Storey Checks - Theoretical Principles#

General Information#

User Story#

For the assessment of structural seismic response, storey-based design criteria are an important ingredient in the engineer’s toolbox. They help to quickly reveal the problematic parts in the layout of the structure and to identify whether a redesign may be required. Storey Checks accordingly provide an overview and additional calculation procedures for the storey results in the form of tables and corresponding plots.

Overview#

Storey checks can be structured into the following groups:

General Tables: Storey Levels (Synopsis Table), Centre of Rigidity and Centre of Mass
Regularity Checks: Soft Storey Check, Weak Storey Check
Advanced Checks or Load Dependent Checks: Second Order Check ( $P{\text -} \Delta \,$ Effects), Storey Shear Forces, Storey Drift Check, Storey Displacements

Note

Prerequisites for the Regularity Checks are the definition of the storey levels at the modelling stage and a subsequent calculation of centre of rigidity and centre of mass. In addition, the Advanced Checks require a load case selection for the total gravity load and a list of load cases for the storey drift and storey shear force selection.

Design Code#

The following two design codes are supported:

EN, acc. Eurocode 1998-1:2004 [1]
US, acc. ASCE/SEI 7-16 [2]

The design code selection has direct influence on the formulas used for the load dependent checks, and on the default values for the following parameters:

Shared Parameters#
Parameter	Description	EN	US
QD	Displacement behaviour factor $q_d$ (EN), Deflection amplification factor $C_d$ (US)	1.500 [-] -	- 1.500 [-]
IMP	Importance factor	1.000 [-]	1.250 [-]

Second Order Check Parameters#
Parameter	Description	EN	US
THT1	First threshold	0.100 [-]	0.100 [-]
THT2	Second threshold	0.200 [-]	0.250 [-]
THTX	Extreme threshold	0.300 [-]	0.250 [-]

Storey Drift Check Parameters#
Parameter	Description	EN	US
D2HX	Allowable threshold for the drift ratio	0.0075 [-]	0.0150 [-]
NRED	Reduction factor	0.500 [-]	1.000 [-]
POI	Point of interest for determining the drifts	COM - Center of mass	MAX - Maximum drifts

General Tables#

Storey Levels (Synopsis Table)#

Storey Levels table offers an overview of all the storeys defined in the project and it is always printed in the output of the Storey Checks.

The following information is available for each storey (row):

Storey ID
Designation
Elevation
Height

Centre of Rigidity and Centre of Mass#

Centre of Rigidity (COR) and Centre of Mass (COM) Check delivers a table with the information about the X and Y coordinates of COM and COR for each storey. The previous calculation of COR and COM is required.

Hint

Ideally, calculation of COR and COM is triggered during analysis in your pre-processing tool. Otherwise it is possible to insert the following Teddy task in the project navigation before calculation of linear loadcases:

+PROG FEABENCH
HEAD
TASK TYPE SLVL
CRTL COR
CRTL COM
END

While COR is static, COM can be recalculated during earthquake design to account for applied loads or lumped masses if desired.

Centre of Rigidity and Centre of Mass table delivers the following information for each storey (row):

Storey extent in X and Y direction
X and Y coordinates of COM
X and Y coordinates of COR
Storey mass
Mass moment of inertia around the Z axis at COM
Radius of gyration

The corresponding plots of the x and y coordinates along the height of the building are also generated.

Regularity Checks#

Soft Storey Check#

Regularity of the storeys’ stiffness and mass along the height of the building can be checked with the Soft Storey Check. Stiffness in x and y direction are checked in 2 separate tables.

