Graphic Statics for Continuous Beams and Frames: A Review of the Fixed-points Method

ABSTRACT

Graphic statics not only applies to statically determinate systems but also extends to indeterminate systems. This paper reviews a historical graphical method tailored for continuous beams and frames: the fixed-points method. Despite its current obscurity, the fixed-points method played a crucial role in the repertoire of graphic statics and enjoyed considerable popularity during the late 19th and early 20th centuries. Our review outlines its historical evolution, explains its principles and techniques, and illuminates Robert Maillart’s applications of this method in the Simme bridge in Garstatt and the Weissensteinstrasse Overpass in Bern.

KEYWORDS:

• Continuous beam
• continuous frame
• graphic statics
• Robert Maillart
• statically indeterminate system
• the fixed-points method

Acknowledgments

The authors would also like to acknowledge the ETHZ for granting access to Maillart’s original working drawings.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

¹ The modern formula for curvature, which is simplified yet sufficiently precise, had already been discovered decades ago of by (Navier 1833, 49) and (Rankine and Roberts 1858, 160–161). However, these findings were regrettably unknown unto Culmann.

² It is worth noting that this analogy can be readily extended to beams with inclined loadings (Wolfe 1921, 26) and moment-resisting three-hinge arches (Saliklis 2019, 70–77).

³ In Mohr’s original paper, this diagram was referred to simply as the ‘auxiliary figure’ (Hülfsfigur) (Mohr 1868). The naming ‘MΔx force polygon’ closely follows the designation provided by (Chalmers 1881, 200), which was one of the earliest introductions of Mohr’s method to English readers. Chalmers termed this diagram the ‘MΔy force polygon’, utilizing ‘y’ as the designation for the beam length dimension.

⁴ It is worth noting that Mohr’s findings cannot be directly applied to structural members with non-rectilinear axis or under inclined loading. This is because secondary moments may occur in such cases, which can affect deflection.

⁵ The ratio between its bending stiffness (EI) and length (L) is commonly referred to as “bending stiffness”, “flexural stiffness/rigidity”, or simply as “stiffness”. The term “rotational stiffness” is chosen to distinguish this ratio from these more general terms.

⁶ Mohr’s Second Theorem: the linear deflection of an elastic curve at any given point, relative to the slope line projected from another point, multiplied by EI, equals to the moment of the area, about the first point, of the bending moment diagram between the two points (Mohr 1868).

⁷ To comprehend the principle behind the crossing line, suppose the moment at S₂ is given (represented by the length of B₂S₂). Connect B₂ to the fixed point I₄ and extend the line until it intersects the vertical line s₁’ at point D. Assume the removal of the load and the imposition of a positive moment at S₁, whose magnitude is represented by the length of line S₁D. According to the concept of the fixed point, this hypothetical moment would propagate through fixed point I₄, inducing a moment at S₂ whose magnitude is represented by the height of B₂S₂. Consequently, the angular deflection at support S₂ induced by this hypothetical moment is equivalent to that of the actual load of the first span. Therefore, the linear deflection at support S₁, relative to the slope line projected from S₂, both equals the slope at support S₂ multiplied by L₁. According to Mohr’s Second Theorem, the moment of the moment area of the parabola S₁M’S₂ about point S₁ must equal that of the ΔS₁DS₂. Therefore, $\frac{1}{2}\ast S_1{S}_2\ast \frac{2}{3}\ast S_1{S}_2\ast \mathit{M}\prime \mathit{M}=\frac{1}{3}\ast S_1{S}_2\ast \frac{1}{2}\ast S_1{S}_2\ast S_1\mathit{D}$. We can deduce that: $S_1\mathit{D}=2\ast \mathit{M}{ }^{\mathrm{\prime }}\mathit{M}$.

The equation holds true regardless of the magnitude of the distributed load, and the entire derivation is reversible. Consequently, we can determine the length of B₂S₂ by measuring off twice the length of MM’ downward from S₁ to D and projecting the length of S₁D via I₄ to the vertical of S₂. As the length of S₁D’ is also double the height of M’M and equals S₁D, the length of B₂S₂ can be more conveniently determined by projecting S₁D’ via I₄’ using the crossing line M’S₂.

