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1Formulas and methods
Vectors: dot & cross product
Magnitude
|a| = √(ax² + ay² + az²)
Unit vector
û = a/|a|
Components from an angle
Fx = F·cos θ, Fy = F·sin θ
Dot product
a·b = axbx + ayby + azbz = |a||b|cos θ
Scalar projection of a onto b
a·b/|b|
Work
W = F·d
Cross product
a × b = (aybz − azby, azbx − axbz, axby − aybx)
Areas
parallelogram |a × b|, triangle ½|a × b|
Moment about O
MO = r × F, r = A − O
Volume of a parallelepiped
V = |a·(b × c)|
Method
Write every vector in components; for a force given by a magnitude and an angle, use F·cos θ and F·sin θ.
For lengths and directions use the magnitude √(x² + y² + z²) and the unit vector a/|a|.
Use the dot product for angles, projections, perpendicularity and work: a·b = |a||b|cos θ.
Use the cross product for perpendicular vectors, areas and moments: write the three components carefully, keeping the order a × b.
Beam support reactions
Equilibrium
ΣFx = 0, ΣFy = 0, ΣM = 0
Uniform load
resultant w·L at the middle of the load
Moment
M = F·d
Method
Draw a free-body diagram and replace each support by its reactions.
Replace every distributed load by a single force at its centroid.
Take moments about one support to find the reaction at the other.
Use ΣFy = 0 to find the remaining reaction, and check with a second moment equation.
Truss: method of joints
Method of joints
ΣFx = 0, ΣFy = 0 at each joint
Method of sections
ΣM = 0 about a cut point
Member force
positive = tension, negative = compression
Method
Find the support reactions for the whole truss.
Method of joints: start at a joint with at most two unknown member forces and use ΣFx = 0 and ΣFy = 0.
Method of sections: cut through the member you need and take moments about a point where the other unknown members meet.
Assume tension; a negative answer means compression.
Frames & machines
Equilibrium of each member
ΣFx = 0, ΣFy = 0, ΣM = 0
Two-force member
force along the line through its two pins
Pin force
equal and opposite on the two members it joins
Magnitude of a pin force
F = √(Fx² + Fy²)
Lever, pliers, bell crank
Fin·a = Fout·b
Toggle
Flink = F/(2 sin θ), Q = F/(2 tan θ)
Uniform load
resultant w·L at the middle of the load
Method
Draw the whole structure and find the support reactions if there are no more than three unknowns.
Spot the two-force members: loaded only at two pins, their force acts along the line between the pins.
Take the frame apart and draw every member on its own, with equal and opposite pin forces where members meet.
For each member, take moments about a pin with unknown forces, then use ΣFx = 0 and ΣFy = 0; combine components into a magnitude at the end.
Shear force & bending moment
Shear and moment
V = ΣF on one side, M = ΣF·d on one side
Bending stress
σ = M·y/I
Rectangle
I = bh³/12
Section modulus
W = I/ymax
Simply supported, uniform load
Mmax = wL²/8
Method
Find the support reactions from equilibrium of the whole beam.
Cut the beam at the section you need and keep one side.
Shear force: sum of the vertical forces on that side. Bending moment: sum of their moments about the cut.
For the maximum moment, find where the shear force is zero.
Centroid & second moment of area
Centroid
ȳ = ΣAiyi/ΣAi
Rectangle
I = bh³/12
Parallel-axis theorem
I = Ic + A·d²
Composite section
I = Σ(Ic,i + Aidi²)
2Practice problems
1.Vectors: dot & cross productMedium
Find the angle θ between the vectors a = (5, 6, 5) and b = (6, −4, −3), in degrees.
a = (5, 6, 5)b = (6, −4, −3)
θ = °
2.Vectors: dot & cross productHard
Find the volume V of the parallelepiped whose edges from one corner are a = (5, −2, 1), b = (5, −3, 4) and c = (4, −1, −3).
a = (5, −2, 1)b = (5, −3, 4)c = (4, −1, −3)
V =
3.Beam support reactionsMedium
The simply supported beam has a pin at A and a roller at B. Find the vertical support reaction RA.
