Chapter 17Lamella Roofing

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):

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:

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,

where

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

and $ds = Rd\theta$

Substitute the values in expression

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

Again for denominator,

Therefore,

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

$H_{1} = 0.923 wR$

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.

Differentiating the above expression for BM, we get

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 $\theta = 0$,

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

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.

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:

(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:

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.

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 $\mathrm{N}$

$T_{1}$ = 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)

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