and give the number of imperfection shape (simultaneous or previous calculation for imperfection is needed) you would like to consider for the 2nd order analysis.

If the loads are too large when using second order analysis, the program stops the calculations with an error message.

#### Diaphragm calculation

If at least onw diaphragm is defined int he stucture, User can choose one of the following diaphragm calculation options:

• None
• Rigid membrane
• Fully rigid

The difference between them is demonstrated by the following pictures. See details in the documentation.

### Analysis for Construction stages

User can start the construction stage calculation at Analysis/Calculation/Construction stages. There is two calculation method, so called Incremental “Tracking” method and “Ghost” structure method.

When incremental method is chosen, the model is built stage-by-stage. In case of “ghost” structure method the full structure is in the calculation, but stiffness of those structural parts which aren’t in the specific stage is highly reduced.

Incremental “Tracking” method

“Ghost” structure method

### Analysis for Load Cases and Combinations

Analysis calculations can be done by load case and/or load combination. The next table summarizes the results available for load cases and load combinations by FEM-Design modules.

 Analysis result Translational displacements (Plate/Beam) (Wall) (Wall) (Wall/Column) Rotational displacements (Plate/Beam) (Wall) (Wall) (Wall/Column) Reactions Connection forces Bar internal forces (Beam) (Column) Shell internal forces (Plate) (Wall) (Wall) (Wall) Bar stresses (Beam) (Column) Shell stresses (Plate) (Wall) (Wall) (Wall)

Table: Basic analysis results by FEM-Design Modules

#### Displacements

Depending on the current FEM-Design module, the program calculates and displays the model displacement from linear or non-linear (for RC elements: cracked-section analysis) analysis. There are two types of displacement results: translational or rotational. For bar elements, the motion and rotation components can be displayed separately (Detailed result) by direction (local axis).

In  Plate, the displacements are calculated for the plate regions and beam elements, and the motion is parallel with the global Z direction, so perpendicular to the plate regions. Only reactions can be asked for columns (point reaction) and walls (line reaction).

In  Wall and  Plane Strain, the motion is parallel with the calculation plane of the wall regions.

In  PreDesign, although the 3D model contains all types of elements, displacements are calculated for the vertical elements such as columns and walls.

#### Reactions

Depending on the support types, the program calculates the reaction forces and/or moments in the supports by direction component, their resultants and the resultant at the support’s center of gravity of line and surface supports.

The  Plate module calculates reactions in columns and walls too above the point/line and surface supports.

The available result components:

Fx’ / Fy’ / Fz’                                            - reaction force in the local x’/y’/z’ axis of the support (group);

Fr                                                                                                                             - resultant of the reaction force components (support group);

F                                                                                                                                                            - reaction force of the single support;

Mx’ / My’ / Mz’          - reaction moment around the local x’/y’/z’ axis of the support (group);

Mr                                                                                                                           - resultant of the reaction moment components (support group);

M                                                                                                                             - reaction moment of the single support.

#### Connection Forces

Similarly to reactions, the program calculates the forces and/or moments in the connection objects (Edge connection, Point-point connection and/or Line-line connection) by direction component and their resultants.

The available result components:

Fx’ / Fy’ / Fz’                                            - connection force in the local x’/y’/z’ axis of the connection;

F                                                                                                                                                            - resultant of the connection force components;

Mx’ / My’ / Mz’          - connection moment around the local x’/y’/z’ axis of the connection;

M                                                                                                                             - resultant of the connection moment components.

The figure shows an example for displaying connection forces at the three connection types. The Fy' is displayed at the line (line-line and edge) connections, and the Fx' and Fz' for the point-point connection. (The color of a result component (e.g. Fy') is the same with the color of the local axis (e.g. y') associated to the component direction) . On the next figure shows the resultants.

Figure: Connection forces by connection type

Figure: The resultants (force and moment) for edge connection and line-line connection forces

#### Local stability results

After calculating the load combinations the Local stability results (Overturning of walls and Sliding) are available in Display results dialog.

