# How is biaxial bending considered for reinforced masonry design?

I am faced with a problem where I have to design a reinforced masonry beam for biaxial bending. The governing code is ACI 530-11. I cannot find a provision in this Code for biaxial bending. The only part that addresses this issue is in Section 2.2.3.1 which states:

The unity formula can be extended when biaxial bending is present by replacing the bending stress quotients with the quotients of the calculated bending stress over the allowable bending stress for both axes

Unfortunately this section deals with unreinforced masonry. It is very strange to me that reinforced masonry is not addressed. Any ideas?

• What's a biaxially bent "beam"? – Paul Uszak Feb 2 '16 at 22:27
• A beam that is subject to moments aboutboth its weak axis and its strong axis – user32882 Feb 4 '16 at 0:01
• A horizontal biaxially loaded bit of masonry? Wall loading and wind? I've never come across that, sorry. If it's reinforced with bars and concrete, I might be tempted to cheat and call it an RC column and design as such ignoring the masonry altogether. This probably isn't the answer you were looking for. – Paul Uszak Feb 5 '16 at 2:33
• No, that is not the answer. Never having come across a problem does not mean it doesn't exist. – user32882 Feb 5 '16 at 21:41

## 2 Answers

### Unity Equation

The Unity Equation is a very standard method of analyzing a section under combined loads. Even though the masonry code doesn't specifically call it out in the reinforced pages, it would be hard to argue that it wasn't a reasonable assumption.

$$\frac{f_b^1}{F_b^1}+\frac{f_b^2}{F_b^2}\le1$$

The equation could even be $$\le\frac{4}{3}$$ if stress increases were acceptable.

### Code

I agree that I couldn't find any other reference to biaxial bending in the masonry code. I also didn't find any more clear discussion in the masonry design handbooks that I had available.

• I will accept this answer since it corresponds with results I am obtaining from the CMD12 program. Furthermore, I do not see why we should not be able to use this equation for reinforced masonry. MSJC will certainly have to clarify this issue in future editions.... – user32882 Feb 18 '16 at 5:51

I think that the MSJC (ACI530) actually contradicts itself, a little, with regard to biaxial bending of reinforced masonry.

As you pointed out, Section 2.2.3 (unreinforced masonry) points to the use of the Unity Equation. However, the commentary of Section 2.3.4.2.2 (reinforced masonry) explicitly says,

The interaction equation used in Section 2.2.3 is not applicable for reinforced masonry and is therefore not included in Section 2.3.

So it seems that the omission of the unity equation from Section 2.3 is intentional. Further, it isn't clear what that statement means, exactly. Is it implying that the equation explicitly written in Section 2.2.3 ( $\frac{f_a}{F_a} + \frac{f_b}{F_b} \leq 1$ ) is not applicable to reinforced masonry, or that the Unity Equation in general is not applicable?

However, I think that the language provided in Section 2.3.4.2.2 is fairly clear, or at least clear enough, to indicate how one might approach biaxial bending or (biaxial) bending and compression. It states,

The compressive stress in masonry due to flexure or due to flexure in combination with axial load, shall not exceed 0.45$f'_m$ provided that the axial compressive stress due to axial load component, $f_a$, does not exceed the allowable stress, $F_a$, in Section 2.2.3.1.

This seems to imply that, for axial force with bending, one would need to satisfy both,

$$\frac{f_a}{F_a} \leq 1 \tag{1}$$

and,

$$\frac{f_a + f_b}{0.45f'_m} \leq 1 \tag{2}$$.

This would then lead me to rationally assume that for pure biaxial bending we'd simply need to satisfy,

$$\frac{f_b^1 + f_b^2}{0.45f'_m} \leq 1 \tag{3}$$

which is just a form of the Unity Equation. Hence, the code seems to contradict itself, kinda.

A few final things worth noting:

• I have a couple references, written to ACI380-11, that qualify masonry beams for biaxial bending using equation 3 (published by PPI),
• I see all sorts of stuff online, from reputable schools and organizations, using equation 3 to qualify reinforced masonry beams for biaxial bending,
• Don't forget that you need to check the combined stress in your steel as well. This conversation has largely focused on the stress in the masonry, but ensure that $$\frac{f_s^1 + f_s^2}{F_s} \leq 1 \tag{4}$$