03-16-2010, 12:46 PM
In my penultimate post (post #7), I hinted that one of the best options for the column section is a circular one.
Why? -
This is due to the fact that torsion tends to assume a concentric form or radiates from a common center as such the most convenient form to suit its distribution is that of a circle represented as a circular cross section.
· The direction of the horizontal load is unpredictable as such a structural form that has an indefinite numbers of edges as represented by a circular cross section is the most ideal to counter its effect.
· The seismic load is assumed to occur in 2-orthogonal directions. This direction is not defined as such it could be at any of the infinite combinations of points in space. The best structural form that could adjust to any of the directions that this load could decide to occur is that of a circular cross section (since it has an indefinite number of edges as such could accommodate the force equally, no matter the direction of its application).
· Cylindrical members tend to have uniform structural strength distribution (homogeneity) due to the fact that it does not have blind spots (unlike other structural forms such as the polygons that tend to have blind spots due to their edges as such the strength distribution is most none uniform as concrete tend to segregate around the edges-thus lose their consistency/strength).
Consider the equation of the stress on a structural member:
= p/A MY/I. This implies that the stress experienced diminishes with increased moment of inerter. So for cross sections of the same cross sectional area, the difference could only be made by the difference in the moment of inerter (I) and Y which is the distance of the most stressed fiber from the neutral axis of the member. If we combine Y/I, we have 1/Z, where Z is the section modulus, as such the difference could only be made by the section modulus of the individual cross sections chosen. Let’s compare the section modulus of some common structural cross sections (the rectangle, square and circular cross sections) of the same cross sectional area 300mm x 450mm in torsion: -
For gravitational loads only (in this case the direction of application of the load is defined), the square, then the rectangular sections are surely the ideal ones as there is no doubt that they have greater section module when compared to the circular one (i.e. d^3/3 (= 1.65x10^7mm^3) for the square section , bd^2/6 = 300x450^2/6 (= 1.0x10^7mm^3) for the rectangular section and PiR^3/4 (= 7.0x10^6 mm^3)for the circular section). But if the direction of the application of the load is not known, then we have to consider all possible critical directions, including: -
Rectangle:-
Area: - 135000 mm^2
Dimension: - 300mm x 450mm
Z: - bd^2/6 = 450x300^2/6 (=6.75 x 10^6)
Square:-
Area: - 135000 mm^2
Dimension: 367.42mm x 367.42mm
Z: - d^3/3 ( =1.65x10^7)
Circular:-
Area: - 135000 mm^2
Dimension: - R = 207.3mm
Z: - PiR^3/4(=7.0x10^6)
When oriented such that the axis of bending passes through the diagonal of the structural element, we have:-
Rectangle:-
Area :- 135000 mm^2
Dimensione:- 300mm x 450mm
Z :- b^2d^2/6 SQR(d^2 + b^2)
= 300^2x450^2/3245
= 5.61x10^6
square:-
Area :- 135000 mm^2
Dimensione:- 367.42mm x 367.42mm
Z :- d^3/6SQR2 = = 0.117x367.42^3 = =5.84X10^6
Circular:-
Area :- 135000 mm^2
Dimensione:- R = 207.3mm
Z :- Pi R^3/4 = = 7.0x10^6
Since design is usually made considering the most critical condition, it implies that we have to use the least section modulus in our design (which will give the highest stress) i.e.:-
Rectangle:-
5.61x10^6mm^3
square:-
5.84X10^6mm^3
circular:-
7.0x10^6 mm^3
The above results give us the ratio of 1 : 1.04 : 1.25
Note that, apart from the fact that the circular section provided the greatest modulus of elasticity as such experienced the least stress, also in all the possible situations(i.e. no matter the direction of application of the load), section modulus for the circular section remained same while it varied for all other sections considered. This implies that circular cross section is more reliable as a structural section for column design in situations in which the actual direction of application of the load could not be predicted as is the case in which torsion is to play a considerable part (we can not predict with certainity, the angle of attach of wind load and earthquake).
