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| The Gherkin - London |
Compression members in concrete structures is considered the most important members in the structure due to its nature of failure. In the same time these type of members considered as the most complicated members in its behavior to understand, analysis and design.
Through this article and subsequent articles we will try to overcome that major obstacles to understand compression members in practical way.
Theoretical Concept:
Practically all compression members in concrete structures are subjected to moments in addition to axial loads. These may be due to:
- Misalignment of the load on the column.
- The column resisting a portion of the unbalanced moments at its ends transferred directly from supported slabs or beams.
In figure(1) the distance e is referred to as the eccentricity of the load. The two cases in fig.1-b and fig.1-c are the same because the eccentric load P can be replaced by a load P acting along the centroidal axis, plus a moment M = P.e about the centroid.
To illustrate conceptually the interaction between moment and axial load in a column, an idealized homogeneous and elastic column with a compressive strength fcu, equal to its tensile strength ftu, (this means -|ftu| = |fcu|) will be considered. For such a column, failure would occur in compression when the maximum stresses reached fcu, as given by the known formula for normal stresses:
Where:
| A | = area of the cross section. |
| I | = moment of inertia of the cross section. |
| y | = distance from the centroidal axis to the most highly compressed surface. |
| P | = axial load, positive in compression. |
| M | = moment, positive in the shown direction. |
Dividing both sides of stress equation by fcu gives:
The maximum axial load the column can support occurs when M = 0
And the maximum moment that can be supported occurs when P = 0
Substituting Pmax and Mmax gives:
This equation is known as an interaction equation, because it shows the interaction of, or relationship between, P and M at failure.
This equation is plotted as line AB in figure(2) and a similar equation for tensile load P governed by the tensile stress ftu plotted as line BC in the same figure, and lines AD, DC result from the moment in opposite direction.
Points on lines AB, BC, CD and DA represent combinations of P and M at failure, So any load combination of P and M inside the area bounded by these lines will not fail the column section, on the other hand, if any load combination of P and M outside this rigion or on the boundaries the column will fail.
Figure(3) shows the interaction diagram for elastic material with -|ftu| = 0.5|fcu|.
Figure(4) shows the interaction diagram for elastic material with tensile stress |ftu| = 0.
In figures 2, 3, 4 lines AB and AD indicate load combinations corresponding to failure initiated by compression (governed by fcu), while lines BC and DC indicate failures initiated by tension (governed by ftu). In each case, the points B and D represent balanced failures, in which the tensile and compressive resistances of the material are reached simultaneously on opposite edges of the column.
Reinforced concrete is not elastic and has a tensile strength that is much lower than its compressive strength. An effective tensile strength is developed, however, by reinforcing bars on the tension face of the member. For these reasons, the calculation of an interaction diagram for reinforced concrete is more complex than that for an elastic material. However, the general shape of the diagram resembles figure(3).
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