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The Modern Müller-Breslau (MMB) Method is a new, purely geometric means of establishing equilibrium in two-dimensional and three-dimensional structures. It builds off the work by Heinrich Müller-Breslau (1851–1925), which is well known by civil engineers around the world as the Influence Line Method.

The impetus of creating the new MMB theory was driven by the author’s experience teaching many architecture and engineering students at California Polytechnic State University (Cal Poly), while trying to address the curious phenomenon of a growing gap between design students’ knowledge of form creation, and their limited understanding of how load flows through the forms they create. The MMB Method demonstrates visually, with no statics at all, how load flows to a support or how an internal force or moment equilibrates external loads. While it can be drawn by hand, even on the proverbial “napkin sketch,” the method lends itself best to precise simple line drawings done parametrically on a computer.

Several examples of the new MMB method will be shown in this article, along with a brief review of the classic Müller-Breslau (MB) Method.

In the classic Müller-Breslau (MB) Method, to find an Unknown reaction or internal force or moment, Heinrich Müller-Breslau first removed the Unknown, be it an external or internal force or moment equilibrator, then he “perturbed” the structure a unit displacement or rotation amount Δ in the assumed direction, or sense, of the Unknown.

The MB and the MMB Methods agree perfectly for any scale of Δ as shown in Figures 1, 2, and 3. In Figure 1, a simply supported beam contains point loads at each of the free cantilever tips. To seek either Unknown Reaction, such as the Right Reaction (RR in the classic MB Method), the RR is removed and a geometric perturbation of 1 unit is applied vertically, because the reaction is vertical. Since the perturbation is purely vertical, the entire beam stretches, i.e. it elongates. This is known as the “Influence Line” for the Right Reaction.

In Figure 2 for the MB method, the geometric displacement known as the perturbation is 1 unit and the “loft” or the movement of each load is -0.3 and 1.1 for F1 and F2 respectively. In Figure 3, the MMB Method does not stretch the beam. Rather, it respects all boundary conditions after the perturbation is applied. While the perturbation Δ is still measured in the direction of the Unknown, the path of the entire perturbed structure is circular, as shown in Figure 3. Notice in Figure 3 that the loft of each load is measured solely in the direction of the load. Both methods agree with theory, and the result is not earth-shattering, so why go through the trouble of drawing the perturbed shape without stretching the beam? The answer arises with lateral loads.

Consider now the slanted, simply supported beam shown in Figure 4. To seek the Unknown Right Reaction RR using the classic MB Method, the “loft” or movement of the applied load is zero in the perturbed configuration, regardless of the size of Δ, because the beam stretches. This would imply that the Right Reaction RR is zero, which is incorrect. If there is no horizontal movement of the external load, then there is no work done by it, so there would be a zero reaction. But, there is indeed a vertical reaction on the slanted beam, induced by the horizontal load.

Using the MMB Method, one simple work equation can be applied to any structural problem. Work is force multiplied by a distance. The distance is either Δ for an external reaction, or it is a Loft for an applied external force.

**Equation 1: Unknown ∙∆+ ∑F _{i} ∙ Loft_{i}=0**

Figure 6 shows the solution to this problem for a large perturbation Δ, and Figure 7 shows the solution approaching the theoretical value as Δ gets small. The circular path is evident, yet the perturbation Δ of the Unknown and the Loft of any load is only in direction of those loads.

Consider now, a slightly more complicated structure, a three-hinged arch subjected to a partial live load projected uniformly along a horizontal line, as shown in Figure 8. Suppose the Unknown moment in the left “elbow” shown in Cartesian Coordinates at (2,5) is sought. Most students would balk at this problem if asked to solve it algebraically. Yet, with the MMB Method, the problem requires as minimal effort as the previous examples show. The Unknown, which here is the moment in the elbow, is removed. That juncture is perturbed some amount Δ. All pieces must remain straight, and all other boundary conditions must be respected. Then, the Loft of the loads is simply measured. One necessary rule is to break a distributed load into separate parts if the load passes over a kink in the perturbed structure. Figure 9 shows the perturbed structure for a large Δ.

Reducing Δ by an order of magnitude shows that the solution rapidly approaches theory. This is demonstrated in Figure 10. Notice that the right elbow does not get distorted in the perturbed configuration. All boundary conditions are respected, and only one violation occurs at the Δ for the Unknown being sought. The first part of the distributed live load passes through Ld1 in the original configuration and through Ld1’ in the perturbed configuration. The second part of the distributed live load passes through Ld2 in the original configuration and through Ld2’ in the perturbed configuration. Since those loads are vertical, the lofts associated with each load are vertical. If the sign of the load vector agrees with the sign of the loft, that work, or product of two terms, is positive. If the signs disagree, that product of two terms is negative. The Unknown internal moment is a double negative, so Equation 1 can be used as-is.

Finally, Figure 11 shows a problem that many engineering students would most likely answer incorrectly if attempting to solve it algebraically via statics. A frame is pinned at the lower left end, roller supported at the lower right end, hinged at the Crown connecting the two frame members, and a tensile tie cable that begins at a distance of 1/3 of each of the lengths of the frame members, starting from the base supports. The load is a dead load, uniformly applied along the length of each frame member and downward. The Unknown tensile force in the cable is sought.

The power and the efficacy of the MMB Method is shown in this example because once again, Equation 1 finds the solution. Since the Unknown force in the tensile tie is sought, it must be perturbed some amount Δ in the direction of the tie itself. The perturbation Δ here means that the tensile tie gets longer. Yet all other boundary conditions must be respected, and the frame members do not stretch. Students delight in solving the purely geometric puzzle of understanding the perturbed shape. The free children’s software GeoGebra, which is meant to teach children Geometry and Algebra, is ideally suited for such problems and available in many languages. The lofts are immediately found using GeoGebra as a easy to find measurement in the Geometry window of the program.

Figure 12 shows the solution for a large Δ, and Figure 13 shows the solution approaching theory for a small Δ. Δ is the difference in lengths of the perturbed tensile tie and the original tensile tie. The lofts are always in the direction of the load. The only difference is that for an internal force, we slightly modify Equation 1 to account for the fact that any Δ is always opposed by the internal force, i.e. that product is always negative. If we stretch the cable, the internal force pulls back in the opposite direction of the Δ. The same phenomenon happens with internal moment, but those details are not shown here, and Equation 1 can solve for an internal moment because the Unknown product has a double negative.

**Equation 2: Unknown ∙∆+ ∑F _{i} ∙ Loft_{i} = 0**

Many other types of problems can be quickly solved using this method, including trusses and three dimensional simple spatial structures. In each case for any statically determinate problem, the MMB Method approaches theoretical values as Δ gets small. The method also can be applied to indeterminate problems, albeit in an approximate manner.

The method’s power arises from the fact that it is purely geometrical. No algebraic or graphic statics are used. The perturbed shapes clearly demonstrate that some loads have significant flow to a reaction, whereas other loads have negligible flow.

Edmond Saliklis, Ph.D, PE, is Professor of Architectural Engineering, California Polytechnic State University, San Luis Obispo, CA. He teaches structural engineering courses to architectural engineering students and to architecture students using geometric methods.