Chapter 17Lamella Roofing

NBC 2016 (SP 7:2016) Volume 1

2016 · PDF page 658

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braced and stiffened members (see Fig. 19 and Fig. 20).

17.1 General

The lamella roofing offers an excellent architectural edifice in timber, amenable to prefabrication, light weight structure with high central clearance. It is essentially an arched structure formed by a system of intersecting skewed arches built-up of relatively short timber planks of uniform length and cross-section. Roof is designed as a two hinged arch with a depth equal to the depth of an individual lamella and width equal to the span of the building. The curved lamellas (planks) are bevelled and bored at the ends and bolted together at an angle, forming a network (grid) pattern of mutually

The design shall be based on the balanced or unbalanced assumed load distribution used for roof arches. Effect of deformation or slip of joints under load on the induced stresses shall be considered in design. Thrust components in both transverse and longitudinal directions of the building due to skewness of the lamella arch shall be adequately resisted. Thrust at lamella joints shall be resisted by the moment of inertia in the continuous lamella and roof sheathing (decking) of lamella roofing. The interaction of arches in two directions adds to the strength and stability against horizontal forces. For design calculations several assumption tested and observed derivations, long-duration loading factors, seasoning advantages and effects of defects are taken into account.

PART 6 STRUCTURAL DESIGN — SECTION 3 TIMBER AND BAMBOO: 3A TIMBER

17.2Lamellas

Planking shall be of a grade of timber that is adequate in strength and stiffness to sustain the assumed loads, forces, thrust and bending moments generated in lamella roofing. Lamella planks shall be seasoned to a moisture content approximating that they will attain in service. Lamella joints shall be proportioned so that allowable stresses at bearings of the non-continuous lamellas on the continuous lamellas or bearings under the head or washer of bolts are not exceeded.

17.2.1 Design Sequence and Methods

A lamella roof is essentially an arch (segmental, parabolic or elliptical) formed by a system of intersecting skewed arches, built up of relatively short members of small dimensional planks called 'lamellas'. The arch is designed as a two hinged arch with a depth equal to the depth of the individual lamella and a width equal to the span of the building of certain length. For design purposes, a unit length of the given span forming a single arch is considered loaded with the given loads. For the particular arch, curvature offers resultant outward suction on the roof to be less than its self weight to facilitate design and construction for economy. (This is possible only when the semi-angle of arch is less than $42^{\circ}$ ).

Some design assumptions are:

a) Self weight of timber components (in particular species), namely lamella planks, roof boarding, wall plates, etc is assumed for the particular timber section.

b) Imposed loads as per Part 6 ‘Structural Design, Section 1 Loads, Forces and Effects’ of the Code for curved roofs where slope at springing point is greater than $10^{\circ}$ . Take the reduced worked out values of imposed loads for actual design for timber components as per this Subsection.

c) Wind load is taken as per Part 6 ‘Structural Design, Section 1 Loads, Forces and Effects’ of the Code for structures with large openings, that is, openings larger than 20 percent of wall area (or as the case may be).

d) Stresses developed in curved lamella planks should not exceed the allowable safe stresses of timber species used.

e) Range of various sizes of timber planks for lamellas shall be as under:

Length — 1.0 m to 2.5 m
Section:

i) Thickness — 25 mm to 50 mm

ii) Width of uncurved plank — 100 mm to 250 mm

f) Limitations to be observed for spacing of lamellas over wall plate (or bressummer):

i) For skew angle of $21^{\circ}$ (diamond angle $42^{\circ}$ ).

ii) For 2.5 m length of plank, spacing shall be 870 mm.

iii) For $1.0\mathrm{m}$ length, spacing shall be $360~\mathrm{mm}$ .

i) For skew angle of $19^{\circ}$ (diamond angle $38^{\circ}$ ).

ii) For 2.5m length of plank, spacing shall be 800 mm.

iii) For 1.0 m length of plank, spacing shall be 320 mm.

g) Value of horizontal thrust, resultant thrust and maximum bending moment, are taken on per metre length of structure in case of transverse direction of arch.

h) Bending moment may be suitably reduced, if so desired by the designer, by applying stability factor if the ratio of the length of structure to the curved length of arch so warrants (as this ratio increases, the stability of the structure decreases).

j) The thrust in the skewed direction of arch, within one spacing of lamellas is shared by two lamella-lines in skewed direction.

k) More than one bolt (MS) shall be used for nodal joint. For design check of bolt against tension, half the value of thrust in skewed direction will be considered for calculations purposes in light of (j) [in case of stiff roof covering material (that is, CGI sheet, close boarding, etc) all lamella planks will be held jointly-behaving as a monolithic unit to a large extent and the actual stresses developed in bolt will be reduced resulting in lesser actual tension in bolts than the calculated design value].

