What a cut reveals, and why it was there all along

Stand on a plank across a stream and something is happening inside the plank. It is not visible, nothing was applied to the middle of it, and yet the fibres near the top are being squeezed and the fibres near the bottom stretched.

Making that visible takes one move: cut the plank in half and ask what the missing half was doing. Whatever it was, it was exactly enough to keep the remaining half still — and that is the definition of the internal forces rather than a calculation of them.

Three quantities, and what each resists #

A plane cut through a beam can transmit exactly three things, because the piece on either side has exactly three equations to satisfy.

Axial force — the tendency of the two pieces to slide apart along the beam’s length, resisted by pulling or pushing.

Shear force — the tendency of one piece to slide past the other across the section, resisted by a force in the plane of the cut.

Bending moment — the tendency of one piece to rotate relative to the other, resisted by a couple.

For a horizontal beam under vertical load the axial force is zero and the other two carry everything. That is why beam theory is written in terms of two quantities rather than three, and why an inclined member or a portal frame needs all three.

Each is a stress resultant: not a force in the ordinary sense but a summary of a distribution of stress across the whole cut face, reduced to a single number. The shear force is the integral of the shear stress over the section; the bending moment is the integral of the direct stress times its distance from the neutral axis. Three numbers standing in for a field, which is a compression that works well away from the load and badly near it.

The cut is not a location, it is a boundary #

Move the cut and the answers change, which is not because the beam changed but because the free body did.

The left-hand piece at the first cut contains one reaction and one load; at the second it contains only the reaction. Summing them gives different totals, and the different totals are the internal forces.

That is the sense in which the shear at a station is not a local property. It is a property of everything on one side, and a load placed at the far end of a beam changes the shear at every station between it and the nearer support. The way to see that is to hold the cut still and move the load instead, which is the opposite experiment to the one just run.

Three pictures of one beam, and no two agree. That is not an instability in the method; it is the method working. A stress resultant is a property of a free body, and the free body has been changed twice — once by moving its boundary and once by moving what is inside it.

Once this is clear, the shape of the diagrams becomes obvious in advance. Between loads, nothing is being added to the free body, so the shear does not change and its diagram is flat. At a point load, a whole force enters at once, so the shear steps. Under a uniform load, force is added continuously, so the shear slopes. The diagram is the running sum, and every feature of it is a feature of what has been passed.

Which piece to sum #

The two pieces give the same answer, so the choice between them is free — and it is worth spending, because they are rarely equally easy.

The rule is to sum the piece with less on it. On the beam above, the cut two units in exposes a left-hand piece carrying one reaction and nothing else, and a right-hand piece carrying a reaction and a load; the left is one multiplication and the right is two.

That figure also settles the shape of the moment diagram between the load and the support without any further arithmetic. At x = 5 the moment was 22.5; at x = 7 it is 7.5; the drop is 15.0 over two units, and the shear on both faces is 7.5. The moment is falling at the rate the shear says it should, which is the differential relation stated as two pictures rather than as an equation.

The case where that becomes a method rather than a convenience is the cantilever. Cut one anywhere and the piece toward the free end contains no reactions at all — only the loads, which were given. So the internal forces at every station follow directly from the applied loading, and the fixed end’s reaction and moment are never needed. A cantilever is the one common structure whose analysis does not begin by solving for its supports, and the reason is a choice of free body rather than anything about the member.

The same trick is what makes the method of sections worth having on a truss. A cut chosen to leave the simpler half exposed, with moments taken about the point where two of the three cut members meet, answers one question in one line — and the art of it is entirely in where the cut goes and which piece is kept.

The sign convention, and why it is odd #

The convention used almost everywhere is that a bending moment is positive when it makes the beam sag, and it is worth explaining because it looks arbitrary.

The reason is that sagging is the common case. A simply supported beam under gravity sags everywhere, so the convention keeps most numbers positive and — more usefully — ties the sign to a physical fact: sagging puts the bottom fibres in tension. Positive moment means tension underneath, which is exactly what a reinforced concrete designer needs to know, because that is where the steel goes.

Hogging is negative, and hogging puts the top in tension. Over the support of a continuous beam the moment is negative and the reinforcement moves to the top of the slab. Anyone who has seen bars in a bridge deck rising over a pier has seen a sign convention made physical.