Stiffness tables deliver the following information for each storey (row):

Stiffness of a corresponding storey (S)
Stiffness of a storey above (S1)
Average stiffness of 3 storeys above (S3)
Stiffness ratio S/S1
Stiffness ratio S/S3
Check (possible options are: Regular, Soft, Extreme Soft)

The check options are determined based on the soft storey thresholds acc. ASCE/SEI 7-16 [2], Table 12.3-2, since EN 1998-1:2004 [1] doesn’t provide any quantitative specifications. The default values are set as follows:

Soft Storey Check - Stiffness Parameters#
Parameter	Description	Default value [-]
SR1	Threshold for soft storey classification for one storey above	0.700 [-]
SRX1	Threshold for extreme soft storey classification for one storey above	0.600 [-]
SR3	Threshold for soft storey classification for the average of three storeys above	0.800 [-]
SRX3	Threshold for extreme soft storey classification for the average of three storeys above	0.700 [-]

If either the ratio S/S1 or S/S3 falls between the values SR1 and SR1X, i.e. SR3 and SR3X, respectively, the storey will be classified as Soft. In the case when the ratios are above the extreme thresholds, the storey will be classified as Extreme soft. Alternatively, it is classified as a Regular storey. The stiffness S, and the ratios S/S1 and S/S3 with the thresholds are plotted based on the information from the table.

Mass regularity is checked in a separate table with the following information for each storey (row):

Mass of a corresponding storey (M)
Mass of a storey above (M1)
Mass ratio M/M1
Check (possible options are: Regular, Irregular)

The mass check options are determined based on the thresholds defined in ASCE/SEI 7-16 [2], Table 12.3-2, with the following default values:

Soft Storey Check - Mass Parameters#
Parameter	Description	Default value
MR1L	Lower threshold for mass ratio regularity	0.677 [-]
MR1U	Upper threshold for mass ratio regularity	1.500 [-]

If the mass ratio falls either below the MR1L or above the MR1U, the corresponding storey is classified as Irregular.

Additionally, mass regularity check generates the plot of the storey mass and shear mass ratio, including the thresholds, along the height of the structure.

Weak Storey Check#

The regularity of the shear capacity can be checked with the Weak Storey Check.

Storey shear capacity is calculated by multiplying the storey’s concrete and steel shear areas with the maximum concrete shear stress, and maximum steel shear stress, respectively:

$SC_{i} = {A_S}_{i,C} \: \tau_C + {A_S}_{i,S} \: \tau_S$

where

$SC_{i}$	is the storey shear capacity in $i \;$ direction (X or Y)
${A_S}_{i,C}$	is the storey concrete shear area in $i \;$ direction (X or Y)
$\tau_C$	is the maximum shear stress of reinforced concrete
${A_S}_{i,S}$	is the storey steel shear area in $i \;$ direction (X or Y)
$\tau_S$	is the maximum shear stress of structural steel

The material information can be given explicitly with the following two parameters:

Weak Storey Check - Shear Stress Parameters#
Parameter	Description	Default value
TAUC	Maximum shear stress of reinforced concrete	0.600 [MPa]
TAUS	Maximum shear stress of structural steel	150.000 [MPa]

The shear area $A_S$ of a storey is the total cross sectional area of its supporting members, i.e. walls and columns, which is effective in resisting shear deformation. Therefore, it consists of two components, ${A_S}_x$ and ${A_S}_y$ , corresponding to the shear deformation with respect to the global directions X and Y. These two components sum up to the total cross sectional area of the supporting members, considering a cut on the horizontal X-Y plane, positioned below the Z-level of the storey.

Initially, the shear area of a supporting member is calculated in the local coordinate system of the member. For a supporting wall, the cross sectional area is considered to be effective exclusively along the length of the wall, assuming no shear area across the wall thickness.

For a supporting column, the cross sectional area is initially distributed to the local Y and Z directions of the member, proportionally to the second moment of inertia components, according to the following equations:

${{A_S}_y}^l = \frac{{I_z}^l}{{I_y}^l + {I_z}^l} {A_S}^l$

${{A_S}_z}^l = \frac{{I_y}^l}{{I_y}^l + {I_z}^l} {A_S}^l$

Alternatively, with the use of record SVAL in AQUA, a user defined cross sectional shear area of a supporting column is possible. This is done by specifying a total cross sectional area A, and using parameters AY and AZ as distribution factors of the total area, along the local Y and Z axis of the cross section.