L = 8 mP1 = 25 kN at 3 mP2 = 25 kN at 7 m
RA = kN
4.Beam support reactionsHard
The cantilever beam is fixed to the wall at A. It carries a linearly varying load that is zero at the wall and rises to 4 kN/m at the free end B, plus a point load P at 3 m from the wall. Find the magnitude of the reaction moment MA at the wall.
L = 5 mw0 = 4 kN/mP = 40 kN at 3 m
MA = kN·m
5.Truss: method of jointsMedium
Exactly one member of this truss carries no force under these loads. Which is the zero-force member? Click it in the diagram.
B: 30 kN ↓C: 10 kN ↓G: 35 kN ↓
1member AB
2member BC
3member CD
4member DE
5member FG
6member GH
7member AF
8member HE
9member BF
10member CG
11member DH
12member FC
13member HC
6.Truss: method of jointsHard
Use the method of joints to find the force FAB in member AB. Give tension as positive and compression as negative.
span = 5 mheight = 4 mB is 2 m from Avertical load = 55 kNhorizontal load = 25 kN ←
FAB = kN
7.Frames & machinesMedium
A horizontal beam AB is pinned to a wall at A and propped by the strut CD, pinned to the wall at C, 0.6 m below A. A load of 5 kN hangs at B. Neglect the weights. Find the magnitude of the pin force A on the beam at A.
P = 5 kNAD = 1.6 mAB = 2.4 mAC = 0.6 m
A = kN
8.Frames & machinesHard
The A-frame shown stands on a pin at A and a roller at B, 5 m apart. The legs meet at the pin C, 2.5 m high, and are tied by the horizontal bar DE at 1.2 m. A horizontal force of 15 kN to the right acts at C. Find the tension T in the tie.
P = 15 kNAB = 5 mheight of C = 2.5 mheight of DE = 1.2 m
T = kN
9.Shear force & bending momentMedium
The simply supported beam carries a clockwise couple M0 = 20 kN·m at x = 2.5 m from A, and no other load. Find the bending moment Mx just to the left of the couple. Use sagging moments positive and upward shear on the left face positive.
L = 4 mM0 = 20 kN·m clockwise, at 2.5 m
Mx = kN·m
10.Shear force & bending momentHard
Find the bending moment Mx at the section x = 2 m from support A. Use sagging moments positive and upward shear on the left face positive.
L = 5 mP = 40 kN at 3 mw = 10 kN/m from 0 to 5 mx = 2 m
Mx = kN·m
11.Centroid & second moment of areaMedium
Find the second moment of area Ix of the T-section about the horizontal axis x through its centroid.
bf = 120 mmtf = 15 mmtw = 16 mmhw = 240 mm
Ix = × 10⁶ mm⁴
12.Centroid & second moment of areaHard
A welded I-section has a top flange 150 × 20 mm, a web 210 mm high and 12 mm thick, and a bottom flange 190 × 16 mm (width × thickness). Find its second moment of area Ix about the horizontal axis x through its centroid.
top flange: 150 × 20 mmweb: 12 × 210 mmbottom flange: 190 × 16 mm
Ix = × 10⁶ mm⁴
3Solutions
1.Vectors: dot & cross product
Answer: θ = 97.14 °
Dot product:a·b = axbx + ayby + azbz = |a|·|b|·cos θ, a scalar. It gives the angle between two vectors, cos θ = a·b/(|a|·|b|), and it is zero when they are perpendicular. The scalar projection of a onto b is a·b/|b|, and a constant force F moving through a displacement d does work W = F·d.
Dot product
a·b = 5·6 + 6·(−4) + 5·(−3) = −9
Magnitudes
|a| = √(5² + 6² + 5²) = 9.274
|b| = √(6² + (−4)² + (−3)²) = 7.81
Angle
cos θ = −9/(9.274·7.81) = −0.1243
θ = arccos(−0.1243) = 97.14°
Answer
θ = 97.14°
2.Vectors: dot & cross product
Answer: V = 10
Cross product:a × b = (aybz − azby, azbx − axbz, axby − aybx) is perpendicular to both vectors, and |a × b| = |a|·|b|·sin θ is the area of the parallelogram they span (the triangle has half of it). Order matters: b × a = −(a × b). The moment of a force F applied at point A about point O is MO = r × F, with r = A − O.
