design of mansonry arches

ENCI 595.05- Masonry Arches M. M. Reda Taha

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Design of Masonry Arches

Masonry arches have been used for about 5000 years beginning with the Chinese and passing through the Babylonians, Egyptians, Greeks, Romans and Normans to modern times. The ubiquitous presence of the masonry arch in vaulted roofs, aqueducts, and bridges attests to its inherent strength, tolerance of movement, and ease of construction. Many examples, ancient, medieval and relatively modern, exist in many areas of the world. The arch is a natural structural solution that is besides being aesthetically pleasing is a robust system with excellent observed functionality. The versatility of the arch can be seen in the many shapes and sizes seen over time. For example Roman semicircular, Norman pointed, Tudor and segmental arches all have distinct, differing shapes. Hence engineers should be able to design an arch to satisfy architectural needs. It is very unfortunate therefore that modern structural education typically emphasizes rectangular frames and space structures with beams and columns. The masonry arch has declined rapidly as a form of construction in the last few decades. This is in part due to perceived cost of initial construction, but also in part because of the lack of education in arch design. This lecture highlights the main considerations in design of masonry arches. The robust performance of an arch is attributed to the fact that an arch optimally utilizes the property of its constituent materials by making use of the strong compressive strength of masonry and transmits through its geometry the non-favourable tension stresses into compression stresses. An arch bridge can be easily integrated into the environment and always meets public approval and acceptance.

Figure [1]: Victorian masonry arch bridge, Manchester, UK [Courtesy of N. G. Shrive]


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1. MASONRY ARCH: TERMINOLOGY

Figure [2]: Basic components/terminology of a masonry arch 2. STRUCTURAL PRINCIPLES FOR ARCH ANALYSIS

Before we proceed to analysis of masonry arches we need to establish a few principles that are needed to understand the behaviour of the arch 1- Downward loads create thrust in the arch from the keystone to the abutment. If this thrust line coincides with the centre of the arch, the arch ring will be under uniform compressive stresses. 2- If the line of thrust deviates from the centre of the arch as shown if Figure [3] , the arch will be subjected to stresses governed by equation [1]

ZeP

AP±=σ [1]

where P is the thrust force, e is eccentricity of the force from the centre of the arch, A is the arch cross section area and Z is the section modulus.

Fig. [3]: The line of thrust and its location in the arch

Arch centreLine of thrust

Rise ( f )

Span (L)

Keystone

Intrados

Extrados

Abutment

Crown


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3- Even if the thrust line deviates from the centre line of the arch, the arch section will stay under compression if the line of thrust stays in the kern distance (zone). Figure [4] shows the kern distance (zone) for a rectangular cross section.

Figure [4]: The kern distance (zone) in a rectangular section 4- When the line of thrust goes out of the kern, this means tensile stresses will be developed in the arch, but it does not mean the arch will collapse. 5- A Few assumptions are made to allow the analysis (a) The abutments do not move. A major concern usually in design and construction of masonry arches is the abutment. The abutment should be able to restrain the arch from movement. The following effects of the abutment movement can happen - If the abutment moves outward due to excessive the arch thrust. This will result in developing 3-pin masonry arch (4 pins necessary for collapse) - If the abutment moves inward due to excessive earth pressure resulting in developing 3-pin arch - If a differential settlement takes place, extra stresses are developed in the arch (b) The arch profile does not change. What are the effects of the accuracy of the construction; thermal, moisture and creep movements? (c) Masonry is homogenous and isotropic. What are the effects of masonry anisotropy and non-homogeneity? Masonry is known to be orthotropic (d) Arch loading is uniform. What are effects of other loading cases? If an optimum arch geometry is used, uniform loading should not result in moment in the arch but only in a thrust force. Therefore, the worst case of loading for an arch is not uniform loading but unbalanced loading which is usually due to live load on only ½ the span

HH/3

KERN


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3. FAILURE MECHANISM OF A MASONRY ARCH

In 1936 and 1938, A.J.S. Pippard published the first analysis of masonry arches

suggesting the possible failure mechanisms of the arch. It was then established that an arch will fail if four hinges are formed. This pioneer work predicted the location of those hinges and calculated the possible failure load. Therefore, the load carrying capacity of an arch depends on its failure mechanism. Numerous structural models using finite elements have since been developed to predict the collapse load of arches. Most models still underestimate the inherent capacity of the arch that is attributed to its geometry. Figure [5] shows the mechanism of failure of a fixed end masonry arch.

