"""This module provides the beam class implementation for the ``eu_cdm``
design class in the BDIM layer.
"""
# Imports from installed packages
import numpy as np
# Imports from the design class (eu_cdm) library
from .materials import Steel, Concrete
# Imports from bdim base library
from ..baselib.beam import BeamBase
# Imports from units library
from ....utils.units import MPa, m
# Constants
ECONOMIC_MU_EB: float = 0.25
"""Maximum mu value considered for the economic emergent beam design."""
ECONOMIC_MU_WB: float = 0.25
"""Maximum mu value considered for the economic wide beam design."""
TAU_C_VECT = np.array(
[0.5, 0.6, 0.65, 0.75, 0.85, 0.90, 1.00, 1.10, 1.15]) * MPa
"""Vector of allowable shear stresses that carried by the concrete or
vector of the design shear strength values of concrete."""
TAU_MAX_VECT = np.array([2.4, 3.2, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0]) * MPa
"""Vector of allowable shear stresses that can be carried by
the beam section."""
FCK_VECT = np.array([12, 16, 20, 25, 30, 35, 40, 45, 50]) * MPa
"""Vector of characteristic concrete compressive strength values."""
[docs]
class Beam(BeamBase):
"""Beam implementation for design class ``eu_cdm``.
This class extends ``BeamBase`` by narrowing the attribute types
and overriding design methods per REBAP (1983).
Attributes
----------
steel : ~simdesign.rcmrf.bdim.eu_cdm.materials.Steel
Steel material assigned to the beam.
concrete : ~simdesign.rcmrf.bdim.eu_cdm.materials.Concrete
Concrete material assigned to the beam.
MIN_B_EB: float : float
The default minimum breadth (width) of emergent beams.
See Also
--------
:class:`~bdim.baselib.beam.BeamBase`
Base class defining the core behaviour and configuration.
References
----------
REBAP (1983). Regulamento de Estruturas de Betão Armado e Pré-Esforçado.
Decreto-Lei N.° 349-C/83, Lisbon, Portugal.
d'Arga e Lima, J., Monteiro, V., Mun, M. (2005).
Betão armado: esforços normais e de flexão: REBAP-83.
Laboratório Nacional de Engenharia Civil, Lisboa.
"""
steel: Steel
concrete: Concrete
MIN_B_EB: float = 0.25 * m
@property
def rhol_min_tens(self) -> float:
"""
Returns
-------
float
Minimum longitudinal reinforcement ratio in tension zone.
"""
# Art 90.1 in REBAP (1983)
if self.steel.grade == "A500":
return 0.12 / 100
elif self.steel.grade == "A400":
return 0.15 / 100
elif self.steel.grade == "A235":
return 0.25 / 100
@property
def rhol_max_tens(self) -> float:
"""
Returns
-------
float
Maximum longitudinal reinforcement ratio in tension
and compression zones.
"""
# Art 90.2 in REBAP (1983)
return 0.04
@property
def rhoh_min(self) -> float:
"""
Returns
-------
float
Minimum transverse reinforcement ratio.
"""
# Art 94.2 in REBAP (1983)
if self.steel.grade == "A500":
return 0.08 / 100
elif self.steel.grade == "A400":
return 0.10 / 100
else:
return 0.16 / 100
[docs]
def verify_section_adequacy(self) -> None:
"""Verify the beam section dimensions for design forces.
"""
# Dimensionless mu values for economic section from REBAP book
# This can be considered as an engineering practice
if self.typology == 1: # Wide Beam
mu_max = ECONOMIC_MU_WB
elif self.typology == 2: # Emergent Beam
mu_max = ECONOMIC_MU_EB
# Allowable shear stress that can be carried by the beam
tau_max = np.interp(self.concrete.fck, FCK_VECT, TAU_MAX_VECT)
# Distance from extreme compression fiber to centroid of longitudinal
# tension reinforcement.
d = 0.9 * self.h
# Maximum of envelope forces
max_shear = max(self.envelope_forces.V1, self.envelope_forces.V5,
self.envelope_forces.V9)
max_moment = max(
self.envelope_forces.M1_pos,
self.envelope_forces.M5_pos,
self.envelope_forces.M9_pos,
abs(self.envelope_forces.M1_neg),
abs(self.envelope_forces.M5_neg),
abs(self.envelope_forces.M9_neg)
)
# Verify the adequacy of the section dimensions
tau = max_shear / (self.b * d) # for max. shear force
mu = max_moment / (self.fcd * self.b * d**2) # for max. bending moment
if mu < mu_max and tau < tau_max:
self.ok = True # Ok
else:
self.ok = False # Not ok
[docs]
def compute_required_longitudinal_reinforcement(self) -> None:
"""Compute the required longitudinal reinforcement for design forces.
