Analyse the statically determinate bar. Analyse the statically determinate bar i

Question

Analyse the statically determinate bar.

Analyse the statically determinate bar illustrated below by expressing the loading as a single function using Macaulay brackets and the Dirac delta, integrating to find the axial force and integrating again to find the displacements, applying the boundary conditions appropriately. Find the axial force in the bar at point A and the displacement at point B. The cross section of the bar is constant with EA = 13000 kN. a = 3.0 m,b = 2.5 m, c = 3.0 m and d = 0.5 m.w1 = 6 kN/m, w2 = 6 kN/m, PI = 19 kN and P2 = 18 kN. Axial force at point A (kN, tension positive): Displacement at point B (mm, positive to right):

Explanation / Answer

The firststep in analyzing a trussisto determine ifthe trussisstable or unstable. The trussin

Figure 1a below is notstable, and istherefore not a structure. The joints of an unstable

structure can be displaced without causing any ofthemembersto deform(lengthen orshorten

in this case), asshown in Figure 1b.

Figure 1a. Unstable Truss Figure 1b.

In Figure 2a, a diagonalmember has been added to the truss of Figure 1. Thistrussisstable: a

the horizontal load shown in Figure 2b causesthe diagonalto stretch and themembersshown

in blue to shorten. The joints cannot be displaced without causingmembersto deformand

therefore to resistthe attempted displacement. The barforcesin the truss of Figure 2b can

easily be calculated using the equations ofstatic equilibriumand the trussissaid to be

statically determinate.

Figure 2a. Stable&Determinate Truss Figure 2b.

In Figure 3 below we have added anothermemberto the truss of Figure 2. The barforces of

thistruss cannot be calculated using the equilibriumequations alone and the trussissaid to be

statically indeterminate. The forces can be calculated by hand forthe trussin Figure 3, butitis

an involved procedure. Most practicing structural engineers would calculate the barforces

using a structural analysis computer program.

Figure 3. Indeterminate Truss

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