Structural Engineering & Statics
where the load goes
Internal forces

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

Cut a beam anywhere and two quantities appear on the face — a shear force and a bending moment. Nothing was applied there. They are what the material was already doing.
10 min read 7 figures The load must go somewhereThe statics of things that do not move

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.

The same beam, cut at x = 5A beam separated at one station. On the exposed face a shear force and a bending moment appear, equal and opposite on the two pieces, with values obtained by summing the forces on whichever piece is easier.2012.57.5shear 7.5moment 22.5the cut, at x = 5nothing was applied here — the internal forces are what the left-hand piece needs
Fig. 1 A beam separated at one station. A shear force and a bending moment appear on the exposed face, equal and opposite on the two pieces, with values obtained by summing whichever piece is easier.

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 same beam, cut at x = 2A beam separated at one station. On the exposed face a shear force and a bending moment appear, equal and opposite on the two pieces, with values obtained by summing the forces on whichever piece is easier.2012.57.5shear 12.5moment 25.0the cut, at x = 2nothing was applied here — the internal forces are what the left-hand piece needs
Fig. 2 The same beam cut on the other side of the load. The shear has reversed sign and the moment is smaller, because the piece being summed now contains different things.

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.

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.

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.

Load, shear and moment — a simple spanThe applied load, the shear force it produces and the bending moment that follows, drawn one above another to the same horizontal scale. Shear is the integral of the load and moment is the integral of shear.4 per unit lengthshear16.0moment32.0 at x = 4.00the moment peaks exactly where the shear passes through zero
Fig. 3 The load, the shear it produces, and the moment that follows, drawn one above another. Every foot of load that enters the beam has to leave through a support, and the diagrams record the journey.

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.

A beam, its loads and its reactionsA free body: the beam cut away from everything it touches, with the forces that were doing the touching drawn on it. The reactions are computed from the loads, so their arrows are to scale relative to each other.1289.510.5ΣM about one support gives the other reaction; ΣF then gives the first
Fig. 4 The whole beam as a free body. Every internal force has vanished, because internal forces occur in equal and opposite pairs and cancel inside any body they are internal to.

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.

Joint 0 of the truss, cut outOne joint of the truss with every force acting on it. Two equations — the horizontal and vertical sums — are enough for a joint with no more than two unknown member forces, which is the whole method.HV29.4-38.6reaction 0.0reaction 25.0ΣH = 0 and ΣV = 0, and nothing else is needed
Fig. 5 A joint cut out of a truss, with every force on it. In a truss the cut faces carry axial force only, because a pin joint transmits no moment — so one quantity per member rather than three.

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.

Bending is a push and a pullA section carrying a bending moment, with the stress at every height computed as the moment times the distance from the neutral axis divided by the second moment of area. It is compression above and tension below, and zero exactly at the neutral axis.neutral axiscompressiontensionI = 29.97 × 10⁶Z = 299.7 × 10³peak stress 200.2σ = M y ÷ I, at every height
Fig. 6 The stress at every height in a section carrying a bending moment, computed from the moment, the distance from the neutral axis and the second moment of area.

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. Everything about which section to use follows from that conversion.

Load, shear and moment — a simple spanThe applied load, the shear force it produces and the bending moment that follows, drawn one above another to the same horizontal scale. Shear is the integral of the load and moment is the integral of shear.1515shear15.0moment37.5 at x = 4.32the moment peaks exactly where the shear passes through zero
Fig. 7 Two point loads on a simple span. Between them the shear is zero and the moment is constant — pure bending, with no shear at all, which is the state a laboratory bending test is built to create.

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.

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.

Where this goes next

2012.57.5shear 7.5moment 22.5the cut, at x = 5nothing was applied here — the internal forces are what the left-hand piece needs The free body is a choice, and choosing it well is the whole skill Cutting a structure open is not a step in the method. It is the method — and where the cut is made decides whether the answer takes one line or twenty. 4 per unit lengthshear16.0moment32.0 at x = 4.00the moment peaks exactly where the shear passes through zero The diagram is an integral, and that is why it can be drawn by eye Load, shear and moment are one function and its two integrals. Once that is seen, the diagrams stop being things to calculate and become things to sketch. neutral axiscompressiontensionI = 29.97 × 10⁶Z = 299.7 × 10³peak stress 200.2σ = M y ÷ I, at every height Bending is a pair of forces, pushing and pulling A bending moment is not a mysterious twisting. It is a push near the top of a section and a pull near the bottom, separated by a lever arm — a couple, made out of stress.