Figure: Display local stability results

• Overturning of walls

Only those walls can be calculated which have at least one horizontal edge in the bottom and edge connection is defined for it. The result is expressed as a percentage:

• 0% belongs to the case when the vertical force acts at the centre of bottom edges,
• 100% belongs to the case when the vertical force acts at one of the corners,
• 1000% belongs to the case when the resultant is outside the wall edge.

Figure: Overturning of walls

Overturning of walls results are informative. Without accurate modelling it may lead to incorrect results!

• Sliding of edge connections

The result is the ratio of the design force and the friction capacity. The friction factor can be set in the edge connection dialog.

Figure: Sliding of a wall

Numerical example below will illustrate the Local Stability.

Properties of edge connections

Non-linear calculation (which allows uplift) is recommended to get correct result for local stability.

Displacement graph (as well as connection force) is the easiest way to check the uplift.

Overturning of wall

With the help of resultants of edge connections, wall’s overturning can be examined as below.

F=i=1nFi= -190-110= -300 kN

MC=i=1nMi+Fi* vi= -429+190*2-53-110*4= -542 kNm

e= MCF=-542-300=1.81 m

Utilization= ed*100 %=1.815.00*100=36%

Sliding of edge connections

Edge connection’s sliding is calculated in each edge connection separately by comparing the x’ component of the connection force as design force, and the limit  force calculated by the y’ components of the connection forces and the friction coefficient of the edge connection.

Flim=μ*i=1nfi*dxi= 0.3*(15*0.4+50*0.4+83*0.4+126*0.4)= 32.9 kN

Utilization= FFlim*100 %=51.632.9*100=157.1%

#### Bar Internal Forces

The program calculates internal forces and/or moments in the bar elements depending on the applied FEM-Design module.

The available result components:

N                                                                                                                              - normal (axial) force (local x’ axis of the bar element);

Ty’ / Tz’                                                                                                                - shear force in the local y’/z’ axis direction of the bar element);

Mt                                                                                                                            - torsion moment (around the local x’ axis of the bar element);

My’ / Mz’                                                                                - bending moment around the local y’/z’ axis of the bar element.

Truss members bear only normal forces (N).

The  Plate module calculates internal forces only for beams. Columns are point supports.

In  PreDesign, although the 3D model contains all types of elements, internal forces are calculated in columns.

#### Shell Internal Forces

Depending on the current FEM-Design module, the program calculates internal forces and/or moments in the planar structural elements

The  Plate module calculates internal forces in the plate regions and in the Global coordinate system:

Mx’/ My’                - bending moment around the global Y / X axis;

Mx’y’                       - torsion moment;

Tx’ / Ty’                 - shear force for the global X / Y normal and in the Z direction;

M1 / M2                 - principal moments;

M1/M2                     - principal moment directions.

Although Analysis calculations give results for the global Descartes system, internal forces can be asked and displayed in arbitrary (reinforcement) directions by checking design forces in case of RC design.

The  Wall module calculates internal forces in the wall regions and in the Global coordinate system:

Nx’ / Ny’               - normal force in the global X / Y direction;

Nx’y’                       - shear force in the global X-Y plane;

N1 / N2                  - principal normal forces;

N1/N2                     - principal normal directions.

The  Plane Strain module calculates only the shear stresses in the wall regions and in the Global coordinate system.

The  3D Structure module calculates internal forces and moments in the planar object regions (plate and wall) in their local coordinate system:

Mx’ / My’              - bending moment around the local y’ / x’ axis of the region element;

Mx’y’                       - torsion moment;

Nx’ / Ny’               - normal force in the local x’ / y’ axis of the region element;

Nx’y’                       - membrane shear force;

Vx’ / Vy’                - shear force for the local x’ / y’ normal and in z’ direction;

M1 / M2                - principal moments;

M1/M2                    - principal moment directions;

N1 / N2                  - principal normal force;

N1/N2                     - principal normal directions.

The  PreDesign module calculates the previous internal forces and moments but only walls.

#### Bar Stresses

FEM-Design calculates the normal stress in bar elements (beams, columns and/or truss members) with the following meaning:

Sigma x‘(max)        - maximal normal stress (tension);

Sigma x’(min)        - minimal normal stress (compression).