Regards
Teddy
Why? -
This is due to the fact that torsion tends to assume a concentric form or radiates from a common center as such the most convenient form to suit its distribution is that of a circle represented as a circular cross section.
· The direction of the horizontal load is unpredictable as such a structural form that has an indefinite numbers of edges as represented by a circular cross section is the most ideal to counter its effect.
· The seismic load is assumed to occur in 2-orthogonal directions. This direction is not defined as such it could be at any of the infinite combinations of points in space. The best structural form that could adjust to any of the directions that this load could decide to occur is that of a circular cross section (since it has an indefinite number of edges as such could accommodate the force equally, no matter the direction of its application).
· Cylindrical members tend to have uniform structural strength distribution (homogeneity) due to the fact that it does not have blind spots (unlike other structural forms such as the polygons that tend to have blind spots due to their edges as such the strength distribution is most none uniform as concrete tend to segregate around the edges-thus lose their consistency/strength).
Consider the equation of the stress on a structural member:
= p/A MY/I. This implies that the stress experienced diminishes with increased moment of inerter. So for cross sections of the same cross sectional area, the difference could only be made by the difference in the moment of inerter (I) and Y which is the distance of the most stressed fiber from the neutral axis of the member. If we combine Y/I, we have 1/Z, where Z is the section modulus, as such the difference could only be made by the section modulus of the individual cross sections chosen. Let’s compare the section modulus of some common structural cross sections (the rectangle, square and circular cross sections) of the same cross sectional area 300mm x 450mm in torsion: -
For gravitational loads only (in this case the direction of application of the load is defined), the square, then the rectangular sections are surely the ideal ones as there is no doubt that they have greater section module when compared to the circular one (i.e. d^3/3 (= 1.65x10^7mm^3) for the square section , bd^2/6 = 300x450^2/6 (= 1.0x10^7mm^3) for the rectangular section and PiR^3/4 (= 7.0x10^6 mm^3)for the circular section). But if the direction of the application of the load is not known, then we have to consider all possible critical directions, including: -
Rectangle:-
Area: - 135000 mm^2
Dimension: - 300mm x 450mm
Z: - bd^2/6 = 450x300^2/6 (=6.75 x 10^6)
Square:-
Area: - 135000 mm^2
Dimension: 367.42mm x 367.42mm
Z: - d^3/3 ( =1.65x10^7)
Circular:-
Area: - 135000 mm^2
Dimension: - R = 207.3mm
Z: - PiR^3/4(=7.0x10^6)
When oriented such that the axis of bending passes through the diagonal of the structural element, we have:-
Rectangle:-
Area :- 135000 mm^2
Dimensione:- 300mm x 450mm
Z :- b^2d^2/6 SQR(d^2 + b^2)
= 300^2x450^2/3245
= 5.61x10^6
square:-
Area :- 135000 mm^2
Dimensione:- 367.42mm x 367.42mm
Z :- d^3/6SQR2 = = 0.117x367.42^3 = =5.84X10^6
Circular:-
Area :- 135000 mm^2
Dimensione:- R = 207.3mm
Z :- Pi R^3/4 = = 7.0x10^6
Since design is usually made considering the most critical condition, it implies that we have to use the least section modulus in our design (which will give the highest stress) i.e.:-
Rectangle:-
5.61x10^6mm^3
square:-
5.84X10^6mm^3
circular:-
7.0x10^6 mm^3
The above results give us the ratio of 1 : 1.04 : 1.25
Note that, apart from the fact that the circular section provided the greatest modulus of elasticity as such experienced the least stress, also in all the possible situations(i.e. no matter the direction of application of the load), section modulus for the circular section remained same while it varied for all other sections considered. This implies that circular cross section is more reliable as a structural section for column design in situations in which the actual direction of application of the load could not be predicted as is the case in which torsion is to play a considerable part (we can not predict with certainity, the angle of attach of wind load and earthquake).
Regards
Teddy