Architectural Structures

Visualizing Load Flow Geometrically

Edmond Saliklis

• Publisher: Routledge
• Year: 2022
• ISBN:
  9781000578638
• Edition: 1
• Topic: Architecture
• Subtopic: Architecture General

Architectural Structures presents an alternative approach to understanding structural engineering load flow using a visually engaging and three-dimensional format. This book presents a ground-breaking new way of establishing equilibrium in architectural structures using the Modern Müller-Breslau method.

While firmly grounded in principles of mechanics, this method does not use traditional algebraic statics, nor does it use classical graphic statics. Rather, it solely uses new geometric tools. Both statically determinate and statically indeterminate structures are analyzed using this graphic method to provide a geometric understanding of how load flows through architectural structures. This book includes approachable coverage of parametric modeling of two-dimensional and three-dimensional structures, as well as more advanced topics such as indeterminate structural analysis and plastic analysis. Hundreds of detailed drawings created by the author are included throughout to aid understanding. Architecture and structural engineering students can employ this novel method by hand sketching, or by programming in parametric design software.

A detailed yet approachable guide, Architectural Structures is ideal for students of architecture, construction management, and structural engineering, at all levels. Practitioners will find the method extremely useful for quickly solving load tracing problems in three-dimensional grids.

1 Introduction

DOI: 10.4324/9781003180913-1

Imagine that there was a way of establishing equilibrium of structural elements without any tedious calculations. Imagine there was a way of accurately establishing how much load flows to a support, or how much bending occurs in a particular spot in a beam or frame, all by a quick hand sketch. Imagine that there was a method that could establish load flow in complicated reinforced concrete beams that have changing cross sections. Imagine a method of analysis that is fun, engaging, and truly provides insight into how load flows through a structure.

Your dreams have come true! There is such a technique, and it is the Modern Müller-Breslau Method.

The Modern Müller-Breslau Method is a means of establishing equilibrium of structural elements and assemblies of elements. It does not use traditional algebraic statics, nor does it use graphic statics. Rather, it is an energy method. By perturbing a system slightly, i.e. moving one specific constraint from its original configuration a small amount, the entire structure moves to accommodate that single, small perturbation. As the entire structure moves, so do the external loads applied to the structure move, in accordance with the constraints of the problem. The method hinges on the fact that the work done by the unknown force or moment being sought moving through the perturbation is equal to the work performed by all of the external loads moving through that lofted configuration.

For structural elements, classified as “determinate”, the external equilibrating reactions are unambiguous and do not depend on the material properties of the element, nor do they depend on the cross-sectional properties (round, square, and rectangular) of the elements. This is what is meant by “determinate”, i.e. the reactions are unambiguously determined. For structural elements classified as “indeterminate”, the external equilibrating reactions are not immediately found because the material properties of the element, as well as the cross-sectional shape, do indeed come into play. This is what is meant by “indeterminate”, i.e. the deformations and stiffness of the structure must be taken into account when establishing the external equilibrating reactions. The Modern Müller-Breslau Method applies to both determinate and indeterminate structures.

The geometric technique described in this textbook is the Modern Müller-Breslau Method. It has roots in the nineteenth century, but has been expanded in the twenty-first century to include lateral loads as well as gravity loads. While elements of the method are extremely well known, Civil Engineers will recognize the Classical Müller-Breslau Method as the “Influence Line” technique, the modern perturbation technique will be new to readers. The method’s application to two-dimensional grids of beams, to lateral loads, and to indeterminate structures, all of which can be performed on a table napkin sketch, is certainly novel. This new method has not received scholarly attention, with very limited literature devoted to the topic. Detailed examples will be shown using elementary structural elements, for gravity as well as for lateral loading.

Just as Form and Forces by Edward Allen and Waclaw Zalewski sparked a revolution in graphic statics, so it is hoped that a similar paradigm shift will occur in the way we teach structures to architecture and engineering students, with the Modern Müller-Breslau Method. While the Classical Müller-Breslau Method is well known to most Civil Engineers around the world, the secrets of this method have been hidden in plain sight for over 150 years. Both the Modern as well as the Classical Müller-Breslau Method can find external equilibrating reactions, in addition to internal axial, shear, and bending moments. These solutions are theoretically exact for any determinate problem subjected to gravity loads only, including complicated grids of horizontal beams. Furthermore, the Modern Müller-Breslau Method can asymptotically approach theoretically correct answers for structures subjected to lateral as well as gravity loading. This textbook will show detailed examples of these methods. Then, this textbook will demonstrate new techniques for analyzing indeterminate structures subjected to gravity loads. The book closes with a few tantalizing areas of projected study such as buckling and plastic analysis.