17.2.1.1 Horizontal thrust

Any of the known methods for solving a statically indeterminate structure is used to evaluate horizontal thrust, bending moment, etc, to which a segmental arch will be subjected under the given condition of loadings. For $60^{\circ}$ segmental arch, the radius of an arch will be equal to the span of the arch. Horizontal thrust can be calculated by the following two methods:

a) Method 1

Horizontal thrust ( $H_{\mathrm{B}}$ ) per length of one bay (that is, distance between ‘tie-rod spacing’ or ‘column centres’):

H _ {\mathrm{B}} = (D L + I L) \times \left[ \frac {l ^ {2} + s _ {\mathrm{t}} {} ^ {2}}{8 r} \right]

PART 6 STRUCTURAL DESIGN — SECTION 3 TIMBER AND BAMBOO: 3A TIMBER

NOTES

1 When the ratio of the length of the structure $(L)$ to the circumference of the arch $(B)$ increases, stability of the structure decreases. The roof should be designed as,

a) short roof (say, $L / B = 1$ ), or

b) long roof (say, L/B = 1.5).

Shorter roofs have an advantage over long roofs for economical design. However, there is no such advantage beyond $L / B$ of 4. 2 Due to stiff roof covering material, stiffness is induced in an arch, which results in safer design.

FIG. 20 STABILITY OF LAMELLA ARCH ROOF

where

$H_{\mathrm{B}} =$ horizontal thrust, in N;

$DL =$ dead load, in N;

IL = imposed load, in N;

$l =$ span of arch, in m;

$s_{\mathrm{t}} =$ tie-rod spacing, in m; and

$r =$ rise of roof arch, in m.

b) Method 2

Horizontal thrust $(H_{1})$ per metre length of structure (in N):

H _ {1} = C \times w \times R

where

C = coefficient varying with the angle of arch and radius of arch;

w = equivalent distributed load per metre of span, in N/m; and

$R =$ radius of circular arch, in m.

Horizontal thrust $(H_{1})$ is evaluated as:

H _ {1} = \frac {\int \mu y d s}{\int y ^ {2} d s}

where

$H_{1}$ = horizontal thrust (per metre length of structure), in N.

$\mu$ = bending moment at any point ‘P’ distance ‘x’ metre from one end of springing point of arch and for vertical loads only (consider arch as a beam simply supported carrying the same load).

y = height of point 'P' above springing line of arch.

$\theta =$ angle which the point $P$ makes at the centre of arch with the vertical.

$ds = Rd\theta$

Now solution for numerator,

\mu = \frac {w l}{2} x - w x \frac {x}{2}

\left| = \frac {w x}{2} (l - x) \right|

where

\begin{array}{r l} x & = \frac {l}{2} - R \sin \theta \\ & = \frac {R}{2} - R \sin \theta \\ & = R \left(\frac {1}{2} - \sin \theta\right) \end{array}

and $y = R\cos \theta -R\cos \alpha$

y = R (\cos \theta - \cos \alpha)

and $ds = Rd\theta$

Substitute the values in expression

\int \mu y d s = \int \frac {w x (l - x)}{2} R (\cos \theta - \cos \alpha) R d \theta

Solve for limits of $\theta$ from $-\alpha$ to $+\alpha$

= \int_ {0} ^ {\alpha} w R ^ {4} (\frac {1}{4} - \sin^ {2} \theta) (\cos \theta - \cos \alpha) d \theta

Again for denominator,

\begin{array}{r l} \int y ^ {2} d s & = \int_ {- \alpha} ^ {+ \alpha} R ^ {2} (\cos \theta - \cos \alpha) ^ {2} R d \theta \\ & = 2 R ^ {3} \int_ {0} ^ {\alpha} (\cos \theta - \cos \alpha) ^ {2} d \theta \end{array}

Therefore,

\begin{array}{r l} \text {erefore,} H _ {1} & = \frac {\int \mu y d s}{\int y ^ {2} d s} \\ & = \frac {w R ^ {4} \int_ {0} ^ {\alpha} \left(\frac {1}{4} - \sin^ {2} \theta\right) (\cos \theta - \cos \alpha) d \theta}{2 R ^ {3} \int_ {0} ^ {\alpha} (\cos \theta - \cos \alpha) ^ {2} d \theta} \end{array}

On simplification of the above equation for a particular arch, $\alpha = 30^{\circ}$ , $R = l$ ( $\alpha$ in radian $= 0.5$ approx),

\sin \alpha = 1 / 2

\cos \alpha = \sqrt {3} / 2

$H_{1} = 0.923 wR$

Vertical reaction, $V_{1}=\frac{Total load \times Span}{2}$ at each end

of arch due to dead and live loads per metre length (curved strip) of arch roofing.