The  Plate module calculates stresses only in beams. Columns are point supports.

In  PreDesign, although the 3D model contains all types of elements, stresses are calculated in columns.

#### Shell Stresses

The program calculates stresses in the top, bottom and middle (so called “membrane”) planes of the planar elements. The meaning of top and bottom side depends on the position ( Plate module) or the local coordinate system (3D modules) of a region element.

Figure: The meaning of planes depending on region position

So, depending on the current FEM-Design module, we get results from the following:

Sigma x’, top / membrane / bottom                      - normal stress from Nx’ in top/membrane/bottom plane;

Sigma y’, top / membrane / bottom                      - normal stress from Ny’ in top/membrane/bottom plane;

Tau x’y’, top / membrane / bottom                      - shear stress from Nx’y’ in top/membrane/bottom plane;

Tau x’z’                                                                       - shear stress (x’ normal and z’ direction);

Tau y’z’                                                                       - shear stress (y’ normal and z’ direction);

Sigma vm, top / membrane / bottom                   - von Mises stress in top/membrane/bottom plane;

Sigma 1/Sigma 2, top / membrane / bottom       - principal stresses and directions in top/membrane/ bottom plane.

The x’, y’ and z’ directions are valid in the global coordinate system at  Plate and in the local coordinate system of planar elements in the 3D modules.

In  PreDesign, although the 3D model contains all types of elements, stresses are calculated in walls.

In  Wall and  Plane Strain, stresses are calculated only in the membrane plane.

#### Equilibrium Check

The program automatically checks the equilibrium of the analysis calculations. Statical equation is written to the origin [0; 0; 0] of the Global Coordinate System. It compares the sum of the reactions and the sum of applied loads. Equilibriums can be asked by load case and load combination.

Just click the  Equilibrium icon (in Analysis or Design mode), choose a load case or load combination to see the equilibrium check results.

Figure: Equilibrium check of Analysis calculations

If equilibrium error derives from an analysis calculation, the error will be appeared in percentage in the Error column by equation types (force (F) and moment (M)) and directions (x, y and z directions of the global coordinate system). “Error” shows the differences between the resultants of the queried loads and the calculated reactions.

### Analysis for Maximum of Load Combinations and Groups

Choosing Load combinations for Analysis automatically generates results for the maximum of load combinations too.

If you define Load groups and choose Maximum of load groups for Analysis, FEM-Design calculates maximum/minimum results (in all finite element nodes) from the most unfavorable combinations of the load groups according to the applied code.

So, maximum and simultaneous results of displacements, reactions, connection forces, internal forces (bar and/or shell) and stresses (bar and/or shell) can be calculated for maximum of load combinations and groups.

The symbol “+” and “–“ sign the direction of the maximal value in the valid systems: local or global coordinate systems (depend on the current FEM-Design module). Some examples for the meaning of “+” and “-“:

Displacement:

ez’                  - maximal uplift in global z’ direction in  Plate,

- maximum motion in the positive direction of element’s local system in 3D modules;

ez’                   - maximal depression in global z’ direction in  Plate,

- maximum motion in the negative direction of element’s local system in 3D modules;

Internal forces:

Mx’               - maximal bending moment around the y’ axis (global/local) in positive direction (= same direction with the axis direction).

Mx’                - maximal bending moment around the y’ axis (global/local) in negative direction (= opposite direction to the axis direction).

The next figure shows the meaning of simultaneous results.

Maximal Mx’                                                                  Nx’ belongs to maximal Mx’

Figure: The meaning of simultaneous results belong to a maximal value

Combination of Load cases that gives the maximum analysis results in Maximum of load groups can be listed in tables. Just use the  List command (Tools menu) for the Maximum of load groups result data.

### Deflection check for RC, steel and timber bars

A new checking criteria is available for reinforced concrete, steel and timber bars. Deflection utilization is calculated for load combinations, Maximum of load combinations and Maximum of load groups according to the user defined serviceability limit states.

