Self-weight gravity load

The main load case of construction sequencing is self-weight, i.e. the gravity force that acts along the span of the frame element, which is a distributed load. One common idea in Finite Element Analysis (FEM) is to “lump” these distributed load to its boundary. In our context, the boundary of our elements is two end points, so we calculate the equivalent point load acting at the element end points.

We lump the uniformly distributed load caused by gravity to the two ends of the beam. The load density is \(\rho A g_{z} = \frac{\rho A L \cdot g_{z}}{L}\) in the global z axis. We want to use the fix-ended uniform load beam results above to lump the load to the two beam ends.

In this document, \(\rho\) indicates the density of the element’s material; \(A\) indicates the element’s cross section area, \(L\) indicates the length of the element. For the subsribe convention, \(F_{0,G,x}^e\) means the value is element e ‘s end point 0 ‘s force in G lobal coordinate system’s x axis. Another example here: \(M_{L,L,z}^e\) means the value is element e ‘s end point L ‘s moment in L ocal coordinate system’s z axis.

For the force part, we can simply divide them by two, as the load itself is given in the global coordinate system initially:

\[\begin{pmatrix} F_{0x}, F_{0y}, F_{0z} \end{pmatrix}^T = \begin{pmatrix} w_x \frac{L}{2}, w_y \frac{L}{2}, w_z \frac{L}{2} \end{pmatrix}^T\]

\[\begin{pmatrix} F_{Lx}, F_{Ly}, F_{Lz} \end{pmatrix}^T = \begin{pmatrix} w_x \frac{L}{2}, w_y \frac{L}{2}, w_z \frac{L}{2} \end{pmatrix}^T\]

Where \(w_{x,y,z} = \rho A g_{x,y,z}\). Usually, \(w_{x}, w_{y} = 0\), and \(g_z\) represents the gravitational acceleration.

For the moment part, we need to first transform the across-span load to the element’s load coordinate system, and then we can use the fixed-end beam equation’s results.

First, we have the element’s local-to-global 3x3 transformation matrix \(R_{LG}^e\).

Then the load density in the local coordinate is:

\[\begin{split}\begin{pmatrix} w_{Lx}\\ w_{Ly}\\ w_{Lz} \end{pmatrix} = (\rho A) R_{LG} \cdot \begin{pmatrix} g_x\\ g_y\\ g_z \end{pmatrix}\end{split}\]

Then, because the moment is activated by loads that are perpendicular to the local x axis, we only need to consider the \(w_{Ly}, w_{Lz}\)’s contribution to the moment reaction.

Loads along the positive local y axis direction will activate positive moment reaction at end point 0 and negative moment reaction at end point L around local z axis.

TODO: picture

Loads along the positive local z axis direction will activate negative moment reaction at end point 0 and positive moment reaction at end point L around local y axis.

TODO: picture

According to the fixed-end beam equation, the reaction moment’s value is \(\frac{w_{local} L^2}{12}\). The reaction moment caused by \(w_{Lxyz}\) is

\[\begin{split}\begin{pmatrix} M_{0,Lx}\\ M_{0,Ly}\\ M_{0,Lz}\\ M_{L,Lx}\\ M_{L,Ly}\\ M_{L,Lz} \end{pmatrix} = \begin{pmatrix} 0\\ -w_{Lz} L^2/2\\ w_{Ly} L^2/2\\ 0\\ w_{Lz} L^2/2\\ -w_{Ly} L^2/2 \end{pmatrix}\end{split}\]

Transforming it back to the global coordinate:

\[\begin{split}\begin{pmatrix} M_{0,Gx}\\ M_{0,Gy}\\ M_{0,Gz}\\ M_{L,Gx}\\ M_{L,Gy}\\ M_{L,Gz} \end{pmatrix} = \begin{pmatrix} R_{LG}^T &0\\ 0 &R_{LG}^T\\ \end{pmatrix} \begin{pmatrix} 0\\ -w_{Lz} L^2/12\\ w_{Ly} L^2/12\\ 0\\ w_{Lz} L^2/12\\ -w_{Ly} L^2/12 \end{pmatrix}\end{split}\]