The resultant of a line load acts at the centroid of the load area. Calculate its distance xs from the start of the loaded region from the two edge ordinates and the loaded length. Only with this line of action can the moment equilibrium be written and the load correctly distributed to the supports.
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The resultant of a line load acts at the centroid of the load area. Calculate its distance xs from the start of the loaded region from the two edge ordinates and the loaded length. Only with this line of action can the moment equilibrium be written and the load correctly distributed to the supports.
A trapezoidal load rises over l = 4 m from q₁ = 5 kN/m to q₂ = 15 kN/m. This gives xs = 4 m/3 · (5 + 2·15)/(5 + 15) = 1.3333 m · 35/20 = 1.3333 m · 1.75 = 2.333 m from the start. The 40 kN resultant therefore acts 2.33 m behind the start of the load and thus clearly behind mid-length at 2.0 m – as befits the load being larger towards the end. Check against the limiting cases: a triangular load with q₁ = 0 and q₂ = 10 kN/m over l = 6 m gives xs = 6/3 · 20/10 = 4.0 m, exactly 2l/3. A uniform load with q₁ = q₂ always gives l/2, here 2.0 m.
A straight-line load variation between the two edge ordinates is assumed, with the load acting in one consistent direction along the length. Together with the resultant, the line of action applies only in rigid-body statics, i.e. for equilibrium of the overall system and for internal forces outside the loaded region; inside the loaded region the actual distribution must be used. Both edge ordinates must have the same sign – if the load changes sign within the length it must be split into sections, because the centroid of the load area can then fall outside the loaded region. Nonlinear variations such as parabolic or stepped loads, and area or volume loads that must first be converted into a line load, are not covered.
The total force of a line load alone is not enough to determine support reactions – you also need to know where it acts. This calculator determines that line of action for a straight-line load variation and shows that it lies at mid-length only for a uniform load.
In rigid-body statics a line load may be replaced by a single substitute force, its resultant. That force acts at the centroid of the load area – the point where the area under the load diagram would balance if you cut it out of cardboard. Formally the position follows from xs = ∫x·q(x)dx / ∫q(x)dx, i.e. the coordinate averaged with the load ordinate as weight. For a straight-line variation between the edge values q₁ and q₂ this gives the centroid of a trapezoid: xs = l/3 · (q₁+2q₂)/(q₁+q₂), measured from the end where q₁ acts.
A tray whose plates stand ever more densely towards the back has to be gripped further back, not in the middle, to keep it from tipping. The grip point is exactly the centroid of the loading, and that is precisely the point this formula computes for a line load.
xs = l/3 · (q₁ + 2·q₂)/(q₁ + q₂)
General: xs = ∫x·q(x)dx / ∫q(x)dxLinear variation, measured from the q₁ end: xs = l/3 · (q₁+2q₂)/(q₁+q₂)Uniform load (q₁ = q₂): xs = l/2Rising triangular load (q₁ = 0): xs = 2l/3Falling triangular load (q₂ = 0): xs = l/3Measured from the other end: l − xsValidity band: l/3 ≤ xs ≤ 2l/3| Symbol / input | Meaning |
|---|---|
| Distance xs of the resultant | The distance of the resultant's line of action from the start of the loaded region, i.e. from where q₁ acts. It always lies between l/3 and 2l/3: at l/3 when the load falls to zero towards the end, at l/2 for a uniform load, and at 2l/3 when it rises from zero. This distance is also the centroid of the load area – the resultant always acts there. |
| Edge ordinate q₁ at the start | The line load at the start of the loaded region, given as force per unit length, for example in kN/m. The distance xs is measured from exactly this side – swapping q₁ and q₂ means measuring from the other end, and then xs(swapped) = l − xs. |
| Edge ordinate q₂ at the end | The line load at the end of the loaded region, in the same unit as q₁, with a straight-line variation in between. The larger q₂ is relative to q₁, the further the resultant moves towards the end – the factor 2 in front of q₂ in the formula produces exactly that weighting. |
| Loaded length l | The length of the loaded region, not the beam length. The calculated distance refers to the start of that region; if it does not coincide with the start of the beam, the distance from the support to the start of the load must be added afterwards. |
q₁ is the line load at the start and q₂ at the end of the loaded region, both as force per unit length in the same unit. l is the length of the loaded region. The counting direction is decisive: xs is measured from the end where q₁ acts. Swapping q₁ and q₂ gives the distance from the other end, and the two values add up to l.
