A line load is a force distributed over a length, given as force per unit length. For calculating support reactions it may be replaced by a single substitute force – its resultant. For a linear distribution this is simply the area under the load ordinate: R = (q₁+q₂)/2 · l. The same formula covers rectangular, triangular and trapezoidal loads.
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A line load is a force distributed over a length, given as force per unit length. For calculating support reactions it may be replaced by a single substitute force – its resultant. For a linear distribution this is simply the area under the load ordinate: R = (q₁+q₂)/2 · l. The same formula covers rectangular, triangular and trapezoidal loads.
A trapezoidal load rises over l = 4 m from q₁ = 5 kN/m to q₂ = 15 kN/m. The resultant is R = (5 + 15)/2 kN/m · 4 m = 10 kN/m · 4 m = 40 kN. The mean of the two edge ordinates is therefore 10 kN/m, so in total the trapezoidal load behaves like a uniform load of 10 kN/m. Limiting cases: a triangular load with q₁ = 0 and q₂ = 10 kN/m over l = 6 m gives R = 5 kN/m · 6 m = 30 kN, exactly half the rectangular load of the same height. A uniform load of 8 kN/m over 3 m gives R = 24 kN.
A straight-line variation of the line load between the two edge ordinates is assumed, with the load acting in one consistent direction along the length. The resultant replaces the distributed load 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 resultant must not be used, because the shear-force and bending-moment diagrams there depend on the actual distribution; deformations and deflections likewise require the true load distribution. Nonlinear load shapes such as parabolic or stepped loads, loads changing sign within the length, and area or volume loads that must first be converted into a line load are not covered.
A load distributed over a length can be replaced by a single substitute force for equilibrium calculations. This calculator determines that resultant for the most common case – a straight-line load variation – and thereby covers rectangular, triangular and trapezoidal loads in one formula.
A line load, also called a distributed load, is a force spread over a length. It is given not as a force but as force per unit length, usually in kN/m. To determine support reactions it is replaced by its resultant R – the total force with the same static effect. Mathematically R is the integral of the load ordinate over the length, i.e. visually the area under the load diagram. If the load varies linearly between the edge values q₁ and q₂, that area is a trapezoid and the formula reduces to R = (q₁+q₂)/2 · l: mean load ordinate times length.
A blanket of snow on a flat roof presses over the whole area. To ask how heavily the two supports are loaded, however, it is enough to know the total weight of the snow and its centre of gravity – just as one treats a bag of cement as one force rather than millions of grains. That is exactly the simplification the resultant provides.
R = (q₁ + q₂)/2 · l
General: R = ∫q(x)·dx, i.e. the area under the load diagramLinear variation (trapezoid): R = (q₁+q₂)/2 · lUniform load (q₁ = q₂ = q₀): R = q₀·lTriangular load (q₁ = 0): R = q₂·l/2Required peak ordinate of a triangular load: q₂ = 2R/l| Symbol / input | Meaning |
|---|---|
| Resultant R | The total force equivalent to the distributed load. In rigid-body statics – i.e. when calculating support reactions and internal moments outside the loaded region – the line load may be fully replaced by this single force. Inside the loaded region it may not: there the shear-force and bending-moment diagrams need the actual distribution. |
| Edge ordinate q₁ at the start | The line load at the left edge of the loaded region, i.e. the force per unit length there. The unit is force divided by length, for example kN/m – not the total force. For a triangular load starting at zero, enter q₁ = 0; for a rectangular (uniform) load, q₁ and q₂ are equal. Source: self-weight per metre, snow load per square metre times the load width, or liquid pressure times width. |
| Edge ordinate q₂ at the end | The line load at the right edge of the loaded region, in the same unit as q₁. A straight-line variation is assumed between the two edge values. A typical example of q₁ ≠ q₂ is the water pressure on a tank wall, which increases linearly from top to bottom. |
| Loaded length l | The length of the region over which the line load acts – not the total length of the beam. If the load acts over only part of the beam, enter only that loaded section here; the unloaded sections enter the moment equilibrium later, through the position of the resultant. |
q₁ is the line load at the start and q₂ at the end of the loaded region, both as force per unit length. For a uniform load set q₁ = q₂; for a triangular load starting at zero set q₁ = 0. l is the length of the loaded region, not the beam length. Where the ordinates come from: self-weight per metre from a section table and density, an area load times the load width, or liquid pressure times wall width.