L.O.TL.O.T

Line of thrust out of kernLine of thrust in kern

Failure Mechanism

Figure [5]: Failure mechanism of a masonry arch

4. ARCH SHAPE AND GEOMETRY

The first step in designing a masonry arch is to choose the geometry of the arch

(span, rise and shape).Then a first estimate on the arch thickness is obtained from the various formulae developed by Victorian-era engineers. Usually the rise (f) is taken (1/3 to 1/2) of the span (S). The following equations are empirical relations which can estimate the arch crown thickness (h) in mm based on the arch span (S) or its rise (f) in meters.

S36650h= (Rankine) [2]

f220h = (Hurst) [3]

f2S18282h ++= (Troutwine) [4]

S40000h= (Depuit - Semi-circular) [5]


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S20000h= (Depuit - Segemental) [6]

)S1(150h += (Sejourne- Semi-circular) [7] Case 1 For a 12 m span semi circular arch bridge, suggest a suitable arch dimension. The rise (f) can be assumed to be 6 metre. The thickness of the arch h can be estimated from the following h = 663 mm (Rankine), h = 539 mm (Hurst), h = 712 mm (Troutwine), h = 692 mm (Depuit) or h = 670 mm (Sejourne). An average value of h = 650 - 700 mm may be a useful start for a design assumptions. 5. THE THEORY OF ARCH

The geometry of an arch should be chosen to ensure that the structure is subjected to predominant axial compression under permanent loads. Thus, it is important to realize that it is the behaviour under permanent loads that governs the serviceability as well as the durability of the arch. It should also be noted that the ultimate goal is to limit the deformations and to reduce the tensile stresses developed in the arch. As the expected bending moment due to permanent gravity loads is the shape of a parabola, the optimal geometry of the arch will also be a parabola (or semi-circular).

f

Mke

y

M(x)

Arch

Self-Weight

y(x)x

y

n

xn

Figure [6]: Determining the arch geometry as a function of permanent loads.

The optimal geometry of the arch can be obtained if we assume the arch is a

statically determinate simply supported beam of a similar span. The maximum moment


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of the beam is denoted Mkey An arch geometry that matches equation [8] will result in a minimum or zero bending moment dependent on the arch end conditions being fixed or hinged.

)x(MM

f)x(ykey

= [8]

where f is the arch rise, M(x) is the bending moment at any point at distance (x) from the abutment and y(x) is the y coordinate of the same point at distance (x) from the abutment as shown in Figure [6]. Equation [8] represents a parabolic geometry as both f and Mkey are constants for a specific arch and M(x) is a parabola. Semi-circular, elliptical or parabolic arch

Although, circular and elliptical arches are also used, they will have bending moments larger than those expected in a parabolic arch. Circular arches are recommended for cases where the load is applied in a radial fashion rather than with gravity. However, a semi-circular arch bridge is very common. Tensile stresses developed in a semi-circular arch can also be limited by using the same principles used with parabolic arches. Moreover, elliptical arches are more efficient when the radial pressure varies with the arch curvature with the highest pressure applied at the arch crown. 6. ANALYSIS OF ARCHES

[a] 3 Pin - Arch [b] 2 Pin - Arch

Figure [7]: 2 and 3 pin arches.

3-Pin Arch

As shown in Figure [7a], a 3-pin arch is a statically determinate structure. Force equilibrium equations at the pins can determine all unknowns. 2-Pin Arch

As shown in Figure [7b], a 2-pin arch is a statically indeterminate structure. For the structure shown with both abutments at the same level, force equilibrium equations at the pins can determine the vertical components. The horizontal force H can be determined using Castigliano’s theorem (neglecting shear and axial deformation) as presented in equation [9].