⁸ KS₄ is the height of the projection of vertex O’ from point K’ to the verticals of support S₄. Due to the similarity between ΔK’KS₄ and ΔK’O’O, KS₄ equals $\frac{(L_3+\mathit{b})\mathit{h}}{L_3}$, in which L₃ is the span, b the length of OS₄, and h the length of OO’. Consequently, the moment of the area of triangle ΔKS₃S₄ about support S₃ equals $\frac{L_3(L_3+\mathit{b})\mathit{h}}{6}$. This moment is the same as the moment of the area of triangle ΔO’S₃S₄ about point S₃. According to Mohr’s Second Theorem, the bending moment area of ΔKS₃S₄ and ΔO’S₃S₄ would induce the same deflection at the spans on the left side of S₃. Similarly, ΔJS₃S₄ and ΔO’S₃S₄ would induce the same deflection at the span on the right side of S₄ (Chalmers 1881, 215–218).

⁹ The length W’W” is the relative deflection induced by the elastic weight of the moment area of △BB’C at the vertical w. According to Mohr’s second theorem and the explanation in 3.2, $W^{\prime }\mathit{W}=\frac{n}{\mathit{EI}}\cdot A_{\mathit{BB}{\mathit{ }}^{\mathrm{\prime }}\mathit{C}}\cdot \frac{1}{3}{L}_1$. In this equation n is the scale factor for the abstract elastic curve (all the y-coordinates is n times the corresponding actual deflection), $\frac{1}{3}{L}_1$ is the distance between the triangle centroid and the vertical w, $A_{\mathit{BB}{\mathit{ }}^{\mathrm{\prime }}\mathit{C}}$ is the area of △BB’C and equals $\frac{1}{2}\cdot \mathit{B}{B}^{\prime }\cdot L_2$. Likewise, $W^{\prime }\mathit{W}=\frac{n}{\mathit{EI}}\cdot A_{\mathit{ABB}}\cdot \frac{1}{3}{L}_2$, in which $A_{\mathit{BB}{\mathit{ }}^{\mathrm{\prime }}\mathit{C}}$ equals $\frac{1}{2}\cdot \mathit{BB}\cdot L_1$. It follows $\mathit{WW}=W^{\prime }\mathit{W}-W^{\prime }\mathit{W}=$ $\frac{n}{6\mathit{EI}}\cdot L_1\cdot L_2\cdot B^{\prime }\mathit{B}$

Meanwhile, the end stiffness of the column head is denoted as ε, the rotation at joint B as τ, which equals $\frac{\mathit{B}{\mathit{ }}^{\mathrm{\prime }}\mathit{B}}{\mathit{\epsilon }}$. The slope of line UV is n times τ. As point V’ trisects the span, $\mathit{V}{V}^{\prime }=\mathit{n}\cdot \mathit{\tau }\cdot \frac{1}{3}{L}_2=\frac{n}{3\mathit{\epsilon }}\cdot B^{\prime }\mathit{B}\cdot L_2$.

Finally, due to the similarity between ΔEWW” and ΔEV’V, $\mathit{a}:\mathit{b}=\mathit{WW}:\mathit{V}{V}^{\prime }=\frac{1}{2}\left(\mathit{\epsilon }:\frac{\mathit{EI}}{L_1}\right)$

¹⁰ Like in the moment distribution method (Cross 1932), the end stiffness of a structural member is the moment that need to be applied to an end of the member to induce a unit rotation of that end. it equals $\frac{3\mathit{EI}}{L}$ for members pinned at the other end like AB, CD and columns, $\frac{4\mathit{EI}}{L}$ for members fixed at the other end, in which L is the length of the member in question. For members rigidly connectedly to other member(s) at other end, like the member BC, its end stiffness can be numerically determined with the rotation angle method.

¹¹ Given its asymmetrical beam spans, this frame would undergo lateral displacement under the vertical load. To evaluate the effect of the lateral displacement, a virtual lateral constraint was added to support D (red roller support in Figure 14). The moment induced by lateral displacement equals that by the counterforce of the retaining force imposed by the virtual constraint. This moment was determined with the procedure explained in 6.2. However, the resulting moment was found to be negligible.

Additional information

Funding

This research is funded by China Scholarship Council (CSC, No. 201706190234) and the supporting grants from UCLouvain.