This new result type is based on the displacement of the bars and the deflection limitation settings which can be defined by the so-called deflection lengths.

In the  Deflection configuration dialog, we can specify the types of load combinations/groups for which the deflection check is performed.

Deflection lengths are used to define those bar segments, where the deflection checking criteria/limitations are coincide. The “Simply supported” deflection lines are denoted with blue arcs below the bar, the ”Cantilever and column” types are orange and the “not relevant” types are black. Relative and/or absolute limit can be set for each length individually. If both are requested the dominant one will be calculated and displayed.

The first option we can set here is the behaviour of the lengths which affects the calculation method of the deflection. If we choose not relevant for a specific length, it will be excluded from the checking process.

For the better understanding of the next two options, namely the Simply supported and Cantilever mode let us consider the following example, a cantilever frame structure.

In the midspan we should use the Simply supported option, where we eliminate the rigid body motions in such a way that we connect the endpoints of the length, and measure the deflections of the middle sections from this imaginary line (red skew line on the picture above).

 As a consequence of this method, the deflections of the endpoints are zero, the dominant section is usually at the middle of the length.

On the cantilever, we would like to use the cantilever mode, where the dominant value of deflection on this length will be the difference between the maximum and minimum absolute deflection (in this example the largest distance from the red horizontal line). For columns the same calculation method is used, the only difference is that the deflection is measured in the horizontal plane (from the green lines).

 For columns only this (“Cantilever and column”) option is available.
 Deflection lengths

As deflection lengths correspond to only specific bar segments, they are independent on the bars in such a way that they can be longer or shorter than the bar itself. But why is this differentiation so important? The answer can be demonstrated with the following two examples. On the left of the picture below, only one beam is drawn, thus if the Relative limit would be calculated directly from the length of this beam, the results would be misleading.

In other words, in the L/? formula, instead of the length of the midspan or the cantilever, the whole length of the beam would be substituted. Therefore, we need two Deflection lengths to differentiate the L in the Relative limit formulae on the midspan and on the cantilever. In addition, the limit value also can differ for the two types of structure according to the National Annexes.

In the second case (to the right on picture below) imagine that our aim is to design a beam splice and check deflections. Two beams need to be drawn for the steel design, but of course during the deflection check we want to use the summed length of the two beams for the calculation of the relative limit value. For this purpose, we define one Deflection length over the two bars - this way we make correct calculations in both cases without any additional modification on the structure.

It is worth to note that in the second case we had two beams, but in contrary to the buckling lengths, definition and editing of deflection lengths can be performed on such set of beams, which are both parallel and continuous.

By default, deflection lengths are generated automatically. This procedure first search all the previously mentioned parallel and continuous beams sets, then intersect them with the edges/axes of the structural elements (beams, columns, trusses, plates, walls, line and surface supports) and point supports. In the majority of the cases deflection lengths obtained by this way are reasonable from engineering point of view, but in some cases we may want to modify them. A good example can be a structure consisting of two beams with a horizontal support, which should not be considered in the deflection checking process. The following flow diagram illustrates the modification of the two beams step by step. By default, as we can see in the upper picture, the automatically generated deflection lengths coincide with the beams because they are intersected with the horizontal point support. If we would like to have one deflection length over the two beams, we can draw it between the support groups using the Define tool, similarly to the buckling lengths. By this way, the new length substitutes the original ones!

 Deflection length has its own layer.

Deflection check button becomes active if Load combinations and/or Load groups are already calculated. The utilization results can be displayed from the New result dialog.

 The Deflection checking process considers only the straight beams and columns. For beams the deflection is measured along their own local z’ axis, for columns it is measured in the global horizontal x-y plane.

Results requested for a Load combination can be displayed both on the deformed and undeformed shape.

Due to the fact that the limit values of the calculation are controlled by the Deflection lengths, the result is constant along them. In other words we have one (dominant) utilization value for each Deflection length. Results for Maximum of load combinations and Maximum of load groups are only displayed on the undeformed shape of the structure.

