Free Body Diagrams - Engineering Mechanics - Civil Engineering (CE) PDF

Equilibrium of Bodies in 3D space

Purpose of a Free Body Diagram (FBD)

A Free Body Diagram (FBD) isolates a body or a portion of a structure and shows all external forces and moments acting on it. An accurate FBD is essential to write the correct equilibrium equations and to determine unknown reactions or internal forces.

Steps to draw a correct FBD

1. Identify the body or portion of structure to be isolated.
2. Sketch the body in a simple, deformed-free shape (ignore small geometric details not needed for statics).
3. Show and label all external forces and moments acting on the body: concentrated forces, distributed loads (replace by resultant), and body forces (weight, gravity) with their points/lines of action.
4. Replace supports and connections by their equivalent reaction forces and/or moments, indicating unknown reaction components clearly with symbols.
5. Indicate a convenient coordinate system (x, y, z) and mark distances and angles required to compute moments or resolve forces.
6. Include dimensions and where the load resultants are applied; show directions of unknowns consistently.

Equations of equilibrium in three dimensions

For a rigid body in general three dimensional equilibrium the following six scalar equations must be satisfied:

∑F_x = 0    ∑F_y = 0    ∑F_z = 0

∑M_x = 0    ∑M_y = 0    ∑M_z = 0

These are the three force equilibrium equations and the three moment equilibrium equations about the chosen coordinate axes. Using these, up to six independent unknown reaction components can be solved for, provided the support reactions are statically determinate and no redundant supports exist.

Typical unknown reactions in 3D (support types and reaction components)

Common support types and their reaction components (examples):

• A fixed support (encastre) provides three force components and three moment components (6 unknowns).
• A pinned support provides three force components but no moment components (3 unknowns).
• A roller support provides one force component normal to the surface; orientation depends on contact (1 unknown).
• A hinge on a pin or ball joint in 3D gives reaction components as appropriate (usually 3 force components for a pin; a spherical joint gives 3 reaction components but no moments).

Worked Examples

Examples 4.7

Given W = Ladder + Person

= 100 * 9.81 = 981 N

The wheels at A & B are flanged while the wheel at C is unflanged.

Determine reactions at A, B and C

Solution (method and steps)

Identify unknown reaction components and count them. For the given wheel supports:

• Flanged wheels at A and B restrain lateral motion in the plane of flange and provide vertical support; model reactions at A and B with vertical force and lateral force components as required by geometry.
• Unflanged wheel at C can provide vertical reaction and may allow lateral movement along its rolling direction; model accordingly (e.g., vertical reaction only if free to roll horizontally).

Write equilibrium equations with the chosen origin and coordinate axes.

Set up three force equilibrium equations:

∑F_x = 0

∑F_y = 0

∑F_z = 0

Set up three moment equations about appropriate points to eliminate as many unknowns as possible:

∑M_x = 0

∑M_y = 0

∑M_z = 0

Solve the resulting linear system for reaction components.

Notes:

• Choose moments about a support point to eliminate its reaction components where convenient.
• If geometry or directions of flange constraints are provided in the figure (angles, offsets), resolve forces along coordinates and substitute numerical distances to obtain numeric reactions.
• If the problem is statically determinate, these six scalar equations suffice to find the unknowns; if redundant, additional compatibility equations or methods (e.g., deformation) are needed.

Example 4.8

Given W = 270 lbs

Determine

• Tensions in AE and BD.
• Reactions at A.

Solution (procedure)

Identify the free body containing the weight W and the cables AE and BD.

Resolve forces at the connection points into components along the chosen orthogonal axes.

Write the equilibrium equations for forces:

∑F_x = T_AE·cos(α_AE) + T_BD·cos(α_BD) + A_x = 0

∑F_y = T_AE·cos(β_AE) + T_BD·cos(β_BD) + A_y = 0

∑F_z = T_AE·cos(γ_AE) + T_BD·cos(γ_BD) - W + A_z = 0

Use moment equations if necessary to eliminate unknown reaction components at A and to relate tensions.

From these equations, solve for T_AE, T_BD, and the reaction components A_x, A_y, A_z.

Notes:

• If AE and BD are symmetrical or lie in principal planes, simplifications occur by symmetry.
• Use geometry from the figure to determine direction cosines (cosα, cosβ, cosγ) for each cable.

Example 4.9

Given mass of the cover: 30 kg Assume no axial reaction at B.

Find Tension in CD and reactions at A & B.

Solution (method)

Compute the weight:

W = m·g = 30·9.81 N

Isolate the cover as a free body and show the applied weight W, the tension in CD (T_CD), and support reactions at A and B (with the assumption that axial reaction at B is zero, i.e., B has only transverse/vertical reactions as stated).

Write equilibrium equations:

∑F_x = 0

∑F_y = 0

∑F_z = 0

∑M_{about appropriate point} = 0 to relate T_CD to W and to reactions.

Solve the linear system for T_CD, A (components), and B (components consistent with the no axial reaction constraint).

Remarks:

• Selecting the moment equation about A or B often eliminates unknowns and simplifies computing T_CD.
• Check sign conventions and directions of reaction components carefully when writing equations.

Example 4.10

Given W = 450 lb.

Find

• Location of G so that the tension EG is minimum
• This minimum value of tension

Solution (general approach)

Model the system with variable location coordinate for point G (for example, x measured along a bar or rope). Express the tension T_EG as a function of that coordinate using equilibrium of the free body containing W and tension EG.

Typical steps:

Express geometry-dependent direction cosines of the rope EG in terms of the coordinate of G.

Write the equilibrium equations relating components of T_EG to W and any other reactions.

Solve algebraically to obtain T_EG = f(x), where x is the variable location of G.

Find the minimum by differentiating with respect to x and setting the derivative to zero:

dT_EG/dx = 0

Check the second derivative or examine behaviour to confirm it is a minimum.

Substitute the optimal x back into T_EG to obtain the minimum tension value.

Remarks:

• This approach converts the statics problem into an optimisation problem subject to equilibrium constraints.
• If the figure provides constraints (e.g., G must lie on a fixed link or surface), include those constraints when forming f(x).

Practical notes and tips for solving 3D FBD problems

• Always sketch a clear 3D coordinate system and label axes; indicate positive directions for forces and moments.
• Replace distributed loads by a single resultant force located at the centroid of the distribution before writing equilibrium equations.
• Use vector notation for clarity when dealing with non-orthogonal forces; convert to scalar components when writing the six scalar equilibrium equations.
• Count unknown reaction components first; compare with available independent equilibrium equations to check determinacy.
• Choose moment equations about points that remove most unknowns from those equations, reducing algebra effort.
• When geometry is complex, compute direction cosines from the coordinates of the points defining the line of action.
• If the system is statically indeterminate, identify the degree of indeterminacy and use methods of structural analysis or compatibility (deflection methods) to solve.