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HM y ds

EI

y dsEI

P=− ∫

∫ 2 [9]

Mp is the moment effect at a general point P on the arch profile due to the loading between the abutment and P, y is the y coordinate of the point P and (ds/EI) is the rate of change of the arch length to its stiffness. Further explanation of this equation will be presented for the case of fixed end masonry arches. Fixed end arches

When a fixed end arch is loaded, the abutment will restrain the arch from shortening due to the load. This restraint is the source of the indeterminate moment and forces at the abutment in fixed end masonry arches. Generally, arches are statically indeterminate and therefore are subjected to bending moments under permanent loads. In other words, the resultant line of thrust in the arch is eccentric as shown on Figure [3]. These moments caused by the shortening of the arch under permanent loads can not be avoided but its distribution can be optimized if the arch geometry follows the geometric rules discussed in section 5.0.

xH

V

M

A

P

y0

Figure [8]: A fixed end symmetrical arch with an infinitely rigid arm joining the

abutment A to the elastic centroid (Origin of coordinates x and y) A fixed end arch can be analyzed using the configuration shown in Figure [8]. An infinitely rigid arm is assumed to connect the left hand abutment “A” to the elastic centre, “the centroid” of the arch. The three load actions at the centroid are then defined by equations [10] to [12]

MM P ds

EIdsEI

=∫

∫ [10]


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VM x ds

EI

x dsEI

P=∫

∫ 2 [11]

HM y ds

EI

y dsEI

P=− ∫

∫ 2 [12]

The formulae are accurate as long as axial and shear deformations can be ignored. If the axial deformation is included, equation [12] will be re-written as equation [13].

HM y ds

EI

y dsEI

LE A

P

M

=−

+

∫

∫ 2 [13]

* note that axial deformation is important for flat arches E is the modulus of elasticity of masonry, I is the moment of inertia of the arch cross section and A is the arch cross sectional area. Mp is the moment effect at a general point P on the arch profile due to the loading between A and P. The elastic centroid is determined using the integration in equation [14].

yy ds

EIdsEI

0 =∫

∫

'

[14]

where in this case, y’ is a coordinate system defined from any point on the line through the springing points. A different horizontal line could be used (e.g. through the crown) with yo then defining the distance from that line to the level of the elastic centre. 7. FIXED END MASONRY ARCHES: METHODS OF ANALYSIS

The unknown parameters M, V and H in equations 10 to 13 can be evaluated either by performing a mathematical integration of the above equations or by dividing the arch into segments each of length ∆s and performing summation of the segments. The summation is then taken over the complete length of the arch where x and y are coordinates within the system shown in Figure [8]. This summation can be done by a number of different methods dependent on the level of accuracy required for the analysis 1- Graphical Analysis

The graphical method is similar to the graphical method you learned in analysis of trusses. The arch is analyzed as a series of rods joined by pins where loads are applied. The method is an inversion of the “funicular polygon” used for cable analysis. Further details of this method can be found in structural masonry design manuals. 2- Analysis Using Spreadsheets

For a higher level of accuracy and simplicity, the arch can be divided into segments of equal horizontal length. The (x’,y’) coordinates of the centre and edge points of each segment are then defined: again for simplicity, it is convenient to use the


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left springing point as the origin of the (x’,y’) axes. The first part of the spreadsheet can then be established with ∆x’ and ∆y’ (the increment in x’ and y’ coordinates for each segment) required as input. A sample spreadsheet is presented in the appendix. ∆x’ and ∆y’ can be calculated from the arch equation if the shape can be expressed mathematically, with input therefore being just the span-rise and shape of the arch.

The increment in arc length is ∆ ∆ ∆s x y= +( ) ( )' '2 2 , h0 is the arch thickness which may vary around the arch. Varying h will imply varying I, required for the calculations of ∆s/EI. y’ is the location of the centre of the segment in the vertical direction. For some masonry arches the change of arch section may be assumed to follow equation [15] which assumes that the section varies such that ∆s/EI = ∆x/EIo. An example masonry arch bridge of cross sectional change similar to that described in equation [15] is presented in Figure [9]

h h sx

= 03∆∆

[15]

Figure [9]: West Gloucester masonry arch Bridge, UK. A cross section variation similar to Eqn. [15] (Courtesy of Derek Locke, UK).

The dead weight should include the weight of the masonry plus the weight of any fill above that segment of the arch. Summations are performed on the ∆s/EI and y’∆s/EI columns. The latter sum divided by the former provides the required value of y0 as in equation [14]. An example spread sheet is given with Example 1. Considering x and y are now the coordinates of the centres of the segments defined in the coordinate system with the elastic centre as origin. Mp the moment effect at the nth segment centroid due to the loading between A and that centroid, is given by equation [16]. In simpler words, Mp is the moment at point P due to ALL loads acting on the arch between the left hand abutment and point P. Thus the moments will not be symmetrical in terms of Mp as the


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moments on the right hand half will be greater compared to the left hand half for equivalent points P. A schematic representation explaining this fact is presented in Figure [10].

