6.11:
Space Trusses: Problem Solving
Consider a tripod consisting of a tetrahedral space truss with a ball-and-socket joint at C.
If the height and lengths of the horizontal and vertical members are given, what force is acting on members BC and BA, considering that a tensile force is applied at D?
A free-body diagram is drawn, including all the reaction forces at A, B and C.
The moment equilibrium condition at joint C is applied. The distances are expressed in terms of position vectors in three dimensions.
Simplifying further and using the force equilibrium conditions, the vector components along i yield FAy. Similarly, equating the k coefficients to zero, gives FBx.
Finally, equating the j coefficients, give FAx.
Now, a free-body diagram at joint B is considered to calculate the forces FBC and FBA.
The forces FBD, FBC and FBA can be expressed using the position vectors.
Applying the force equilibrium condition at joint B, and equating the coefficients of i, j and k unit vectors to zero, yields the forces along BC and BA.
6.11:
Space Trusses: Problem Solving
A space truss is a three-dimensional counterpart of a planar truss. These structures consist of members connected at their ends, often utilizing ball-and-socket joints to create a stable and versatile framework. Due to its adaptability and capacity to withstand complex loads, the space truss is widely used in various construction projects.
Consider a tripod consisting of a tetrahedral space truss with a ball-and-socket joint at C. Suppose the height and lengths of the horizontal and vertical members are known.
Assuming that a tensile force is applied at joint D, a free-body diagram that includes all the reaction forces at A, B, and C joints can be drawn to determine the force acting on members BC and BA.
The moment equilibrium condition at joint C is applied, considering the distances expressed in position vectors in three dimensions.
Simplifying further and using the force equilibrium conditions, the vector components along i yield FAy as 6 N and along k give FBx as -7.2 N. Finally, equating the j coefficients gives the value of FAx as 6 N.
Now, consider the free-body diagram at joint B to calculate the forces FBC and FBA. The forces FBD, FBC, and FBA can be expressed using position vectors. The force equilibrium condition at joint B is applied.
Equating the coefficients of the i, j, and k unit vectors to zero yields the forces along BC and BA as zero.
Suggested Reading
- Hibbeler, R.C. (2016). Engineering Mechanics ‒ Statics and Dynamics. Hoboken, New Jersey: Pearson Prentice Hall. pp 301.
- Beer, F.P.; Johnston, E.R.; Mazurek, D.F; Cromwell, P.J. and Self, B.P. (2019). Vector Mechanics for Engineers ‒ Statics and Dynamics. New York: McGraw-Hill. pp 306
- Meriam, J.L., Kraige, L.G., and Bolton, J.N. (2020). Engineering Mechanics ‒ Statics. Hoboken, New Jersey: John Wiley. pp 193