Calculator
A plane frame of straight members joined at nodes, rigidly or with a hinge at either end, held by fixed, pinned, roller, guided or elastic supports. Load its nodes, load its members, warm them, or let a support settle; read the bending moment, shear and axial force diagrams, the reactions and the deflected shape. Start from a portal, a gable, an L-frame, a column or a beam on columns, or draw your own, up to 50 nodes.
- Frame
- Rigid or hinged joints
- Loads
- Nodal, member, temperature, settlement
- Results
- Diagrams, reactions, deflections
Calculation sheet
Plane frame analysis with diagrams, reactions and deflections
ttcivilstructural.com/resources/plane-frame-analysis
The frame
Statically indeterminate to degree 3
degree = 3m + r − e − h = 3 × 3 + 6 − 12 − 0 = 3
Equilibrium alone is not enough to find the forces: they depend on member stiffness, E·I and E·A. m is the number of members, r the reaction components (fixed 3, pinned 2, roller 1, guide 2, each spring 1), e the equations at the nodes (3 each, or 2 where every member is hinged) and h the hinges.
- Largest bending moment, M
- −50.44 kip·ft - Member 3, at node 4
- Largest shear force, V
- −13.87 kip - Member 3, at node 4
- Largest axial force, N
- −13.87 kip - Member 2, at node 2
- Largest node displacement
- 0.09287 in - node 3
Frame, loads, reactions and diagrams · ft, kip
Moment diagram drawn at 1 ft = 15 kip·ft, measured at the scale of the frame.
- Member
- Node
- Fixed support
- Load, with its size
- Distributed load, with its intensity
- Reaction, with its size
- Reaction moment
- Bending moment M: diagonal hatching, drawn on the tension side
- + and − are the signs of the values
Reading the drawing: bending moment, drawn on the side of each member that is in tension. The largest bending moment is −50.44 kip·ft, in member 3, at node 4.
Point at a member in the drawing, or choose one here, to read N, V and M at any place along it.
Reactions
| Support | Node | Type | Rx (kip) | Ry (kip) | Moment (kip·ft) | Resultant (kip) |
|---|---|---|---|---|---|---|
| 1 | 1 | Fixed | −0.3109 | +10.13 | +16.11 | 10.13 |
| 2 | 2 | Fixed | −9.689 | +13.87 | +46.45 | 16.92 |
A reaction is what a support does to the frame: Rx is positive to the right, Ry upward, and a counter-clockwise moment is positive. A roller or guide has just one force, which acts along the line it holds and counts as positive along its angle, measured from +x.
Member end forces
| Member | Nodes | Length (ft) | N at start (kip) | V at start (kip) | M at start (kip·ft) | N at end (kip) | V at end (kip) | M at end (kip·ft) |
|---|---|---|---|---|---|---|---|---|
| 1 – 3 | 10 | −10.13 | +0.3109 | −16.11 | −10.13 | +0.3109 | −13.01 | |
| 2 – 4 | 10 | −13.87 | +9.689 | −46.45 | −13.87 | +9.689 | +50.44 | |
| 3 – 4 | 20 | −9.689 | +10.13 | −13.01 | −9.689 | −13.87 | −50.44 |
These are the internal forces just inside each end of the member, in the member’s own axes (first node towards second). Axial N is positive in tension. Shear V counts as positive if the loads and reactions acting on the part before a cut push that part upward, towards the member’s left. Moment M is positive when it stretches the right-hand side of the member’s direction (the underside of a level member drawn left to right). Turn a member round and V and M change sign.
Extremes along each member
| Member | Largest M (kip·ft) | Smallest M (kip·ft) | M is zero at (ft) | Largest |V| (kip) | Largest |N| (kip) | Largest deflection (in) |
|---|---|---|---|---|---|---|
| −13.01 at 10 | −16.11 at 0 | none | +0.3109 at 0 | −10.13 at 0 | +0.02241 at 4.911 | |
| +50.44 at 10 | −46.45 at 0 | 4.794 | +9.689 at 0 | −13.87 at 0 (largest in the frame) | −0.01168 at 7.688 | |
| +29.74 at 8.44 | −50.44 at 20 (largest in the frame) | 1.4, 15.48 | −13.87 at 20 (largest in the frame) | −9.689 at 0 | −0.1146 at 8.926 |
Each value is followed by where it occurs, in ft from the member’s first node. A deflection is the gap between the bent member and the straight line joining its two ends, positive towards the member’s left. The mark ● shows the largest in the frame.
