In step 1, we looked at a beam bending about one axis, so the normal stress had only one term. Real members are rarely that tidy. A column in a building carries axial load from the floors above and bends at the same time, from wind or from beams connected to one face of it. A corner column, with beams framing in from two directions, often bends about both of its axes at once.
In this step, we'll build up the normal stress one term at a time on a 3D column. As each term is added, we'll watch the neutral axis move and see where the worst stress ends up.
Axial force and bending: a 4 m column. Orbit, read the results and hover the members; nothing you do here changes the model.
Members can copy this model into their own projects and work the step in it. Join or sign in.
1.0 Adding the stresses from axial force and bending
The axial force and each of the two bending moments produce their own normal stress on the section. While the steel is elastic, the response is linear, so we can simply add them together. This is the principle of superposition, applied to stresses on a section.
With y and z measured from the centroid along the section's two principal axes, the normal stress at any point on the section is given by,
ฯ(y,z)=ANโโIzโMzโyโโIyโMyโzโ
The first term is the axial stress, the same at every point on the section. The second is bending about the strong axis (z), which we met in step 1. The third is bending about the weak axis (y).
The minus signs come from the sign convention. A positive moment - Structure Bench calls it sagging, even in a column - puts tension on the โy or โz face. Don't worry too much about the signs. In practice, you'll always know which face a moment stretches just by looking at the structure.
Now, each term varies linearly over the cross-section, so their sum does too. Plotted as a height above the section, the stress is a flat, tilted surface. The neutral axis is the line where that surface crosses zero. In pure bending, it passes through the centroid. Once an axial force is added, it no longer has to.
2.0 The model
Our model is a UC 203 x 203 x 46 column in S355 steel, standing 4 m tall. It's fixed at its base and free at its top, so it behaves as a vertical cantilever. There are three load cases, each applying a single load at the top of the column:
Axial (300 kN), 300 kN acting downwards,
Along X (10 kN), 10 kN acting along X, which bends the column about its strong axis,
Along Y (5 kN), 5 kN acting along Y, which bends it about its weak axis.
Two combinations add the loads in one at a time: Axial + along X, then Axial + along X + along Y.
The axial force is the same all the way up, but the moment grows from zero at the top to a maximum at the base, where each horizontal load acts on the full 4 m lever arm. So the base is where we'll look. The moments there are,
Mzโ=10ร4=40ย kNย m,Myโ=5ร4=20ย kNย m
Things to try
Press Load the model and check that Axial (300 kN) is the result set shown (a model opens on its first load case). Switch on the Member stress layer and select the ฯ chip.
Zoom in on the base of the column, where the moments are largest, and hover over the corners of the flanges. In 3D, the card lists ฯ, ฯ, ฯvMโ and U together, whichever chip is on, and the colour bar shows the range. Then do the same for Axial + along X, and then for Axial + along X + along Y.
With Axial + along X selected, click FRONT on the view cube to look along Y, side-on to the bending. Find where the colour changes sign across the depth. That's the neutral axis - notice it isn't at the middle of the section.
Switch to the U chip with the last combination selected and find the most heavily utilised corner.
3.0 What you'll see
What you should see at the base
Compression is negative.
Result set
Most compressive
Most tensile
Axial (300 kN)
-51.11 MPa everywhere
none
Axial + along X
-140.04 MPa
37.82 MPa
Axial + along X + along Y
-271.39 MPa
169.18 MPa
The utilisation under the last combination peaks at about 0.76, at one corner of a flange.
The axial load alone paints the column a single colour. Every fibre carries the same 51 MPa of compression, so there's no neutral axis on the section at all. Of course, a perfectly central load is an idealisation - a real column almost always has some eccentricity.
Adding the X load bends the column about its strong axis. The flange on the side the load pushes from goes into tension, while the compression in the far flange grows.
Because the axial compression is still there, the zero line doesn't sit at the centroid. The bending stress has to cancel the axial compression before any tension appears, and that only happens a little over halfway out to the tension flange. So the neutral axis is pushed towards the tension flange, about 58 mm from the centre of a 203 mm deep section. In other words, the axial compression shrinks the tension zone.
Adding the Y load tilts the stress surface the other way too. Now the worst stress isn't on a whole flange but at a single corner, where the compression from all three terms lines up. The opposite corner has the most tension.
This is why biaxial bending catches people out. Each moment on its own might look modest, but on an I-section the extreme fibres for both axes meet at the flange tips. Notice too that the weak-axis term (131.35 MPa) is larger than the strong-axis term (88.93 MPa), even though the Y load is half the X load. That's because the weak-axis section modulus is roughly a third of the strong-axis one (152.26 against 449.8 cmยณ). So don't dismiss a small weak-axis load, and always check the corners.
Hand check for the worst corner
The section properties. From the section tables, A = 58.7 cm2, Izโ = 4,570 cm4 with h = 203.2 mm, and Iyโ = 1,550 cm4 with b = 203.6 mm.
The section moduli. The bending stress at an extreme fibre is the moment divided by the elastic section modulus, W. This is the second moment of area divided by the distance to that fibre, which is half the depth or half the flange width. So,
Wzโ=20.32/24,570
The three terms. Converting to N and mm, the three terms at the base are,
ANโ=
The two corners. At the corner where all three compress, ฯ=โ51.11โ88.93โ131.35=โ271.39 MPa. At the opposite corner, both bending terms are tensile, so ฯ= MPa, or 169.18 with the three terms unrounded. These agree with the model's โ271.39 and +169.18 MPa.
How much bending before any tension?
The condition for zero stress. Under the axial load and the X load alone, the tension flange is at zero stress when the bending stress just cancels the axial stress. A moment Mzโ with an axial force N has the same effect as N acting at an eccentricity e=Mzโ/N, so,
WzโM
The core. The core of a section is the zone around the centroid within which a compressive load causes no tension anywhere on the section. We've just found its edge along the strong axis: as long as the 300 kN acts within 77 mm of the centroid in that direction, the whole section stays in compression. Along the weak axis the edge is much closer, at Wyโ/A = 152.26 / 58.7 = 2.59 cm, or 26 mm. For a solid rectangle, the same calculation gives the familiar middle third rule.
The horizontal load. In our column, the eccentricity comes from the horizontal load acting over the 4 m height, so the load that just reaches the core's edge is 300ร0.0766/4=5.75 kN. If you're a member, try it in your copy: change the load in the Along X (10 kN) case to 5.75 kN, solve, pick Axial + along X, and the tension flange will read zero.
4.0 What to take from this step
We've seen that the normal stresses from axial force and from bending about each axis simply add, as long as the steel stays elastic, giving a flat, tilted stress distribution over the section. With an axial force present, the neutral axis moves away from the centroid, and it moves further as the axial force grows. With bending about both axes, the worst stress sits at a corner of the section, and that's the point to check.
Remember that the simple addition only holds while the whole section is elastic. Once the worst corner yields, the distribution is no longer a flat plane, and the capacity under combined axial force and bending has to be found another way. To see how that's done, take a look at this EngineeringSkills tutorial. Also, these are stresses on the section only - a slender column in compression must also be checked for buckling.
So far, every stress we've looked at has acted along the member. Next, shear stress in a beam: the stress that acts across it.