How does a truss calculator find member forces?

Each member is treated as a two-force bar pinned at both ends, so it carries only axial tension or compression. This calculator assembles the axial stiffness EA/L of every member into a global stiffness matrix, solves for the joint displacements under the panel-point loads, and recovers each member force from the change in its length. For a statically determinate truss the forces are identical to the method of joints and do not depend on EA; for an indeterminate truss the stiffness method also distributes force by relative stiffness, which hand methods cannot do directly.

What is the difference between a Pratt and a Howe truss?

Both have verticals and diagonals between parallel chords. In a Pratt truss the diagonals slope down toward the center, so under gravity load the diagonals are in tension and the verticals in compression. In a Howe truss the diagonals slope the other way, so the diagonals are in compression and the verticals in tension. Pratt suits steel, where long members are better in tension; the Howe was developed for timber diagonals with iron-rod verticals.

Why can the bottom chord of a roof truss go into compression?

Wind uplift on the roof can exceed the dead load. When the net load on the top chord points upward, every member force reverses sign, so the bottom chord, normally in tension, carries compression with a long unbraced length. Enter a net uplift pressure and the calculator runs the ASCE 7 combinations 0.6D + 0.6W (ASD) or 0.9D + 1.0W (LRFD) and checks the bottom chord for buckling in that case.

What is a zero-force member?

At an unloaded joint where only two non-collinear members meet, both members carry zero force. At an unloaded joint where three members meet and two are collinear, the third member carries zero force. The king post of a king-post truss with no ceiling load is an example. Zero-force members still brace other members and carry force under other load cases, so they are not removed.

Does this design a wood roof truss to ANSI/TPI 1?

No. It checks each member's axial stress, and the chords for combined axial load and panel bending, against the NDS with load-duration factors per combination. ANSI/TPI 1 adds metal-plate connection design, continuous-chord bending moments, plate and heel checks and bracing requirements that are not included. Use the results for preliminary sizing, checking a supplier's design, or a site-built or heavy-timber truss you will detail yourself.

Is my truss statically determinate?

For a plane truss with m members, r reaction components and j joints, m + r = 2j is the necessary condition for a determinate truss, m + r > 2j means indeterminate, and m + r < 2j means it is a mechanism. The count is necessary but not sufficient: a badly arranged truss can satisfy it and still be unstable. The calculator reports the count and also checks the stiffness matrix, so it flags geometric instability the count misses.

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Truss Calculator

Member forces, support reactions and joint deflections for any plane pin-jointed truss. Start from a roof or parallel-chord template, or edit the joints and members directly. Dead, live and wind-uplift loads are combined per ASCE 7, and each member can be checked as sawn lumber (NDS, with the load-duration factor for each combination and chord bending between panel points) or as steel (AISC 360). The diagram shows tension, compression, demand/capacity or the deflected shape.

Geometry

Units

Truss type

Span, Lft

Top chord pitchin 12

Bottom chord pitch (scissor)in 12

Truss depth, Hft

Number of panels—

Truss spacing (tributary width)ft

Loads (on horizontal projection, applied at panel points)

Top chord dead, TCDLpsf

Top chord live / snow, TCLLpsf

TCLL is

Bottom chord dead, BCDLpsf

Bottom chord live, BCLLpsf

Net wind uplift on top chord, Wpsf (up +)

Member self-weight

Unit weight of materialpcf

Load combinations

Members

Member check

Species & grade (NDS Supp. Table 4A)

F_b / F_t / F_c (reference)psi

E / E_minpsi

Top chord size × pliesnominal

Bottom chord size × pliesnominal

Web size × pliesnominal

Top chord out-of-plane bracing

Bottom chord out-of-plane bracing

Web out-of-plane bracing

Top chord A / r_minin² / in

Bottom chord A / r_minin² / in

Web A / r_minin² / in

Modulus of elasticity, Eksi

Yield strength, F_yksi

Effective length factor, K— (pinned ends: 1.0)

Deflection limits, total / liveL / —

Edit joints, members and point loads (editing switches the type to Custom)

Joint	x	y	Support

Mem	i	j	Group

Joint	F_x	F_y (up +)	Case

Coordinates in ft, point loads in lb. Support: pin = x and y restrained, roller = y only, x-roller = x only. Group sets which section and which distributed load a member gets.