### Imperfections

Imperfection calculation is run only for steel bar elements of the structure. Users can add imperfections to a structure in two ways:

• Imperfection modeled by defining loads (manual)

Place for example horizontal point and line loads on a multi-storey building to model imperfection manually.

• Imperfection calculation according to the formula EC3: 1-1 (automatic)

For load combinations, the program can calculates the probable imperfect shapes in real dimensions from the mode shapes (get from stability analysis) according to Eurocode. Second order analysis must be run by using imperfection. To do automatic imperfection calculations, activate Imperfections and set the required number of the imperfect shapes (Rqd. cell) for the load combination which you would like to run imperfection for.

Figure: Imperfection calculation by load combination

For the automatic imperfection calculation you got the buckling shape of the structure with real size in real dimension. Critical parameter assigned to a buckling shape is also available with the following meaning:

critical parameter = critical buckling force/actual load

or in other words:

if the critical parameter is bigger than 1, the structure or a part of it is sufficient to perform the stability analysis; if it is smaller it is not.

If the critical parameters differ a lot between the buckling lengths, the first buckling shape is the critical. If the critical parameter values are close to each other, it is your decision what structural part you check by its shape.

The factor defines the real imperfect shape, so:

imperfect shape in real dimension = factor * buckling shape

Figure: Automatic imperfect shape calculation

Before imperfection calculation, it is recommended to set minimum 4-5 division numbers (finite elements) for bars.

### Stability Analysis

In 3D modules, global stability of the structure can be analyzed automatically if it is requested. Similarly to Imperfections, the program calculates buckling shapes together with their critical parameters for selected load combinations.

To do stability analysis, activate Stability analysis and set the required number of the buckling shapes (Rqd. cell) for the load combination which you would ask stability results for.

If the Rqd. as positive is checked, program will calculate as many stability shapes as necessary to get required number of shapes with positive critical factor. Since it is an iterative method, maximum number of iteration steps can be set by the User in Max no. of iteration cell.

Figure: Stability analysis by load combination

As result, you got the buckling shape(s) of the structure with unit dimension. The greatest displacement value of the buckling shape is 1 and the others are the ratio of that.

Critical parameter assigned to a buckling shape is also available with the following meaning:

critical parameter = critical buckling force/actual load

or in other words:

if the critical parameter is bigger than 1, the structure or a part of it is sufficient to perform the stability analysis; if it is smaller it is not.

Figure: Buckling shape calculation

The last three columns shows the probability of the buckling shapes are global or local, where eH meant for horizontal displacement, eV for vertical displacement (global Z direction) and rZ for rotaion around global Z axis.

 1st shapeGlobal in horizontal direction
 3rd shapeGlobal in rotational direction
 4th shapeLocal in rotational direction

In the example below, the eH value of the first shape is 89%, which means it is probably a global buckling shape with horizontal displacement.

Displaying the result (see the leftmost inset above) and examining the buckling shape shows that this is indeed a case of global buckling with the horizontal displacement of the frame’s top.

The same structure’s second shape possesses a very high rZ value (99%), meaning this almost certainly is a global torsional buckling shape (shown in the middle inset).

The fourth shape’s eH, eV and rZ values are significantly lower, which implies it is a local buckling shape. As the rightmost inset shows, the assumption was correct (local buckling of both columns).

 Higher probability values shows high probability that the shape is global. If there are not enough shapes calculated, none might be global.

Before stability analysis, it is recommended to set minimum 4-5 division numbers (finite elements) for bars.

### Eigenfrequencies

#### Mass/Vibration shape

FEM-Design can do dynamic analysis by calculating vibration shapes of the structural model and the belonging eigenfrequencies and free vibration time values (periodic time).

To do dynamic calculation, activate Eigenfrequencies and just set the required number of the vibration shapes (Number of shapes cell), the active masses in X, Y or Z direction and the Top of substructure.
There is an option to semi-automate calculate the number of shapes to fulfill the 90% horizontal effective mass criteria with the Try to reach 90% of horizontal effective mass. One has to set the starting number of shapes, then the program will calculate them and will check if 90% rule is passed. If it’s not, the program will calculate more shapes and checks the rule again in another iteration. There are two possible ways to end the calculation:

• The 90% total effective mass is reached in horizontal direction
• The maximum iteration number is reached

Figure: Dynamic calculation

Dynamic calculation requires masses to be predefined.