Finally, the local shear area of all supporting members is redistributed to the global X and Y directions of the system, in order to form the shear area components, ${A_S}_x$ and ${A_S}_y$ , of the storey. The shear area of supporting members made of different materials, i.e. reinforced concrete and steel, is calculated and stored separately in the system.

Weak check delivers 2 separate tables for the x and y direction, with the following information for each storey (row):

capacity of the corresponding storey (SC)
capacity of the storey above (SC1)
capacity ratio SC/SC1
(possible options are: Regular, Weak, Extreme Weak)

The check options are determined based on the soft storey thresholds acc. ASCE/SEI 7-16 [2], Table 12.3-2, since EN 1998-1:2004 [1] does not provide any quantitative specifications. The default values are set as follows:

Weak Storey Check - Threshold Parameters#
Parameter	Description	Default value
CR1	Weak storey threshold	0.800 [-]
CR1X	Extreme weak storey threshold	0.650 [-]

If the ratio SC/SC1 falls between the values CR1 and CR1X, the storey will be classified as Weak. In the case when the ratio is above the extreme threshold, the storey will be classified as Extreme weak. Alternatively, the storey is classified as a Regular storey.

Two plots will be generated for each direction, which include the plot of the shear capacity and shear capacity ratio with the thresholds along the height of the structure.

Advanced Checks (Load-dependent Checks)#

Advanced checks require a loadcase selection for the total gravity load. Furthermore, for the extraction of storey drifts and storey shear forces a list of load cases or a single load case needs to be specified. Total gravity load case and the list of selected load cases are printed in two additional tables if any of the advanced checks is selected.

Generated tables and plots will only display the results of the decisive load cases per default. Load case is considered decisive if it delivers the extremal target value for at least one storey. Decisive target values for each advanced check are listed in the following table:

Advanced Checks - Decisive Target Values#
Check	Target values	Description
Second Order Check ( $P {\text -} \Delta \,$ Effects)	${\theta}$	Interstorey drift sensitivity coefficient (EN), Stability coefficient (US)
Storey Shear Forces	Shear force ratios : X, Y	Shear force ratios in X and Y direction
Storey Drifts	dr-mod	Modified storey drift
Storey Displacements	Displacement : X-mod, Y-mod	Modified storey displacement in X and Y direction

Second Order Check ( $P {\text -} \Delta \,$ Effects)#

Second Order Check calculates the storey drift sensitivity coefficient, i.e. the stability coefficient acc. EN 1998-1:2004 [1] and ASCE/SEI 7-16 [2], respectively. Additionally, the check determines if the second order effects need to be considered, based on the check’s threshold values.

Storey drift sensitivy coefficient is calculated acc. EN 1998-1:2004 [1], 4.4.2.2 :

$\theta = \frac{P_{tot} \: d_{r{\text -} mod}} {V_{tot} \: h}$

where

$\theta \quad$	is the storey drift sensitivity coefficient
$d_{r{\text -} mod}\quad$	is the modified storey drift
$h \quad$	is the storey height
$P_{tot} \quad$	is the total gravity load at and above the storey
$V_{tot} \quad$	is the total storey shear force

Storey drift $d_{r}\:$ is calculated as the difference of the displacements at the top and at the bottom of the considered storey. Projection of its Centre of Mass (COM) to the current storey level is taken as the position for determining the displacement at the top of the storey, while the displacement at the COM’s projection to the storey level below is taken as the bottom value.

