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Three-bar truss by the direct stiffness method: displacements, reactions and bar forces (McGuire, Gallagher and Ziemian Examples 3.1 and 3.2)

Tool	Truss and Frame Analysis
Source	McGuire, W., Gallagher, R. H., and Ziemian, R. D. (2000). Matrix Structural Analysis, 2nd ed. John Wiley and Sons, New York. Examples 3.1 and 3.2 (pp. 35–37).
Level	Undergraduate

Problem

A three-bar truss has its base cb 10.928 m long with node c pinned and node b on a roller, and its apex a 4 m above the base where the bars make 30° (at c) and 45° (at b) with the base, so a = (6.928, 4.000) m. E = 200,000 MPa; A = 20,000 mm² for ab, 18,000 mm² for bc and 15,000 mm² for ac. A force of 500 kN acts at a at 40° above the horizontal (383.0 kN right, 321.4 kN up). Find the joint displacements, the reactions and the bar forces.

Inputs to Enter

Nodes: a 6.928 4.000; b 10.928 0; c 0 0 (m). Members: ab a b 200000 20000; bc b c 200000 18000; ac a c 200000 15000 (E in MPa, A in mm²)
Supports: c pin; b roller-x. Joint load: a 383.022 321.394 (kN). Structure: truss; units m, kN, kN-m, MPa, mm²

Open the Truss and Frame Analysis tool (loaded with this example)

Answers

Displacements: u_a = 0.8702 mm, v_a = 1.2429 mm, u_b = −0.1929 mm (book 0.871, 1.244, −0.193 mm from four-digit matrix entries)
Reactions: R_xc = −383.02 kN, R_yc = −257.84 kN, R_yb = −63.56 kN (book −383.4, −258.1, −63.6 kN); equilibrium residual 1e-16
Bar forces: F_ab = +89.88 kN (tension), F_bc = −63.56 kN (compression), F_ac = +515.67 kN (tension) (book +90.0, −63.6, +516.2 kN); σ_ac = 34.38 MPa

Notes

The tool's default inputs are this truss; press Submit. The book rounds its stiffness coefficients to four digits, so its printed values differ from the exact solution by up to 0.15 %; the exact values are confirmed by an independent numpy solve (References/Truss-Frame/frame_check.py) and the test script test_truss_frame.php. The example select also offers the book's Example 5.7 frame and a fixed-base portal frame checked against Kleinlogel's closed forms.