Seismic analysis needs the eigenfrequencies calculations.

In Calculation / Eigenfrequencies dialog the user can set the level of top of the substructure. The masses will be neglected at and under this level.

In the mass centre of the masses the total mass is displayed with red circle.

To get the whole structure’s mass centre position set the level of the Top of the substructure a bit under the structure.

This function is useful to neglect the foundation mass in the eigenfrequency calculation so the total mass contribution in Modal analysis can reach >=90%.

Results of Eigienfrequencies calculation:

Masses                   - mass matrix of point masses and/or masses calculated from load cases converted into finite element nodes;

Vibration shape   - vibration shape and associated eigeinfrequency (Frequency) and periodic time (Period).

Figure: Results of dynamic calculations

Before dynamic analysis, it is recommended to set minimum 4-5 division numbers (finite elements) for bars.

#### Shear center result

FEM-Design can calculate Shear centers for each storey of a building. The figures below show a shear center result of an Eigenfrequency calculation.

 For displaying shear center, diaphragms are needed for every storey.

 Each displayed shear center represents the result of a calculation based on the storeys below that storey. For example, the calculation of the center displayed on “Storey 2” takes also “Storey 1” and “Foundation” into account.
 Values in the Tooltip:alpha: the angle between the X axis of the global coordinate system and stiffness directions,EI1, EI2: stiffnesses in principal directions of strains,x, y, z: the global coordinates of the shear center.

Shear center results can be listed in List tables dialog/Analysis/Eigenfrequencies/Shear center.

### Seismic Analysis

#### Methods

In the   and   modules, seismic calculation offers the following methods to the users according to Eurocode 8.

• Modal response spectrum analysis (“Modal analysis”)
• Lateral force method / Equivalent static load method

This method can be used to calculate the seismic effect in horizontal plan, x’ and/or y’ direction. The main point is to calculate “base shear force” taking into account the base vibration period and design spectrum in x’ or y’ direction, which is distributed into those nodes of the structure where there are nodal masses. The “base shear force” formula is taken from EC-8 4.3.3.2.2(1)P. The “base shear force” is nothing else than the total seismic force of inertia that acts between the ground and the structure, and it can be distributed in two ways:

• Linear shape method (Static, linear shape)

The distribution of the “base shear force” happens according to a simplified fundamental mode shape, which is approximated by horizontal displacements that increased linearly along the height.

• Mode shape method (Static, mode shape)

See the detailed description and the applied theory of all calculation methods in the Theory book. This guide introduces only the user interface and the steps of seismic analysis.

#### Steps of Seismic Calculation

The suggested steps of seismic calculation are the followings:

1. Mass definition

To calculate the seismic effect, it is necessary to know the vibration shapes and corresponding periods (except the Static, linear shape method). To perform dynamic calculations, it is necessary to define mass distribution which can be defined as concentrated mass or load case-mass conversion.

1. Design spectrum definition

The program contains predefined design spectra according to EC8, but you can also define your own spectra. Use the command Seismic load (Loads menu).

1. Dynamic calculation

Dynamic calculation should be done before performing seismic calculation, which gives sufficient vibration shapes of the structure. Although setup for the seismic calculation can be done at any time, but the seismic calculation could be performed only after Eigenfrequency calculation. Run dynamic calculation under Analysis by setting the required number of vibration shapes.

It is suggested to set the finite element number bigger than 1 at bars (Finite elements/ Division number).

1. Settings of seismic calculation

A national code always provides which seismic calculation method has to be performed for different structure, where and when it should be performed and what other effects to be considered (e.g. torsional effect, P-∆ effect). FEM-Design provides three types of calculation methods (depending on the applied code):

Figure: Settings of seismic analysis

• Static, linear shape

As a matter of fact, eigenfrequency calculation is not necessary for this method, because giving the base period time in x’ and y’ directions (Tx’ and Ty’) is enough for the calculation. Practically, eigenfrequency calculation performs before setting this data, but these data can be defined using experimental formulas as well. Investigation can be done in x’ or y’ direction, or both together.

