The Warren truss
An engineering truss is essentially a type of frame designed to transmit loads in structures to the supports. In this way the static loads (mass), and live loads (traffic, wind) can be transmitted safely to the piers and abutments which in turn transfer the loads to the ground.
Trusses were usually made of timber, iron, or steel. They are not as common as they used to be because modern bridges tend to be steel box girder, post-stressed concrete, or cable-stayed structures.
A typical truss is made up of members that are joined together to form triangular sections. Members of a truss may be in tension, compression, bending or shear. There are a number of different ways in which members can be arranged to form a truss, and over the years different arrangements have been assigned different names for ease of identification. Truss types include the Lattice truss, the McDonald truss (there is an example at Galston Gorge), the Allan truss, the Pratt truss (Singleton), the Whipple truss (Lewisham and Nowra), the K truss (Hawkesbury River), and the Warren truss (Sydney Harbour Bridge approach spans).
Truss joints can be pinned, or fixed (bolted, riveted, welded). The load analysis for a pin-jointed truss is reasonably straightforward (members are either in tension or compression), but much more complex in a fixed joint truss where the members may be in tension or compression and/or bending.
The five approach spans on each side of Sydney Harbour are of the Warren type and each weighs approximately 1200 tons (1219 tonnes).
The Warren truss approach spans use mild steel which, while not as strong as silicon steel, is more than capable of carrying the loads on the approach spans.
Approach truss problem
Shown below is a simplified representation of a Warren truss approach span similar to that used on the approaches to the Sydney Harbour Bridge.
If the truss is loaded by the three forces shown, determine:
- the reactions at the supports
- the magnitude and nature of the forces in members X and Y using the Method of Joints
- The magnitude and nature of the forces in members P and Q using the Method of Sections.
- It should be noted that in Engineering Studies we only consider simple pin-jointed trusses for analysis. This is not what the approaches to the Harbour Bridge use; it is a simplification so the question is appropriate to the course. Moreover the author has simplified the truss by removing the vertical member from each truss and making all angles in the truss 60° to better facilitate solution using the methods permitted in the course. Moreover this arrangement obviates the need for any specific dimensions.
- Reactions at supports: First we shall take moments about the support A which is considered as a fixed support. The reaction force at A has an unknown direction. Also since we have simplified the truss into a Warren truss, we shall consider each member to be 1 unit long. We also need to break up the 15 kN force into its vertical and horizontal force components, 15sin75° kN and 15cos75° kN respectively as shown below:
- To find the force in members X and Y, look at joint A at the left-hand support:
- To determine the forces in P and Q using the Method of Sections, imagine cutting the truss with a cutting plane that passes through P and Q. We can then ignore one side of the cutting plane, in this case the left-hand side, which has more forces acting on it.
Now we can determine the resultant reaction magnitude and direction RA by adding the vertical and horizontal force vector components:
Summarising, the reactions at the supports are:
RA = 24·676 kN inclined at 80.94° to the horizontal
RA = 24.676 kN
RB = 18.746 kN
There are three ways this can be solved. Here we shall sum the vertical and horizontal forces. Alternatively, you could draw a force triangle to scale as a graphical solution or, you can use trigonometric ratios to determine the lengths of the sides, and hence the value of the forces.