Node displacements
| Node | x displacement (in) | y displacement (in) | Total (in) | Rotation (rad) |
|---|---|---|---|---|
| 1 | 0 | 0 | 0 | 0 (0°) |
| 2 | 0 | 0 | 0 | 0 (0°) |
| 3 | +0.09282 | −0.002972 | 0.09287 | −0.001494 (−0.0856°) |
| 4 | +0.08713 | −0.004071 | 0.08723 | +0.0002045 (+0.0117°) |
Positive to the right, upward and counter-clockwise; rotations in radians, with degrees beside them. A rotation shown as free belongs to a node where every member is hinged. Displacements come from the stiffness of the members.
Equilibrium check
- Loads added up
- Fx +10 · Fy −24 kip · M −50 kip·ft
- Reactions added up
- Fx −10 · Fy +24 kip · M +50 kip·ft
- Σ Fx, loads and reactions
- 0 kip
- Σ Fy, loads and reactions
- 0 kip
- Moments about the point (10, 5), Σ M
- 0 kip·ft
- Biggest leftover force at any node
- 0 kip
In balance: loads and reactions cancel, and so does every node, to within a millionth of the loads.
First-order linear analysis of straight members with bending and axial deformation; shear deformation, P-Δ effects and buckling are not included. The page gives analysis numbers only: no design check, no factors, no verdict.
The calculation runs in your browser. Nothing you enter is sent to us.
Method
How the frame is solved
Direct stiffness, the displacement method in matrix form, solves the frame. Every node has three displacements: one along x, one along y and a rotation. Every member contributes a 6 × 6 stiffness matrix written in its own axes. EA/L ties the two ends axially. 12EI/L³ is the transverse force at one end for a unit transverse displacement of that end, 6EI/L² the transverse force for a unit rotation (or the moment for a unit displacement), 4EI/L the moment at an end for a unit rotation of the same end, and 2EI/L the moment that rotation induces at the far end. The matrix is turned to the global axes by the member’s direction cosines and added into the frame’s matrix at the rows of the member’s two nodes.
A load along a member is first replaced by its fixed-end forces, the forces and moments that would hold both ends of the member against any movement: wL²/12 at each end for a uniform load w over the length L, PL/8 for a point load P at mid-length, and, for P at distances a and b from the first and the second end, Pab²/L² and Pa²b/L² there. Their opposites are applied to the nodes with the nodal loads. A temperature change, a gradient and a couple are carried to the nodes in the same way.
A hinge at the end of a member is handled by static condensation: the member’s own rotation at that end is eliminated from its matrix and from its fixed-end forces, so the member carries no moment into that node and turns there independently of the node. When every member at a node is hinged there, the node’s rotation is undetermined and the analysis warns of it.
Supports are eliminated, not imitated with very stiff springs. A fixed support removes the three displacements of its node, a pin removes the two translations, and a roller removes the translation along one line, the line of its reaction, at any angle you give it: the node’s displacement is rewritten in the direction left unrestrained, which makes an inclined roller as exact as a horizontal one. A guide removes that translation and the rotation, and slides across. A spring adds its stiffness to the frame’s matrix, and a settlement is a displacement prescribed at the node, whose effect on the rest of the frame enters as load.
The reduced system is solved for the node displacements, the end forces of every member follow from its end displacements and its loads, and the reactions come from the equilibrium of the supported nodes. The diagrams are not interpolated between nodes: N, V and M along a member are written from its end forces and its own loads, piece by piece, so that a point load or a couple appears as a jump and a distributed load as a line or a curve of the right degree. The deflected shape of a member is found by integrating M/EI twice along it from its end displacements.