17.2.1.2 Resultant thrust, $T\{1\}$

Due to horizontal thrust and the vertical reaction, resultant transverse thrust, in kg per metre length on the arch can be worked out:

T _ {1} = \sqrt {H _ {1} ^ {2} + V _ {1} ^ {2}}

17.2.1.3 Maximum bending moment

Bending moment at any point ‘P’ on arch making an angle of $\theta$ with the vertical line OC is, $BM = \mu - (H \times y)$

= \frac {w R ^ {2}}{2} \left(\frac {1}{4} - \sin^ {2} \theta\right) - 0. 9 2 3 w R ^ {2} \left(\cos \theta - \frac {\sqrt {3}}{2}\right)

Differentiating the above expression for BM, we get

\frac {d M}{d \theta} = \left(- \sin \theta \cos \theta + 0. 9 2 3 \sin \theta\right) w R ^ {2}

It is well known that $\frac{dM}{d\theta} =$ shear force

This shear is zero when BM is maximum, that is,

either $\theta =$ zero

or $\cos \theta = 0.923$

Therefore, $\theta = \cos^{-1}0.923 = 22.6^{\circ}$

Substituting these values in the above equation of bending moment, we get maximum BM, that is, the maximum BM occurs at,

a) crown of arch, and

b) at $22.6^{\circ}$ from the central line OC.

Thus when $\theta = 0$ ,

\begin{array}{r l} B M & = \frac {w R ^ {2}}{2} \times \frac {1}{4} - 0. 9 2 3 w R ^ {2} \left(1 - \frac {\sqrt {3}}{2}\right) \\ & = w R ^ {2} (0. 1 2 5 - 0. 1 2 3 6 8) \\ & = 0. 0 0 1 3 2 w R ^ {2} \end{array}

When $\theta = 22.6^{\circ}$ , $\sin \theta = 0.384$

\begin{array}{r l} B M & = \frac {w R ^ {2}}{2} \left(\frac {1}{4} - 0. 3 8 4\right) ^ {2} - 0. 9 2 3 w R ^ {2} (0. 9 2 3 - 0. 8 6 6) \\ & = w R ^ {2} (0. 1 2 5 - 0. 0 7 3 7 2 8) - 0. 9 2 3 \times 0. 0 5 7 w R ^ {2} \\ & = - 0. 0 0 1 3 3 9 w R ^ {2} \text {or say} - 0. 0 0 1 3 4 w R ^ {2} \end{array}

The arch has to be designed for the greater of the two values (subject to the longitudinal stiffness coefficient on the basis of ratio $L / B$ of length of the structure to the circumference of the arch) for economical design.

PART 6 STRUCTURAL DESIGN — SECTION 3 TIMBER AND BAMBOO: 3A TIMBER

NOTE — Value of horizontal thrust, $H_{1}$ , resultant thrust, $T_{1}$ and bending moment, $M_{1}$ will be for 1 m length of building.

17.2.1.4 Spacing of lamellas

Lengthwise spacing of lamellas depends upon the angle of skew lamellas and the length of the planks used for lamellas (As a general guidance it may be stated that the angle enclosed between the intersecting skewed arches should not exceed $45^{\circ}$ . An angle between $38^{\circ}$ and $40^{\circ}$ is always preferred.)

17.2.1.5 Design of lamella sizes

Net section of each of lamellas is assumed in curved profile conforming to the curvature of the roof surface.

Maximum $\mathrm{BM} = M_{\mathrm{s}}$ over $600~\mathrm{mm}$ spacing of lamella in the direction of skew angle, for example

= \frac {0 . 6 M _ {1} \times 1 0}{\cos 1 9 ^ {\circ}} \mathrm{Nm}

where

$M_{1} =$ maximum bending moment per metre length of structure in transverse direction.

And maximum thrust $N$ (in Newton) on this strip of $600\mathrm{mm}$ in skewed direction is:

N = \frac {0 . 6 T _ {1}}{\cos 1 9 ^ {\circ}} \times \frac {1}{2}

(as the force $T_{1}$ is shared by two lines of lamellas)

where

N = maximum thrust on 600 mm strip in skewed direction.

