Constraints, rigid diaphragm

Problem. A Rigid Diaphragm makes the nodes of a storey move together in their plane. The tests check that columns sharing the same drift take shear in proportion to their stiffness.

Reference document. Load sharing of parallel springs in proportion to stiffness is used.

The constraint transformation and the distribution of shear among the columns are given by these relations.

u = T\,u_{ana} + u_0,\qquad V_i = \dfrac{k_i}{\sum k}\,P

The translation of every node of the diaphragm is found from the translation and rotation of the master node by these relations.

u_{x,i} = u_{x,m} - (y_i - y_m)\,r_{z,m},\qquad u_{y,i} = u_{y,m} + (x_i - x_m)\,r_{z,m}

Numerical model. The scope is a horizontal diaphragm, with its normal along global Z.

Two columns, rigid diaphragm, centred horizontal load

Parameters. stiffness ratio = 8 · P = 10,000 · H = 3

Closed form. In columns sharing the same translation the shear is distributed in proportion to the stiffness

V_i = k_i P / \sum k

Quantity	Reference	Computed	Deviation
Ratio of the base shears at a stiffness ratio of 8	8	8	0%

Two columns, rigid diaphragm, centred horizontal load#

Two columns, rigid diaphragm, centred horizontal load