Stability—

Max tension (D + L, service)—lb

Max compression (D + L, service)—lb

Total vertical load (D + L)—lb

Max joint deflection, D + L—in

Max joint deflection, live only—in

Span / deflection (total · live)—

Max demand / capacity—

Axial force (D + L) Demand / capacity Deflected shape Loads & numbering

Blue = tension, red = compression, gray = zero force. Line weight is proportional to force. Click a member to find it in the table.

Mem	Joints	Group	Length	D + L force	Max T (combo)	Max C (combo)	D/C	Governs

Support joint	R_x (D + L)	R_y (D + L)	R_y min, any combo	R_y max, any combo

Forces in lb, lengths in ft. Positive force = tension. Negative R_y = uplift; the connection must be anchored for it.

$$ \mathbf{K}\,\mathbf{u} = \mathbf{F}, \qquad \mathbf{k}_e = \frac{EA}{L}\begin{bmatrix} c^2 & cs & -c^2 & -cs\\ cs & s^2 & -cs & -s^2\\ -c^2 & -cs & c^2 & cs\\ -cs & -s^2 & cs & s^2\end{bmatrix}, \qquad N = \frac{EA}{L}\left[c\,(u_j-u_i) + s\,(v_j-v_i)\right] $$

c, s direction cosines of the member · u, v joint displacements · N axial force, tension positive · Panel-point load at a joint = w × spacing × tributary horizontal length (half-way to the neighbouring joints on the same chord).

How the calculator works

Every member is a pin-ended bar that carries only axial force. The page assembles the 4 × 4 member stiffness matrices into a global matrix, removes the restrained degrees of freedom, and solves for the joint displacements by Gaussian elimination with pivoting. If a pivot collapses to zero, the truss is a mechanism, and the page names the joint and direction that can move freely. Member forces come from the change in each member's length, and reactions from the restrained rows of K·u. For a statically determinate truss this gives exactly the method-of-joints answer, independent of the member areas. For an indeterminate truss (m + r > 2j) the areas and E matter, and the stiffness solution distributes the force correctly.

Distributed loads are converted to panel-point loads. Each joint on the top chord collects the load from half-way to the neighbouring top-chord joints, measured horizontally, times the truss spacing. The bottom chord works the same way. Dead, live and wind are solved as three separate load cases and then combined, so every combination costs nothing extra. Each member's maximum tension and compression over the combinations are reported, along with the combination that produces them.

Wood check (NDS, ASD). Each combination is checked with its own load-duration factor: C_D = 0.9 for dead load alone, 1.0 / 1.15 / 1.25 for floor live, snow or roof live, and 1.6 for the wind combinations. Tension uses F_t′ = F_tC_DC_F. Compression uses F_c′ = F_cC_DC_FC_P, where the column stability factor C_P (NDS Eq. 3.7-1, c = 0.8) is taken about the weaker of the two axes. In the plane of the truss, l_e/d uses the member length over the wide face. Out of plane, it uses the member length over the total ply thickness, unless the chord is continuously braced or a web has a lateral restraint at mid-length (then half the length). Chords that carry distributed load are also checked for panel bending, M = w_hΔx²/8 between panel points, with C_r = 1.15 when spacing ≤ 24 in, using NDS Eq. 3.9-3 (compression + bending) or Eqs. 3.9-1 and 3.9-2 (tension + bending).

Steel check (AISC 360). Tension is gross-section yielding, P_n = F_yA_g (D2). Compression is flexural buckling with F_e = π²E/(KL/r)², and F_cr = 0.658^F_y/F_eF_y when F_y/F_e ≤ 2.25, otherwise 0.877F_e (E3). Use r_min of the section, for example r_z for a single angle or r_y for a double angle. φ = 0.90 (LRFD) or Ω = 1.67 (ASD). The default areas and radii are placeholders, not a specific shape: enter the A and r_min of the sections you are using.

