pin joints and trusses PDF

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Pin jointed Historical Context Before the 17th century many great structures were built. However, although the builders who constructed these buildings knew qualitatively how the components of the structure would behave, they could not quantify the magnitude of the forces. For arch structures, quantifying the forces is not vitally important once you can build adequate foundations. Masonry arches will fail if they are the wrong shape (this is also a statics problem) but once the shape is correct then the scale of the structure is not vital. The great masons of the great gothic cathedrals built larger and larger cathedrals by judiciously copying and modifying earlier designs. Unfortunately this approach doesn’t work with structures that fail due to material failure. Pin-jointed Problems Pin-jointed trusses where among the first structures in which the forces in the elements of the structure were calculated. The members in such trusses were connected via pin-joints to ensure that the forces applied to the members were purely axial. Many complex structures can be analysed as pin-jointed sturctures even when the joints clearly are not pinned. Pin-jointed analysis is valid so long as the loads in the structure's members are predominantly axial. All structures, however complex, must satisfy the equations of statics. Pin- joints The photograph on the left shows a real pin-joint while the diargram on the right shows an exploded view of another pin-joint. Pin-joints are deliberately constructed such that the truss members are free to rotate at the joints, as a result only axial forces are transferred into the truss members.

In order to size the members of a pin-jointed truss the designer must be able to calculate the forces in each member, so that the members are capable of carrying the forces. In many cases the equations of static equilibrium are sufficient to calculate the member forces Note: Not all pin-jointed trusses can be analysed using statics alone. If the forces in the members are dependent on the relative stiffness of the various members, then the truss is said to be statically indeterminate and cannot be analysed using statics alone. Support and symbols used to represent them In the analysis of trusses we will use a number of standardised symbols to indicate different types of support. The symbol on the left below will be used to indicate a pinned support that resists lateral and vertical movement. The figure on the right gives some indication of what such a support would look like.

Similarly the symbol show on the left below indicates a support that resists vertical loading but which allows lateral movement. Again the Figure on the right hand side indicates what such a support might look like

Trusses Consider the example shown below. This truss is supporting a load W, which is applied at one of the nodes (name used for the joints) of the truss. In a pin jointed truss the loads are applied at the nodes.

A free body diagram of the truss showing the forces acting on it is shown below. The truss is in equilibrium, therefore the sum of all the forces acting on the truss must sum to zero.

The sum of the horizontal forces acting on th truss is zero.

Similarly the sum of the vertical forces on the truss is also zero.

This equation is not sufficient on its own to solve for RA and RB. However, the third equation of static equilibrium must also be satisfied. The sum of the moments about any point must also equal zero. We have control over the point about which the moments are being calculated. It makes sense to choose the point so as to reduce the number of calculations required. By taking moments about the left hand support there is no need to include a moment due to RA, because RA acts through the left hand support.

Therefore putting this value for RB back into the previous vertical equilibrium equation yields

Analysis of Trusses – Equilibrium of Joints approach When a rigid body is in equilibrium the sum of the external forces and moments acting on the body must sum to zero. In the same way if any section of a rigid body is in equilibrium then the sum of the forces and moments acting on that section of the body must also sum to zero. The forces resisted by individual members within a statically determinate truss can be identified by carefully constructing appropriate free body diagrams. One of the best procedures is called the method of joints. This method involves considering the forces acting at the nodes of the truss. For example, consider the joint shown below

Since this section of the truss is in equilibrium we have;

Thus we have two unknown variables and two constraint equations the hence we can solve for FA and FB. It won’t always be as easy as in the example given in general we will end up with two simultaneous equations with two unknown variables. Two comments are worth making, 1. The equilibrium equation isn’t of use for this case because the forces acting on the pin all intersect at the pin’s centre. 2. This means that (for the 2D case) when we construct a free-body diagram of the pin we have two potential equilibrium equations. Therefore it is best if we attempt to analyses the joints in an order such that there are only two unknown forces at each time....