First sketch the load diagram and read off both edge ordinates and the loaded length, making sure q₁ and q₂ use the same unit and really are force per unit length. After computing the resultant comes the second step: the moment equilibrium additionally needs its line of action, which lies at mid-length only for a uniform load. With the resultant and its position, support reactions follow just as for a point load. Important: for the shear-force and bending-moment diagrams inside the loaded region, and for deflections, the actual distribution must be used again.
A trapezoidal load rises over l = 4 m from q₁ = 5 kN/m to q₂ = 15 kN/m. The mean ordinate is (5+15)/2 = 10 kN/m, so R = 10 kN/m · 4 m = 40 kN. Limiting cases as a check: a triangular load from 0 to 10 kN/m over 6 m gives R = 5 · 6 = 30 kN – exactly half the rectangular load of 10 kN/m over the same length, matching the half-triangle area. A uniform load of 8 kN/m over 3 m gives R = 24 kN. And in reverse: for a triangular load over 6 m to carry 30 kN in total, its peak ordinate must be 10 kN/m.
Because only the mean of the edge ordinates enters, every linearly varying load behaves in total like a uniform load of that mean value. For the total force it therefore makes no difference whether the load rises from left to right or the other way round – the resultant is the same. Only its position distinguishes the cases, and with it the split between the two supports. This is why the resultant alone is never enough: it answers the force balance, not the moment balance.
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, the length in mm, cm, m, km or inches, and the resultant in N, kN, lbf or kip. A useful check: kN/m times m gives kN – if the units do not combine into a force, one of the inputs is not a genuine line load.
Support reactions of beams under self-weight, snow or imposed load; water pressure on tank and dam walls, which increases linearly with depth; earth pressure on retaining walls; wind pressure on facades and signs; load determination for brackets, rails and crane runways; converting an area load onto a beam via the load width.
A straight-line load variation between the edge ordinates is assumed, with the load acting in one consistent direction over the whole length. The resultant replaces the distributed load only in rigid-body statics – for equilibrium of the overall system and for internal forces outside the loaded region. Inside the loaded region this is inadmissible, because the shear-force and bending-moment diagrams there depend on the true distribution; the same applies to deflections. Nonlinear variations such as parabolic or stepped loads, loads changing sign within the length, and area or volume loads that must first be converted to a line load are not covered.
Common mistake: The most common error is entering the total force as a line load or vice versa – a mix-up the units reveal: q is given in kN/m, R in kN. Second, the beam length is often used instead of the loaded length when the load covers only part of the beam. Third, a triangular load is frequently computed as peak ordinate times length, which doubles the resultant; the correct value is the half area. And fourth, the resultant is used to compute moments inside the loaded region, where it gives a wrong answer.
Support reactions of beams under self-weight, snow or imposed load; water pressure on tank and dam walls, which increases linearly with depth; earth pressure on retaining walls; wind pressure on facades and signs; load determination for brackets, rails and crane runways; converting an area load onto a beam via the load width.
q₁ is the line load at the start and q₂ at the end of the loaded region, both as force per unit length. For a uniform load set q₁ = q₂; for a triangular load starting at zero set q₁ = 0. l is the length of the loaded region, not the beam length. Where the ordinates come from: self-weight per metre from a section table and density, an area load times the load width, or liquid pressure times wall width.
A straight-line load variation between the edge ordinates is assumed, with the load acting in one consistent direction over the whole length. The resultant replaces the distributed load only in rigid-body statics – for equilibrium of the overall system and for internal forces outside the loaded region. Inside the loaded region this is inadmissible, because the shear-force and bending-moment diagrams there depend on the true distribution; the same applies to deflections. Nonlinear variations such as parabolic or stepped loads, loads changing sign within the length, and area or volume loads that must first be converted to a line load are not covered.