Design of flexural members

2. Design of Compression Members.

Members subject to axial compression only should be designed according to the following expression provided there is no tendency for buckling to occur.

σ_{c,o d} ≤ fc,o,d where σc,o d = N/A; N – Axial factored load and A – cross-sectional area.

fc,o,d is the design compressive strengthparallel to the grain obtained from Eq. 11.3.

3. Design of flexural members.

It involves principally:

1. Bending.

2. Deflection.

3. Shear.

4. Bearing.

5. Vibration.

6. Lateral buckling.

Description of methods.

3.1 Bending

If member is not to fail in bending, the following conditions should be satisfied:

k f f

f k

f

m m y d m y d

m z d m z d m y d

m y d m

m z d m z d

σ σ

σ σ

, , , ,

, , , , , ,

, ,

, , , ,

+ ≤

+ ≤

1

1

Where σm,y,d and σm,z,d are the design bending stresses about axes y-y and z-z.

fm,y,d and fm,z,d are the design strengths from equation 11.3 and km the bending factor as follows: For rectangular sections km = 0.7

For other cross – sections km = 1.0

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For a beam whit rectangular cross-section:

σ

_{m y d} ^y

σ

y

y

m z d z z

M z

Z

M

bh and M

Z

M

, , = = ₂ , , = = hb₂

6 6

My and Mz are the design bending moments about axes y-y and z-z and Zy and Zz the moduli of elasticity about axes y-y and z-z.

3.2. Deflection.

The components of the deflection are:

Limiting values.

1. Instantaneous deflection due to variable load, u2,inst, should not exceed:

u2,inst ≤ 1/300 x span.

u2,inst ≤ 1/150 x span (for cantilever) 2. Final deflection due to variable load only u2,fin, should not exceed:

u2,fin ≤ 1/200 x span.

u2,fin ≤ 1/100 x span (for cantilever)

3. Final deflection due to all the loads and any precamber, unet, fin

u2,net,fin ≤ 1/200 x span.

u2,net,fin ≤ 1/100 x span (for cantilever).

The instantaneous deflection due to the variable loads, u2,inst, and the final deflection due to the total load, u2,net,fin, can be calculated using the formulae given in Table 6.9 and should be based on E0,mean or E90,mean. The final deflection due to variable loading, u2,fin, is derived from the instantaneous deflection using the following expression:

( )

u_fin =u_inst 1+k_def

Where kdef is the deformation factor which takes into account the increase in deformation with time due to the combined effect of creep and moisture. Values of kdef are given as follow.

Load duration class. Service class 1 2 3 Permanent 0.80 0.80 2.00 Long term 0.50 0.50 1.50 Medium term 0.25 0.25 0.75 Short term 0.00 0.00 0.00

u0 – Precamber (if applied)

u1 – deflection due to permanent loads.

u2 – deflection due to variable loads.

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3.3 Shear.

In flexural members are not to fail in shear, the following condition should be satisfied:

τ

_d ≤ f_{v d,}

where τd is the design shear stress and fv,d the design shear strength.

For beam with a rectangular cross-section, the design shear stress occurs at the neutral axis and is given by:

τ

_d ^Vd

=3 A

2 ; where Vd is the design shear force and A the cross-sectional area.

f k f

v d

v k m ,

mod ,

=

γ

^{; where fv,k} is the characteristic shear strength.

For beam notched at the ends as shown in Fig. Below, the following condition should be checked:

τ

_d ≤k f_{v v d,} ; where kv is the shear factor which may attain the following values:

a). For beams notched on the unloaded side kv = 1

b). For beams of solid timber notched on the loaded side kv is taken as the lesser of kv = 1 and

( )

k

i h

h x

h

v =

⎛ +

⎝⎜ ⎞

⎠⎟

− + ⎛ −

⎝⎜ ⎞

⎠⎟

⎡

⎣⎢ ⎤

⎦⎥ 5 1 1 1

1 0 8 1

1 5

2

.

.

.

α α

α α

,

where α = he/h and x is the distance from line of action to the corner.

3.4 Bearing. (Compression perpendicular to grain).

For compression perpendicular to grain the following condition should be satisfied:

σ

_c,90,d ≤k_c,90f_c,90,d

where: σc,90,d is the design compressive stress perpendicular to grain

fc,90,d is the design compressive strength perpendicular to grain from equation 11.3 kc,90 is the compressive strength factor.

Here kc,90 takes into account that the load can increased if the loaded length, l in Fig.belows is short.

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Values for kc,90 for various combination of a, l and l1 are given in the following table.

l1 > 150 mm

l1 ≤ 150 mm a ≥ 100 mm a < 100 mm L l ≥ 150 mm 1 1 1

150 > l ≥ 15 mm 1 1 + (150 – l)/170 1 + a(150 – l)/ 17000 15 mm > l 1 1.8 1 + a/125

3.5 Vibration. (Applied to residential floors).

The method given in E.C. # 5 assumes that the floor is supported on four edges.

The fundamental frequency of vibration of a rectangular residential floor supported on four edges, fI can be estimated using:

f

( )

l

EI l

1 2 m

=2

π

=

; where m is the mass equal to the self-weight of the floor and other permanent actions per unit area(kN/m²)

l is the floor span (m).

(EI) l is the equivalent bending stiffness in the beam direction.

Unit (Nm²/m).

For residential floors with a fundamental frequency greater than 8 Hz the following conditions should be satisfied:

( )

uF≤1 5. mm kN/ and

υ

≤100 ^f1ξ−1

where ξ is the damping coefficient, normally taken as 0.01

u is the maximum vertical deflection caused by a concentrated static force F = 1 kN and ν is the unit impulse velocity.

The transverse distribution of load can be taken as 50%, i.e., 0.5 kN on the loaded joist and 25

% on the adjacent ones.

The value of the unit impulse velocity ν may be estimated from:

( ) ^{( )}

ν

=4 0 4. +0 6. n₄₀ / mbl+200 mN⁻¹S⁻²; where b is the floor width (m) and n40 the number of first – order modes with natural frequencies below 40 Hz given by.

( )

n

( )

f

b l

EI l

40 EI b

1

2 4

40 1

= ⎛

⎝⎜ ⎞

⎠⎟ −

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

⎛

⎝⎜ ⎞

⎠⎟

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪; where (EI)l is the equivalent plate bending stiffness parallel to the beam.

3.6. Lateral buckling.

The following condition shall be satisfied.

σ

_{m d,} ≤k_inst f_{m d,}

where σm,d is the design bending stress fm,d is the design bending strength kinst is the instability factor, given by:

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k for

k for

k for

inst rel m

inst rel m rel m

inst

rel m rel m

= ≤

= − ≤

=

1 0 75

1 56 0 75 0 75 1 4

1

2 1 4

λ

λ λ

λ λ

,

, ,

, ,

.

. . . .

.

<

<

where λrel,m is the relative slenderness ratio for bending.

For beams with rectangular cross-section, λrel,m can be calculated from the following expression:

λ

^{rel m}

π

ef m k k

mean mean

l hf b E

E

, G

, ,

= ⎡ ,

⎣⎢

⎢

⎤

⎦⎥

⎥

2 0 05

0 where l_efis the effective length of the beam and is obtained from the figure below.

b is the width of the beam h is the depth of the beam

fm,k is the characteristic bending strength (table 11.3)

E0,k05 is the characteristic modulus of elasticity parallel to the grain (Table 11.3) E0,mean is the mean modulus of elasticity parallel to the grain (Table 11.3) Gmean is the mean shear modulus = E0,mean/16.

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In document Design of Steel and Timber Structures -Examples (Page 73-78)