Before any solving the frame is tested for stability, which depends on geometry, hinges and supports and never on E, A or I, and its degree of indeterminacy is counted: 3 for each member plus the reaction components, less 3 equations for each node (2 at a node whose members are all hinged) and less 1 for each hinge. A degree of 0 is determinate: equilibrium alone gives every force, none of them depends on E, A or I, and so without section properties the forces and reactions are still given and the displacements are not. A degree above 0 is indeterminate, and then the forces depend on the stiffness of each member relative to the others. After the solution, the sums ΣFx, ΣFy and ΣM of loads and reactions and the equilibrium of every joint are checked, and a frame whose check is not met is not shown.
Four groups of frames with textbook answers, to compare the calculator’s numbers with.
A beam of length L, held fixed at one end and by a roller at the other, under w per unit length downward: the roller takes 3wL/8, the wall 5wL/8, the moment at the wall is wL²/8 hogging, and the largest sagging moment, 9wL²/128, lies 3L/8 from the roller. With both ends fixed the end moments are wL²/12 hogging and the moment at mid-length wL²/24 sagging.
A portal with fixed feet, equal columns of height h, a span L and a beam so stiff, bending and axially, that its joints do not turn, under a horizontal load H at the beam: each column takes H/2 as shear, the moment is Hh/4 at the foot and at the top of each, and the feet carry vertical reactions of Hh/(2L), one up and one down.
A beam fixed at both ends with one support settled by Δ and no load: the end moments are 6EIΔ/L² and the shear 12EIΔ/L³. A member held at both ends by supports that cannot move, warmed uniformly by ΔT, takes the compression N = −EAαΔT and no bending.
A three-hinged gable, with pins at both feet and a hinge at the crown, of span L and rise f, carrying w on each unit of its horizontal projection: each foot pushes inward with a horizontal thrust of wL²/(8f) and a vertical reaction of wL/2.
What it assumes
- The response is linear elastic and the analysis is first order: equilibrium is written on the undeformed frame and displacements are taken as small. Stiffness does not change with the axial force, so there is no P–Δ effect and no buckling load, and a slender column under a large compression is not flagged.
- Members are Euler–Bernoulli beam-columns: plane sections stay plane and perpendicular to the axis, and shear deformation is not included, which matters for deep members on short spans.
- Members are straight and prismatic, with one E, A and I over the whole length. A tapered or haunched member is modelled as several members of different sections.
- Axial stiffness is included, so members shorten and lengthen under axial force. A hand solution that treats members as inextensible, as moment distribution does, differs slightly. For the frame shown when the page first loads, the solver gives horizontal reactions of 0.311 and 9.689 kip where the inextensible hand solution gives 0.2 and 9.8; with the area made very large (10⁷ in², say) the calculator reproduces the hand answer.
- Loads act on nodes and on members, as entered. A member’s self-weight is not added: enter it as a uniform load on the member. Nothing is multiplied by a factor and no combination is formed, so run each combination as a case of its own.
- Connections are rigid or hinged. A semi-rigid connection, a panel zone, a joint of finite size and a base with flexibility of its own are not modelled; a spring support is the way to put in a stiffness known at a support.
- A temperature change acts as the unrestrained thermal strain of the member, α times the change, and produces force only where supports and the other members hold it back. A gradient needs the depth of the section. A determinate frame moves under temperature or a settlement and develops no force.
- The wide-flange sections and the two materials offered are typical values for a first estimate: use the properties of the section and the material actually specified.
- A frame whose members are all hinged at both ends is a truss, which has its own calculator in the catalogue, Truss member forces.
What it leaves out
- The results are forces, moments, reactions and displacements, and nothing beyond them: it applies no design code, multiplies no load by a factor and does not size a member or a connection. Whether a member will do is for you to decide.
- Up to 50 nodes, 100 members, 40 supports, 60 nodal loads and 120 member loads.