You may set the calculation direction to be performed by selecting the desired direction. To set the desired x’-y’ direction, you should give Alfa ( is the angle between the global X and x’; see Direction of the horizontal effect).  =0.0 means x’-y’ directions coincide with global X-Y directions.

This method is unusable, if the whole foundation is not in same plane or the horizontal foundation is elastic. In these cases, Static, mode shape or Modal analysis should be used.

• Static, mode shape

In this method the distribution of “base shear force” happens according to fundamental mode shapes (base vibration shapes).

The table shows how to select the base vibration shapes. It contains all mode shapes (No), the vibration time (T(s)) and effective masses of the mode shapes in x’ and y’ directions (mx’(%) and my’(%)). The effective masses are given in a relative form to the total or reduced mass of the structure. The reduced mass means the total mass above the foundation or above the rigid basement. The value of the effective mass refers to how the mode shape responds to a ground motion direction, so the effective mass shows the participation weight of the mode shape.

Select (or double-click on it) one mode shape in x' or/and y' direction(s) (mx’ /my’). (Yellow field color shows the activation.)

It is recommended to select that mode shape which gives the largest effective mass as the fundamental mode shape.

The calculation of “base shear force” is performed according to the total mass of the structure and not to the effective mass.

• Modal analysis

The essence of the method is the calculation of the structural response for different ground motions by the sufficient summation of more vibration shapes. Method gives possibility to take into account full x', y' and z (=global Z) direction investigation.

In the table, more vibration mode shape could be selected in x’, y’ and directions if necessary. The last row (orange cells) of the table shows that how large is the sum of the considered effective masses compared to the total or reduced mass of the structure in a given ground motion direction.

According to EC8, sum of the effective mass of the choosen mode shapes (at least in horizontal direction) should reach 90% of total mass. Additionally every mode shape has to be taken into account where effective mass is larger than 5%.

If the sum of the effective mass is much smaller than 90%, eigenfrequency calculation should be done for more shapes in order to reach 90%.

Lots of mode shapes should be ensured to reach the 90% of total mass in vertical direction. It is highly recommended to check the national code, whether it is necessary to examine the vertical effect or it is not.

The mode shapes which have small effective mass may be neglected, because their effect in result is very small, but calculation time increases.

Summation rule by directions

According to the EC8, the summation rule in the individual directions can be selected. In all other codes always the SRSS rule is used for summation (there is no choice). Read more about SRSS and CQC summation rules in Theory book. If the Automatic is selected, the rule selection procedure is as follow:

-     Always three directions are investigated (if more than one mode shape is selected in a column), where all mode shape is independent from each other or not.

-     If at least one dependent situation exists in a direction, the program automatically uses the CQC rule for all mode shape in that direction, otherwise SRSS rule is used.

Direction of the horizontal effect

Codes generally speak about seismic calculation in X-Y directions. These directions give the maximum effect, if the mass and elastic properties of the structure ensure that the calculated mode shapes lay in X-Z or Y-Z plane. But it is not always achieved in practice.

To achieve the unfavorable direction, where the results of ground motion are maximum, the program gives the possibility to set x'-y' direction for the seismic horizontal effect (Alpha). The program suggests the Alfa value, if you click on Auto button. It finds the most unfavorable direction, where any of the mx’ and my’ is zero and the other is maximum in the same row (same shape). But, there is a rule: the direction can be ensured only for one mode shape, so the program selects the row where the effective mass is the maximum. If manually definition is chosen, give an angle for Alfa and press the button Set.

On the left hand side figure you can see a badly adjusted x’-y’ direction (Alpha = 0). Appling Auto button, the program arranges the direction for the 58.5% effective mass my’ and correct it to 78%.