Choosing a truss type

Common truss forms and where they are used
Type	Typical span	Behaviour under gravity load	Typical use
King post	up to ~16–20 ft	Two top chords and a tie; the post is zero-force unless the ceiling hangs from it	Sheds, garages, short residential spans
Queen post	~20–30 ft	Two verticals and two diagonals to the apex	Residential and timber-frame roofs
Fink (W)	~20–40 ft	Webs short and in a W; efficient, short compression webs	The most common light-frame residential roof truss
Howe roof	~30–60 ft	Verticals in tension, diagonals in compression	Heavy timber, longer roof spans
Pratt roof	~30–80 ft	Diagonals in tension, verticals in compression	Steel roof trusses
Scissor	~20–40 ft	Raised bottom chord; large horizontal deflection at the roller	Vaulted ceilings
Mono-pitch	~8–30 ft	Single slope with a tall vertical at one end	Lean-tos, additions
Pratt (flat)	~30–250 ft	Diagonals in tension, verticals in compression	Steel bridges, floor trusses, open-web joists
Howe (flat)	~30–150 ft	Diagonals in compression, verticals in tension	Timber bridges
Warren	~50–300 ft	Diagonals alternate tension and compression; no verticals	Bridges, long-span roofs, joists

Worked example — king-post truss by the method of joints

24 ft span, 6/12 pitch, 24 in spacing, TCDL 10 psf, BCDL 10 psf

Given: rise h = 12 × 6/12 = 6 ft, top chord length 13.42 ft, sin θ = 0.4472, tan θ = 0.5.

Apex panel load: tributary 6 ft each side → P_top = 10 psf × 2 ft × 12 ft = 240 lb. The heel joints take the outer 6 ft, which goes straight into the supports.

Bottom-centre joint: tributary 12 ft → P_bot = 10 × 2 × 12 = 240 lb, so the king post carries T = 240 lb up to the apex.

Apex: 2C sin θ = 240 + 240 → C = 480 / (2 × 0.4472) = 536.7 lb compression in each top chord.

Heel: T = C cos θ = 536.7 × 0.8944 = 480 lb tension in the bottom chord.

Top chords −537 lb · bottom chord +480 lb · king post +240 lb · reactions 480 lb each (960 lb total). Select "King post", span 24, pitch 6, TCLL 0 to reproduce this.

Limits

Pin-jointed, linear-elastic, small-displacement analysis. Joints are frictionless pins; secondary moments from rigid plates or welded joints are not computed.
Loads act at joints. Chord bending between panel points is checked for wood with a simple-span moment, not the continuous-chord moments that ANSI/TPI 1 uses. Steel members are checked for axial load only.
Connections, metal plates, bearing, heel shear and permanent lateral bracing are not designed.
Wind is a net vertical uplift on the horizontal projection of the top chord. Pressures normal to each roof plane, internal pressure and unbalanced snow are not generated; add them as point loads in the editor.
Wood: dry service, normal temperature, no incising (C_M = C_t = C_i = 1). Multi-ply members are assumed fastened to act as one. Size factors are those of NDS Supplement Table 4A. Check the reference values against the current NDS Supplement before relying on them.

References: Hibbeler, R.C., Structural Analysis, Ch. 3 (trusses) and Ch. 14 (truss stiffness method). ASCE/SEI 7-22 §2.3.1 and §2.4.1 (load combinations). ANSI/AWC National Design Specification for Wood Construction (2018 / 2024), §2.3.2 and Table 2.3.2 (C_D), §3.7.1 (column stability, Eq. 3.7-1), §3.9 (combined loading), §4.3.9 (repetitive member factor), and NDS Supplement Table 4A (reference design values and C_F). ANSI/AISC 360-22 §D2 and §E3. ANSI/TPI 1 for metal-plate-connected wood truss design.

Related tools

3D frame analysis — beams, columns and braced frames with bending, in 2D or 3D
Continuous beam — multi-span girders and purlins carrying truss reactions
Euler buckling — check a long compression web by hand
Section properties — A and r for the member groups
AISC LRFD steel design — design the steel chords in more detail

Worked examples and guides

Fink roof truss, NDS check — the default inputs, solved by hand step by step
Roof truss wind uplift — force reversal, bottom-chord and web bracing
Pratt truss footbridge — method of sections and AISC compression checks
How to analyze a roof truss — inputs, bracing and reading the results
Truss member force coefficients — printable reference card
All structural analysis tools and examples