Holooly Plus Logo
Holooly Plus Logo

Question 24.17: Determine the position of the shear center of the channel se...

Determine the position of the shear center of the channel section beam of Example 24.13.

The Blue Check Mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.

As in Example 16.2 the shear center lies on the horizontal axis of symmetry so that we only need to apply an arbitrary shear load, S_{Y}, through the shear center and calculate the shear flow distribution in, say, the bottom flange. In this case, S_{X}=0, I_{X X}^{\prime}=0 and Eq. (24.40) reduces to

 

N_{x}=\sum\limits_{ p =1}^{ N }\left[\begin{array}{lll}\bar{k}_{11}&\bar{k}_{12}&\bar{k}_{13}\end{array}\right]\left\{\begin{array}{c}\varepsilon_{x}\\\varepsilon_{y} \\\gamma_{x y}\end{array}\right\}\left(z_{ p }-z_{ p -1}\right) (24.40)

 

q_{s}=-E_{Z, i }\left(S_{Y} / I_{X X}^{\prime}\right) \int_{0}^{s} t_{ i } Y d s  (i)

 

Referring to Fig. 24.17 and Example 24.13

 

I_{X X}^{\prime}=2 \times 60,000 \times 100 \times 2 \times 75^{2}+20,000\left(1.0 \times 150^{3} / 12\right)=14.1 \times 10^{10} N / mm ^{2}

 

On the bottom flange Y=-75 mm and t_{i}=2.0 mm so that Eq.(i) becomes

 

q_{s}=60,000\left(S_{Y} / 14.1 \times 10^{10}\right) \int_{0}^{s} 2 \times 75 ds

 

where s is measured from the free edge of the flange. Then

 

q_{s}=64 \times 10^{-5} S_{Y S}  (ii)

 

Suppose that the shear center is a distance \xi_{ S } to the left of the vertical web. Then, taking moments about the midpoint of the web

 

S_{Y} \xi_{ S }=2 S_{Y} \int_{0}^{100} 6.4 \times 10^{-5} \times 75 s d s

 

which gives

 

\xi_{ S }=48.0 mm
Capture

Related Answered Questions

A beam has the singly symmetrical composite section shown in Fig. 24.17. The flange laminates are identical and have a Young’s modulus, EZ, of 60,000 N/mm^2 while the vertical web has a Young’s modulus, EZ, of 20,000 N/mm^2. If the beam is subjected to an axial load of 40 kN, determine the axial

Verified Answer:
For each flange,   b_{i} t_{i} E_{Z, i...

In the box section beam shown in Fig. 24.18 the covers 1 and 3 are identical laminates each comprising four orthotropic plies, 0.15 mm thick, and for which the ply angle is 45°. The ply elastic constants are El=140 000 N/mm^2, Et=10 000 N/mm^2, Glt=5000 N/mm^2 and νlt=0.3. The webs 2 and 4 are also

Verified Answer:
The equivalent elastic constants for the covers ha...

The symmetric four-ply laminate whose cross-section is shown in Fig. 24.14 comprises plies of the same thickness but of two different materials with ply angles of 30° and 45°. For plies 1 and 4, El=200 000 N/mm^2, Et=15 000 N/mm^2, Glt=10 000 N/mm^2 and νlt=0.3. For plies 2 and 3, El=140 000 N/mm^2

Verified Answer:
Consider plies 1 and 4. The minor Poisson’s ratio ...

If the ply angle in the laminate of Example 24.9 is 45° calculate the equivalent elastic constants.

Verified Answer:
The minor Poisson’s ratio has the same value, 0.02...

If the laminate in Example 24.8 comprises plies of the same thickness but which are orthotropic not isotropic determine the equivalent elastic constants. The ply angle θ=0° and the ply elastic constants are El=140 000 N/mm^2, Et=10 000 N/mm^2. Glt=5000 N/mm^2 and νlt=0.3.

Verified Answer:
Initially we calculate the minor Poisson’s ratio [...

Figure 24.13 shows the cross-section in the xz plane of a symmetric laminate comprising four plies which are singly orientated and isotropic; all the plies have the same thickness of 0.15 mm. If Young’s modulus for the plies is 70 000 N/mm^2 and Poisson’s ratio is 0.3 calculate the equivalent

Verified Answer:
Since the laminate is symmetric and isotropic the ...

A generally orthotropic ply is subjected to direct stresses of 60 N/mm^2 parallel to the x reference axis and 40 N/mm^2 perpendicular to the x reference axis. If the longitudinal plies are inclined at an angle of 45° to the x axis and the elastic constants are El=150,000 N/mm^2, Et=90,000 N/mm^2,

Verified Answer:
We note that there is no applied shear stress, so ...

A composite channel section has the dimensions shown in Fig. 17.12 and is subjected to a torque of 10 Nm. If the flanges have a laminate shear modulus of 20,000 N/mm^2 and that of the web is 15,000 N/mm^2 determine the maximum shear stress in the beam section and the distribution of warping,

Verified Answer:
The torsional stiffness of the section is obtained...

If the reduced stiffnesses in a unidirectional ply are k11=40,000 N/mm^2, k12 = k21 = 2,000 N/mm^2, k22 = 8,000 N/mm^2 and k33 = 5,000 N/mm^2, determine the elastic constants of the ply.

Verified Answer:
From Eqs (24.27) and (24.32)   \left\{...

A thin ply consists of an isotropic material having a Young’s modulus of 80,000 N/mm^2 and a Poisson’s ratio of 0.3. Determine the terms in the reduced stiffness and compliance matrices.

Verified Answer:
The reduced stiffnesses are   k_{11}=k...