DIFFERENT ARCH SHAPES:
STRUCTURAL BRICK ARCHES
Architects always wish to add beauty and elegance to buildings. Also the public like the way in which it looks as attractive and beautiful with respect to the surroundings. It became customary to add arches for the openings not only outside the building but also for the door and other openings. When these add shapes are added it becomes a challenge to structural engineers with respect the dimensional design aspect.
In olden days the arches are made use in building bridges as at that time neither the concrete nor reinforcement were invented. It is the idea that these arches are invert of bending moment diagram wherein there is no tension in the member. So by inverting the BM diagram the structural engineer can either design an arch or a roof truss where the member will have no tension but only compression forces only. The shape of the member can be circular, parabola, catenaries or an ellipse. In the olden days many masonry structures were either one of the shapes mentioned.
As there are no texts explaining or covering the detailed design of a Reinforced Concrete Arches an attempt is made to explain the design by referring many books.
Extrados: The curve which bounds the upper edge of the arch.
Intrados: The curve which bounds the lower edge of the arch. The distinction between soffit and intrados is that the intrados is a line, while the soffit is a surface.
Crown: The apex of the arch’s extrados. In symmetrical arches, the crown is at the midspan.
Rise: The maximum height of the arch soffit above the level of its spring line.
Soffit: The surface of an arch or vault at the intrados.
Span: The horizontal clear dimension between abutments.
Spandrel: The masonry contained between a horizontal line drawn through the crown and a vertical line drawn through the upper most point of the skewback.
Springing: The point where the skewback intersects the intrados.
Springer: The first voussoir from a skewback.
Spring Line: A horizontal line which intersects the springing.
Voussoir: One masonry unit of an arch.
CHARACTERISTICS OF ARCH ACTION:
The main characteristics of arches common to all is the presence at the support of Horizontal thrust induced there, because of unyielding supports prevent the curved beams from straightening under vertical loads. The horizontal thrust acts towards the arch. It produces compression stresses at all sections of the arch.
The horizontal thrust also produces -ve BMs at all sections of the arch which counteract the +ve BMS due to the loads. Thus the second characteristic of arch action is that in addition to the BMs each section is subjected to a direct thrust and also that at all sections the static BMS due to load is considerably reduced by the BM due to horizontal thrust.
The above characteristics are common to FIXED as well as HINGED arches.
Design of R.C.C..Arch
Load on arch:
Reference is made to IS4090-1967 in which the clause 220.127.116.11 says that “when the ratio of fill above the crown to span of the arch less than 1, full weight of fill will be borne by arch. This gives an idea that when the arch is used at lintel level the weight of the brick masonry above the lintel and the floor loads is carried by the arch. This cane be calculated and applied as UDL over the span.
Rise of arch:
The rise of the arch generally be between 1/3 to 1/4 6 of the span for economy; the smaller value being applicable to relatively larger span and large value for relatively smaller spans.
The cross-sectional area of longitudinal reinforcement should not be less than 0.8% of the area of arch section.
The practice of placing bars only on the tension face and bending them to other face where the BM changes sign is not recommended.
This is due to the fact that the arch is subject to only compressive forces.
In arch slabs the transverse reinforcement shall be provided for distribution temperature and shrinkage. Min area of 0.2% of sectional area.
The above clauses of IS code offers a fair idea of the arches. Concrete arches may be built either plain concrete or reinforced. But referring to the book “ Concrete plain and reinforced Vol-II- by late Frederic and Thampson” it is important that plain concrete arches should never be used unless rest directly on rock foundation. Reinforcement is particularly necessary for FLAT ARCHES because there the effect of rib shortening ans change of temperature is especially large.
The area of tensile reinforcement must not be less than 0.25% of the largest gross cross section of the arch.
Placing of reinforcement:
Reference is made to the text book ”Concrete plain and reinforced Vol-II- by late Frederic and Thampson” in which it is illustrated that:
The main reinforcement runs longitudinal with the arch. Since for different conditions, tension can near intrados just as well as near the extrados, reinforcement is usually placed near both faces of the arch. In addition to the longitudinal bars, cross bars are used which tie the main bars and also prevent any longitudinal cracks. To prevent buckling of longitudinal bars hoops are used running around the top and bottom bars.
The amount of longitudinal reinforcement usually 0.5% to 1.0% of cross section of arch.
To take care of tensile stresses which may develop on top at the crown and at the bottom at the springing, usually the reinforcement is placed symmetrically about the arch axis, one half of the total area being used near each face.
Instead of rectangular stirrups if it is spiral it increases the allowable compressive strength of the concrete.
MINOR AND MAJOR ARCHES:
(Ref:BIA TECH.NOTES 31A)
Depending upon the span of the arches they are classified as MINOR and MAJOR arches.
