Page 258 - Design of Reinforced Masonry Structures

DESIGN OF REINFORCED MASONRY BEAMS 4.121

Consequently, its location is determined using the concept of transformed section. A trans-
formed section is a fictitious section in which the cross-sectional area of one material is
converted (transformed) into an equivalent area of the other. In general, this concept can be
used for sections consisting of any dissimilar materials such as concrete and steel, concrete
and wood, wood and steel, composites, etc.
In the analysis of transformed sections, one material is selected as a reference material,
and other materials are transformed into an equivalent area of the reference material. This
transformation is achieved by multiplying the areas of the other materials by the respective
ratios of the moduli of elasticity of the other materials to that of the reference material. The
centroidal distance of the transformed area measured perpendicular to a given axis, as well
as the effective height or length of the element, remain unchained.
The ratio of moduli of elasticity of two materials is called modular ratio and denoted
by n. For example, the modular ratio n of two materials having different moduli of elastic-
ity, E and E (E > E ) respectively, can be expressed as Eq. (4.136):
1
2
1
2
n= E 2 (4.136)
E 1
where E = modulus of elasticity of the reference material
1
E = modulus of elasticity of the material whose area is to be transformed to the
2
equivalent area of the reference material
The transformation to an equivalent area is accomplished in one of the two ways depend-
ing on the reference material:
1. The cross-sectional area of the stiffer material (i.e., one with larger modulus of elasticity)
is multiplied by the modular ratio, the product being the transformed area equivalent to
the less stiff material (i.e., one with smaller value of modulus of elasticity). This method
is followed for allowable stress design (ASD, also called working stress design, WSD)
of reinforced concrete and masonry. In the latter case, the modular ratio is defined as
the ratio of moduli of elasticity of reinforcing material to that of masonry. The cross-
sectional area of reinforcement is multiplied by the modular ratio, the result being the
transformed area of reinforcing material (steel in our case). For example, in the design
of reinforced masonry structures, the modular ratio for reinforcing steel is expressed
as the quotient of moduli of elasticity of steel and masonry, E and E , respectively.
m
s
Thus,
n= E s (4.137)
E
m

The transformed area of reinforcing steel is then expressed as nA , with its centroid
s
located at the location of the centroid of reinforcement.
2. Conversely, the area of the less stiff material can be divided by the modular ratio, the
quotient then being the transformed area equivalent to that of the stiffer material. This
method is followed in composite steel design. The width of concrete cross section, b,
is divided by the modular ratio, resulting in a new width called the effective width b e
(= b/n), the resulting area (effective width times the depth of concrete cross section)
being the transformed area of concrete.

In reinforced masonry, the transformed section concept is applied to sections consisting
of masonry and steel, in which it is assumed that masonry in the tension zone of the beam
is cracked and structurally absent. The area of steel reinforcement is transformed into an