SME
Mechanics of Materials (Solid Mechanics I) · Problem type
Two materials sharing a load
Use this for “A concrete post with 2,000 mm² of steel bars carries 1,000 kN. How much does the steel take?”
Diagram
Bonded rods and concrete shorten alike, so each takes load in proportion to its EA.
The
rods’
share:
Pₛ
P
=
Eₛ
Aₛ
Eₛ Aₛ + Ec Ac
=
0.1538.
The section is to scale; the post’s height is not.
This is an example. Type your own values to start a new problem.
f
=
Eₛ (GPa)
×
Aₛ (mm²)
×
+
×
Pₛ (kN)
=
f
×
P (kN)
Pc
Concrete’s share
calculated
σₛ
Steel stress
calculated · MPa
σc
Concrete stress
calculated · MPa
Formulas
f
=
Eₛ
×
Aₛ
Eₛ × Aₛ + Ec × Ac
0.1538
=
200,000
×
2,000
200,000 × 2,000 + 25,000 × 88,000
Pₛ
=
f
×
P
153,846.1538 = 0.1538 × 1,000,000
P
=
Pₛ
+
Pc
1,000,000 = 153,846.1538 + 846,153.8462
σₛ
=
Pₛ
Aₛ
76.9231
=
153,846.1538
2,000
σc
=
Pc
Ac
9.6154
=
846,153.8462
88,000
σₛ
Eₛ
=
σc
Ec
76.9231
200,000
=
9.6154
25,000
Formulas are worked in N, MPa. The step-by-step shows the conversions.
Assumptions
•
Steel and concrete are bonded and loaded together, so both shorten by the same strain.
•
Both stay elastic; P is a squeeze through the post’s center, and stresses are given as sizes.
•
Ac is the concrete alone (the post less its bars); worked in N, mm and MPa.
Step-by-step
WE KNOW
P = 1,000 kN, Eₛ = 200 GPa, Ec = 25 GPa, Aₛ = 2,000 mm², Ac = 88,000 mm²
FIND
steel’s share of EA (f), steel’s share (Pₛ), concrete’s share (Pc), steel stress (σₛ), concrete stress (σc)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to N, MPa).
P = 1,000 kN = 1,000,000 N (1 kN = 1000 N)
Eₛ = 200 GPa = 200,000 MPa (1 GPa = 1000 MPa)
Ec = 25 GPa = 25,000 MPa (1 GPa = 1000 MPa)
Step 2 · Find steel’s share of EA (f)
Use f = Eₛ × Aₛ ÷ (Eₛ × Aₛ + Ec × Ac)
Equal strain makes each part carry load in proportion to its EA: the steel’s share of the total EA.
f
=
200,000
×
2,000
200,000 × 2,000 + 25,000 × 88,000
f
=
200,000
×
2,000
400,000,000 + 2,200,000,000
f
=
200,000
×
2,000
2,600,000,000
f
=
400,000,000
2,600,000,000
f
=
0.1538
Step 3 · Find steel’s share (Pₛ)
Use Pₛ = f × P
The
steel
carries
its
share
f
of
the
whole
load.
Pₛ = 0.1538 × 1,000,000
Pₛ = 153,846.1538 N
Step 4 · Find concrete’s share (Pc)
Use P = Pₛ + Pc
Whatever the steel doesn’t carry, the concrete does.
Pc
=
P
−
Pₛ
Pc = 1,000,000 − 153,846.2
Pc = 846,153.8462 N
Step 5 · Find steel stress (σₛ)
Use σₛ = Pₛ ÷ Aₛ
Spread the steel’s share over the steel area.
σₛ
=
153,846.2
2,000
σₛ = 76.9231 MPa
Step 6 · Find concrete stress (σc)
Use σc = Pc ÷ Ac
Spread the concrete’s share over the concrete area.
σc
=
846,153.8
88,000
σc = 9.6154 MPa
Step 7 · Convert the answers
Change each answer back to the unit you chose.
Pₛ = 153,846.1538 N = 153.8462 kN (1 kN = 1000 N)
Pc = 846,153.8462 N = 846.1538 kN (1 kN = 1000 N)
Check
Put the numbers back in. Both sides should match.
0.1538
=
200,000
×
2,000
200,000 × 2,000 + 25,000 × 88,000
✓
153,846.2 = 0.1538462 × 1,000,000 ✓
1,000,000 = 153,846.2 + 846,153.8 ✓
76.9231
=
153,846.2
2,000
✓
9.6154
=
846,153.8
88,000
✓
76.9231
200,000
=
9.6154
25,000
✓
Topic check
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