UNDERSTAND IT. WORK IT OUT.

Learn: EC8 floor force distribution

Equivalent seismic base shear is distributed among floors using their masses and assumed modal displacement shape. This calculation finds one floor’s share of a supplied total base shear.

Beginner-friendlyFree · No accountTwo worked examples + separate practice
01

What the formula is saying

Multiply floor shape ordinate si by floor mass mi. Divide by the sum Σsjmj for all participating floors, then multiply the fraction by Fb. The denominator must include this floor and every other floor in the same model.

Fi = Fb (si mi) / Σ(sj mj)

Read the symbols in plain language

Fb
Base shear

Base shear. Multiply total base shear by the dimensionless floor share.

kN

Kilonewtons measure force; 1 kN = 1000 N.

si
Mode displacement / height proxy

First-mode displacement ordinate (or permitted height-proportional proxy) using the same scale at every floor.

shape ordinate

Use shape ordinate as the base unit shown here. Fb and Fi use kN. mi and all masses inside the sum must use the same unit, here tonnes. The shape ordinates can use any consistent normalization because that common scale cancels between numerator and denominator.

mi
Floor mass

Floor mass. The modal ordinate determines how strongly this floor contributes in the assumed displacement pattern.

t

Use t as the base unit shown here. Fb and Fi use kN. mi and all masses inside the sum must use the same unit, here tonnes. The shape ordinates can use any consistent normalization because that common scale cancels between numerator and denominator.

Σ(sj mj)
Sum over floors

Sum of sj × mj over every participating floor, including this one. Each mass must use the same tonne unit and each ordinate the same scale.

t × shape

Use t × shape as the base unit shown here. Fb and Fi use kN. mi and all masses inside the sum must use the same unit, here tonnes. The shape ordinates can use any consistent normalization because that common scale cancels between numerator and denominator.

Fi
Result to find

EC8 floor force distribution. Multiply total base shear by the dimensionless floor share.

kN

Sort out the units first

Fb and Fi use kN. mi and all masses inside the sum must use the same unit, here tonnes. The shape ordinates can use any consistent normalization because that common scale cancels between numerator and denominator.

Assumptions before calculating

Assume a valid equivalent lateral-force model with nonnegative same-direction floor shape ordinates and positive participating masses. The total weighted sum is positive and cannot be smaller than the selected floor contribution.

This is a first-generation Eurocode teaching relationship or an explicitly simplified coefficient calculation. The numbers supplied here are exercise data, not a recommendation for any country. Check the adopted edition, relevant clause, National Annex, applicability conditions and all other limit states before any real design.

02

Let’s solve one together

Read the given values, follow each operation, then check what the result means.

Read the supplied values as one complete study case. Find Fi and explain the result in the stated output unit.

Fb · Base shear
500 kN
si · Mode displacement / height proxy
3 shape ordinate
mi · Floor mass
100 t
Σ(sj mj) · Sum over floors
1500 t × shape

  1. Weight the selected floor mass by its shape

    The modal ordinate determines how strongly this floor contributes in the assumed displacement pattern.

    (3) × (100) = 300 t × shape

  2. Find this floor’s share of the total

    Divide by the same weighted mass sum used for all floors, including this one.

    (300) ÷ (1500) = 0.2

  3. Allocate the corresponding base-shear share

    Multiply total base shear by the dimensionless floor share.

    (500) × (0.2) = 100 kN
Answer100 kN

Does this worked answer make sense?

The floor fraction must lie between 0 and 1 in this positive-shape model. Calculating every floor with the same denominator should make their forces sum to Fb.

A second worked example — different values

A second case uses different data. Predict which way the answer will change, then calculate it without reusing the first answer.

Fb · Base shear
600 kN
si · Mode displacement / height proxy
2 shape ordinate
mi · Floor mass
120 t
Σ(sj mj) · Sum over floors
1200 t × shape

  1. Weight the selected floor mass by its shape

    The modal ordinate determines how strongly this floor contributes in the assumed displacement pattern.

    (2) × (120) = 240 t × shape

  2. Find this floor’s share of the total

    Divide by the same weighted mass sum used for all floors, including this one.

    (240) ÷ (1200) = 0.2

  3. Allocate the corresponding base-shear share

    Multiply total base shear by the dimensionless floor share.

    (600) × (0.2) = 120 kN
Answer120 kN
03

Now try your own values

Change a value or its unit. The same method will show your calculation, step by step.

Base shear. Multiply total base shear by the dimensionless floor share.

First-mode displacement ordinate (or permitted height-proportional proxy) using the same scale at every floor.

Floor mass. The modal ordinate determines how strongly this floor contributes in the assumed displacement pattern.

Sum of sj × mj over every participating floor, including this one. Each mass must use the same tonne unit and each ordinate the same scale.

English, Arabic and Persian digits are supported. The steps convert inputs to the formula’s base units.

Results update only when you calculate. The lesson example above stays unchanged.

04

Your turn — check your understanding

Solve this separate case yourself. Use only the values below; the two worked examples use different data. Give the requested result in the selected unit.

Fb · Base shear
450 kN
si · Mode displacement / height proxy
4 shape ordinate
mi · Floor mass
80 t
Σ(sj mj) · Sum over floors
1600 t × shape

Find: Learn: EC8 floor force distribution

For repeating decimals, use at least four significant figures. Accepted rounding tolerance: 0.05% of the expected value; zero uses an absolute tolerance of 10⁻¹².

A hint, not the answer

Multiply floor shape ordinate si by floor mass mi. Divide by the sum Σsjmj for all participating floors, then multiply the fraction by Fb. The denominator must include this floor and every other floor in the same model.

Fb and Fi use kN. mi and all masses inside the sum must use the same unit, here tonnes. The shape ordinates can use any consistent normalization because that common scale cancels between numerator and denominator.

Show the full practice solution

Compare the steps with your work; revealing a solution does not mark the lesson complete.

  1. Weight the selected floor mass by its shape

    The modal ordinate determines how strongly this floor contributes in the assumed displacement pattern.

    (4) × (80) = 320 t × shape

  2. Find this floor’s share of the total

    Divide by the same weighted mass sum used for all floors, including this one.

    (320) ÷ (1600) = 0.2

  3. Allocate the corresponding base-shear share

    Multiply total base shear by the dimensionless floor share.

    (450) × (0.2) = 90 kN
Answer90 kN

Avoid the common trap

Do not put the unweighted mass total in the denominator when different shape ordinates are used. Do not confuse a floor’s applied force with the cumulative story shear.

When this method applies — and when it does not

This distributes a known base shear; it does not determine modal shapes or whether the lateral-force method is permitted. A height-proportional shape is an additional assumption, not an automatic truth. Story shear requires summing the forces above the story, not using one floor force alone.

For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.

One idea understood. Keep going.

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Sources & further reading

References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.

Reading focus: EC8 floor force distribution. First-generation EN 1998 lateral-force teaching method: design spectrum, eligible structural model, base shear and distribution to floors. This is not a site-specific seismic design.

Lesson updated: · Both examples and the separate practice case are checked against an independent high-precision numerical implementation. This verifies arithmetic for the stated model, not engineering certification.

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