Minor arches are those whose spans do not exceed 6’-0” and with maximum rise to span ratios of 0.15.
Major arches are those with spans in excess of 6 ft or rise to span ratios greater than 0.15
THRUST COEFFICIENTS FOR SEGMENTAL ARCHES
The above graph is the representation of thrust coefficients (H/W) for segmental arches subjected to uniform load over the entire span. Once the thrust coefficient is determined for a particular arch, the horizontal thrust (H) may be determined as the product of the thrust coefficient and total load(W). To determine the proper thrust coefficient, one must first determine the characteristics of the arch, S/r and S/d. r-rise and d-depth of arch.
S- the clear span,
r- the rise of the soffit and
d- the depth of the arch.
In these ratios and in the ratios and equations that follow, all terms of length must be expressed in the same. For example, in computing S/r and S/d if S is in feet, r and d must be in feet also.
Minor arch loadings:
The loads falling upon a minor arch may consists of LIVE loads and DEAD loads from floors, roofs, walls and other structural elements. These are applied as point loads or as uniform loads fully or partially distributed.
The Dead load of a wall above an arch may be assumed to be the weight of wall contained within a triangle immediately above the opening. The sides of this triangle are at 45-degree angles to the base. Therefore, its height is ½ of the span. Such triangular loading may be assumed to be equivalent to a uniformly distributed load of 1/3 times the times the triangular load.
Superimposed uniform loads above this triangle may be carried out by arching action of the masonry wall itself. Uniform live and dead loads occurring below the apex of the triangle are applied directly upon the arch for deign purposes. Heavy concentrated loads should not be allowed to bear directly on minor arches. This is specially true of JACK arches. Minor concentrated loads bearing on or nearly directly on the arch may safely be assumed to be equivalent to a uniformly distributed load to twice the concentrated load.
Major arch loadings:
The principal forces acting upon arches in buildings are the result of vertical dead and live loads and wind loads. It often assumed that the entire weight of masonry, above the soffit, presses vertically upon the arch. This certainly is not accurate, since even with dry masonry a part of the wall will be self-supporting.
The designer must rely on empirical formulae, based on the performance of existing structures, to determine the loads on an arch. The dead load of masonry wall supported by an integral arch depends upon the arch rise and span and the wall height above the arch. It may be considered to be either uniform (rectangular) or variable (complementary parabolic) in distribution or combination thereof.
In the reference “FRAMES AND ARCHES” by Valerian Leontovich solutions for arches with rise-to-span ratios(f/L) ranging from 0.0 to 0.6, the following recommended assumptions for loading of such arches are believed to be safe:
For low rise arches, f/L=0.2 or less, a uniform load may be assumed. This load will be weight of wall above crown of the arch up to a maximum height of L/4.
For higher rise arches a dead load consisting of uniform plus complementary parabolic loading may be assumed. The maximum ordinate of the parabolic loading will be equal to a weight of wall whose height is the rise of the arch. The minimum ordinate of the parabolic loading will be zero. The uniform loading will be weight of the wall above crown of the arch up to a maximum height of L^2/100.
Uniform floor and roof loads are applied as a uniform load on the arch. Small concentrated loads may be treated as uniform loads of twice the magnitude. Large concentrated loads may be treated as point loads on the arch.
Formulae for Arches:
Ref: “FRAMES AND ARCHES” by Valerian Leontovich
M and Q are zero at any section of the arch. V=W/2
When X?L/2, Nc=H1cosØ+W/2(1-2X/L) sinØ
When X?L/2, Nc=H1cosØ-W/2(1-2X/L) sinØ
For Vertical complementary Parabolic lo adding for a parabolic arch is,
J =1+ ( ?/ ?) ;
F =?- ?J ;
K =S ?/?;
? =2(? + ?)
For vertical uniform load over the entire arch:
M and Q are zero at any section of the arch.
When x ? L / 2:
Notation: In these equations, the subscripts 1 and 2 denote the left and right supports respectively. The subscript x denotes values at any horizontal distance, x, from the origin. ? is the angle, at any point, whose tangent is the slope of the arch axis at that point. (See Table4.)
M = moment
N = axial force
Q = shearing force
f = rise of the arch
W = total load under consideration
H = horizontal thrust
V = vertical reaction
L = span of the arch
S and T are load constants (see Table 5).
Establish principal dimensions of the arch.
Depending upon the established shape and f/L ratio of the arch, obtain the corresponding K value of the arch (Table 3).
Obtain the elastic parameters (? ,?,?,?) from Table 6.
Perform the algebraic operations with the given equation as given above.
By.T.Rangarajan,B.E,M.Sc(struct.engg),F.I.E,C.E.,Consulting Structural engineer,Coimbatore,Tamilnadu.