Structural Analysis

Truss analysis

A plane truss is an assembly of straight members joined at pin joints, so each member carries only axial force, tension or compression. Before analysing a truss, check its determinacy and stability. For a plane pin-jointed truss with m members, r support reactions and j joints, the truss is statically determinate when m + r = 2j. It is stable only if its geometry prevents internal and external mechanisms. Two standard methods are used. The method of joints applies the two equilibrium equations at each joint in turn. The method of sections cuts three members that are not all concurrent and takes moments about a convenient joint, which gives a member force directly.

Zero-force members should be identified first, because they simplify every later step. A joint with no external load and only two non-collinear members forces both members to zero. At a loaded-free joint where two members are collinear, a third member that is perpendicular to them carries zero force. Sign convention is standard: tension is positive and compression is negative. Once a member force F is known, the axial stress is \sigma = F/A, so mm^2 areas must be converted to m^2 before stress is quoted in kN/m^2. For a truss whose members all have the same axial rigidity AE, each bar acts as an axial spring of stiffness AE/L, and the joint displacements follow from assembling these springs.

Indeterminate frame moment analysis

A frame with more unknown actions than equilibrium equations is indeterminate. Its analysis uses compatibility of displacements together with equilibrium. In the force (flexibility) method, one support reaction or internal action is taken as the redundant and found from the condition that the displacement at its location is zero. The displacement at that point is written as the sum of the displacement from the applied loads and the redundant times its flexibility coefficient. Cantilever-type flexibility coefficients are obtained from unit-load integrals such as \delta = \int Mm\,dx/EI.

In the slope-deflection and moment-distribution methods, the member end moments depend on joint rotations. For a member with a fixed far end and rigid near end, the near-end moment is 4EI\theta/L, the far-end moment is 2EI\theta/L, and the carry-over factor is 1/2. The distribution factor for a member at a joint is its stiffness divided by the sum of stiffnesses of all members meeting there, with a free-ended cantilever contributing no stiffness. Symmetry of geometry and loading reduces the work: a symmetric frame with symmetric loads has no sway, and the rotations at mirror-image joints are equal or opposite. A rigid joint with axially inextensible members cannot translate, so only its rotation is unknown.

Influence lines for beams

An influence line gives the variation of a response, such as a reaction, shear force, bending moment or member force, at a fixed section as a unit load moves across the structure. The ordinate at any point is the response produced by a unit load at that point. The Müller-Breslau principle states that the influence line for a force response (reaction, shear or moment) equals, to a scale, the deflected shape of the structure obtained by removing the restraint corresponding to that response and applying a unit displacement in its direction. This is the usual method for sketching the qualitative shape of an influence line for a statically indeterminate beam. For a shear force at a section, the sign convention must be followed carefully, because the positive shear on the left and right faces are opposite in direction.

For a simply supported beam, the influence line for bending moment at a point is triangular, and the peak ordinate at a section located a from one end of a span L is a(L-a)/L. For a moving train of loads, the maximum moment at a section is found by placing the heaviest load close to the section and checking the position for which the resultant lies symmetrically with the section. Maxwell's reciprocal theorem gives the rotation at a support from a unit load at a point as the deflection at that point due to a unit couple at the support. For a truss, the force in a member is found from the vertical displacement of the loaded chord under a kinematic mode.

Propped cantilever with internal hinge

A propped cantilever is fixed at one end and supported by a simple support or roller at the other. It is indeterminate to the first degree, but an internal hinge releases the bending moment at its location and reduces the indeterminacy by one. At the hinge the bending moment is zero, while shear force and axial force can be transmitted. Taking moments of all forces on one side of the hinge about the hinge is therefore a direct way to find the reaction at the far support. The overall vertical equilibrium and the moment equilibrium of the whole beam then give the remaining reactions. If the hinge and supports make the structure statically determinate, the reactions do not depend on the stiffness.

When the propped support settles, the reaction is found by releasing the prop and applying compatibility. The tip of a cantilever of length L and flexural rigidity EI deflects by RL^3/(3EI) under a force R and by ML^2/(2EI) under an end moment M. Setting the net upward displacement at the prop equal to the prescribed settlement gives the reaction. Sign conventions must be fixed before computing, since upward displacements and downward displacements change the sign of the result. Also, a hinge does not alter the sign of shear, so the reactions computed on each side of the hinge must balance the total external load.

Stiffness matrix method

The stiffness matrix method expresses the forces at the element ends as a product of element stiffness and end displacements. For an axial element of length L, area A and modulus E inclined at angle \theta to the global x-axis, the global stiffness in terms of the direction cosines c = \cos\theta and s = \sin\theta is \frac{AE}{L}\begin{bmatrix} c^2 & cs \\ cs & s^2 \end{bmatrix} for the two translational degrees of freedom at each joint. Element matrices are assembled by the degrees of freedom they share. Boundary conditions remove the rows and columns for restrained displacements, and the reduced matrix is then inverted to obtain the unknown displacements.

When two members inclined in opposite directions meet at a joint, their off-diagonal contributions cs can have opposite signs and cancel on assembly. Each diagonal term then receives a sum of c^2 and s^2 from every member. For a determinate beam, the reactions depend only on equilibrium, and the deflection at a point can be found by the unit-load integral \delta = \int Mm\,dx/EI, where M is the actual moment and m is the moment due to a unit load at the point.

Truss joint deflection by virtual work

The deflection of a truss joint can be found by the unit-load method, which is a form of virtual work. A unit load is applied at the joint in the direction of the required deflection, with no other loads, and the member forces n_i are found. The deflection is then \delta = \sum n_i N_i L_i/(A_iE_i), where N_i are the member forces due to the actual loads, L_i are member lengths, and A_iE_i are the axial rigidities. For members of equal rigidity, the sum reduces to \delta = (1/AE)\sum n_i N_i L_i. Members with zero force in either system contribute nothing, so zero-force identification saves work.

For an indeterminate truss, the redundant member force is found from compatibility. The elongation of the member that is cut must be zero when the actual loads and the redundant force act together. Using the unit-force member forces u_i, the redundant is X = -\sum F_i u_i L_i/\sum u_i^2 L_i, where F_i are the forces from the external loads on the released truss. Once X is known, all member forces are found by superposition, and reactions follow from the equilibrium of the whole truss.

Indeterminate structure behavior

A statically indeterminate structure responds to loading and support changes differently from a determinate one. Its internal forces depend on the relative stiffness of its members, and temperature changes, settlements or fabrication errors can produce internal forces that a determinate structure would not experience. Symmetry is a powerful tool for judging qualitative behaviour. If both geometry and loading are symmetric about an axis, the response is symmetric, so shear forces and horizontal displacements are zero on that axis, and rotations at points on the axis are zero. Antisymmetric loading produces the opposite pattern.

For a beam resting on flexible supports, such as rods, the rod forces depend on both equilibrium and compatibility. The deflection of the beam at the rod position must equal the elongation of the rod. The rod forces are therefore not simply shared in proportion to tributary areas. A support with a larger stiffness takes a larger share of the load, which is the central idea of indeterminate behaviour. Also, a structure with a stable configuration can still exhibit internal force redistribution under symmetric or asymmetric loads.