Cross product b × c
(b × c)x = bycz − bzcy = (−3)·(−3) − 4·(−1) = 13
(b × c)y = bzcx − bxcz = 4·4 − 5·(−3) = 31
(b × c)z = bxcy − bycx = 5·(−1) − (−3)·4 = 7
Dot product with a
a·(b × c) = 5·13 + (−2)·31 + 1·7 = 10
V = |10| = 10
Answer
V = 10
3.Beam support reactions
Answer: RA = 18.75 kN
Static equilibrium: a body at rest has ΣFx = 0, ΣFy = 0 and ΣM = 0 about any point. Taking moments about one support removes that support’s reaction from the equation.
Free-body diagram
Replace the supports with reaction forces. A roller gives one vertical reaction; with only vertical loads the pin gives one vertical reaction too.
Equilibrium
Moments about A, counter-clockwise positive. Then vertical force balance.
ΣMA = 0: RB·8 = 25·3 + 25·7
RB = 250/8 = 31.25 kN
ΣFy = 0: RA = 50 − 31.25 = 18.75 kN
Answer
RA = 18.75 kN
4.Beam support reactions
Answer: MA = 153.3 kN·m
Cantilever equilibrium: a fixed support can push up and also resist rotation, so it gives a vertical reaction RA and a reaction moment MA. ΣFy = 0 gives RA, and moments about the wall give MA = Σ(load × distance from the wall). A distributed load acts as its resultant at the centroid of the load diagram.
Resultant of the triangular load
W = ½·4·5 = 10 kN
x̄ = 2L/3 = 3.333 m
Moment about the wall
MA = W·x̄ + P·a = 10·3.333 + 40·3 = 153.3 kN·m
Answer
MA = 153.3 kN·m
5.Truss: method of joints
Answer: member DH
Zero-force members: at a joint, the forces must balance in every direction. If a joint has no load and no support, and two of its three members are in line, then the balance across that line involves only the third member, so its force is zero. If only two members meet at an unloaded joint and they are not in line, both are zero.
Joint D
CD and DE lie on one straight line and the joint carries no load. Balancing forces across that line leaves only DH.
ΣF⊥ = 0 → FDH = 0
Check with the method of joints
Solving all joints gives these member forces (T = tension, C = compression).
Method of joints: every pin of a truss is in equilibrium, so ΣFx = 0 and ΣFy = 0 at each joint. Start at a joint with at most two unknown member forces. A negative result means the member is in compression.
Support reactions
Equilibrium of the whole truss. Moments about A, counter-clockwise positive.
ΣMA = 0: Cy·5 = 55·2 − 25·4 → Cy = 2 kN
ΣFy = 0: Ay = 55 − 2 = 53 kN
ΣFx = 0: Ax = 25 kN
Joint A
Assume every member is in tension (pulling away from the joint). Only AB has a vertical component at A.
sin α = 4/4.472 = 0.8944
ΣFy = 0: Ay + FAB·sin α = 0
FAB = −53/0.8944 = −59.26 kN
Answer
FAB = −59.26 kN (compression)
7.Frames & machines
Answer: A = 20.16 kN
Frames and machines: unlike a truss, a frame contains at least one member that carries forces at more than two points, or a load between its pins. First look for two-force members: a member loaded only at two pins has its force along the line through them, so it adds one unknown, not two. Then use the whole structure for the support reactions when there are no more than three unknowns. Finally take the frame apart: draw each member separately, with the pin forces equal and opposite on the two members they connect, and write ΣFx = 0, ΣFy = 0 and ΣM = 0 for it. Taking moments about a pin removes both components of that pin force from the equation.