In case of the EN design code selection the modified storey drift is determined by multiplying the drift values with the displacement behaviour factor:

$d_{r{\text -} mod}= d_r \: q_d$

where

$d_{rmod} \quad$	is the modified storey drift
$d_r \quad$	is the storey drift from the selected load cases
$q_d \quad$	is the displacement behaviour factor

For the US design code selection, the stability coefficient is determined acc. ASCE/SEI 7-16 [2], 12.8.7 :

$\theta = \frac{P_{tot} \: d_{r} \: I_{e}} {V_{tot} \: h \: C_{d}}$

where

$\theta \quad$	is the stability coefficient
$I_{e} \quad$	is the importance factor
$C_{d} \quad$	is the deflection amplification factor

In case of the US code selection the modified storey drift is calculated acc. ASCE/SEI 7-16 [2], 12.8.6:

$d_{r{\text -} mod}= \frac {C_{d} \: d_r} {I_{e}}$

Default values of the shared parameters used in the previous formulas are listed in the following table:

Shared Parameters#
Parameter	Description	EN	US
QD	Displacement behaviour factor $q_d$ (EN), Deflection amplification factor $C_d$ (US)	1.500 [-] -	- 1.500 [-]
IMP	Importance factor	1.000 [-]	1.250 [-]

Default value of the deflection amplification factor corresponds to the building frame system with ordinary concrete shear walls acc. ASCE/SEI 7-16 [2], Table 12.2-1, while the default value of the importance factor is taken for the Risk category III acc. ASCE/SEI 7-16 [2], Table 1.5-2.

Second order check (P- ${\Delta \,}$ effects) delivers a table with the following information for each storey:

height of the storey
gravity load, i.e. the selected load case number for the total gravity load
LC, selected load case numbers
shear force from the selected load case
storey drift, $d_{r{\text -} mod}$ , from the selected load cases
modified storey drift
storey drift sensitivity coefficient, i.e. stability coefficient ${\theta}$
check (possible options are: OK, Simplified TH2, TH2 and Redesign)
$P {\text -} \Delta \,$ factor

The possible check options are determined based on the check’s thresholds and the calculated ${\theta}$ value:

Check Options#
Condition	Check	Description
${\theta \leq}$ THT1	OK	Second order effects do not need to be considered.
THT1 ${< \theta \leq }$ THT2	Simplified TH2	Second order effects can be considered by multiplying the seismic effects with a $P {\text -} \Delta$ factor.
THT2 ${< \theta \leq }$ THTX	TH2	Analysis acc. second order theory needs to be performed.
THTX ${< \theta}$	Redesign	Maximum allowed value is exceeded, and a redesign is required.

Note

ASCE design code specifies only 3 check ranges: OK, Simplified TH2 and Redesign. This can be attained by setting the second threshold, THT2, equal to the extreme threshold, THTX.

Default values for the P- ${\Delta \,}$ thresholds are:

Second Order Check Parameters#
Parameter	Description	EN	US
THT1	First threshold	0.100 [-]	0.100 [-]
THT2	Second threshold	0.200 [-]	0.250 [-]
THTX	Extreme threshold	0.300 [-]	0.250 [-]

Second Order Check plots the storey drift sensitivity coefficient, i.e. the stability coefficient from the decisive load cases along the height of the structure, including the thresholds.

When ${\theta}$ value is greater than the first threshold, THT1, $P {\text -} \Delta$ factor will be calculated as:

$P{\text -}\Delta= \frac {1} {1 - \theta }$

For TH2 and Redesign check results, $P {\text -} \Delta \,$ factor is only informatively printed in the table. If the consideration of the $P {\text -} \Delta \,$ factor is selected for other advanced checks, a constant $P {\text -} \Delta \,$ factor along the height of the building will be used. This value is then determined as a maximum of all $P {\text -} \Delta \,$ factors that meet the condition for simplified account of second order effects (Simplified TH2).

Note

If other advanced checks are selected with the consideration of the $P {\text -} \Delta \,$ factor, Second Order check will be always calculated, even if the check is not directly selected. Printing of the check’s table and the corresponding plot in the output requires an explicit selection of the Second Order check.

