Creep–recovery design of a polypropylene water tank: 23-Chem-B8 Polymer Engineering, December 2015, Q6

Question 6: Creep–recovery design of a polypropylene water tank (25 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. PP creep law (above), $\varepsilon$ in %, $t$ in s, $\sigma$ in MPa. Cylinder $D=3$ m, $H=2.5$ m. Load history: full (stressed) 4 months, then empty (unloaded) 8 months. Residual strain limit $\varepsilon_{res}\le0.05\%$ at $t=12$ months.

Find. Required wall thickness $t_w$.

The maximum hoop stress $\sigma_\theta=pD/2t$ occurs at the base where the water head, and hence pressure $p=\rho g H$, is greatest. The wall must be thick enough that the strain remaining after the load-then-recovery cycle stays below 0.05%.

Approach. Use Boltzmann superposition: loading $+\sigma$ at $t=0$ and unloading ($-\sigma$) at 4 months. The residual strain at 12 months is the creep function at 12 months minus the creep function at the 8-month recovery time; set that to 0.05%, solve for the hoop stress $\sigma$, then back out the wall thickness from $\sigma=pD/2t$.

Design pressure and hoop stress. Maximum water pressure at the base: $p=\rho g H=1000(9.81)(2.5)=24.5$ kPa $=0.02453$ MPa. For a thin cylinder the hoop stress is $$\sigma=\frac{pD}{2t}=\frac{0.02453\times3}{2\,t}=\frac{0.0368}{t}\ \text{MPa}\quad(t\ \text{in m})$$
Residual strain by superposition. With the stress removed at $t_1=4$ months, the strain remaining at $t=12$ months is $\varepsilon_{res}=\varepsilon(t,\sigma)-\varepsilon(t-t_1,\sigma)$. The constant $0.022\cdot3.5\,\sigma$ term cancels, leaving (with $t_a=1$ yr $=3.15\times10^7$ s and $t_b=8$ mo $=2.10\times10^7$ s): $$\varepsilon_{res}=0.022\,\sigma\,(t_a^{0.16}-t_b^{0.16}) +1.5\times10^{-6}\,e^{0.9\sigma}\,(t_a^{0.33}-t_b^{0.33})$$ $$\varepsilon_{res}=0.0219\,\sigma+5.60\times10^{-5}\,e^{0.9\sigma}$$ (the nonlinear term carries its stress dependence entirely in $e^{0.9\sigma}$; as printed it is not multiplied by a further $\sigma$).
Solve for the allowable stress. Setting $\varepsilon_{res}=0.05\%$ and solving the (mildly nonlinear) equation gives $$\sigma=\boxed{2.26\ \text{MPa}}$$ The linear recovery term supplies almost all of it ($0.0219\times2.26=0.0496$); the nonlinear term adds only $\approx0.0004$.
Wall thickness. From $\sigma=pD/2t$, $$t_w=\frac{pD}{2\sigma}=\frac{0.02453\times3}{2\times2.26}=0.0162\ \text{m}=\boxed{16.2\ \text{mm}}$$

Quantity	Value
Maximum base pressure $p=\rho g H$	24.5 kPa (0.0245 MPa)
Allowable hoop stress $\sigma$ ($\varepsilon_{res}=0.05\%$)	2.26 MPa
Required wall thickness $t_w$	$\approx16.2$ mm

Check

The residual (not the peak) strain governs, so the constant instantaneous term $0.022\times3.5\,\sigma$ correctly cancels in the superposition. The design uses the maximum hoop stress at the base (full-head pressure); using the mean head would roughly halve the required thickness. One year is taken as $3.15\times10^7$ s with months as equal twelfths.

Topic: Linear-viscoelastic creep–recovery (Boltzmann superposition) applied to pressure-vessel wall design.

Key relations: thin-cylinder hoop stress $\sigma_\theta=pD/2t$; hydrostatic $p=\rho g H$; superposition $\varepsilon_{res}(t)=\varepsilon_{creep}(t)-\varepsilon_{creep}(t-t_1)$.

Why this works: Boltzmann superposition treats each stress change as launching an independent creep response; removing the load at $t_1$ is equivalent to adding a step of $-\sigma$ whose creep partially cancels the original. What survives at 12 months is the difference between the full-history creep and the creep that the removal step has accumulated over its 8-month life — the residual (permanent-set-like) strain. Equating it to the serviceability limit sizes the wall.

Common pitfall: Applying the creep strain at 12 months directly (ignoring recovery), which grossly over-designs the wall; or forgetting that $t$ must be in seconds to match the empirical exponents. The nonlinear $e^{0.9\sigma}$ term is small here but must be retained for consistency.

Source: Crawford, Plastics Engineering 3rd ed., §2.6–2.9 (creep, recovery and design with viscoelastic data); Sperling, Physical Polymer Science 4th ed., §8.

Quantity	Value
Maximum base pressure \(p=\rho g H\)	24.5 kPa (0.0245 MPa)
Allowable hoop stress \(\sigma\) (\(\varepsilon_{res}=0.05\%\))	2.26 MPa
Required wall thickness \(t_w\)	\(\approx16.2\) mm

Question 6 of 6: Creep–recovery design of a polypropylene water tank

Question 6: Creep–recovery design of a polypropylene water tank (25 marks)