Strut force
FCD = 5·2.4/(0.3511·1.6) = 21.36 kN
Components at A
Ax = FCD·cos α = 20 kN (towards the wall)
Ay = 5 − FCD·sin α = −2.5 kN
Magnitude
A = √(20² + −2.5²) = 20.16 kN
Answer
A = 20.16 kN
A negative Ay means the wall pulls the beam down at A.
8.Frames & machines
Answer: T = 14.42 kN
Frames and machines: unlike a truss, a frame contains at least one member that carries forces at more than two points, or a load between its pins. First look for two-force members: a member loaded only at two pins has its force along the line through them, so it adds one unknown, not two. Then use the whole structure for the support reactions when there are no more than three unknowns. Finally take the frame apart: draw each member separately, with the pin forces equal and opposite on the two members they connect, and write ΣFx = 0, ΣFy = 0 and ΣM = 0 for it. Taking moments about a pin removes both components of that pin force from the equation.
Whole frame: moments about A
By·5 = 15·2.5
By = 7.5 kN
Leg BC: moments about C
7.5·2.5 = T·1.3
T = 14.42 kN
Answer
T = 14.42 kN
A positive T means the tie is in tension, as assumed.
9.Shear force & bending moment
Answer: Mx = −12.5 kN·m
Method of sections: cut the beam where you want the internal forces and keep one side. The shear force V balances the vertical forces on that side, and the bending moment M balances their moments about the cut. An applied couple makes the bending moment diagram jump by the size of the couple.
Support reactions
Moments about A, counter-clockwise positive: RB·L − M0 = 0. Vertical balance: RA + RB = 0.
RB = M0/L = 20/4 = 5 kN (upward)
RA = −RB = −5 kN (downward)
Cut just left of the couple
The left part contains only RA.
Mx = RA·a = −5·2.5 = −12.5 kN·m
Answer
Mx = −12.5 kN·m
10.Shear force & bending moment
Answer: Mx = 62 kN·m
Method of sections: cut the beam where you want the internal forces and keep one side. The shear force V balances the vertical forces on that side, and the bending moment M balances their moments about the cut.
Support reactions
Equilibrium of the whole beam.
UDL resultant: W = w·ℓ = 10·5 = 50 kN, acting at x̄ = 2.5 m
ΣMA = 0: RB·5 = 40·3 + 10·5·2.5
RB = 245/5 = 49 kN
ΣFy = 0: RA = 90 − 49 = 41 kN
Cut at x = 2 m and keep the left part
Only forces left of the cut appear. A UDL portion acts as one force at the middle of that portion.
M = RA·x − (each load left of the cut × its distance to the cut)
Mx = 41·2 − 10·2·1 = 62 kN·m
Answer
Mx = 62 kN·m
11.Centroid & second moment of area
Answer: Ix = 38.39 × 10⁶ mm⁴
Composite areas: split the section into rectangles. The centroid is the area-weighted average ȳ = ΣAᵢyᵢ/ΣAᵢ. The second moment of area about the centroidal axis adds each part’s own bh³/12 and its parallel-axis term A·d², where d is the distance from the part’s centroid to the overall centroid. Bending stress is σ = M·y/I.
Split into two rectangles
Measure every y from the bottom edge.
flange: Af = 120·15 = 1800 mm², yf = 240 + 15/2 = 247.5 mm
Composite areas: split the section into rectangles. The centroid is the area-weighted average ȳ = ΣAᵢyᵢ/ΣAᵢ. The second moment of area about the centroidal axis adds each part’s own bh³/12 and its parallel-axis term A·d², where d is the distance from the part’s centroid to the overall centroid. Bending stress is σ = M·y/I.
Split into three rectangles
Measure every y from the bottom edge.
top flange: A = 3000 mm², y = 236 mm
web: A = 2520 mm², y = 121 mm
bottom flange: A = 3040 mm², y = 8 mm
Centroid
ȳ = (3000·236 + 2520·121 + 3040·8)/8560 = 121.2 mm
Parallel-axis theorem
top flange: 150·20³/12 + 3000·114.8² = 3.966 × 10⁷ mm⁴