building.prediction.masonry_cavity_wall

Wall ties in masonry cavity walls: the structural bridge across the cavity (Hopkins 2007, Sections 3.11.3.2 and 4.3.5.4.1).

A masonry cavity wall is a double leaf, and the textbook double-leaf prediction (phonometry.double_wall_transmission_loss) treats the cavity as pure air. Real cavity walls are stitched together by wall ties every few courses. Those ties do two things the air-only model cannot see: they add a mechanical spring in parallel with the air spring, which pushes the mass-spring-mass resonance up, and they open a structure-borne path from one leaf to the other, which caps the insulation the pair can reach.

Dynamic stiffness of a tie (Section 3.11.3.2). A tie is characterised by a single number sX mm, its dynamic stiffness at a cavity width X, measured on two nominally identical 100 mm concrete cubes from the mass-spring-mass resonance of the pair, $s_{X\mathrm{mm}}=2{\pi }^2{f}_{\mathrm{msm}}^2{m}_{\mathrm{av}}$ (Eq. 3.202). WALL_TIE_STIFFNESS carries Hopkins’ Table A4, whose 50 mm rows come from Hopkins, Wilson & Craik (1999) and whose 100 mm row comes from Hall & Hopkins (2001).

The tie array as a spring in parallel with the air (Eq. 4.89). N ties over a plate of area S add $Nk/S$ to the cavity air stiffness s_a:

$$f_{\mathrm{msm}}=\frac{1}{2\pi }\sqrt{\frac{s_{\mathrm{a}}+Nk/S}{{\rho }_{\mathrm{s}1}{\rho }_{\mathrm{s}2}/({\rho }_{\mathrm{s}1}+{\rho }_{\mathrm{s}2})}}\text{\hspace{1em}(Eq. 4.89)}$$

Below fmsm the two leaves move as one plate of the combined mass. Stiff ties are therefore doubly bad: they raise the resonance into the rating range and bridge the cavity. Use wall_tie_stiffness_per_area and pass the result as tie_stiffness_per_area to phonometry.mass_spring_mass_resonance or phonometry.double_wall_transmission_loss.

The structure-borne path (Eqs. 4.84 to 4.88). Each tie is a point connection between two plates. With the driving-point mobilities Yi, Yj of the two leaves (infinite thin plates, $Y=1/(8\sqrt{B^{\prime }{m}^{\prime \prime }})$, Eq. 2.190) and the connector mobility of a linear spring $Y_{\mathrm{c}}=i\omega /k$ (Eq. 4.88), N identical uncorrelated connections give the coupling loss factor

$${\eta }_{ij}=\frac{N}{\omega m_i}\frac{\mathrm{Re}\{Y_j\}}{|Y_i+Y_j+Y_{\mathrm{c}}{|}^2}\text{\hspace{1em}(Eq. 4.87)}$$

The plate area cancels ($N/m_i=n/{\rho }_{\mathrm{s}1}$ with n ties per m2), so wall_tie_coupling_loss_factor needs only the tie density. A rigid connection (screw, nail, bolt, or a tie so stiff it never yields) is the limit $Y_{\mathrm{c}}=0$, where the only frequency dependence left is the $1/\omega$ and ${\eta }_{ij}$ falls as $1/f$. Once $|Y_{\mathrm{c}}|=\omega /k$ overtakes the plate mobilities a resilient tie adds two more powers, so ${\eta }_{ij}$ falls as $1/f^3$ and the ratio to the rigid ceiling as $1/f^2$. That is exactly why a butterfly tie at 1.7 MN/m and a vertical-twist tie at 94 MN/m behave so differently: the soft one enters that regime inside the building acoustics range, the stiff one stays on the rigid ceiling for another two octaves.

Note

The inputs of this model are printed data: Table A4 here, confirmed value for value by Hopkins, Wilson & Craik (1999) Table 1, which prints the same 1.7 / 16.1 / 94.0 MN/m at a 50 mm cavity. Craik & Wilson (1995) Table 1 measures the same tie types at an 85 mm cavity and reports 1.1 and 4.3 MN/m for the butterfly and double-triangle ties, so it corroborates the ordering and the order of magnitude but not the values; the dynamic stiffness is defined at a given cavity width and changes with it. The output is not printed anywhere: every published sound reduction index of a bridged masonry cavity wall is a figure, so the per-band transmission-loss penalty from wall ties has no numeric oracle. The resonance shift does: Hopkins Fig. 4.35 prints $f_{\mathrm{msm}}=26$ Hz without ties and $f_{\mathrm{msm}}=50$ Hz with them for the same wall.