Notes
-----
- Top reinforcement is calculated as the maximum of required
reinforcement in tension for maximum of negative bending moments
and required reinforcement in compression for maximum of positive
bending moments.
- Bottom reinforcement is calculated as the maximum of required
reinforcement in compression for maximum of negative bending moments
and required reinforcement in tension for maximum of positive
bending moments.
- Required reinforcement is computed at three different sections:
start, middle, end.
"""
# Distance from extreme compression fiber to centroid of longitudinal
# tension reinforcement.
d = 0.9 * self.h
# Dimensionless limit values
mu_lim = 0.31 # mu limit defined in REBAP-83
omega_lim = 0.41 # omega limit defined in REBAP-83
# Design forces
moment_pos = np.array([self.envelope_forces.M1_pos,
self.envelope_forces.M5_pos,
self.envelope_forces.M9_pos])
moment_neg = np.array([abs(self.envelope_forces.M1_neg),
abs(self.envelope_forces.M5_neg),
abs(self.envelope_forces.M9_neg)])
# Reinforcement area computation for positive moment envelope (+)
# REBAP pp. 33
mu_pos = moment_pos / (self.fcd * self.b * d**2)
# REBAP pp. 35, eq 11
omega_pos_prime = (mu_pos - mu_lim) / (1 - (self.cover / (d)))
# REBAP pp. 35, eq 10
omega_pos_prime[mu_pos <= mu_lim] = 0
# REBAP pp. 35, eq 11
omega_pos = omega_lim + omega_pos_prime
# REBAP pp. 35, eq 10
omega_pos[mu_pos <= mu_lim] = (
mu_pos[mu_pos <= mu_lim] * (1 + mu_pos[mu_pos <= mu_lim]))
# Reinforcement area computation for negative moment envelope (-)
# REBAP pp. 33
mu_neg = moment_neg / (self.fcd * self.b * d**2)
# REBAP pp. 35, eq 11
omega_neg_prime = (mu_neg - mu_lim) / (1 - (self.cover / (d)))
# REBAP pp. 35, eq 10
omega_neg_prime[mu_neg <= mu_lim] = 0
# REBAP pp. 35, eq 11
omega_neg = omega_lim + omega_neg_prime
# REBAP pp. 35, eq 10
omega_neg[mu_neg <= mu_lim] = (
mu_neg[mu_neg <= mu_lim] * (1 + mu_neg[mu_neg <= mu_lim]))
# Prime is used for compression reinforcement.
# It can be both at top and bottom due to seismic loading
omega_pos = np.maximum(omega_pos, omega_neg_prime)
omega_neg = np.maximum(omega_neg, omega_pos_prime)
# Determine required reinforcement at top and bottom
Asl_top = omega_neg * self.b * d * self.fcd / self.fsyd
Asl_bot = omega_pos * self.b * d * self.fcd / self.fsyd
# Check against minimum longitudinal reinforcement area
Asl_min_top = self.rhol_min_tens * self.b * d # REBAP 90.1
Asl_min_bot = self.rhol_min_tens * self.b * d # REBAP 90.1
Asl_top = np.maximum(Asl_top, Asl_min_top)
Asl_bot = np.maximum(Asl_bot, Asl_min_bot)
# Save required longitudinal reinforcement area
self.Asl_top_req = Asl_top
self.Asl_bot_req = Asl_bot
[docs]
def compute_required_transverse_reinforcement(self) -> None:
"""Compute the required transverse reinforcement for design forces.
Notes
-----
Reinforcement is computed at three sections: start, mid, and end.
"""
# Allowable shear stress that can be carried by the beam
tau_c = np.interp(self.concrete.fck, FCK_VECT, TAU_C_VECT)
# Distance from extreme compression fiber to centroid of longitudinal
# tension reinforcement.
d = 0.9 * self.h
# lever arm, i.e., distance between comp. and tens. forces
z = 0.9 * d
# Design forces
shear = np.array([self.envelope_forces.V1,
self.envelope_forces.V5,
self.envelope_forces.V9])
# Calculate the minimum shear reinforcement
Ash_sbh_min = self.rhoh_min * self.b # REBAP 1983 - Art 94.2
# REBAP 1983 - Art 53.1 V < Vcd + Vwd
Vcd = tau_c * self.b * d # Article 53.2
Ash_sbh = (shear - Vcd) / (z * self.fsyd) # Art 53.3
Ash_sbh = np.maximum(Ash_sbh, Ash_sbh_min)
# Save required transverse reinforcement area to spacing ratio
self.Ash_sbh_req = Ash_sbh