Storey Shear Forces#

Storey Shear Forces Check delivers a table with the information about the storey shear forces in X and Y direction, and also their combined value from the selected load cases. If the consideration of the $P {\text -} \Delta \,$ factor is selected, these values will be multiplied with the factor calculated acc. Second Order Check .

Additionally, the shear force ratio is determined for each direction as:

$V_r= \frac{P_{tot}} {V_{tot}}$

where

$V_r$	is the shear force ratio
$P_{tot}$	is the total gravity load at and above the storey
$V_{tot}$	is the total shear force at the top of the storey, directly beneath the storey level itself

The ratio offers an orientational value for comparison of the shear force to the vertical gravity load.

The table generated with the check provides the following information for each storey (row):

gravity load, i.e. the selected load case number for the total gravity load
$P {\text -} \Delta \,$ factor
LC, selected load case numbers
shear storey force in X and Y direction, and their combined value (multiplied with $P {\text -} \Delta \,$ factor)
shear force ratio in X and Y direction, and their combined value

In addition to the table, the plots of the shear storey force and the shear force ratio along the height of the structure are provided for X and Y direction.

Storey Drift Check#

Storey Drift Check delivers the information about the storey drifts, and checks their values in relation to the storey height. The check generates a table with the following information for each storey (row):

Height
$P {\text -} \Delta \,$ factor
LC, selected load case numbers
Storey drift, $d_{r}$
Modified storey drift, $d_{r{\text -} mod}$
Storey drift ratio, $d_{r {\text -} ratio}$

For each load case, storey drift is calculated as the difference of the displacements of the considered storey and the storey below. Drift can be either determined from displacements at the Centre of Mass (COM) or from maximum displacements.

In case of the Centre of Mass, the storey drift is calculated as follows:

$d_{r,COM}= u_{i, COM} - u_{i, COM-B}$

where

$u_{i, COM}$	is the displacement of the considered storey $i \;$ calculated at the storey’s COM projection to the current storey level
$u_{i, COM-B}$	is the displacement calculated at the storey’s COM projection to the storey level below

In case of the maximum displacement method, the storey drift is calculated as follows:

$d_{r,MAX}= max \begin{cases} \lvert u_{j, i, max} - u_{j, i{\text -}1, max} \rvert \\ \lvert u_{j, i, min} - u_{j, i{\text -}1, min} \rvert \end{cases}$

where

$u_{j, i, max}, u_{j, i, min} \quad$	are the maximum and minimum displacement, respectively, of the considered storey $i \;$ in $j \;$ direction (see the picture below)
$u_{j, i{\text -}1, max}, u_{j, i{\text -}1, min} \quad$	are the maximum and minimum displacement, respectively, of the storey below $i{\text -}1 \; ` in :math:`j \;$ direction (see the picture below)

../../../_images/maximum_displacements.svg

Type of a storey drift can be explicitly selected with the Point of Interest (POI) option in the Storey Drift Check parameters. Furthermore, the default selection is design code dependent, listed in Table: Storey Drift Check Parameters.

In case of the EN design code selection the modified storey drift is determined acc. EN 1998-1:2004 [1], 4.3.4 and 4.4.3.2:

$d_{r{\text -} mod}= d_{r{\text -} COM} \: P {\text -} \Delta \: n_{red} \:q_d$

where

$d_{r {\text -} COM} \quad$	is the storey drift from the selected load case, determined at the Centre of Mass (COM)
$P {\text -} \Delta \quad$	is the $P {\text -} \Delta \,$ factor, calculated acc. Second Order Check (if selected)
$q_d \quad$	is the displacement behaviour factor
$n_{red} \quad$	is the reduction factor

For the US design code selection the modified storey drift is calculated acc. ASCE/SEI 7-16 [2], 12.12.3:

$d_{r{\text -} mod}= \frac {d_{r{\text -} MAX} \: P {\text -} \Delta \: C_{d}} {I_{e}}$

where

$d_{r {\text -} MAX} \quad$	is the storey drift from the selected load case, determined as the maximum drift
$I_{e} \quad$	is the importance factor
$C_{d} \quad$	is the deflection amplification factor

Storey drift ratio is calculated and checked in relation to the allowable threshold as:

$d_{r {\text -} ratio} = \frac{d_{r{\text -} mod}} {h} \leq d2hx$

where

$h \quad$	is the storey height
$d2hx \quad$	is the allowable threshold for the drift check

In addition to the table, Storey Drift Check generates a plot of the modifed storey drifts and a plot of the storey drift ratios, including the threshold, along the height of the structure.

Default values of the reduction factor and the allowable threshold for the drift check are listed in the following table:

Storey Drift Check Parameters#
Parameter	Description	EN	US
D2HX	Allowable threshold for the drift ratio	0.0075 [-]	0.0150 [-]
NRED	Reduction factor	0.500 [-]	1.000 [-]
POI	Point of interest for determining the drifts	COM - Center of mass	MAX - Maximum drifts

Shared parameters used in the previous formulas are listed as follows:

Shared Parameters#
Parameter	Description	EN	US
QD	Displacement behaviour factor $q_d$ (EN), Deflection amplification factor $C_d$ (US)	1.500 [-] -	- 1.500 [-]
IMP	Importance factor	1.000 [-]	1.250 [-]

Storey Displacements#

Storey Displacements Check provides an ovierview of the displacements in X and Y direction, and also their combined value from the selected load cases.

The table generates the following information for each storey (row):

Storey height
$P {\text -} \Delta \,$ factor
LC, selected load case numbers
Displacement and its modified value in X direction
Displacement and its modified value in Y direction
Combined displacement value, based on the modified values in X and Y direction

Modified storey displacement value is calculated acc. EN 1998-1:2004 [1], 4.3.4:

$u_{mod}= u \: P {\text -} \Delta \:q_d$

where

$u$	is the storey displacement from the selected load case, determined at the Centre of Mass (COM)
$P {\text -} \Delta \quad$	is the $P {\text -} \Delta \,$ factor, calculated acc. Second Order Check (if selected)
$q_d \quad$	is the displacement behaviour factor

In case of the US code selection the modified displacement is calculated in accordance with ASCE/SEI 7-16 [2], 12.8.6:

$u_{mod}= \frac {d \: {\text P \text -} \Delta \: C_{d}} {I_{e}}$

where

$I_{e} \quad$	is the importance factor
$C_{d} \quad$	is the deflection amplification factor

Shared parameters used in the previous formulas are listed as follows:

Shared Parameters#
Parameter	Description	EN	US
QD	Displacement behaviour factor $q_d$ (EN), Deflection amplification factor $C_d$ (US)	1.500 [-] -	- 1.500 [-]
IMP	Importance factor	1.000 [-]	1.250 [-]

Modified storey displacements along the height of the structure are plotted separately for X and Y direction.

Literature#

[1] ASCE standard, ASCE/SEI 7-16 Minimum design loads for buildings and other structures, 2017

[2] EN 1998-1:2004 Eurocode 8: Design of structures for earthquake resistance – Part 1: General rules, seismic actions and rules for buildings, 2004

Storey Checks - Theoretical Principles#

General Information#

User Story#

Overview#

Design Code#

General Tables#

Storey Levels (Synopsis Table)#

Centre of Rigidity and Centre of Mass#

Regularity Checks#

Soft Storey Check#

Weak Storey Check#

Advanced Checks (Load-dependent Checks)#

Second Order Check ( Effects)#

Storey Shear Forces#

Storey Drift Check#

Storey Displacements#

Literature#

Second Order Check ( $P {\text -} \Delta \,$ Effects)#