Learning topic
Statically Determinate Frames
Analyze statically determinate plane frames: calculate support reactions and member internal forces, handle rigid joints and hinges, and construct N, V, M diagrams.
Frame analysis determines the reactions and internal forces that develop in a structural frame under applied loads. Unlike a truss idealization, a frame member generally carries not only axial force but also shear force and bending moment. This page gives a practical introduction to the analysis of plane frames, with emphasis on statically determinate systems.
What is a structural frame?
A frame is an assemblage of members connected at joints. A rigid joint can transmit force and moment between connected members, so bending is normally an essential part of frame behavior. Internal hinges, when present, release bending moment locally and can divide a frame into parts that are convenient for equilibrium analysis.
In a plane-frame model, external loads and the main structural response lie in one plane. The usual internal force resultants in a member are:
- $N$ — axial (normal) force;
- $V$ or $Q$ — shear force;
- $M$ — bending moment.
Statically determinate frame analysis
For a statically determinate plane frame, reactions and internal forces can be obtained from equilibrium alone. For the whole frame, use:
$$\sum F_x=0,\qquad \sum F_y=0,\qquad \sum M=0.$$
The useful feature of frame analysis is that equilibrium can then be applied again to individual members or isolated parts of the structure. Forces acting at a common joint must be equal and opposite on the connected member free-body diagrams.
Step-by-step frame analysis
- Idealize the structure. Identify members, rigid joints, internal hinges, supports, dimensions, and applied loads.
- Draw the free-body diagram of the entire frame. Replace supports by their reaction components.
- Calculate support reactions. Apply global equilibrium and use internal hinges or other releases when they provide additional useful equilibrium conditions.
- Separate the frame into members or convenient parts. Draw a free-body diagram for each part and show the joint forces and moments acting on it.
- Determine $N$, $V$, and $M$. Use sections or member equilibrium. Keep one sign convention throughout the calculation.
- Construct the internal-force diagrams. Plot axial-force, shear-force, and bending-moment distributions along the members.
- Check equilibrium and compatibility at joints. Member-end actions at a joint must balance, and an ideal internal hinge must have zero bending moment.
Internal-force diagrams in frames
The diagrams are drawn along the local axis of each member. This is particularly important at corners: the global horizontal and vertical directions change their role relative to a member's local axial and transverse directions.
For a member without a distributed axial load, $N$ is constant between concentrated axial actions. The shear-force diagram changes according to transverse loading, while the bending-moment diagram is related to shear by the usual beam relations. Concentrated forces cause jumps in the corresponding force diagrams; an applied concentrated couple causes a jump in the bending-moment diagram.
Rigid joints and internal hinges
A rigid joint does not imply that the bending moment is zero. It transfers end forces and moments between members. By contrast, an ideal internal hinge cannot transmit bending moment, so $M=0$ at the hinge. The hinge may still transmit force components, and those forces appear with opposite directions on the two separated free bodies.
Common mistakes in analysis of frames
- treating every joint as a pin and therefore incorrectly setting member-end moments to zero;
- using only the free-body diagram of the complete frame when member equilibrium is also required;
- mixing global $x$-$y$ directions with a member's local axial and transverse directions;
- changing the sign convention for $N$, $V$, or $M$ midway through the solution;
- forgetting that an internal hinge gives a zero-moment condition but can transmit forces;
- drawing an internal-force diagram that does not satisfy the calculated member-end actions.
Determinate and indeterminate frames
Equilibrium equations are sufficient only for a statically determinate frame. A statically indeterminate frame has additional unknown reactions or internal actions, so deformation compatibility and member stiffness must also be considered. Common approaches include force methods, displacement methods, slope-deflection and matrix stiffness methods. Those methods are beyond the scope of this introductory page; the equilibrium and member free-body concepts developed here remain their foundation.
Frame deflections
When a displacement or rotation is required, energy methods can be applied after the internal-force state has been established. For example, the unit-load method evaluates the contribution of the real and auxiliary internal-force diagrams:
The displacement at a specified point and direction can be determined by the unit-load method. For bending of a beam or frame:
$$\delta=\int_0^L\frac{M(x)\,\bar M(x)}{EI}\,dx,$$
where $M(x)$ is the bending moment from the real loading and $\bar M(x)$ is the moment from a unit force applied at the point and in the direction of the required linear displacement.
For a required rotation, apply a unit moment.
For a general member, applicable contributions may be summed:
$$\delta=\int\frac{N\bar N}{EA}\,dx+\int\frac{M\bar M}{EI}\,dx+\int\frac{T\bar T}{GJ_p}\,dx+\int\frac{Q\bar Q}{\kappa GA}\,dx,$$
- $N$, $M$, $T$, $Q$ — internal force resultants from the real loading;
- $\bar N$, $\bar M$, $\bar T$, $\bar Q$ — corresponding resultants from the unit-load state;
- $E$, $G$, $A$, $I$, $J_p$, $\kappa$ — stiffness and section parameters.
The sign of $\delta$ indicates whether the actual displacement agrees with the direction of the introduced unit load.
Practical checks
A completed structural frame analysis should satisfy global equilibrium, equilibrium of every isolated member or subassembly, action-reaction at connected member ends, zero moment at ideal hinges, and consistency between loads and the shapes of the $N$, $V$, and $M$ diagrams. These checks often reveal sign errors before a numerical result is used for design.
About this topic
Frame analysis determines support reactions and internal axial force, shear force, and bending moment in structural frames. This introduction focuses on statically determinate plane frames, member-by-member equilibrium, rigid-joint behavior, N-V-M diagrams, essential checks, and how the same ideas extend to indeterminate frames.