Mq n x x

D L x xP i centroid ni

n

centroid i=− +

+ −

=∑

( )( . .) * ( )' '

12

2

2

1

∆∆

[16]

where q is the uniformly distributed live load. It is clear that the first part of equation [16] depends on the load case being analyzed. In the spreadsheet, the following columns are summed: y2 (∆s/EI), x2 (∆s/EI), Mp (∆s/EI), Mpy(∆s/EI), and Mpx(∆s/EI). Values of M, V and H at the elastic centroid are then determined according to equations [10,11 and 12] presented earlier but in the summation form. As M, V, and H are determined at the elastic centroid, the moment at the left-hand abutment can be determined using equation [17].

A

P

x'

y' y'

A x'

P

MM

PP

Figure [10]: Calculation of MP at any section on the arch

M M V x H y MA A A PA= + − − [17]

where MpA is zero in this case (no moment between A and A). In the mean time, as no loads are acting on the rigid arm both H and V are actually acting at the left hand abutment. Therefore, the straining actions at the left hand abutment are determined and the straining actions at the right hand abutment can be determined from equilibrium equations. A simple check can be done for a symmetrically loaded arch, V should be half the total vertical force. H can be checked against the value obtained by assuming the arch is 3-pin. The moment at any point of the arch can be found from equation [18] M x M V x H y Mi i i P i( ) = + − − [18]


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3- Analysis Using Computer Program (MathCad®) A computer program using MathCad® was developed to provide exact analysis

of the fixed end masonry arches. Here data entry involves the span and rise of the arch, together with the arch type (i.e. parabolic or semi-circular). As above, the self weight of the arch and fill will be required, together with the live load. The integral equations to determine y0, M, V and H can be programmed directly. The integration will depend on the function governing the change of the arch thickness along the arch length. This function is involved when relating ∆s to ∆x and when determining moments. The relation described in equation [15] was used to describe the change of thickness of parabolic arches. Circular arches are assumed to have constant thickness along its span in the current version of the program. The current program allows two cases of live loading: uniformly distributed load on either the full arch span or on half the span. If super imposed dead load above the self-weight exists (e.g. spandrel walls) the program can include these loads as well. The program also allows the user to incorporate earth pressure loads directly applied to the arch in conjunction with any surcharge loads. If the arch is exposed to external wind loads the program can also consider these loads. The effect of axial deformation is also considered in the program. Once calculations are complete, the bending moment diagram and the line of thrust are plotted. The line of thrust is shown with respect to the kern distance of the arch section and the stress distribution of the top and bottom fibres of the arch are also provided. The advantage of the MathCad® analysis can be seen in its accurate calculations and visual aids. 4- Analysis using commercial FE programs

Arch analysis can also be performed using commercial finite element (FE) programs used for analyzing plane frame structures. The analysis using these programs will be highly dependent on the number of divisions (elements) used in the arch model. As most commercial FE programs are intended for analysis of prismatic members only, a very fine mesh of the model may be needed to reach a solution close to that determined by the Mathcad program®. The following example illustrates the difference in the results of arch analysis when the three methods are used: Example 1 Using the three methods of analysis described above: Determine the reactions for a 16 m span with 4 m rise parabolic arch. For one metre width of the arch the live load is 12 kN/m, the weight density of the masonry is 23 kN/m3, and the arch thickness at the crown (h0) is 0.4 m. The section of the arch is to change according to the following equation (∆s/EI)=(∆x/EI0) where EI0 is constant. The described arch is analyzed by the three methods, first by using the spread sheet, second by using the MathCad® program and finally by using a FE commercial program. The end reactions under self-weight as well as the uniform live load at the abutment using the three methods are presented in Table [1]. In the spread sheet as well as the FE methods the arch was divided into 16 segments. The calculations are given in the Appendix. The location of the line of thrust with respect to the kern as provided by the Mathcad® program is given in Figure [11].