The shear convention is less intuitive and matters less. What matters is consistency: a shear diagram drawn with one convention and read with another gives the wrong direction for everything.

The load path, made visible #

Following the internal forces from where a load is applied to where it reaches the ground is following the load path, and it is the most useful habit in structural thinking.

A single ordinate of that middle diagram is one cut, and it is worth drawing one to make the connection rather than asserting it.

A load on a floor slab goes into a beam, from the beam into a column, from the column into a foundation, from the foundation into the ground. At every stage there are internal forces, and at every stage the same question applies: what is holding this up, and what is holding that up.

Structures fail where the path is interrupted, and the interruption is often a detail rather than a member — or a restraint that exists on the drawing and not in the building. A beam adequate in bending can fail at its connection; a column adequate in compression can punch through a slab. The internal forces say what has to be transferred; whether the transfer has been detailed is a separate question that the diagram cannot ask.

The contrast with the cut diagram is the point. The same beam under the same load has internal forces at every station and none in this picture, because where the boundary is drawn decides what is internal.

The same idea in a truss #

A truss cut behaves identically, and the method of sections is exactly this argument applied to a frame.

In a truss the cut faces carry axial force only, because a pin joint transmits no moment, so a cut delivers one quantity per member rather than three. That is the whole of the simplification, and it is a simplification of the same sum.

Cutting a truss through three members and taking moments about the intersection of two of them gives the third member’s force in one line. That is the method of sections, and it answers one question without solving the structure — which for a fifty-member truss where only the worst chord is wanted is a very large saving.

The reason it works is that a pin-jointed member carries only axial force, so a cut through three members exposes three unknowns rather than nine, and three equations settle them.

What the section does with it #

A cut gives a moment. What resists it is a distribution of stress across the face, and the two are connected by the section’s geometry alone.

The moment becomes a push and a pull separated by a lever arm, and the section modulus is the ratio that converts one into the other. The stress at the extreme fibre is the moment divided by that modulus, the stress everywhere else is that value scaled by the distance from the neutral axis, and everything about which section to use follows from the conversion rather than from the cut. The cut supplies one number; the section decides what that number costs.

Which means the cleanest possible test of the whole theory is a cut with a moment on it and nothing else — and that state can be arranged.

That middle region is the cleanest case there is: a constant moment with no shear, which is why four-point bending is the standard test arrangement and why the theory is validated there rather than under a single central load. A single central load would put the peak moment at the same station as the largest shear, and a specimen that failed would have been asked two questions at once.

The two numbers, computed #

The hero figure quotes values on its cut face, and they are worth deriving, because the derivation is three lines and it makes the claim that the internal forces are summed rather than measured concrete.

The beam spans eight units with a single load of $20$ at three units from the left. Moments about the right support give $R_1 = 20\times5/8 = 12.5$, and the vertical sum gives $R_2 = 7.5$.

At the cut five units in, the free body to the left contains the reaction $12.5$ and the load $20$, which is two units to the left of the cut. So

$$V = 12.5 - 20 = -7.5,\qquad M = 12.5\times5 - 20\times2 = 22.5.$$

At the cut two units in, the same free body contains only the reaction, because the load has not been passed yet:

$$V = 12.5,\qquad M = 12.5\times2 = 25.$$

Which contains a small surprise worth not glossing over. Moving the cut past the load reverses the shear, as expected. It does not reduce the moment — the moment is higher at the earlier cut, 25 against 22.5, because the moment peaks under the load at $12.5\times3 = 37.5$ and both cuts are on the way down from it in one direction or the other. Reading the two pictures side by side and expecting the more heavily loaded free body to give the larger moment gets it the wrong way round, and the only reliable way to know is to do the sum.

There is a check available at no cost, and it is the one the whole subject rests on: the same two quantities computed on the right-hand piece must come out equal and opposite. At the five-unit cut, that piece carries the reaction $7.5$ three units away, giving $M = 7.5\times3 = 22.5$ and $V = -7.5$. Identical. That agreement is not a coincidence to be verified case by case — it is Newton’s third law, and if it ever failed, the failure would be in the arithmetic rather than in the beam.

The cut that is not imaginary #

Every cut so far has been imaginary. Occasionally one is not, and then the three quantities stop being an analytical device and become a schedule of components.

