Design of RCC Rectangular Beam for biaxial bending

[ksreedhar]

Hello,

Any body can help in this issue.
I want details of “ Design of RCC Rectangular Beam for biaxial bending” with basis formulas.

Regards
Sreedhar

 

[ishvaaag]

Biaxial bending of RC beams can be thought as a case of biaxial flexocompression where the compressive force P=0.

Since normally RC rectangular beams are stout and joined in a floor, buckling in most cases is scarcely a concern, and you can simply resource to sectional analysis for the design. This we do for straight RC beams and we can do as well for those subject to biaxial moments, always subject to engineering judgement on the actual possibility of buckling develop.

Sectional analysis in compatibility of deformations is merely finding a distribution of the stresses in steel and concrete meeting the respectively assumed stress-strain laws that equilibrate the forces at the section, in this case the eccentrical in x and y bending.

You have freely downloadable sheets in the Mathcad Collaboratory that solve this and more complex cases of sectional analysis in compatibility of deformations. Since the sheets were conceived to have some P non equal to zero, put a small P there or otherwise you will be getting a division by zero error. This trick will be non-meaningful to rebar design and ascertaining the status of the section.

 

[Ingenuity]

ksreedhar,

I would do it from from first principles, much the same as a rect RC beam under uni-axis bending. If you know the section size and the rebar, trial a neutral axis depth BUT you have to also include another variable, the angle that the neutral axis (NA) makes with the horiz (x) or vertical axis.

so...first select an ORIGIN, 0, say top left corner, and reference all dimensions from this point:

1. trial (basically use an intelligent guess!) a NA angle, theta,
2. trial a NA depth, kud,
2. construct strain diagram - same as uni-axis bending but inclined at angle theta
3. determine T and C in rebar and concrete (using rectangular stress block and concrete strain 0.003)
4. check that sum C = sum T
5. if C not = T, vary NA depth, kud, in 2. above and recycle
6. when C = T, take moments about each axis about the common point, O, on section to get Mx and My
7. calc Mu = (Mx^2+My^2)^0.5
8. calc angle of resulting moment in 7., alpha, as:
alpha = arctan(My/Mx)
9. check that alpha is correct. if not go to 1. and trial a new theta and recycle.

the angle alpha is checked against the angle that the line from the centroid of C to the centroid of T makes. The line from C to T must be perpendicular to the resolved moment Mu, NOT the NA. So note that the angle of the NA, theta, and the angle that Mu subtends (alpha) will not normally be equal - and can vary considerably.

also, a strain of 0.003 is not really correct for a rectangular RC section with flexure about an angle and it probably should be 0.004, but 0.003 is conservative.

the above lends itself to a spreadsheet. you can usually get it in 3 or 4 iterations. It is a lot easier than it looks

alternatively, you can construct a Mu vs alpha interaction diagram for a given section and rebar and use that for "design" check.

HTH