|Influence Lines | Index of Examples | CCE Homepage|
Qualitative Influence Lines for a Statically Determinate Continuous Beam
Draw the qualitative influence lines for the vertical reactions at the supports, the shear and moments at sections s1 and s2, and the shear at the left and right of support B of the continuous beam shown.
Figure 1 - Beam structure to analyze
Qualitative influence lines for the support reactions at A, B, and C are found by using the Müller Breslau Principle for reactions, i.e., apply a force which will introduce a unit displacement in the structure at each support. The resulting deflected shape will be proportional to the influence line for the support reactions.
The resulting influence lines for the support reactions at A, B, and C are shown in Figure 2, below.
Figure 2 - Influence lines for the reactions at A, B, and C
Note: Beam BC does not experience internal forces or reactions when the load moves from A to h. In other words, influence lines for beam hC will be zero as long as the load is located between A and h. This can also be explained by the fact that portion hC of the beam is supported by beam ABh as shown in Figure 3, below.
Figure 3 - Beam hC supported by beam ABh
Therefore, the force Yh required to maintain equilibrium in portion hC when the load from h to C is provided by portion ABh. This force, Yh, is equal to zero when the load moves between A an h, and hence, no shear or moment will be induced in portion hC.
To determine the shear at s1, remove the shear resistance of the beam at the section by inserting a support that does not resist shear, but maintains axial force and bending moment resistance (see the inserted support in Figure 4). Removing the shear resistance will allow the ends on each side of the section to move perpendicular to the beam axis of the structure at this section. Next, apply shear forces on each side of the section to induce a relative displacement between the two ends that will equal unity. Since the section is cut at the midspan, the magnitude of each force is equal to 1/2.
Figure 4 - Structure with shear capacity removed at s1 and resulting influence line
For the moment at s1, remove the moment restraint at the section, but maintain axial and shear force resistance. The moment resistance is eliminated by inserting a hinge in the structure at the section location. Apply equal and opposite moments on the right and left sides of the hinge that will introduce a unit relative rotation between the two tangents of the deflected shape at the hinge. The corresponding elastic curve for the beam, under these conditions, is the influence line for the bending moment at the section.
Figure 5 - Structure with moment capacity removed at s1 and resulting influence line
The value of the moment shown in Figure 5, above, is equal to the value of Ra when a unit load is applied at s1, multiplied by the distance from A to s1. Ms1 = 1/2 x 4 = 2.
The influence lines for the shear and moment at section s2 can be constructed following a similar procedure. Notice that when the load is located between A and h, the magnitudes of the influence lines are zero for the shear and moment at s1. The was explained previously in the discussion of the influence line for the support reaction at C (see Figures 2 and 3).
Figure 6 - Structure with shear capacity removed at s2 and resulting influence line
Figure 7 - Structure with moment capacity removed at s2 and resulting influence line
Since the shear at B occurs on both sides of a support, it is necessary to independently determine the shear for each side.
To plot the influence line for Vb-L, follow the instructions outlined above for plotting the influence line for the shear at s1. To construct the shear just to the left of support B, imagine that the section s1 has been moved to the left of B. In this case, the positive ordinates of the influence line between A and B will decrease to zero while the negative ordinates will increase to 1 (see Figure 8).
Figure 8 - Structure with shear capacity removed at the left of B and the resulting influence line
The influence line for the shear forces just to the right of support B, Vb-R, is represented by the resulting deflected shape of the beam induced by shear forces acting just to the right of support B. Notice that the portion of the beam from B to h moves as a rigid body (see explanation in the Simple Beam with a Cantilever example) while the influence line varies linearly from h to C. This is due to the fact that the deflection at C is zero and the assumption that the deflection of a statically determinate system is linear.
Figure 9 - Structure with shear capacity removed at the right of B and the resulting influence line
Contact Dr. Fouad Fanous for more information.