Figure: Settings of Alpha

Effective mass

The modal effective masses can be compared to the total mass or reduced mass at Eff. mass:

In FEM-DesignReduced mass” means the difference between the total mass of the structure and the basement mass. The basement mass is the sum of all masses which lay on the foundation level (set at Seismic load/ Others).

EC8 defines the total mass without basement (Reduced mass). The effective masses are generally compared to the Reduced mass, but this is not valid for the massive basement with elastic foundation. If the above mentioned situation is the case, it might happen that the sum of the effective masses of a column is larger than the 100%.

It is uninterested in the calculation point of view, that effective masses are compared to the total or the reduced mass, because these values are given in percentage and only gives information, that which mode shape is the fundamental or which shapes are dominant in a given direction.

At Options, more calculation properties can be set:

Combination rule

The combination rule of effects in the x', y' and maybe z directions can be set here. You can choose from two possibilities.

Consider torsional effect / Consider second order effect

Additional effects can be taken into consideration during seismic calculation. See the detailed description of these effects in Theory book.

The calculation of both effects needs the definition of storeys.

1. Seismic calculation

After choosing a calculation method and setting its properties, activate first Seismic analysis under Analysis and then press OK.

#### The Results

Besides displacements, reactions, connection forces and internal forces, the program calculates the Equivalent loads and the “Base shear force”. Results can be displayed by vibration shape (selected at calculation settings), from torsional effect, from sums by direction and from the total sum (Seismic max). If equivalent loads are displayed, also the “base shear force” appears on screen (in grey color). Torsional moment effect on the whole structure can also be displayed, if torsional effect was taken into consideration during calculation.

Figure: Results of Seismic analysis

Because of the square combination rule, the results summed by direction (Sum, x’, Sum, y’, Sum, z) and the total sum (Seismic max) give only positive values, so absolute maximums. Also, because of combination rule, the displacement components and the internal forces in one point are not simultaneous results.

#### Summary of Static and Seismic Effects

Seismic effect can be combined with static loads in two ways:

-     By defining new load cases contain equivalent seismic loads to take them into consideration at analysis or design calculations as real static loads,

The x' and y' directional loads (also torsional moments) equivalent to the horizontal ground motion can be converted to load cases. “Seismic,...”-type load cases behave as static loads: they can be combined, they can be added to groups, and they can be taken into consideration at stability, imperfection and design calculations. As you see in the list of load case types, the seismic effects can be considered with positive and/or negative sign.

Maximum seismic effect in load combinations

The total, the maximum seismic effect (see Seismic max at Equivalent loads) can be added to load combinations. Start the command Load combinations (Loads menu). Apply Insert case(s) on a predefined or new load combination, choose “(Seismic max)”, define a load factor and press OK.

Maximum seismic effect as load group

The maximum seismic effect (Seismic max) can also be added to groups in all codes. Define a group as “Seismic”. The program automatically takes the “(Seismic max)” into consideration with +/- values in the generation of the most unfavorable results.

Figure: Maximum seismic effect defined as load group

### Footfall analysis

This calculation method allows for checking the structure's response for an excited vibration.

The calculation can be started in Analysis/Calculations/Footfall analysis. The settings for the calculation can be found under the Setup.... Here one can select one of the three available methods:

• Self excitation
• Full excitation
• Rhythmic crowd load (Load case shall be selected with this method)

It is important to choose the correct method (Self excitationFull excitation or Rhythmic crowd load), because the analysis will run according to the selected method, even though it’s possible to define both self excitation regions and full excitation points.

Results

After Footfall calculation, one can check the Eigenfrequency results (masses, vibration shapes), the nodal accelerations and the nodal response factors.

In Detailed result one can see the response factor – frequency diagram of a point, if a Response factor or Acceleration result is shown. Every previously placed point is remembered and named. These points can be deleted with Delete option. Their name and font can be set by Properties option.

These results can be listed in Analysis/Footfall analysis.

### Investigate

If a warning message appears during calculation (e.g. Load mismatch or Finite element mesh problem) there is a possibility to check and fix the error by navigating in the model with Investigate.

The following pictures show a badly defined load and how can the user check and fix the error with Investigate function.

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