- A mechanism is refused, and the nodes that move are named. It means that a part of the frame can move with no member deforming, as with a pinned-base portal that has a hinge at the top of both columns. To cure it, brace the part that moves with another member or a support, or take out a hinge.
- A frame its supports cannot hold is refused as well: fewer than 3 reaction components, or 3 whose lines are all parallel or all cross at a single point, leave it able to move as a rigid body.
- An indeterminate frame is refused unless each member has a modulus, an area and an inertia; a determinate one is solved without them, with forces and reactions but no displacements.
- Stiffnesses that differ by many orders of magnitude, such as a nearly rigid member beside very flexible ones, or a very short member beside long ones, can leave the arithmetic with too few figures. The equilibrium check is then not met within its tolerance, and the frame is refused as ill-conditioned instead of being shown. A member shorter than one hundred-thousandth of the frame’s size is refused as too short.
- Two nodes in one place, two members between the same pair of nodes, a node with no member, separate pieces of frame, a load placed outside its member and a gradient on a member with no depth are each refused, with the entry named.
- A couple applied to a node at which all the members are hinged has nothing to resist it and is refused, unless a support holds the rotation there.
Sign conventions, units and reading the output
- Global axes: x to the right, y up. Gravity is a negative y load. An applied moment, a reaction moment and a rotation are positive counter-clockwise. The labels of nodes and members start at 1 and follow the order of their tables.
- Each member has axes of its own: x̄ runs from its first node to its second, and ȳ is x̄ turned 90° counter-clockwise, to its left. For a horizontal member drawn from left to right ȳ points up; drawn the other way round, down.
- N, the axial force, is positive in tension and negative in compression. V, the shear force, is positive when the loads and reactions on the piece of member between its first node and a cut push that piece toward +ȳ. M, the bending moment, is positive when it puts tension on the −ȳ side of the member, which on a horizontal member drawn left to right is its underside: a sagging moment.
- The moment diagram is drawn on the tension side, whichever way the member was drawn. The shear and axial diagrams are drawn on the +ȳ side where positive and on the other where negative.
- Worked example: a member from (0, 0) to (10, 0) ft, pinned at both ends, with 2 kip/ft downward. V is +10 kip at the left end, 0 at mid-length and −10 at the right end; M is 0, +25 and 0 kip·ft, and its diagram hangs below the member. Define the same member from its right end to its left and ȳ points down: M at mid-length reads −25 and V changes sign, while the drawn diagram does not change.
- A reaction is the force and moment the support exerts on the frame, along the global axes, positive toward +x and +y; a reaction moment is positive counter-clockwise. A roller’s reaction is also given along its own line.
- Displacements are positive toward +x and +y and rotations counter-clockwise, in radians. A settlement is entered along the global axes in inches or millimetres: a support that sinks has a negative dy, and dr is a rotation in radians. At a roller only the part of the settlement along its line counts.
- A uniform temperature change is the change in degrees, with α the strain per degree. A gradient is the temperature of the +ȳ face minus that of the −ȳ face, and needs the depth h of the member: a warmer +ȳ face bows a member that can deform toward +ȳ, and, held at both ends, gives the sagging moment EIαΔTg/h.
- US units: feet and kips, with line loads in kip/ft and moments in kip·ft; A in in², I in in⁴, E in ksi; section depths and settlements in inches; temperatures in °F. SI units: metres and kilonewtons, with kN/m and kN·m; A in mm², I in 10⁶ mm⁴, E in GPa; depths and settlements in millimetres; temperatures in °C.
- Everything entered is kept in the address after #f=, so that a copy of the address reopens the same frame.
Use it as an engineer would
This calculator is an aid for people qualified to judge its results. It is not an engineering opinion on any structure, and it comes with no warranty. Check the results by independent means: a decision in a design rests on your own check, never on this calculator. Tercero Tablada Civil and Structural Engineering Inc. did not prepare, review or seal what is computed with it, and accepts no responsibility for its results or for how they are used.
Produced with the plane frame calculator of Tercero Tablada Civil and Structural Engineering Inc. — ttcivilstructural.com/resources/plane-frame-analysis