I have increased the reinforcement in my Hollow reinforced masonry retaining wall. Why is the moment resistance decreasing?

Question

I have increased the reinforcement in my Hollow reinforced masonry retaining wall. Why
is the moment resistance decreasing?

Answer

The Moment resistance for a Hollow reinforced retaining wall is calculated as the lesser of
A_sr,prov x f_yd x z , where A_sr,prov is the reinforcement provided to the wall, f_yd is the Design yield strength

And;

0.3 x f_d x d² , where f_d is the design compressive strength of the masonry and d is the depth to the
tension reinforcement.

This would indicate that increasing the reinforcement should increase the moment resistance in the
wall for the first term. Provided this is less than the second term, any increase in reinforcement
should lead to a lower utilisation ratio.

However, the other part of the first term uses the lever arm. Since the lever arm is calculated as;
Z = d x min(A_sr,prov x f_yd /(d x f_d ) or 0.95.

Since A_sr,prov is now part of a negative term, any increase in reinforcement will lead to a smaller lever
arm. Since this is part of our moment resistance calculation, the lower lever arm will result in a
lower moment resistance.

Example

The wall is 150mm thick and is currently failing using H16 bars at 250mm spacing. This gives us a
moment resistance of 8.5kNm/m

 

Looking through our output, we can see that this gives a lever arm of 25mm

 

85mm x (1 – 0.5 x 804mm²/m x 435 N/mm² / (2.928 N/mm² x 85 mm) = 25mm
So, M_Rd = 804mm²/m x 435 N/mm² x 25mm = 8.743 kNm/m
Or
0.4 x f_d x d² = 0.4 x 2.928 N/mm² x (85 mm)² = 8.462 kNm/m = 8.5kNm to 1 dp.

If we then reduce the spacing of the reinforcement, we can see the resistance moment actually decreases.

 

Using H16 bars @ 200mm spacing increase the area of steel provided from 804mm²/m to 1005mm²/m. This is due to the change in the lever arm.

If we check the output, we can see that the lever arm is now down to 10mm from 25mm.

 

85mm x (1 – 0.5 x 1005mm²/m x 435N/mm² / (85mm x 2.928 N/mm²) = 10mm

Proportionally, the change in lever arm represents a reduction of 250%. The increase in reinforcement provided is only an increase of 25% so the increase in reinforcement will not lead to an increase in moment resistance due to the disproportionate decrease in the lever arm.

This is more of an issue in narrow walls where the depth to reinforcement is likely to be small and so any change in the reinforcement will have a greater effect on the resistance moment. If we were to run the same example using a 450mm thick wall and made the same change in reinforcement, we now see the increase in reinforcement leading to a greater moment resistance.
H16 @ 250mm centres.

 

 

H16 @ 200mm centres

 

 

So just as before, the increase in reinforcement has led to a reduction in the lever arm, but due to the lever arm for the thicker wall being greater, the reduction in lever arm is only an 8% reduction, which is less than the 25% increase in reinforcement, so the change in reinforcement for this wall would result in a greater resistance moment.

The effect of changing the reinforcement specified will depend on the geometry of the wall. The thinner the stem of the wall, the shorter the lever arm is likely to be. If the lever arm is small, any change in the reinforcement provided will result in a greater reduction to the lever arm than the percentage increase in reinforcement, so the resistance moment is more likely to be reduced than increased. In these cases, it would be a more efficient solution to increase the thickness of the wall than to increase the reinforcement as this will increase the lever arm and lead to a greater moment resistance in the wall.