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Table [1]: Comparison between the three methods Straining action Spreadsheet method MathCad® method FE Program

MA (kN.m) 7.15 6.8 12.2 VA (kN) 184.6 184.7 184.6 HA (kN) 176.2 176.3 174.9

The large difference between the FE method and the other methods is attributed to the very rough FE model assumed here. If a finer model with larger number of elements is chosen for analysis the FE model will give a value close to that of the Mathcad®. The difference between the Mathcad® program and the spreadsheet program is also attributed to the small number of elements used in the spreadsheet. Larger number of elements should yield better results. To examine the error in each method, the three methods can be used to find the arch reactions under a unit uniform distributed load (1 kN/m). In doing so the MathCad ® program was found to be the most accurate.

8 6 4 2 0 2 4 6 8

2

0

2

3m

3− m

y x( )h x( )

6+

y x( )h x( )

6−

y x( ) e x( )+

L2

L−2

x

Li f Th t b t li it

Figure [11]: The line of thrust with respect to the kern (Example 1) Example 2

Figure [12]: Proposed geometry for the University of Calgary on campus walkway.


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A proposed on campus feature walkway (pedestrian bridge) that will be constructed at The University of Calgary is presented in Figure [12]. A preliminary conceptual design is required to estimate construction cost. The followings are the bridge design information: 1- For aesthetic purposes as a feature bridge, a parabolic masonry arch bridge is sought. 2- Expected bridge geometry (Span 13 m, mid span clearance should not be less than 3.8 m, road elevation is at (1050.7) and walkway elevation should be finished at (1056.8). Design procedure a- Suggest the arch geometry. b- Suggest a crown thickness using empirical equations and your own engineering judgement. c- Calculate the loads on the arch. d- Perform preliminary analysis using a computer program under gravity loads only. e- Change the crown (arch) thickness if needed. f- Check other load cases for final design. g- Keep in mind (Numbers only can not guarantee aesthetic bridge). Using the previous steps the following suggestions are presented: 1- Check a parabolic arch bridge with 300 thickness, 4.5 clear height. 2- Check a parabolic arch bridge with 500 thickness, 4.5 clear height. 3- The analysis of the suggested arches are presented in the appendix showing how to decide whether this proposed system is suitable or not using the MathCad® program. 4- The accepted system was also checked using the spreadsheet program. CLOSURE

Masonry arches can be designed using software programs available for PCs, providing both numerical and graphical information. Careful analysis of selected points on the arch can be performed. Arch geometry can be assessed through repeated analysis until a suitable design is obtained for the loading expected. An arch bridge is a structure that is built to stay in service as a durable and pleasing structure. REFERENCES FOR READINGS Curtin, W. G., Shaw, G., Beck, J. K., amd Bray, W. A., “Structural Masonry Designers’ Manual” Granada Publishing, London, 1982, 498 pp. Favre, R. and De Castro, San Roman, J, 2001, “The arch: enduring and endearing”, Structural Concrete, Vol. 2, No. 4, pp. 187-200. Gebara, J. M. and Austin, D. P., 1994, “Finite Block Analysis of Masonry Arches”, TMS Journal,, August 1994, pp. 57-69. Hendry, A. W. “Structural Masonry”, 2nd Edition, McMillan Publisher, UK, 1998. Hendry, A. W., Sinha, B. P., and Davis, S. R., “An Introduction to Load Bearing Masonry Brickwork Design”, Ellis Horwood Ltd, John Wiley and Sons, 1981, 184 pp.


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Locke, D., Masonry arch bridge pictures. http://www.brantacan.co.uk/archesm.htm Shrive, N. G., Reda Taha, M. M. and Huizer, A., 2000, “Simple Design Procedures for Masonry Arches”, 12th International Brick/Block Masonry Conference, Madrid, Spain, pp. 1687-1696. Timoshenko, S. P. and Young, D. H., 1965, Theory of Structures, 2nd Edition, McGraw-Hill Book Company, NY, USA (Chapter 8 – Arches and Frames). FOR FURTHER INTERESTS - For inspiring photos for arch bridges and structures

http://nisee.berkeley.edu/godden/godden_b.html - For information and terminology of arches

http://www.bia.org/BIA/technotes/t31.htm - For inspiring photos of masonry arch bridges

http://www.brantacan.co.uk/archesm.htm - For lecture notes and presentation

http://www.reda-taha.com or http://www.unm.edu/~mrtaha

design of mansonry arches

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