Read the edge ordinates and loaded length off the load diagram, keeping track of which end you want to count from – that end is q₁. Then convert the calculated distance into the beam's reference system: if the load does not start at the support, add the distance between the support and the start of the load. With the resultant and this distance you can write the moment equilibrium about one support and determine the reaction at the other. As a plausibility check: the result must always lie between l/3 and 2l/3.
A trapezoidal load rises over l = 4 m from q₁ = 5 kN/m to q₂ = 15 kN/m. Then xs = 4/3 · (5 + 2·15)/(5 + 15) = 1.3333 · 1.75 = 2.333 m from the start of the load. The 40 kN resultant therefore acts behind mid-length, consistent with the load growing to the right. Measured from the other end it would be 4 − 2.333 = 1.667 m, which you also get by swapping q₁ and q₂. The limiting cases confirm the formula: a triangular load with q₁ = 0, q₂ = 10 kN/m over l = 6 m gives xs = 6/3 · 20/10 = 4.0 m, exactly 2l/3; a uniform load always gives l/2; and a triangular load falling from q₁ to zero gives l/3.
The line of action always lies between one third and two thirds of the loaded length – for a straight-line variation it cannot be narrowed further, but also never falls outside. The factor 2 in front of q₂ shows the weighting: the side with the larger load pulls the point of application towards itself. In practice this means a triangular load loads the two supports in a 1:2 ratio, whereas a uniform load loads them equally. Anyone assuming mid-length for a triangular load therefore underestimates the force at the more heavily loaded support by a third.
The calculation runs in newtons per metre and metres internally. Line loads may be entered in N/mm, N/m, kN/m or lbf/ft, and length and distance in mm, cm, m, km or inches. Since only the ratio of the two edge ordinates enters, the result is independent of their unit – but both must use the same one. The calculated distance has the same dimension as the loaded length.
Support reactions of beams under triangular or trapezoidal loads; water and earth pressure on walls, where the load rises linearly with depth and the resultant acts at one third of the height above the base; overturning moment of retaining walls and foundations; wind pressure with a height profile; load assumptions for brackets and cantilevers; checking graphical solutions.
A straight-line load variation between the edge ordinates is assumed. Together with the resultant, the line of action applies only in rigid-body statics – for equilibrium of the overall system and for internal forces outside the loaded region; inside the loaded region the actual distribution must be used. Both edge ordinates must have the same sign: if the load changes sign within the length, the centroid of the load area can fall outside the loaded region and the load must be split into sections. Nonlinear variations such as parabolic or stepped loads, and area or volume loads that must first be converted into a line load, are not covered.
Common mistake: The most common error is assuming mid-length for a triangular or trapezoidal load – for a triangular load the point of application is off by one sixth of the length. Second, it is easy to lose track of which end the distance is counted from: l/3 and 2l/3 sit at different places for the same triangular load depending on the reference end. Third, the calculated distance is used directly as the distance from the support although it refers to the start of the load. And fourth, the formula is applied to loads that change sign, for which it does not hold.
Support reactions of beams under triangular or trapezoidal loads; water and earth pressure on walls, where the load rises linearly with depth and the resultant acts at one third of the height above the base; overturning moment of retaining walls and foundations; wind pressure with a height profile; load assumptions for brackets and cantilevers; checking graphical solutions.
q₁ is the line load at the start and q₂ at the end of the loaded region, both as force per unit length in the same unit. l is the length of the loaded region. The counting direction is decisive: xs is measured from the end where q₁ acts. Swapping q₁ and q₂ gives the distance from the other end, and the two values add up to l.
A straight-line load variation between the edge ordinates is assumed. Together with the resultant, the line of action applies only in rigid-body statics – for equilibrium of the overall system and for internal forces outside the loaded region; inside the loaded region the actual distribution must be used. Both edge ordinates must have the same sign: if the load changes sign within the length, the centroid of the load area can fall outside the loaded region and the load must be split into sections. Nonlinear variations such as parabolic or stepped loads, and area or volume loads that must first be converted into a line load, are not covered.