# Life Cycle Cost Assessment and the Optimum Design of Timber Roofs for Sustainable Construction

• Kamil Krzywiński
Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 98)

## Abstract

This paper describes the optimum design and life cycle cost (LCC) assessment of timber roofs for sustainable construction. For this purpose, collar beam roof construction in a typical single-family house was analyzed. Special focus was placed on the impact of the patch cross-section position for different rafter spacing. The calculations were performed for four roof angles (15°, 30°, 45°, 60°). The main goal was to find the optimum LCC for each angle. It was found that smaller rafter spacing generates a higher assembly cost and takes more time to construct. On the other hand, the wood cost for these elements is lower. The implications of LCC were evaluated to find out which patch and rafter cross-section, as well as rafter spacing for each roof angle is the most economical solution.

## Keywords

Life cycle cost Collar beam roof Patch Rafter Optimum design Sustainable construction

## Latin Upper Case Letters

$$A$$

Cross-sectional area $$\left( {{\text{m}}^{2} } \right),$$

$$E_{mean}$$

Mean value of modulus of elasticity $$\left( {\text{GPa}} \right),$$

$$E_{mean, fin}$$

Final mean value of modulus of elasticity $$\left( {\text{GPa}} \right),$$

$$F$$

Force $$\left( {\text{kN}} \right),$$

$$F_{d}$$

Design force $$\left( {\text{kN}} \right),$$

$$F_{x,Ed}$$

Design value of a force in capacity in x-direction $$\left( {\text{kN}} \right),$$

$$F_{y,Ed}$$

Design value of a force in capacity in y-direction $$\left( {\text{kN}} \right),$$

$$H$$

Overall rise of collar beam $$\left( {\text{m}} \right),$$

$$I_{y}$$

First moment of area about the strong axis $$\left( {{\text{m}}^{4} } \right),$$

$$I_{z}$$

Second moment of area about the weak axis $$\left( {{\text{m}}^{4} } \right),$$

$$L$$

Length $$\left( {\text{m}} \right),$$

$$M_{d}$$

Design moment $$\left( {\text{kNm}} \right),$$

$$M_{y}$$

Moment of y axis $$\left( {\text{kNm}} \right),$$

$$M_{z}$$

Moment of z axis $$\left( {\text{kNm}} \right),$$

$$N$$

Axial force $$\left( {\text{kN}} \right),$$

$$N_{c}$$

Compression force $$\left( {\text{kN}} \right),$$

$$P$$

Human force $$\left( {\text{kN}} \right),$$

$$V$$

Volume $$\left( {{\text{m}}^{3} } \right),$$

$$V_{t}$$

Shear force $$\left( {\text{kN}} \right),$$

$$W_{y}$$

Section modulus about axis y $$\left( {{\text{m}}^{3} } \right).$$

## Latin Lower Case Letters

$$a$$

Distance $$\left( {\text{m}} \right),$$

$$b$$

Width $$\left( {\text{m}} \right),$$

$$f_{c,0,d}$$

Design compressive strength along the grain $$\left( {\text{MPa}} \right),$$

$$f_{c,w,d}$$

Design compressive strength of web $$\left( {\text{MPa}} \right),$$

$$f_{m,k}$$

Characteristic bending strength $$\left( {\text{MPa}} \right),$$

$$f_{m,y,d}$$

Design bending strength about principal y-axis $$\left( {\text{MPa}} \right),$$

$$f_{m,z,d}$$

Design bending strength about principal z-axis $$\left( {\text{MPa}} \right),$$

$$g$$

Weight load $$\left( {{\text{kN}}/{\text{m}}} \right),$$

$$h$$

Height of element; house height $$\left( {\text{m}} \right),$$

$$k_{crit}$$

Factor used for lateral buckling $$\left( - \right),$$

$$k_{m}$$

Factor considering re-distribution of bending stresses in a cross-section $$\left( - \right),$$

$$k_{mod}$$

Modification factor for duration of load and moisture content $$\left( - \right),$$

$$m$$

Mass per unit area $$\left( {{\text{kg}}/{\text{m}}^{2} } \right),$$

$$q_{p}$$

Peak velocity pressure $$\left( {{\text{kN}}/{\text{m}}^{2} } \right),$$

$$s$$

Snow load $$\left( {{\text{kN}}/{\text{m}}} \right),$$

$$s_{k}$$

Characteristic value of snow on the ground at the relevant site $$\left( {{\text{kN}}/{\text{m}}^{2} } \right),$$

$$u_{fin}$$

Final deformation $$\left( {\text{mm}} \right),$$

$$u_{inst}$$

Instantaneous deformation,

$$l_{eff}$$

Rafter spacing distance $$\left( {\text{m}} \right),$$

$$w$$

Wind load $$\left( {{\text{kN}}/{\text{m}}} \right).$$

## Greek Lower Case Letters

$$\alpha$$

Angle between the direction of the load and the loaded edge; Angle between horizontal axis and rafter $$\left(^\circ \right),$$

$$\gamma_{M}$$

Partial factor for material properties, also accounting for model uncertainties and dimensional variations $$\left( - \right),$$

$$\sigma_{c,0,d}$$

Design compressive stress along the grain $$\left( {\text{MPa}} \right),$$

$$\sigma_{m,y,d}$$

Design bending stress about the principal y-axis $$\left( {\text{MPa}} \right),$$

$$\sigma_{m,z,d}$$

Design bending stress about the principal z-axis $$\left( {\text{MPa}} \right),$$

$$\tau_{d}$$

Design shear stress $$\left( {\text{MPa}} \right).$$

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