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The shear line force used to determine the reaction to the overturning moment on a level x is the vertically accumulated shear line force Vx = S Vi , i = x...n. Thus the overturning component is accumulated twice – as a shear line force and as a hold-down/compression force. This seeming duplication is an indirect way of accounting for the increased moment arm of upper story shear line forces in the calculation of lower story overturning moments.
For a 3-level structure, with story shear forces V1, V2, V3, wall heights h1, h2, h3 and wall length L,. Shearwalls calculates the hold-down or compression force reaction to overturning moment due to shear force on each level as
T3 = V3 h1 / L .
T2 = (V2 + V3) h2 / L
T1 = (V1 +V2 + V3) h1 / L
Thus on level one, the total reaction is
T = S Ti = [V3 h3 + (V2 + V3 ) h2 + (V1 +V2 + V3 ) h1] / L
Considering the moment of forces acting on the structure as a whole, it is
(h1 +h2 + h3)V3 + (h1 +h2 )V2 + V1h1
A rearrangement of terms shows this moment is exactly counteracted by the moment due to the reaction at T at the base, i.e. TL.
This can also be shown by considering the free body diagram of the lowest level. It is subject to T1 + T2 upwards at the top of the wall end and T1 + T2 + T3 downwards at the bottom, and to a shear force at the top of V1 + V2 + V3. The net moment is T1L = (V1 +V2 + V3) h1, which is the moment of the accumulated shear force.
Offset Wall Segments
If a wall segment is a different length on an upper level than a lower level, the hold-down/compression force at the end or ends which do not line up is transferred down a vertical element as described in Vertical Transfer of Hold-down and Compression Forces. The simplest case is used to show that moments still balance for this scenario.
Consider a short wall with height h2, length L2 and subject to force V1 on top of a longer wall with length height h1, length L1 and force V1. There is a vertical element at a distance L2 with the force V2h2/L2 transferred down from the end of the upper level shear wall, and a hold-down force (V1 + V1) h1/ L1 at the end of the wall with length V1.
Considering the walls as a unit, the moment of the shear forces is thus
V2 (h1 + h2) + V1h1 .
= V2h1 + V2h2 + V1h1
The moment of the hold-down forces is
(V2h2//L2) L2 + ((V1 + V2) h1 / L1 ) * L1
= V2h2 + V1h1 + V2h1
The moments balance.