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wall_tie_coupling_loss_factor

wall_tie_coupling_loss_factor(
    frequency: ArrayLike,
    mass1: float,
    mass2: float,
    bending_stiffness1: float,
    bending_stiffness2: float,
    *,
    ties_per_area: float,
    tie: str | float | None = None,
) -> WallTieCouplingResult

Coupling loss factor of a wall-tie array (Hopkins Eqs. 4.87 and 4.88).

$${\eta }_{ij}=\frac{N}{\omega m_i}\frac{\mathrm{Re}\{Y_j\}}{|Y_i+Y_j+Y_{\mathrm{c}}{|}^2}\text{\hspace{1em}(Eq. 4.87)}$$

for N identical, uncorrelated point connections, with the leaves modelled as infinite thin plates ($Y=1/(8\sqrt{B^{\prime }{m}^{\prime \prime }})$, Eq. 2.190) and each tie as a linear spring ($Y_{\mathrm{c}}=i\omega /k$, Eq. 4.88). Since $m_i={\rho }_{\mathrm{s}1}S$ and $N=nS$, the plate area cancels and only the tie density n enters.

Parameters

Name	Description
frequency	Frequencies f, in hertz (array, > 0).
mass1	Surface density rho_s1 of the excited leaf, in kg/m^2 (> 0).
mass2	Surface density rho_s2 of the receiving leaf, in kg/m^2 (> 0).
bending_stiffness1	Bending stiffness per unit width B' of leaf 1, in N.m (> 0); see phonometry.vibration.structural.point_mobility.plate_bending_stiffness.
bending_stiffness2	Bending stiffness per unit width of leaf 2, in N.m.
ties_per_area	Number of ties per unit area n, in 1/m^2 (> 0).
tie	A name from WALL_TIE_STIFFNESS, an explicit dynamic stiffness k in N/m, or None for a rigid connection ($Y_{\mathrm{c}}=0$, the screw/nail/bolt limit).

Returns: A WallTieCouplingResult.

Raises

Exception	When
ValueError	for a non-positive input or an unknown tie name.

WALL_TIE_STIFFNESS

Constant (dict).

WALL_TIE_STIFFNESS = {'butterfly': (0.05, 1700000.0), 'double_triangle': (0.05, 16100000.0), 'vertical_twist': (0.05, 94000000.0), 'vertical_twist_100mm': (0.1, 43400000.0)}

wall_tie_stiffness

wall_tie_stiffness(tie: str) -> tuple[float, float]

Look up a wall tie in Hopkins Table A4.

Parameters

Name	Description
tie	One of "butterfly", "double_triangle", "vertical_twist" (all at a 50 mm cavity) or "vertical_twist_100mm".

Returns: (cavity_width, stiffness): the cavity width in m at which the tie was measured, and its dynamic stiffness sX mm in N/m.

Raises

Exception	When
ValueError	for an unknown tie name.

wall_tie_stiffness_per_area

wall_tie_stiffness_per_area(ties_per_area: float, tie: str | float) -> float

Stiffness per unit area of a tie array, $Nk/S$ (Hopkins Eq. 4.89).

The term that acts in parallel with the cavity air stiffness s_a in the mass-spring-mass resonance. Feed it to phonometry.mass_spring_mass_resonance or phonometry.double_wall_transmission_loss as tie_stiffness_per_area.

Parameters

Name	Description
ties_per_area	Number of ties per unit area $n=N/S$, in 1/m^2 (> 0).
tie	A name from WALL_TIE_STIFFNESS, or an explicit dynamic stiffness k of one tie, in N/m (> 0).

Returns: The stiffness per unit area $nk$, in N/m^3.

Raises

Exception	When
ValueError	for a non-positive input or an unknown tie name.

WallTieCouplingResult

WallTieCouplingResult(
    frequencies: np.ndarray,
    coupling_loss_factor: np.ndarray,
    mobility1: float,
    mobility2: float,
    connector_mobility: np.ndarray,
    ties_per_area: float,
    tie_stiffness: float | None,
    rigid_coupling_loss_factor: np.ndarray,
)

Structure-borne coupling of a wall-tie array (Hopkins Eqs. 4.87/4.88).

Attributes

Name	Description
frequencies	Frequencies, in hertz.
coupling_loss_factor	Coupling loss factor eta_ij from leaf 1 to leaf 2 per frequency (dimensionless).
mobility1	Driving-point mobility Yi of leaf 1, in m/(N.s).
mobility2	Driving-point mobility Yj of leaf 2, in m/(N.s).
connector_mobility	Magnitude $|Y_{\mathrm{c}}|=\omega /k$ of the tie mobility per frequency, in m/(N.s); all zeros for a rigid connection.
ties_per_area	Number of ties per unit area n, in 1/m^2.
tie_stiffness	Dynamic stiffness k of one tie, in N/m, or None for a rigid connection.
rigid_coupling_loss_factor	The $Y_{\mathrm{c}}=0$ coupling loss factor per frequency, the ceiling a resilient tie is measured against.

WallTieCouplingResult.plot()

WallTieCouplingResult.plot(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Plot eta_ij(f) against the rigid-connection ceiling.

Requires matplotlib (pip install phonometry[plot]); returns the Axes.