$T_{1}$ = resultant thrust in transverse direction (in N) per metre length of structure.

Now total compressive and flexural stresses should not exceed the allowable safe stress of timber species used for lamella plank. For fulfilling this condition the following equation should be checked:

\frac {N / A}{f _ {\mathrm{cII}}} + \frac {M _ {s} / Z}{f _ {\mathrm{t}}} \leq 1

where

N = maximum thrust in skewed direction on strip of 600 mm (in N) corresponding to the maximum bending moment;

$Z =$ section modulus of lamella plank, in $\mathrm{mm}^3$ ;

$A =$ cross-sectional area of lamella plank, in $\mathrm{mm}^2$

$f_{cII}$ = safe compression strength of timber fibre parallel to grain, in N/mm $^{2}$ ;

$f_{t}$ = safe tensile strength of timber parallel to grain, in N/mm $^{2}$ ; and

$M_{s} =$ bending moment over 600 mm strip of arch in skewed direction, in Nmm;

NOTE — In case the lamella plank is under stressed, the minimum section for practical reasons shall be $25 \, mm \times 100 \, mm$ subject to satisfying minimum edge and intermediate distances for accommodating bolts at nodal joints.

17.2.1.6 Design of bolts for nodal joint

To avoid any possibility of hinge action with the use of bolts, it is imperative to adopt two or more bolts of smaller diameter and adequately spaced in accordance with the good practice [6-3A(9)].

Assume diameter of bolt ‘d’ in mm, and length of bolt in one plank ‘t’ mm, that is, thickness of middle lamella plank.

Area of contact of bolt with timber plank $= d\times t\mathrm{mm}^2$

a) Check against crushing of timber at contact point of bolt — For the bolt to transmit load without crushing the timber fibres, the following condition shall be satisfied:

$t \times d \times$ compressive stress perpendicular to grain for the timber species $>P$

where

$P =$ total load to be transmitted through bolt, in N.

b) Check against tension in bolt — Arrangement of bolts at the junction of lamellas result in pure tension in bolt as bolt line is at right angle to the outer two lamellas.

Therefore, $Z_{\mathrm{t}} = N\times \cot 2\phi$

where

$Z_{t} =$ tension in bolt, in N.

$N =$ normal thrust corresponding to maximum moment $M_{s}$ , in $\mathrm{N}$

= \frac {T _ {1}}{\cos \phi} \times \frac {1}{2}

$T_{1}$ = resultant thrust in transverse direction (in N) per metre length of structure.

Therefore,

Area of bolt $(A_{\mathrm{t}})\times$ tensile stress of steel $(F_{\mathrm{t}})\geq Z_{\mathrm{t}}$

Area of bolt required for safe design $= Z_{\mathrm{t}} / F_{\mathrm{t}}$ Select the diameter to engage at least two bolts.

(Required area of bolt is the effective area after considering reduction of area at the threaded end, which is generally 17 percent of the actual bolt diameter)

17.2.1.7 Design of steel washer at joints

Area of steel plate $= A_{\mathrm{t}}\mathrm{mm}^2$

Stress perpendicular to grain of wood at the point of contact with steel plate washer $= Z_{\mathrm{t}} / A_{\mathrm{t}}\mathrm{N} / \mathrm{mm}^2$

If $Z_{\mathrm{t}} / A_{\mathrm{t}} \leq$ compressive stress perpendicular to grain of the species of timber used, the assumed section is safe.

17.1 General

17.2.1 Design Sequence and Methods

Some design assumptions are:

17.2.1.1 Horizontal thrust

a) Method 1

b) Method 2

17.2.1.2 Resultant thrust, T{1}T\{1\}T{1}

17.2.1.3 Maximum bending moment

17.2.1.4 Spacing of lamellas

17.2.1.5 Design of lamella sizes

17.2.1.6 Design of bolts for nodal joint

17.2.1.7 Design of steel washer at joints

17.1 General

17.2.1 Design Sequence and Methods

Some design assumptions are:

17.2.1.1 Horizontal thrust

a) Method 1

b) Method 2

17.2.1.2 Resultant thrust, T{1}T\{1\}T{1}

17.2.1.3 Maximum bending moment

17.2.1.4 Spacing of lamellas

17.2.1.5 Design of lamella sizes

17.2.1.6 Design of bolts for nodal joint

17.2.1.7 Design of steel washer at joints

17.2.1.2 Resultant thrust, $T\{1\}$

17.2.1.2 Resultant thrust, $T\{1\}$