Influence Lines for Beams - 1 - Structural Analysis - Civil Engineering

Instructional Objectives:

The objectives of this lesson are as follows:

• How to draw qualitative influence lines?
• Understand the behaviour of the beam under rolling loads
• Construction of influence line when the beam is loaded with uniformly distributed load having shorter or longer length than the span of the beam.

Müller-Breslau Principle for Qualitative Influence Lines

The Müller-Breslau Principle (proposed in 1886 by Heinrich Müller-Breslau) provides a quick method to draw qualitative influence lines for any function (reaction, shear, or moment) on a structure. The principle states that the ordinate of an influence line for a particular function is proportional to the deflected shape obtained when the restraint corresponding to that function is removed and the structure is subjected to a unit displacement or rotation in the positive direction of the function.

To illustrate, consider the influence line for the support reaction at A for a beam (see the figure reference below). Remove the support at A (remove the restraint that produces R_A) and apply a unit vertical displacement in the positive direction of R_A. The resulting deflected shape of the beam is proportional to the influence line for R_A.

Following the procedure, remove the support at A and apply a unit upward displacement at that location. The deflected shape produced by this unit displacement is a straight line (rotation of rigid segments) for a statically determinate beam and matches the qualitative influence line for the reaction at A.

Note: for statically determinate systems the deflected shape produced by removing a support and applying a unit displacement is linear (segments rotate as rigid bodies) and therefore directly gives a linear influence line. For statically indeterminate systems the deflected shape may have curvature and proper static or structural analysis is required to obtain the exact influence line.

Examples using Müller-Breslau Principle

Influence line for shear at a section (overhang beam)

Consider an overhanging beam and the shear at section C. To obtain the qualitative influence line for shear at C, introduce a vertical release (a roller) at the section C to provide vertical freedom and then apply a unit vertical displacement upward at that location. The resulting deflected shape is proportional to the influence line for shear at C.

Introduce the roller at C and apply a unit upward displacement to get the deflected shape.

The deflected shape obtained by these steps is linear for a determinate beam and corresponds to the qualitative influence line for shear at C.

Influence line for bending moment at a section

To draw the qualitative influence line for the bending moment at a section C, release the bending restraint at C by introducing a hinge so that rotation is permitted at C. Then apply a unit rotation in the positive sense of the moment M_C. The deflected shape produced by this unit rotation is proportional to the influence line for bending moment at C.

Introduce the hinge at C and apply the unit rotation; the resulting deflected shape gives the qualitative influence line.

Maximum Shear in Beam Supporting Uniformly Distributed Loads (UDLs)

When a UDL of constant intensity w (force per unit length) rolls across a beam, the maximum shear at any section depends on whether the UDL length is longer than the span or shorter than the span. For the maximum shear corresponding to a moving load, the maximum value occurs when the loaded portion covers the maximum area under the influence line in the sign that gives the extreme (positive or negative) shear.

UDL longer than the span

Consider a simply supported beam AB with span L and a UDL of intensity w whose loaded length is longer than the span (so the beam is completely covered by the load at some positions as it moves). Let C be a section located at distance x from support A. The influence lines for the reactions R_A, R_B and for the shear at section C determine how much of the distributed load contributes to those functions as it moves.

To obtain shear due to a moving UDL, multiply the load intensity w by the area of the influence line that is covered by the loaded length at the given position. For a UDL longer than the span, the beam can be entirely under the load; thus the relevant area of the influence line will often be the full triangular or trapezoidal area shown in the shear influence diagram.

For a section at C, the maximum negative shear occurs when the head of the moving UDL is located at section C; the maximum positive shear occurs when the tail of the moving UDL is located at section C. Thus the extreme shear magnitudes are

and

UDL shorter than the span

When the UDL length is less than the span (y < L), the moving UDL can only load a portion of the beam at any instant. Maximum negative shear at a section occurs when the head of the loading reaches the section, and maximum positive shear occurs when the tail passes the section. The shear value is evaluated by multiplying the intensity w by the area of the influence line covered by the loaded length for that position. The generic procedure is identical to that used for longer UDLs but the covered area is smaller and depends on the relative positions.

Worked examples and diagrams for this case are demonstrated in the illustrative figures.

Maximum Bending Moment at Sections in Beams Supporting UDLs

The maximum bending moment at a given section due to a moving UDL also depends on whether the UDL length is longer than or shorter than the span. The bending moment at a section for any position of the moving UDL equals the load intensity w multiplied by the area under the bending-moment influence line covered by the loaded portion.

UDL longer than the span

When the moving UDL is longer than the span, at some positions the beam is fully covered by the load. Let C be the section at distance x from support A. The maximum bending moment at C is obtained by multiplying w by the full area under the influence line for moment at C (the entire span may contribute). The expression for bending moment at C for a completely covered span is:

If the section C is at mid-span (x = L/2), the maximum moment simplifies to the familiar mid-span value under full UDL coverage:

UDL shorter than the span - detailed derivation for maximum moment at a section

Consider a beam AB of span L, with a UDL of length y (< L) and intensity w. Let the midpoint of the UDL be at D located at distance z from support A. Let C be a section at distance x from A. We want the position of the UDL (value of z) that gives the maximum bending moment at section C and the value of that maximum moment.

The procedure is to compute support reactions for the currently positioned UDL, then compute the bending moment at C, and finally maximise that moment with respect to z.

The total load on the beam from the UDL equals w y. Taking moments about A to find reaction at B:

Reaction at B, R_B = (w y × z) / L

Moment at C due to the UDL and reactions (taking counterclockwise positive) can be written as the algebraic sum of moments due to the reaction at B and the portion of the UDL to one side or the other of C. The general expression for the bending moment at C is:

Substitute the expression for R_B into the moment equation to obtain M_C as a function of z:

To find the position z that gives maximum M_C, differentiate M_C with respect to z and set the derivative to zero.

Differentiate M_C with respect to z and set equal to zero:

From the differentiation we obtain the condition for maximum moment:

Using geometric relations between the distances on the beam (relating x, y, z and L as needed for the particular configuration), we can express the optimum z in terms of x and L or other known distances. For the usual symmetric placement that gives the maximum moment at C, the result reduces to the geometric statement that the section C divides the moving load in the same proportion as it divides the span when the maximum occurs.

The final interpretation is important for practical placement: for a UDL shorter than the span, the moving load should be positioned such that the section divides the loaded length and the span in the same ratio; when this condition is satisfied the bending moment at the section reaches its maximum.

Notes on Practical Use and Summary

• Use the Müller-Breslau Principle for quick qualitative influence lines: remove the restraint corresponding to the function, apply a unit displacement/rotation, and sketch the deflected shape; for determinate structures this gives a linear (or piecewise linear) influence line.
• For moving concentrated loads, place the load where the influence line attains its maximum (ordinate) to produce the extreme effect.
• For moving uniformly distributed loads, compute the effect by multiplying the load intensity w by the area of the influence line covered by the loaded length at each position; extreme values correspond to positions that maximise this covered area for the required sign.
• Distinguish two cases for UDLs: UDL longer than span (beam may be fully covered) and UDL shorter than span (only a portion of the influence line is covered). The algebraic procedures are the same; only the covered area differs.

Careful sketching of influence lines and correct evaluation of the area under these lines for the moving load cases yields the maximum reactions, shears and moments required for design checks and influence-based analysis of beams carrying moving loads.