The method always uses the same fundamental work equation, shown as in Equation 1.1. Note that the signs of the work done by forces (or moments) are indeed important.

(Equation 1.1)$$Unknown\cdot \Delta +{{\displaystyle \sum }}^{\text{}}Forc{e}_i\cdot lof{t}_i=0$$

In Equation 1.1, the Unknown is the force or moment being sought. Δ is a perturbation of the structure, which occurs when employing the method. The symbol Δ is generic; it could be a displacement if seeking an external force or an internal shear, or it could be a rotation if seeking an external equilibrating reaction bending moment or an internal bending moment. Δ is created in the location of the unknown, in some assumed direction. The Forces are the externally applied loads to the structure and the loft is the travel taken by each Force during the perturbation of the entire structure which was caused by Δ. The main advantage of this technique is that the perturbed shape has absolutely nothing to do with the applied loads. Thus, even for an indeterminate element such as a beam fixed at both ends, the analyst would only be responsible for seeing how the beam would deform due to a given rotation of one end if the moment at that end was sought. This is analogous to the steps taken in the Slope-Deflection Method, where effects are looked at due to individual end degrees of freedom.

STEPS

1. Remove the capability of the structure to carry the Unknown you are looking for (reaction or moment).
2. Apply the perturbation Δ in the location of the Unknown. In the Classical Müller-Breslau Method this was a unit perturbation; in the Modern Müller-Breslau Method, this starts out as a visibly large perturbation and is reduced to nearly zero, thereby asymptotically approaching the theoretical answer. Civil Engineers will immediately recognize the Classical Method as the Influence Line problem.
3. Enforce all other boundary conditions.
4. Watch as the external loads are lofted or moved due to the perturbation and measure these lofts. If the method is applied as a hand sketch on determinate structures, the lofts are exactly found by similar triangles. If the method is applied as a hand sketch on indeterminate structures, the lofts are approximated by the sketch which is drawn to some convenient scale.
5. If there is a kink in the structure due to the perturbation, the load must be broken up into sub-portions on either side of the kink.
6. If the structure is determinate, all pieces remain straight after the perturbation.
7. If the structure is indeterminate, most pieces are curved after the perturbation and the load must be distributed reasonably smoothly along the curved structural elements.

To introduce the method, consider a statically determinate beam with an overhang, which will be immediately obvious to Civil Engineers. We seek the vertical left reaction, and Figure 1.1 describes the problem. Any reaction symbolized by a triangle will potentially have a horizontal component as well as a vertical component. Any reaction symbolized by a circle will have only a vertical reaction. Loads can be distributed along the length of a structure, as shown in Figure 1.1, or they could be concentrated at one particular point.

1.1 Example 1 setup

Using the previously stated steps, since we seek the left vertical reaction, it is removed. In the Classical Müller-Breslau Method, the perturbation at this point is a unit vertical movement, while in the Modern Müller-Breslau Method, it is a variable movement that follows a circular path. This circular path establishes one major difference between the Classical and the Modern Method. In the Classical Method, the beam is assumed to stretch so that there is no horizontal movement, only vertical. In the Modern Method, only the member being cut and investigated stretches; all other elements remain as rigid body links which move in a circular path and no elements other than the one being investigated change length. If there are no kinks in the perturbed shape, we replace any distributed loads by a single equivalent load at the distribution’s centroid. If there are kinks in the perturbed shape, the load must be focused, or concentrated, on either side of the kink, in proportion to the distance the distributed load covers on either side of the kink. However, if there are no distributed external loads, and only concentrated or so-called “point loads”, then they do not need to be modified in any way. In Figure 1.1, there are two distinct load distributions, creating two equivalent point loads as shown in Figure 1.2. Figure 1.2 shows the Classical method and Figure 1.3 shows the Modern method of establishing the left vertical equilibrating reaction. Force1 multiplied by Loft1 is negative in Equation 1.1, while Force2 multiplied by Loft2 is positive in Equation 1.1. Note that these work terms are on the left side of Equation 1.1.

1.2 Classical method for RLeft

1.3 Modern method for RLeft

For gravity loads on horizontal structures, the Classical Method and the Modern Method are equivalent. One can immediately see how a simple hand sketch can be used in the method since similar triangles establish the lofts effortlessly. Yet, if the...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• List of Figures
• List of Equations
• Acknowledgments
• 1 Introduction
• 2 Determinate Beams
• 3 GeoGebra for Beams
• 4 Trusses
• 5 GeoGebra for Trusses
• 6 Horizontal Grids
• 7 Arches
• 8 GeoGebra for Arches
• 9 Indeterminate Structures
• 10 Special Topics
• Index