Member End Forces and Joint Equlibrium

Member End Forces

Member end forces and moments in the member result from loads applied to the structure. These forces are in the local member coordinate system.

The end forces (FX, FY and FZ) will be positive if the forces are in the positive local axis. The following figure shows the positive member end forces.

The member end moments (MX, MY and MZ) follow the right hand rule. If the right hand thumb points the positive local axis, then the fingers point in the direction of the positive moment. The following figure shows the positive member end moments.

Joint Equilibrium

The joint equilibrium should be checked in three planes (XY, YZ and ZX plane). For each plane, the axis which is perpendicular to that plane is considered for the joint equilibrium check. The summation of forces along that axis and moments about that axis should be zero.

Example:

In this example, the following plane is considered - member 31501, 31518, 1139 and 1140 is on that plane and member 1032 is perpendicular to this plane. 

Force Equilibrium:

The following forces are considered —

Positive FZ force at node 1015 for member 1139 : 0.02 kN

Negative FZ force at node 1015 for member 31501 : -0.482 kN

Positive FZ force at node 1015 for member 1140 : -0.675 kN

Negative FZ force at node 1015 for member 31518 : 0.797 kN

Negative FX force at node 1015 for member 1032 : 0.34 kN

So the summation of force along the axis perpendicular to the plane is: 0.02+(-0.482)+(-0.675)+0.797+0.34 = 0

Moment Equilibrium:

The moments are acting at the member end. To check the joint equilibrium, the reactions at node 1015 is considered.

MZ force at node 1015 for member 1139 : 4.668 kN-m

MZ force at node 1015 for member 31501 : 45.66 kN-m

MZ force at node 1015 for member 1140 : -116.441 kN-m

MZ force at node 1015 for member 31518 : 66.11 kN-m

MX force at node 1015 for member 1032 : 0.003 kN-m

Summation of moment about the axis perpendicular to the plane is: 4.668+46.66+(-116.441)+66.11+0.003 = 0