A splice is a beam genuinely cut through and rejoined, because the member was longer than a lorry or because the erection sequence required it. Whatever the cut face was carrying has to be carried across by something bolted or welded, and the design of the splice is precisely the transfer of the three resultants, one at a time.

The moment is transferred as a couple: plates on the top and bottom flanges, one in tension and one in compression, separated by the beam’s depth. The force in each is the moment divided by the lever arm, which is the same $M/d$ that governs a truss chord appearing as a bolt count. The shear is transferred by plates on the web, because the web is where the shear was. The axial force, if any, is shared between them.

Two features of that make the abstraction concrete in a way no diagram does. The first is that the splice is usually placed where the moment is small — near a point of contraflexure — which is a design decision taken by reading a diagram whose every ordinate was one of these cuts. The second is that the splice is one of the very few places in a structure where the internal forces have to be believed individually rather than in combination, because each is resisted by different steel with different bolts.

A cut face is an idea until somebody has to buy the bolts for it.

The fourth resultant, and the other two #

The claim near the top of this essay — that a cut transmits exactly three things — is true of a plane problem and is a restriction of a larger fact.

A cut through a three-dimensional member exposes a face on which six quantities can act: three forces and three moments, one of each along and about each axis. Named, they are the axial force, two shear forces, two bending moments and one torsional moment. The plane case keeps three of the six by assuming everything happens in one plane, and the assumption is a decision that something else is looking after the other three.

Torsion is the interesting one, because it does not behave like the others. A bending moment is resisted by direct stress varying linearly across the section, which is why the second moment of area governs it. A torsional moment is resisted by shear stress circulating around the section, and the property that governs it depends on whether the section is closed or open — a hollow tube resists torsion by a shear flow running continuously around the wall, while an open section such as an I-beam has no such loop and can only resist by each plate twisting through its own thickness, which is a hundred times feebler.

Worse, an open section under torsion does something no other stress resultant does: it warps. Plane sections do not stay plane; the flanges of an I-section displace along the beam’s length, one forward and one back. If that warping is restrained — at a fixed end, at a stiffener, at a connection — the restraint generates direct stresses that the ordinary torsion theory does not contain, and warping torsion is a separate calculation added to the first.

The practical residue is a hierarchy worth carrying. Axial force is easy. Bending is well understood and dominates most design. Shear is straightforward at the level of a beam and awkward at the level of a section. Torsion in a closed section is manageable. Torsion in an open section is genuinely difficult and is designed around rather than designed for — which is why a beam expected to twist is specified as a hollow section, and why an eccentric load on an I-beam is usually resolved by adding a member to take the eccentricity out rather than by calculating the twist.

Where the model stops #

Slender members. Beam theory treats a member as a line with section properties. It is accurate when the span is much longer than the depth — say eight times or more — and progressively wrong below that. A deep beam or a short bracket does not have a linear stress distribution and cannot be analysed this way.

Away from the load. Saint-Venant’s principle says the three stress resultants describe the state of stress accurately at a distance of roughly one section depth from any concentrated load or support. Closer than that, the actual distribution matters, and bearing failures and web crippling live there.

Rigid geometry. The sums are taken on the undeformed shape.

Statically determinate. For a redundant beam the cut gives a relationship between the internal forces rather than their values.

The figures have a limitation they cannot escape: a cut face is drawn with a gap so the internal actions are visible, and there is no gap. The shear arrow and the curved moment arrow are summaries of a stress distribution over an area, and drawing them as two arrows implies a concentration that does not exist. Every free-body diagram in every textbook does this, and the honest correction is to remember that the arrow has an area under it.

The ladder from here #

Later rungs on this anchor: the differential relations between load, shear and moment. Sign conventions and the reinforcement they imply. The method of sections. Axial force in inclined members and frames. Torsion, the fourth stress resultant. Saint-Venant’s principle. Deep beams and strut-and-tie modelling, which is what replaces beam theory when the span-to-depth ratio fails. Influence lines, which ask where a moving load does the most damage. And the plastic hinge, which is what happens to a section after the moment it can carry has been reached.

Galileo posed the problem of the cantilever’s strength in 1638 and got the answer wrong, because he assumed the whole section was in tension. The correct distribution — part compression, part tension, with a neutral axis between — waited until Parent in 1713 and was not widely accepted until Navier in 1826.